# Fractional calculus and time-fractional differential equations: revisit and
construction of a theory

1,2,3 M. Yamamoto

###### Abstract.

For fractional derivatives and time-fractional differential equations, we
construct a framework on the basis of the operator theory in fractional
Sobolev spaces. Our framework provides a feasible extension of the classical
Caputo and the Riemann-Liouville derivatives within Sobolev spaces of
fractional orders including negative ones. Our approach enables a unified
treatment for fractional calculus and time-fractional differential equations.
We formulate initial value problems for fractional ordinary differential
equations and initial boundary value problems for fractional partial
differential equations to prove the well-posedness and other properties.
Key words. fractional calculus, time-fractional differential equations,
fractional Sobolev spaces, operator theory
AMS subject classifications. 26A33, 34A08, 35R11, 34A12

dedicated to the memory of Professor Dr. Rudolf Gorenflo who invited the
author to studies of fractional calculus
1 Graduate School of Mathematical Sciences, The University of Tokyo, Komaba,
Meguro, Tokyo 153-8914, Japan
2 Honorary Member of Academy of Romanian Scientists, Ilfov, nr. 3, Bucuresti,
Romania
3 Correspondence Member of Accademia Peloritana dei Pericolanti, Palazzo
Università, Piazza S. Pugliatti 1 98122 Messina, Italy
4 Peoples’ Friendship University of Russia (RUDN University) 6 Miklukho-
Maklaya St, Moscow, 117198, Russian Federation
e-mail<EMAIL_ADDRESS>
##  1\. Motivations

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain with smooth boundary
$\partial\Omega$, and let $\nu=\nu(x)$ be the unit outward normal vector to
$\partial\Omega$ at $x\in\partial\Omega$. We set

$None$
$-\mathcal{A}v(x)=\sum_{i,j=1}^{d}\partial_{i}(a_{ij}(x)\partial_{j}v(x))+\sum_{j=1}^{d}b_{j}(x)\partial_{j}v(x)+c(x)v(x),\quad
x\in\Omega,\,0<t<T,$

where $a_{ij}=a_{ji},b_{j},c\in C^{1}(\overline{\Omega})$, $1\leq i,j\leq d$
and we assume that there exists a constant $\kappa>0$ such that

$\sum_{i,j=1}^{d}a_{ij}(x)\xi_{i}\xi_{j}\geq\kappa\sum_{j=1}^{d}|\xi_{j}|^{2},\quad
x\in\Omega,\,\xi_{1},...,\xi_{d}\in\mathbb{R}.$

Henceforth $\Gamma(\gamma)$ denotes the gamma function for $\gamma>0$:
$\Gamma(\gamma):=\int^{\infty}_{0}e^{-t}t^{\gamma-1}dt$.

Our eventual purpose is to construct a theoretical framework for treating
initial boundary value problems for time fractional diffusion equations with
source term $F(x,t)$, which can be described for the case of $0<\alpha<1$:

$None$
$\left\\{\begin{array}[]{rl}&d_{t}^{\alpha}u(x,t)=-\mathcal{A}u(x,t)+F(x,t),\quad
x\in\Omega,\,0<t<T,\\\ &u(x,0)=0,\quad x\in\Omega,\\\ &u(x,t)=0,\quad
x\in\partial\Omega,\,0<t<T.\end{array}\right.$

Here, for $0<\alpha<1$, we can formally define the pointwise Caputo
derivative:

$None$
$d_{t}^{\alpha}v(t)=\frac{1}{\Gamma(1-\alpha)}\int^{t}_{0}(t-s)^{-\alpha}\frac{dv}{ds}(s)ds,$

as long as the right-hand side exists.

As for classical treatments on fractional calculus and equations, we can refer
to monographs Gorenflo, Kilbas, Mainardi and Rogosin [6], Kilbas, Srivastava
and Trujillo [16], Podlubny [30] for example.

Henceforth let $L^{p}(0,T):=\\{v;\,\int_{0}^{T}|v(t)|^{p}dt<\infty\\}$ with
$p\geq 1$ and $W^{1,1}(0,T):=\\{v\in L^{1}(0,T);\,\frac{dv}{dt}\in
L^{1}(0,T)\\}$. We define the norm by

$\|v\|_{L^{p}(0,T)}:=\left(\int_{0}^{T}|v(t)|^{p}dt\right)^{\frac{1}{p}},\quad\|v\|_{W^{1,1}(0,T)}:=\|v\|_{L^{1}(0,T)}+\left\|\frac{dv}{dt}\right\|_{L^{1}(0,T)}.$

In view of the Young inequality on the convolution, we can directly verify
that the classical Caputo derivative (1.3) can be well-defined for $v\in
W^{1,1}(0,T)$ and $d_{t}^{\alpha}v\in L^{1}(0,T)$. However, everything is not
clear for $v\not\in W^{1,1}(0,T)$, and it is not a feasible assumption that
any solutions to (1.2) have the $W^{1,1}(0,T)$-regularity in $t$. The Young
inequality is well-known and we can refer to e.g., Lemma A.1 in Appendix in
Kubica, Ryszewska and Yamamoto [18].

In (1.3), the pointwise Caputo derivative $d_{t}^{\alpha}$ requires the first-
order derivative $\frac{dv}{ds}$ in any sense. Therefore, in order to discuss
$d_{t}^{\alpha}v$ for a function $v$ which keeps apparently reasonable
regularity such as ”$\alpha$-times” differentiability, we should formulate
$d_{t}^{\alpha}v$ in a suitable distribution space. Moreover, such a
formulation is not automatically unique. There have been several works for
example, Kubica, Ryszewska and Yamamoto [18], Kubica and Yamamoto [19], Zacher
[37]. Here we restrict ourselves to an extremely limited number of references.
In this article, we extend the approach in Kubica, Ryszewska and Yamamoto [18]
to discuss fractional derivatives in fractional Sobolev spaces of arbitrary
real number orders and construct a convenient theory for initial value
problems and initial boundary value problems for time-fractional differential
equations. In [18], the orders $\alpha$ is restricted to $0<\alpha<1$, but
here we study fractional orders $\alpha\in\mathbb{R}$.

The main purpose of this article is to define such fractional derivatives and
establish the framework which enables us for example, to uniformly consider
weaker and stronger solutions with exact specification of classes of solutions
in terms of fractional Sobolev spaces. Thus we intend to construct a
comprehensive theory for time-fractional differential equations within Sobolev
spaces.

In (1.3), we take the first-order derivative and then $(1-\alpha)$-times
integral operator :
$\frac{1}{\Gamma(1-\alpha)}\int^{t}_{0}(t-s)^{-\alpha}\cdots ds$ to finally
reach $\alpha$-times derivative. As is described in Section 2, our strategy
for the definition of the fractional derivative in $t$, is to consider
$\alpha$-times detivative of $v$ as the inverse to the $\alpha$-times
integral, not via $\frac{dv}{dt}$.

Moreover, for initial value problems for time-fractional differential
equations, we meet other complexity for how to pose initial condition, as the
following examples show.
Example 1.1.
We consider an initial value problem for a simple time-fractional ordinary
differential equation:

$None$ $d_{t}^{\alpha}v(t)=f(t),\quad 0<t<T,\qquad v(0)=a.$

We remark that (1.4) is not necessarily well-posed for all $\alpha\in(0,1)$,
$f\in L^{2}(0,T)$ and $a\in\mathbb{R}$, because of the initial condition.
Especially the order $\alpha\in\left(0,\,\frac{1}{2}\right)$ causes a
difficulty: Choosing $\gamma>-1$ and

$f(t)=t^{\gamma}\in L^{1}(0,T),$

in (1.4), we consider

$None$ $d_{t}^{\alpha}v(t)=t^{\gamma},\quad v(0)=a.$

A solution formula is known:

$None$
$v(t)=a+\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}s^{\gamma}ds$

(e.g., (3.1.34) (p.141) in Kilbas, Srivastava and Trujillo [16]). For
$0<\alpha<1$ and $\gamma>-1$, the right-hand side of (1.6) makes sense and we
can obtain

$None$
$v(t)=a+\frac{\Gamma(\gamma+1)}{\Gamma(\alpha+\gamma+1)}t^{\alpha+\gamma},\quad
t>0.$

However, the initial condition $v(0)=a$ is delicate:
(i) Let $\alpha+\gamma>0$. Then we can readily see that $v$ given by (1.7)
satisfies (1.5).
(ii) Let $\alpha+\gamma=0$. Then (1.7) provides that
$v(t)=a+\Gamma(1-\alpha)$. However this $v$ does not satisfy
$d_{t}^{\alpha}v=t^{-\alpha}$.
(iii) Let $\alpha+\gamma<0$. Then for $v(t)$ defined by (1.7), we see that
$\lim_{t\downarrow 0}v(t)=\infty$. Therefore (1.7) cannot give a solution to
(1.5), and moreover we are not sure whether there exists a solution to (1.5).

We have another classical fractional derivative called the Riemann-Liouville
derivative:

$None$
$D_{t}^{\alpha}v(t)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int^{t}_{0}(t-s)^{-\alpha}v(s)ds,\quad
0<t<T$

for $0<\alpha<1$, provided that the right-hand side exists.

Let $f\in L^{2}(0,T)$ be arbitrarily given. In terms of $D_{t}^{\alpha}$, we
can formulate an initial value problem:

$None$ $D_{t}^{\alpha}u(t)=f(t),\quad 0<t<T,\quad(J^{1-\alpha}u)(0)=a.$

The solution formula is

$None$ $u(t)=\frac{a}{\Gamma(\alpha)}t^{\alpha-1}+J^{\alpha}f(t),\quad 0<t<T$

(e.g., Kilbas, Srivastava and Trujillo [16], p.138). Indeed, we can directly
verify that $u$ given by (1.10) satisfies (1.9):

$\displaystyle
D_{t}^{\alpha}J^{\alpha}f(t)=\frac{1}{\Gamma(\alpha)\Gamma(1-\alpha)}\frac{d}{dt}\int^{t}_{0}(t-s)^{-\alpha}\left(\int^{s}_{0}(s-\xi)^{\alpha-1}f(\xi)d\xi\right)ds$
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\alpha)\Gamma(1-\alpha)}\frac{d}{dt}\int^{t}_{0}f(\xi)\left(\int^{t}_{\xi}(t-s)^{-\alpha}(s-\xi)^{\alpha-1}ds\right)d\xi$
$\displaystyle=$
$\displaystyle\frac{d}{dt}\int^{t}_{0}f(\xi)d\xi=f(t)\quad\mbox{for $f\in
L^{1}(0,T)$}.$

and

$D_{t}^{\alpha}\left(\frac{a}{\Gamma(\alpha)}t^{\alpha-1}\right)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int^{t}_{0}(t-s)^{-\alpha}s^{\alpha-1}ds=0,\quad
0<t<T,$

which means that $D_{t}^{\alpha}u(t)=f(t)$ for $0<t<T$. We can similarly
verify

$J^{1-\alpha}\left(\frac{a}{\Gamma(\alpha)}t^{\alpha-1}+J^{\alpha}f(t)\right)(0)=a.$

By $0<\alpha<\frac{1}{2}$, the formula (1.10) makes sense in $L^{1}(0,T)$, but
$u\not\in L^{2}(0,T)$ for each $f\in L^{2}(0,T)$ if $a\neq 0$. This suggests
that the initial value problem (1.9) may not be well-posed in $L^{2}(0,T)$.

Furthermore we can consider other formulation:

$None$ $D_{t}^{\alpha}u(t)=f(t),\quad 0<t<T,\quad u(0)=a.$

We see that $u(t)$ given by (1.10) satisfies $D_{t}^{\alpha}u=f$ in $(0,T)$,
but it follows from $\alpha-1<0$ that $\lim_{t\downarrow 0}|u(t)|=\infty$ by
$\lim_{t\downarrow 0}t^{\alpha-1}=\infty$ if $a\neq 0$, while if $a=0$, then
(1.10) can satisfy $u(0)=0$, that is, (1.11) by $f$ in some class such as
$f\in L^{\infty}(0,T)$.

The delicacy in the above examples are more or less known, and motivates us to
construct a uniform framework for time-fractional derivatives. Moreover, we
are concerned with the range space of the corresponding solutions for a
prescribed function space of $f$, e.g., $L^{2}(0,T)$, not only calculations of
the derivatives of individually given function $u$. Naturally, for the well-
posedness of an initial value problem, we are required to characterize the
function space of solutions corresponding to a space of $f$.

In other words, one of our main interests is to define a fractional
derivative, denoted by $\partial_{t}^{\alpha}$, and characterize the space of
$u$ satisfying $\partial_{t}^{\alpha}u\in L^{2}(0,T)$. Moreover, such
$\partial_{t}^{\alpha}$ should be an extension of $d_{t}^{\alpha}$ and
$D_{t}^{\alpha}$ in a minimum sense in order that important properties of
these classical fractional derivatives should be inherited to
$\partial_{t}^{\alpha}$.

Throughout this article, we treat fractional integrations and fractional
derivatives as operators from specified function spaces to others, that is, we
always attach them with their domains and ranges.

Thus we will define a time fractional derivative denoted by
$\partial_{t}^{\alpha}$ as suitable extension of $d_{t}^{\alpha}$ satisfying
the requirements:

  * •

Such extended derivative $\partial_{t}^{\alpha}$ admits usual rules of
differentiation as much as possible. For example,
$\partial_{t}^{\alpha}\partial_{t}^{\beta}=\partial_{t}^{\alpha+\beta}$ for
all $\alpha,\beta\geq 0$.

  * •

It admits a relevant formulation of initial condition even for
$\alpha\in\mathbb{R}$.

  * •

In $L^{2}$-based Sobolev spaces, there exists a unique solution to an initial
value problem for a time-fractional ordinary differential equation and an
initial boundary value problem for a time-fractional partial differential
equation for $\alpha>0$ and even $\alpha\in\mathbb{R}$.

In this article, we intend to outline foundations for a comprehensive theory
for time-fractional differential equations. Some arguments are based on
Gorenflo, Luchko and Yamamoto [7], Kubica, Ryszewska and Yamamoto [18].

This article is composed of eight sections and one appendix:

  * •

Section 2: Definition of the extended derivative $\partial_{t}^{\alpha}$:
We extend $d_{t}^{\alpha}$ as operartor in order that it is well-defined as as
isomorphism in relevant Sobolev spaces. We emphasize that our fractional
derivative coincides with the classical Riemann-Liouville derivative and the
Caputo derivative in suitable spaces, and we never aim at creating novel
notions of fractional derivatives but we are concerned with a formulation of a
fractional derivative allowing us convenient applications to time-fractional
differential equations, as Sections 5 and 6 discuss.

  * •

Section 3: Basic properties in fractional calculus

  * •

Sectoin 4: Fractional derivatives of the Mittag-Leffler functions

  * •

Section 5: Initial value problem for fractional ordinary differential
equations

  * •

Section 6: Initial boundary value problem for fractional partial differential
equations: selected topics

  * •

Section 7: Applicaion to an inverse source problem:
For illustraing the feasibility of our approach, we conside one inverse source
problem of determining a time-varying function.

  * •

Section 8: Concluding remarks.

##  2\. Definition of the extended derivative $\partial_{t}^{\alpha}$

§2.1. Introduction of function spaces and operators

We set

$None$
$\left\\{\begin{array}[]{rl}&(J^{\alpha}v)(t):=\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}v(s)ds,\quad
0<t<T,\quad\mathcal{D}(J^{\alpha})=L^{2}(0,T),\\\
&(J_{\alpha}v)(t):=\frac{1}{\Gamma(\alpha)}\int^{T}_{t}(\xi-t)^{\alpha-1}v(\xi)d\xi,\quad
0<t<T,\quad\mathcal{D}(J^{\alpha})=L^{2}(0,T).\end{array}\right.$

We can consider $J^{\alpha},J_{\alpha}$ for $v\in L^{1}(0,T)$, but for the
moment, we consider them for $v\in L^{2}(0,T)$. There are many works on the
Riemann-Liouville time-fractional integral operator $J^{\alpha}$, and we here
refer only to Gorenflo and Vessella [11] as monograph, and Gorenflo and
Yamamoto [12] for an operator theoretical approach.

We can directly prove
Lemma 2.1.
Let $\alpha,\beta\geq 0$. Then

$J^{\alpha}J^{\beta}v=J^{\alpha+\beta}v,\quad
J_{\alpha}J_{\beta}v=J_{\alpha+\beta}v\quad\mbox{for $v\in L^{1}(0,T)$}.$

For $0<\alpha<1$, we define an operator $\tau:L^{2}(0,T)\longrightarrow
L^{2}(0,T)$ by

$None$ $(\tau v)(t):=v(T-t),\quad v\in L^{2}(0,T).$

Then it is readily seen that $\tau$ is an isomorphism between $L^{2}(0,T)$ and
itself.

Throughout this article, we call $K$ an isomorphism between two Banach spaces
$X$ and $Y$ if the mapping $K$ is injective, and $KX=Y$ and there exists a
constant $C>0$ such that $C^{-1}\|Kv\|_{Y}\leq\|v\|_{X}\leq C\|Kv\|_{Y}$ for
all $v\in X$. Here $\|\cdot\|_{X}$ and $\|\cdot\|_{Y}$ denote the norms in $X$
and $Y$ respectively.

We set

$\displaystyle{{}_{0}{C^{1}[0,T]}}:=\\{v\in C^{1}[0,T];\,v(0)=0\\},$
${}^{0}{C^{1}[0,T]}:=\\{v\in
C^{1}[0,T];\,v(T)=0\\}=\tau\,({{}_{0}{C^{1}[0,T]}}).$

By $H^{\alpha}(0,T)$ with $0<\alpha<1$, we denote the Sobolev-Slobodecki space
with the norm $\|\cdot\|_{H^{\alpha}(0,T)}$ defined by

$\|v\|_{H^{\alpha}(0,T)}:=\left(\|v\|^{2}_{L^{2}(0,T)}+\int^{T}_{0}\int^{T}_{0}\frac{|v(t)-v(s)|^{2}}{|t-s|^{1+2\alpha}}dtds\right)^{\frac{1}{2}}$

(e.g., Adams [1]).

By $\overline{X}^{Y}$ we denote the closure of $X\subset Y$ in a normed space
$Y$. We set

$None$
$H_{\alpha}(0,T):=\overline{{{}_{0}{C^{1}[0,T]}}}^{H^{\alpha}(0,T)},\quad{{}^{\alpha}{H}}(0,T):=\overline{{}^{0}{C^{1}}[0,T]}^{H^{\alpha}(0,T)}.$

Henceforth we set $H_{0}(0,T):=L^{2}(0,T)$ and ${}^{0}{H}(0,T):=L^{2}(0,T)$.

Then
Proposition 2.1.
Let $0<\alpha<1$.
(i)

$H_{\alpha}(0,T):=\left\\{\begin{array}[]{rl}&H^{\alpha}(0,T),\quad
0<\alpha<\frac{1}{2},\\\ &\left\\{v\in
H^{\frac{1}{2}}(0,T);\thinspace\int^{T}_{0}\frac{|v(t)|^{2}}{t}dt<\infty\right\\},\quad\alpha=\frac{1}{2},\\\
&\\{v\in H^{\alpha}(0,T);\,v(0)=0\\},\quad\frac{1}{2}<\alpha\leq
1.\end{array}\right.$

Moreover, the norm in $H_{\alpha}(0,T)$ is equivalent to

$\|v\|_{H_{\alpha}(0,T)}:=\left\\{\begin{array}[]{rl}\|v\|_{H^{\alpha}(0,T)},&\alpha\neq\frac{1}{2},\\\
\left(\|v\|^{2}_{H^{\frac{1}{2}}(0,T)}+\int^{T}_{0}\frac{|v(t)|^{2}}{t}dt\right)^{\frac{1}{2}},&\alpha=\frac{1}{2}.\end{array}\right.$

(ii) Similarly we have

${{}^{\alpha}{H}}(0,T):=\left\\{\begin{array}[]{rl}&H^{\alpha}(0,T),\quad
0<\alpha<\frac{1}{2},\\\ &\left\\{v\in
H^{\frac{1}{2}}(0,T);\thinspace\int^{T}_{0}\frac{|v(t)|^{2}}{T-t}dt<\infty\right\\},\quad\alpha=\frac{1}{2},\\\
&\\{v\in H^{\alpha}(0,T);\,v(T)=0\\},\quad\frac{1}{2}<\alpha\leq
1.\end{array}\right.$

Moreover, the norm in $\,{}^{\alpha}H(0,T)$ is equivalent to

$\|v\|_{{{}^{\alpha}{H}}(0,T)}:=\left\\{\begin{array}[]{rl}\|v\|_{H^{\alpha}(0,T)},&\alpha\neq\frac{1}{2},\\\
\left(\|v\|^{2}_{H^{\frac{1}{2}}(0,T)}+\int^{T}_{0}\frac{|v(t)|^{2}}{T-t}dt\right)^{\frac{1}{2}},&\alpha=\frac{1}{2}.\end{array}\right.$

We can refer to [18] as for the proof of Proposition 2.1 (i). Since
$\tau:H_{\alpha}(0,T)\longrightarrow{{}^{\alpha}{H}}(0,T)$ is an isomorphism,
we can verify part (ii) of the proposition.

§2.2. Extension of $d_{t}^{\alpha}$ to $H_{\alpha}(0,T)$: intermediate step

We can prove (e.g., [7], [18]):
Proposition 2.2.
Let $0<\alpha<1$. Then

$J^{\alpha}:L^{2}(0,T)\longrightarrow H_{\alpha}(0,T)$

is an isomorphism. In particular, $H_{\alpha}(0,T)=J^{\alpha}L^{2}(0,T)$.

By the proposition, we can easily verify that some functions belong to
$H_{\alpha}(0,T)$, although the verification by (2.3) is complicated.
Example 2.1.
In view of Proposition 2.2, we will verify that

$t^{\beta}\in H_{\alpha}(0,T)\quad\mbox{if $\beta>\alpha-\frac{1}{2}$.}$

Indeed, setting $\gamma:=\beta-\alpha$, we see $\gamma>-\frac{1}{2}$, and so
$t^{\gamma}\in L^{2}(0,T)$. Moreover,

$J^{\alpha}t^{\gamma}=\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}s^{\gamma}ds=\frac{\Gamma(\gamma+1)}{\Gamma(\alpha+\gamma+1)}t^{\alpha+\gamma},$

that is,

$t^{\beta}=t^{\alpha+\gamma}=J^{\alpha}\left(\frac{\Gamma(\alpha+\gamma+1)}{\Gamma(\gamma+1)}t^{\gamma}\right)\in
J^{\alpha}L^{2}(0,T).$

Thus we see that $t^{\beta}\in H_{\alpha}(0,T)$.

In Proposition 2.4 in [18], the direct proof for $t^{\beta}\in
H_{\alpha}(0,T)$ is given for more restricted $\beta>0$ and
$0<\alpha<\frac{1}{2}$, which is more complicated than the proof here.

We set

$None$
$\partial_{t}^{\alpha}:=(J^{\alpha})^{-1}=J^{-\alpha}\quad\mbox{with}\quad\mathcal{D}(\partial_{t}^{\alpha})=H_{\alpha}(0,T)=J^{\alpha}L^{2}(0,T).$

The essence of this extension of $\partial_{t}^{\alpha}$ is that we define
$\partial_{t}^{\alpha}$ as the inverse to the isomorphism $J^{\alpha}$ on
$L^{2}(0,T)$ onto $H_{\alpha}(0,T)$. For estimating or treating
$\partial_{t}^{\alpha}v=g$ later, we will often consider through
$J^{\alpha}g$, as is already calculated in Example 2.1.

From Proposition 2.2 we can directly derive
Proposition 2.3.
There exists a constant $C>0$ such that

$C^{-1}\|v\|_{H_{\alpha}(0,T)}\leq\|\partial_{t}^{\alpha}v\|_{L^{2}(0,T)}\leq
C\|v\|_{H_{\alpha}(0,T)}$

for all $v\in H_{\alpha}(0,T)$.
Example 2.2.
We return to Example 2.1. Let $\beta>\alpha-\frac{1}{2}$. Then $t^{\beta}\in
H_{\alpha}(0,T)$ and

$J^{\alpha}t^{\beta-\alpha}=\frac{\Gamma(\beta-\alpha+1)}{\Gamma(\beta+1)}t^{\beta}.$

Hence, by the definition of $\partial_{t}^{\alpha}$, we obtain

$t^{\beta-\alpha}=\frac{\Gamma(\beta-\alpha+1)}{\Gamma(\beta+1)}\partial_{t}^{\alpha}t^{\beta},$

that is,

$None$
$\partial_{t}^{\alpha}t^{\beta}=\frac{(\beta+1)}{\Gamma(\beta-\alpha+1)}t^{\beta-\alpha}\quad\mbox{if
$\beta>\alpha-\frac{1}{2}$ and $0<\alpha<1$.}$

If $0<\alpha<\frac{1}{2}$, then $\beta<0$ is possible and for $\beta<0$ we
have $t^{\beta}\not\in W^{1,1}(0,T)$. Therefore $d_{t}^{\alpha}t^{\beta}$
cannot be defined directly.

Next we will define $\partial_{t}^{\alpha}$ for general $\alpha>0$. Let
$\alpha:=m+\sigma$ with $m\in\mathbb{N}\cup\\{0\\}$ and $0<\sigma\leq 1$. Then
we define

$None$
$\left\\{\begin{array}[]{rl}&H_{m}(0,T):=\left\\{\begin{array}[]{rl}&\left\\{v\in
H^{m}(0,T);\,v(0)=\cdots=\frac{d^{m-1}v}{dt^{m-1}}(0)=0\right\\},\quad m\geq
1,\\\ &L^{2}(0,T),\quad m=0,\\\ \end{array}\right.\\\
&H_{m+\sigma}(0,T):=\left\\{v\in H_{m}(0,T);\,\frac{d^{m}v}{dt^{m}}\in
H_{\sigma}(0,T)\right\\}\end{array}\right.$

and

$\|v\|_{H_{m+\sigma}(0,T)}:=\left(\left\|\frac{d^{m}}{dt^{m}}v\right\|^{2}_{L^{2}(0,T)}+\left\|\frac{d^{m}}{dt^{m}}v\right\|^{2}_{H_{\sigma}(0,T)}\right)^{\frac{1}{2}}.$

We can easily verify that $H_{m+\sigma}(0,T)$ is a Banach space.

We similarly define ${{}^{\alpha}{H}}(0,T)$ for each $\alpha>0$.

Then, in view of Proposition 2.2, we can prove
Proposition 2.4.
Let $m\in\mathbb{N}\cup\\{0\\}$ and $0<\sigma\leq 1$. Then
$J^{m+\sigma}:L^{2}(0,T)\longrightarrow H_{m+\sigma}(0,T)$ is an isomorphism.
Proof.
We can assume that $m\geq 1$. By the definition, $v\in H_{m+\sigma}(0,T)$ if
and only if

$\frac{d^{m}}{dt^{m}}v\in H_{\sigma}(0,T),\quad
v(0)=\cdots=\frac{d^{m-1}}{dt^{m-1}}v(0)=0,$

which implies that $\frac{d^{m}}{dt^{m}}v=J^{\sigma}w$ with some $w\in
L^{2}(0,T)$. By $v(0)=\cdots=\frac{d^{m-1}}{dt^{m-1}}v(0)=0$, we see that

$v(t)=\frac{1}{(m-1)!}\int^{t}_{0}(t-s)^{m-1}\frac{d^{m}}{dt^{m}}v(s)ds,\quad
0<t<T.$

Therefore, exchanging the order of the integral, we have

$\displaystyle
v(t)=\frac{1}{(m-1)!}\int^{t}_{0}(t-s)^{m-1}\left(\frac{1}{\Gamma(\sigma)}\int^{s}_{0}(s-\xi)^{\sigma-1}w(\xi)d\xi\right)ds$
$\displaystyle=$
$\displaystyle\frac{1}{(m-1)!\Gamma(\sigma)}\int^{t}_{0}\left(\int^{t}_{\xi}(t-s)^{m-1}(s-\xi)^{\sigma-1}ds\right)w(\xi)d\xi=\frac{1}{\Gamma(m+\sigma)}\int^{t}_{0}(t-\xi)^{m+\sigma-1}w(\xi)d\xi$
$\displaystyle=$ $\displaystyle J^{m+\sigma}w(t)\quad\mbox{for $0<t<T$.}$

Hence, $v\in J^{m+\sigma}L^{2}(0,T)$, which means that
$J^{m+\sigma}L^{2}(0,T)\supset H_{m+\sigma}(0,T)$. The converse inclusion is
direct, and we see that $J^{m+\sigma}L^{2}(0,T)=H_{m+\sigma}(0,T)$. The norm
equivalence between $\|v\|_{L^{2}(0,T)}$ and
$\|J^{m+\sigma}v\|_{H_{m+\sigma}(0,T)}$, readily follows from the definition
and Proposition 2.2. $\blacksquare$

For $\alpha=m+\sigma$ with $m\in\mathbb{N}$ and $0<\sigma\leq 1$, we now
define $\partial_{t}^{\alpha}$ as extension of the Caputo derivative

$d_{t}^{\alpha}v(t)=\frac{1}{\Gamma(m+1-\alpha)}\int^{t}_{0}(t-s)^{m-\alpha}\frac{d^{m+1}}{ds^{m+1}}v(s)ds,\quad
0<t<T.$

We note that $d_{t}^{\alpha}$ requires $(m+1)$-times differentiablity of $v$.

By Proposition 2.4, the inverse to $J^{\alpha}$ exists for each $\alpha\geq
0$, and by $J^{-\alpha}$ we denote the inverse:

$J^{-\alpha}:=(J^{\alpha})^{-1}.$

As the extension of such $d_{t}^{\alpha}$ to $H_{m+\sigma}(0,T)$, we define

$None$
$\partial_{t}^{m+\sigma}=J^{-m-\sigma}\quad\mbox{with}\quad\mathcal{D}(\partial_{t}^{m+\sigma})=H_{m+\sigma}(0,T).$

Thus
Proposition 2.5.
Let $\alpha,\beta\geq 0$.
(i)

$\partial_{t}^{\alpha}:H_{\alpha+\beta}(0,T)\longrightarrow H_{\beta}(0,T)$

and

$J^{\alpha}:H_{\beta}(0,T)\longrightarrow H_{\alpha+\beta}(0,T)$

are isomorphisms.
(ii) It holds

$J^{-\alpha}J^{\beta}=J^{-\alpha+\beta}\quad\mbox{on
$\mathcal{D}(J^{-\alpha+\beta})$}.$

Here we set

$\mathcal{D}(J^{-\alpha+\beta})=\left\\{\begin{array}[]{rl}&L^{2}(0,T)\quad\mbox{if
$-\alpha+\beta\geq 0$},\\\ &H_{\alpha-\beta}(0,T)\quad\mbox{if
$-\alpha+\beta<0$}.\end{array}\right.$

Proof.
Part (i) is seen by (2.7) and Proposition 2.4. Part (ii) can be proved as
follows. Let $-\alpha+\beta\geq 0$. If $\alpha=\beta$, then
$J^{-\alpha}J^{\alpha}=I$: the identity operator on $L^{2}(0,T)$ by (2.7) and
the conclusion is trivial. Let $\beta>\alpha$. Set $\gamma:=\beta-\alpha>0$.
Let $v\in L^{2}(0,T)$ be arbitrary. Then, by Lemma 2.1, we have

$J^{-\alpha}J^{\beta}v=J^{-\alpha}(J^{\alpha+\gamma}v)=J^{-\alpha}(J^{\alpha}J^{\gamma})v=(J^{-\alpha}J^{\alpha})J^{\gamma}v=J^{\gamma}v.$

Since $J^{\gamma}v=J^{-\alpha+\beta}v$, we obtain
$J^{-\alpha}J^{\beta}v=J^{-\alpha+\beta}v$ for each $v\in L^{2}(0,T)$.

Next let $-\alpha+\beta<0$. Given $v\in H_{\alpha-\beta}(0,T)$ arbitrarily, by
Proposition 2.4 we can find $w\in L^{2}(0,T)$ such that $v=J^{\alpha-\beta}w$.
Then

$J^{-\alpha}J^{\beta}v=J^{-\alpha}J^{\beta}(J^{\alpha-\beta}w)=J^{-\alpha}(J^{\beta+(\alpha-\beta)}w)=J^{-\alpha}J^{\alpha}w=w$

by $\beta,\alpha-\beta>0$ and Lemma 2.1. Therefore, $J^{-\alpha}J^{\beta}v=w$.
Since $v=J^{\alpha-\beta}w$ implies $w=(J^{\alpha-\beta})^{-1}v$, we have

$w=(J^{\alpha-\beta})^{-1}v=J^{-(\alpha-\beta)}v=J^{-\alpha+\beta}v,$

that is, $J^{-\alpha}J^{\beta}v=J^{-\alpha+\beta}v$ for each $v\in
H_{\alpha-\beta}(0,T)$. Thus the proof of Proposition 2.5 is complete.
$\blacksquare$

Next, for $0<\alpha<1$, we characterize $\partial_{t}^{\alpha}$ as extension
over $H_{\alpha}(0,T)$ of the operator $d_{t}^{\alpha}$ defined on
${{}_{0}{C^{1}[0,T]}}$. Let $X$ and $Y$ be Banach spaces with the norms
$\|\cdot\|_{X}$ and $\|\cdot\|_{Y}$ respectively, and let $K:X\longrightarrow
Y$ be a densely defined linear operator. We call $K$ a closed operator if
$v_{n}\in\mathcal{D}(K)$, $\lim_{n\to\infty}v_{n}=v$ and $Kv_{n}$ converges to
some $w$ in $Y$, then $v\in\mathcal{D}(K)$ and $Kv=w$. By $\overline{K}$ we
denote the closure of an operator $K$ from $X$ to $Y$, that is, $\overline{K}$
is the minimum closed extension of $K$ in the sense that if $\widetilde{K}$ is
a closed operator such that $\mathcal{D}(\widetilde{K})\supset\mathcal{D}(K)$,
then $\mathcal{D}(\widetilde{K})\supset\mathcal{D}(\overline{K})$. It is
trivial that $\overline{K}=K$ for a closed operator $K$. Moreover we say that
$K$ is closable if there exists $\overline{K}$.

Here we always consider the operator $d_{t}^{\alpha}$ with the domain
${{}_{0}{C^{1}[0,T]}}$: $d_{t}^{\alpha}:{{}_{0}{C^{1}[0,T]}}\subset
H_{\alpha}(0,T)\,\longrightarrow\,L^{2}(0,T)$. Then we can prove (e.g., [18]):
Proposition 2.6.
The operator $d_{t}^{\alpha}$ with the domain ${{}_{0}{C^{1}[0,T]}}$ is
closable and $\overline{d_{t}^{\alpha}}=\partial_{t}^{\alpha}$.

Proposition 2.6 is not used later but means that our derivative
$\partial_{t}^{\alpha}$ is reasonable as the minimum extension of the
classical Caputo derivative $d_{t}^{\alpha}$ for $v\in{{}_{0}{C^{1}[0,T]}}$.

Interpretation of $\mathcal{D}(\partial_{t}^{\alpha})=H_{\alpha}(0,T)$.
Let $\alpha>\frac{1}{2}$ and let $v\in H_{\alpha}(0,T)$. By the definition
(2.3) of $H_{\alpha}(0,T)$, we can choose an approximating sequence
$v_{n}\in{{}_{0}{C^{1}[0,T]}}$, $n\in\mathbb{N}$ such that
$\lim_{n\to\infty}\|v_{n}-v\|_{H_{\alpha}(0,T)}=0$. By the Sobolev embedding
(e.g., Adams [1]), we see that $H_{\alpha}(0,T)\subset H^{\alpha}(0,T)\subset
C[0,T]$ if $\alpha>\frac{1}{2}$, so that $v\in C[0,T]$ and
$\lim_{n\to\infty}\|v_{n}-v\|_{C[0,T]}=0$. Therefore,

$\lim_{n\to\infty}|v(0)-v_{n}(0)|\leq\lim_{n\to\infty}\|v_{n}-v\|_{C[0,T]}=0,$

that is, $v(0)=\lim_{n\to\infty}v_{n}(0)$. Therefore, $v\in H_{\alpha}(0,T)$
with $\alpha>\frac{1}{2}$, yields $v(0)=0$. In other words, if
$\alpha>\frac{1}{2}$, then $v\in\mathcal{D}(\partial_{t}^{\alpha})$ means that
$v=v(t)$ satisfies the zero initial condition, which is useful for the
formulation of the initial value problem in Sections 5 and 6. We can similarly
consider and see that $v\in\mathcal{D}(\partial_{t}^{\alpha})$ with
$\alpha>\frac{3}{2}$ yields $v(0)=\partial_{t}v(0)=0$.

As is directly proved by the Young inequality on the convolution, we see that

$D_{t}^{\alpha}v\in L^{1}(0,T)\quad\mbox{if $v\in W^{1,1}(0,T)$}.$

However, we do not necessarily have $D_{t}^{\alpha}v\in L^{2}(0,T)$. Indeed,

$D_{t}^{\alpha}t^{\beta}=\frac{\Gamma(1+\beta)}{\Gamma(1-\alpha+\beta)}t^{\beta-\alpha}\quad\mbox{for
$\beta>0$.}$

If $\beta-\alpha<-\frac{1}{2}$ and $\beta>0$, then
$D_{t}^{\alpha}t^{\beta}\not\in L^{2}(0,T)$, in spite of $t^{\beta}\in
W^{1,1}(0,T)$. This means that if we want to keep the range of
$D_{t}^{\alpha}$ within $L^{2}(0,T)$, then even $W^{1,1}(0,T)$ of a space of
differentiable functions is not sufficient, although $D_{t}^{\alpha}$ are
concerned with $\alpha$-times differentiability.

In general, we can readily prove
Proposition 2.7.
We have

$None$ $\partial_{t}^{\alpha}v=D_{t}^{\alpha}v=d_{t}^{\alpha}v\quad\mbox{for
$v\in{{}_{0}{C^{1}[0,T]}}$}$

and

$None$
$\partial_{t}^{\alpha}v(t)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int^{t}_{0}(t-s)^{-\alpha}v(s)ds=D_{t}^{\alpha}v(t)\quad\mbox{for
$v\in H_{\alpha}(0,T)$.}$

Proof.
Equation (2.8) is easily seen. We will prove (2.9) as follows. For arbitrary
$v\in H_{\alpha}(0,T)$, Proposition 2.2 yields that $v=J^{\alpha}w$ with some
$w\in L^{2}(0,T)$ and $w=\partial_{t}^{\alpha}v$. On the other hand, Lemma 2.1
implies

$D_{t}^{\alpha}v=\frac{d}{dt}J^{1-\alpha}v=\frac{d}{dt}J^{1-\alpha}(J^{\alpha}w)=\frac{d}{dt}Jw=w.$

Therefore, $D_{t}^{\alpha}v=\partial_{t}^{\alpha}v$, and then the proof of
Proposition 2.7 is complete. $\blacksquare$

Equation (2.9) means that $\partial_{t}^{\alpha}$ coincides with the Riemann-
Liouville fractional derivative, provided that we consider $H_{\alpha}(0,T)$
as the domain of $\partial_{t}^{\alpha}$ and $D_{t}^{\alpha}$. This extension
$\partial_{t}^{\alpha}$ of $d_{t}^{\alpha}|_{{{}_{0}{C^{1}[0,T]}}}$ is not yet
complete, and in the next subsection, we will continue to extend.

§2.3. Definition of $\partial_{t}^{\alpha}$: completion of the extension of
$d_{t}^{\alpha}$

By the current extension of $\partial_{t}^{\alpha}$, we understand that
$\partial_{t}^{\alpha}1=0$ for $0<\alpha<\frac{1}{2}$, but
$\partial_{t}^{\alpha}1$ cannot be defined for $\frac{1}{2}\leq\alpha<1$:

$\left\\{\begin{array}[]{rl}&1\in
H_{\alpha}(0,T)=\mathcal{D}(\partial_{t}^{\alpha})\quad\mbox{$0<\alpha<\frac{1}{2}$},\\\
&1\not\in\mathcal{D}(\partial_{t}^{\alpha})\quad\mbox{$\frac{1}{2}\leq\alpha<1$}.\end{array}\right.$

Moreover, we note that $d_{t}^{\alpha}1=0$ and
$D_{t}^{\alpha}1=\frac{t^{-\alpha}}{\Gamma(1-\alpha)}$ for all
$\alpha\in(0,1)$. Therefore, $\partial_{t}^{\alpha}1$ is not consistent with
neither the classical fractional derivative $d_{t}^{\alpha}$ nor
$D_{t}^{\alpha}$, which suggests that our current extension from
$d_{t}^{\alpha}$ to $\partial_{t}^{\alpha}$ is not sufficient as fractional
derivative. Moreover, as is seen in Section 7, the current
$\partial_{t}^{\alpha}$ is not convenient for treating less regular source
terms in fractional differential equations. In order to define
$\partial_{t}^{\alpha}v$ in more general spaces such as $L^{2}(0,T)$, we
should continue to extend $\partial_{t}^{\alpha}$.

We recall (2.1) and (2.3) for $\alpha>0$. We can readily verify that
$\tau:H_{\alpha}(0,T)\longrightarrow{{}^{\alpha}{H}}(0,T)$ is an isomorphism
with $\tau$ defined by (2.2). Then
Proposition 2.8.
Let $\alpha>0$ and $\beta\geq 0$. Then
$J_{\alpha}:\,^{\beta}{H}(0,T)\,\longrightarrow\,^{\alpha+\beta}{H}(0,T)$ is
an isomorphism.

We recall that
$J^{\alpha}:H_{\beta}(0,T)\,\longrightarrow\,H_{\alpha+\beta}(0,T)$ is an
isomorphism by Proposition 2.5 (i). Proposition 2.8 can be proved via the
mapping $\tau:L^{2}(0,T)\longrightarrow L^{2}(0,T)$ defined by (2.2).

When $J^{\alpha}$ remains an operator defined over $L^{2}(0,T)$, the operator
$J^{\alpha}$ is not defined for $f\in L^{1}(0,T)$, but the integral
$\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}f(s)ds$ itself exists as
a function in $L^{1}(0,T)$ for any $f\in L^{1}(0,T)$. This is a substantial
inconvenience, and we have to make suitable extension of
$J^{\alpha}|_{L^{2}(0,T)}$. Our formulation is based on $L^{2}$-space and we
cannot directly treat $L^{1}(0,T)$-space. Thus we introduce the dual space of
$H_{\alpha}(0,T)$ which contains $L^{1}(0,T)$.

The key idea for the further extension of $\partial_{t}^{\alpha}$, is the
family $\\{H_{\alpha}(0,T)\\}_{\alpha>0}$ and the isomorphism
$\partial_{t}^{\alpha}:H^{\alpha+\beta}(0,T)\,\longrightarrow\,H^{\beta}(0,T)$
with $\alpha,\beta>0$, which is called a Hilbert scale. We can refer to
Chapter V in Amann [2] as for a general reference.

Let $X$ be a Hilbert space over $\mathbb{R}$ and let $V\subset X$ be a dense
subspace of $X$, and the embedding $V\longrightarrow X$ be continuous. By the
dual space $X^{\prime}$ of $X$, we call the space of all the bounded linear
functionals defined on $X$. Then, identifying the dual space $X^{\prime}$ with
itself, we can conclude that $X$ is a dense subspace of the dual space
$V^{\prime}$ of $V$:

$V\subset X\subset V^{\prime}.$

By $\,{}_{V^{\prime}}<f,\,v>_{V}$, we denote the value of $f\in V^{\prime}$ at
$v\in V$. We note that

$\,{}_{V^{\prime}}<f,\,v>_{V}=(f,v)_{X}\quad\mbox{if $f\in X$},$

where $(f,v)_{X}$ is the scalar product in $X$.

We note that $H_{\alpha}(0,T)$ and ${{}^{\alpha}{H}}(0,T)$ are both dense in
$L^{2}(0,T)$. Henceforth, identifying the dual space $L^{2}(0,T)^{\prime}$
with itself, by the above manner, we can define $(H_{\alpha}(0,T))^{\prime}$
and $({{}^{\alpha}{H}}(0,T))^{\prime}$. Then

$None$ $\left\\{\begin{array}[]{rl}&H_{\alpha}(0,T)\subset
L^{2}(0,T)\subset(H_{\alpha}(0,T))^{\prime}=:H_{-\alpha}(0,T),\\\
&{{}^{\alpha}{H}}(0,T)\subset
L^{2}(0,T)\subset({{}^{\alpha}{H}}(0,T))^{\prime}=:\,{{{}^{-\alpha}{H}}(0,T)},\quad\alpha>0,\end{array}\right.$

where the above inclusions mean dense subsets. We refer to e.g., Brezis [4],
Yosida [36] as for general treatments for dual spaces.

Let $X,Y$ be Hilbert spaces and let $K:X\longrightarrow Y$ be a bounded linear
operator with $\mathcal{D}(K)=X$. Then we recall that the dual operator
$K^{\prime}$ is the maximum operator among operators
$\widehat{K}:Y^{\prime}\longrightarrow X^{\prime}$ with
$\mathcal{D}(\widehat{K})\subset Y^{\prime}$ such that
${}_{X^{\prime}}{<}\widehat{K}y,x>_{X}\,=\,{{}_{Y^{\prime}}{<}y,Kx>_{Y}}$ for
each $x\in X$ and $y\in\mathcal{D}(\widehat{K})\subset Y^{\prime}$ (e.g.,
[4]).

Henceforth, we consider the dual operator $(J^{\alpha})^{\prime}$ of
$J^{\alpha}:L^{2}(0,T)\longrightarrow H_{\alpha}(0,T)$ and the dual
$(J_{\alpha})^{\prime}$ of
$J_{\alpha}:L^{2}(0,T)\longrightarrow\,^{\alpha}H(0,T)$ by setting
$X=L^{2}(0,T)$ and $Y=H_{\alpha}(0,T)$ or $Y=\,^{\alpha}H(0,T)$.

Then we can show
Proposition 2.9.
Let $\alpha>0$ and $\beta\geq 0$. Then:
(i) $(J^{\alpha})^{\prime}:H_{-\alpha-\beta}(0,T)\longrightarrow
H_{-\beta}(0,T)$ is an isomorphism.
In particular, $(J^{\alpha})^{\prime}:H_{-\alpha}(0,T)\longrightarrow
L^{2}(0,T)$ is an isomorphism.
(ii)
$(J_{\alpha})^{\prime}:\,{{}^{-\alpha-\beta}{H}(0,T)}\longrightarrow\,^{-\beta}{H}(0,T)$
is an isomorphism.
In particular, $(J_{\alpha})^{\prime}:\,{{}^{-\alpha}{H}(0,T)}\longrightarrow
L^{2}(0,T)$ is an isomorphism.
(iii) It holds:

$J^{\alpha}v=(J_{\alpha})^{\prime}v\quad\mbox{for $v\in L^{2}(0,T)$}.$

Henceforth we write $J_{\alpha}^{\prime}:=(J_{\alpha})^{\prime}$. We see

$J_{\alpha}^{\prime}(J_{\alpha}^{\prime})^{-1}w=w\quad\mbox{for $w\in
L^{2}(0,T)$}$

and

$(J_{\alpha}^{\prime})^{-1}J_{\alpha}^{\prime}w=w\quad\mbox{for
$w\in{{}^{-\alpha}{H}}(0,T)$}.$

Proof. From Proposition 2.8, in terms of the closed range theorem (e.g.,
Section 7 of Chapter 2 in [4]), we can see (i) and (ii).

We now prove (iii). We can directly verify
$(J^{\alpha}v,\,w)_{L^{2}(0,T)}=(v,\,J_{\alpha}w)_{L^{2}(0,T)}$ for each
$v,w\in L^{2}(0,T)$. Hence, by the maximality property of the dual operator
$J_{\alpha}^{\prime}$ of $J_{\alpha}$, we see that $J_{\alpha}^{\prime}\supset
J^{\alpha}$. Thus the proof of Proposition 2.9 is complete. $\blacksquare$

Thanks to Propositions 2.5 and 2.8-2.9, for $\beta\geq 0$, we can regard the
operators $J_{\alpha}$ with $\mathcal{D}(J_{\alpha})=\,{{}^{\beta}{H}}(0,T)$
and $J^{\alpha}$ with $\mathcal{D}(J^{\alpha})=H_{\beta}(0,T)$, and
accordingly also the operators $J_{\alpha}^{\prime}$ with
$\mathcal{D}(J_{\alpha}^{\prime})=\,^{-\alpha-\beta}{H}(0,T)$ and
$(J^{\alpha})^{\prime}$ with
$\mathcal{D}((J^{\alpha})^{\prime})=H_{-\alpha-\beta}(0,T)$. We do not specify
the domains if we need not emphasize them.

We can define $J_{\alpha}^{\prime}u$ for $u\in L^{1}(0,T)$ as follows. We note
that $J_{\alpha}^{\prime}$ is the dual operator of
$J_{\alpha}:\,^{\gamma}H(0,T)\,\longrightarrow\,^{\alpha+\gamma}H(0,T)$, where
we choose $\gamma>0$ such that $\mathcal{D}(J_{\alpha}^{\prime})\supset
L^{1}(0,T)$.

To this end, choosing $\gamma>0$ such that $\alpha+\gamma>\frac{1}{2}$, we
regard $J_{\alpha}^{\prime}$ as an operator :
${{}^{-\alpha-\gamma}{H}}(0,T)\,\longrightarrow\,{{}^{-\gamma}{H}}(0,T)$. Then
we can define $J_{\alpha}^{\prime}u$ for $u\in L^{1}(0,T)$. Indeed, the
Sobolev embedding yields that

${{}^{\alpha+\gamma}{H}}(0,T)\subset H^{\alpha+\gamma}(0,T)\subset C[0,T]$

by $\alpha+\gamma>\frac{1}{2}$. Therefore, any $u\in L^{1}(0,T)$ can be
considered as an element in $(^{\alpha+\gamma}{H}(0,T))^{\prime}$ by

${}_{{}^{-\alpha-\gamma}H(0,T)}<u,\,\varphi>_{{}^{\alpha+\gamma}H(0,T)}\,=\int^{T}_{0}u(t)\varphi(t)dt\quad\mbox{for
$\varphi\in\,{{}^{\alpha+\gamma}{H}(0,T)}$},$

which means that $L^{1}(0,T)\subset\mathcal{D}(J_{\alpha}^{\prime})=\
^{-\alpha-\gamma}H(0,T)$ and $J_{\alpha}^{\prime}u$ is well-defined for $u\in
L^{1}(0,T)$.

Henceforth, we always make the above definition of $J_{\alpha}^{\prime}u$ for
$u\in L^{1}(0,T)$, if not specified.

Then we can improve Proposition 2.9 (iii) as
Proposition 2.10.
We have

$None$
$J_{\alpha}^{\prime}u=\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}u(s)ds,\quad
0<t<T\quad\mbox{for $u\in L^{1}(0,T)$}.$

We can write (2.11) as $J_{\alpha}^{\prime}u=J^{\alpha}u$ for $u\in
L^{1}(0,T)$, when we do not specify the domain $\mathcal{D}(J^{\alpha})$.

Proof of Prposition 2.10.
Let $\gamma>\frac{1}{2}$. We consider $J_{\alpha}^{\prime}$ as an operator
$J_{\alpha}^{\prime}:\,{{}^{-\alpha-\gamma}{H}}(0,T)\,\longrightarrow\,{{}^{-\gamma}{H}}(0,T)$.

Let $u\in L^{1}(0,T)$ be arbitrary. Since
$L^{1}(0,T)\subset\,{{}^{-\gamma}{H}}(0,T)$ and $L^{2}(0,T)$ is dense in
$L^{1}(0,T)$, we can choose $u_{n}\in L^{2}(0,T)$, $n\in\mathbb{N}$ such that
$u_{n}\longrightarrow u$ in $L^{1}(0,T)$ as $n\to\infty$. By Proposition 2.9
(iii), we have $J_{\alpha}^{\prime}u_{n}=J^{\alpha}u_{n}$ for
$n\in\mathbb{N}$. By Proposition 2.9 (ii), we obtain

$None$ $J_{\alpha}^{\prime}u_{n}\longrightarrow
J_{\alpha}^{\prime}u\quad\mbox{in ${{}^{-\gamma}{H}}(0,T)$}.$

On the other hand, by $u\in L^{1}(0,T)$, the Young inequality on the
convolution implies

$\|J^{\alpha}u_{n}-J^{\alpha}u\|_{L^{1}(0,T)}\leq\frac{1}{\Gamma(\alpha)}\|s^{\alpha-1}\|_{L^{1}(0,T)}\|u_{n}-u\|_{L^{1}(0,T)},$

which yields
$J^{\alpha}u_{n}=J_{\alpha}^{\prime}u_{n}\,\longrightarrow\,J^{\alpha}u$ in
$L^{1}(0,T)$, that is, $J^{\alpha}u_{n}\,\longrightarrow\,J^{\alpha}u$ in
${}^{-\gamma}{H}(0,T)$. Therefore, with (2.12), we obtain
$J_{\alpha}^{\prime}u=J^{\alpha}u$. Thus the proof of Proposition 2.10 is
complete. $\blacksquare$

Now we complete the extension of $d_{t}^{\alpha}$ with the domain
${{}_{0}{C^{1}[0,T]}}$.
Definition 2.1.
For $\beta\geq 0$, we define

$None$
$\partial_{t}^{\alpha}:=(J_{\alpha}^{\prime})^{-1}\quad\mbox{with}\quad\mathcal{D}(\partial_{t}^{\alpha})=\,H_{\beta}(0,T)\quad\mbox{or}\quad\mathcal{D}(\partial_{t}^{\alpha})=\,^{-\beta}H(0,T).$

Thus
Theorem 2.1. Let $\alpha>0$ and $\beta\geq 0$. Then
$\partial_{t}^{\alpha}:\,{{}^{-\beta}H}(0,T)\,\longrightarrow\,^{-\alpha-\beta}H(0,T)$
and
$\partial_{t}^{\alpha}:H_{\alpha+\beta}(0,T)\,\longrightarrow\,H_{\beta}(0,T)$
are both isomorphisms.

The extension
$\partial_{t}^{\alpha}:\,^{-\beta}H(0,T)\,\longrightarrow\,^{-\alpha-\beta}H(0,T)$
is not only a theoretical interest, but also useful for studies of fractional
differential equations even if we consider all the functions within
$L^{2}(0,T)$, as we will do in Sections 6 and 7.

We consider a special case $\beta=0$. Then
$(J_{\alpha}^{\prime})^{-1}:\mathcal{D}((J_{\alpha}^{\prime})^{-1})=L^{2}(0,T)\,\longrightarrow\,{{}^{-\alpha}{H}}(0,T)$.
Then, since Proposition 2.9 (iii) yields

$None$ $(J_{\alpha}^{\prime})^{-1}\supset(J^{\alpha})^{-1},$

we see that this $(J_{\alpha}^{\prime})^{-1}=\partial_{t}^{\alpha}$ defined on
$L^{2}(0,T)$ is an extension defined previously in $H_{\alpha}(0,T)$. In
particular, this extended $\partial_{t}^{\alpha}$ still satisfies

$None$ $(J_{\alpha}^{\prime})^{-1}H_{\alpha}(0,T)=L^{2}(0,T).$

Our completely extended derivative $\partial_{t}^{\alpha}$ of $d_{t}^{\alpha}$
with $\mathcal{D}(d_{t}^{\alpha})={{}_{0}{C^{1}[0,T]}}$, operates similarly to
the Riemann-Liouville fractional derivative $D_{t}^{\alpha}$, and is
equivalent to $\frac{d}{dt}J^{1-\alpha}$ associated with the domain
$\,{}^{-\beta}H(0,T)$ and the range $\,{}^{-\alpha-\beta}H(0,T)$ with
$\alpha>0$ and $\beta\geq 0$.

Proposition 2.10 enables us to calculate $\partial_{t}^{\alpha}u=f$ provided
that $u,f\in L^{1}(0,T)$, and we show
Example 2.3.
Let $0<\alpha<1$. Choosing $\alpha+\gamma>\frac{1}{2}$ we consider
$\partial_{t}^{\alpha}$ with
$\mathcal{D}(\partial_{t}^{\alpha})=\,^{-\gamma-\alpha}H(0,T)\supset
L^{1}(0,T)$. Then $1\in\mathcal{D}(\partial_{t}^{\alpha})$, and we have

$\partial_{t}^{\alpha}1=\frac{1}{\Gamma(1-\alpha)}t^{-\alpha}.$

Indeed, since $\frac{1}{\Gamma(1-\alpha)}t^{-\alpha}\in L^{1}(0,T)$,
Proposition 2.10 yields

$J_{\alpha}^{\prime}\left(\frac{1}{\Gamma(1-\alpha)}t^{-\alpha}\right)=\frac{1}{\Gamma(1-\alpha)}\left(\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}s^{-\alpha}ds\right)=1.$

Therefore, the definition justifies
$\partial_{t}^{\alpha}1=\frac{1}{\Gamma(1-\alpha)}t^{-\alpha}$.

Before proceeding to the next section, we will provide two propositions.
Proposition 2.11.
Let $0<\alpha<1$. Then

$\partial_{t}^{\alpha}u(t)=\frac{d}{dt}\left(\frac{1}{\Gamma(1-\alpha)}\int^{t}_{0}(t-s)^{-\alpha}u(s)ds\right)\quad\mbox{in
$(C^{\infty}_{0}(0,T))^{\prime}$}$

for $u\in L^{1}(0,T)$, where $\frac{d}{dt}$ is taken in
$(C^{\infty}_{0}(0,T))^{\prime}$.

In the proposition, we note
$\frac{1}{\Gamma(1-\alpha)}\int^{t}_{0}(t-s)^{-\alpha}u(s)ds\in
L^{1}(0,T)\subset(C^{\infty}_{0}(0,T))^{\prime}$. We remark that we cannot
take the pointwise differentiation of $\int^{t}_{0}(t-s)^{-\alpha}u(s)ds$ in
general for $u\in L^{1}(0,T)$.

Proposition 2.11 enables us to calculate $\partial_{t}^{\alpha}u=f$ for $u\in
L^{1}(0,T)$ even if $\partial_{t}^{\alpha}u$ cannot be defined within
$L^{1}(0,T)$, as Example 2.4 (a) shows.

Example 2.4.
(a) Let $\frac{1}{2}<\alpha<1$ and let

$None$ $h_{t_{0}}(t)=\left\\{\begin{array}[]{rl}&0,\quad t\leq t_{0},\\\
&1,\quad t>t_{0}\end{array}\right.$

with arbitrary $t_{0}\in(0,T)$. We can verify that $h_{t_{0}}\in
H_{1-\alpha}(0,T)$ if $\alpha>\frac{1}{2}$. Indeed

$\displaystyle\int^{T}_{0}\int^{T}_{0}\frac{|h_{t_{0}}(t)-h_{t_{0}}(s)|^{2}}{|t-s|^{3-2\alpha}}dsdt=2\int^{t_{0}}_{0}\left(\int^{T}_{t_{0}}\frac{1}{|t-s|^{3-2\alpha}}dt\right)ds$
$\displaystyle=$
$\displaystyle\frac{1}{1-\alpha}\int^{t_{0}}_{0}((t_{0}-s)^{-2+2\alpha}-(T-s)^{-2+2\alpha})ds<\infty$

by $-2+2\alpha>-1$.

Therefore, by Proposition 2.2, we can choose $w\in L^{2}(0,T)$ such that
$J^{1-\alpha}w=h_{t_{0}}$. By Proposition 2.11, we can calculate
$\partial_{t}^{\alpha}w$ because $w\in L^{2}(0,T)\subset L^{1}(0,T)$:

$\partial_{t}^{\alpha}w=\frac{d}{dt}\left(\frac{1}{\Gamma(1-\alpha)}\int^{t}_{0}(t-s)^{-\alpha}w(s)ds\right)=\frac{d}{dt}J^{1-\alpha}w=\frac{d}{dt}h_{t_{0}}=\delta_{t_{0}},$

which is a Dirac delta function satisfying

$\,{}_{(C^{\infty}_{0}(0,T))^{\prime}}<\delta_{t_{0}},\,\psi>_{C^{\infty}_{0}(0,T)}\,=\psi(t_{0})\quad\mbox{for
all $\psi\in C^{\infty}_{0}(0,T)$}.$

(b) We have

$None$
$\partial_{t}^{\alpha}t^{\beta}=\frac{\Gamma(1+\beta)}{\Gamma(1-\alpha+\beta)}t^{\beta-\alpha}\quad
0<\alpha<1,\,\beta>-1$

in ${}^{-\alpha-\gamma}H(0,T)$ with some $\alpha+\gamma>\frac{1}{2}$. See
Example 2.3 as (2.17) with $\beta=0$. We remark that $\beta-\alpha<-1$ is
possible, and so $t^{\beta-\alpha}\not\in L^{1}(0,T)$ may occur.

Indeed, by $\beta>-1$, we can see

$\frac{1}{\Gamma(1-\alpha)}\int^{t}_{0}(t-s)^{-\alpha}s^{\beta}ds=\frac{\Gamma(1+\beta)}{\Gamma(2-\alpha+\beta)}t^{1-\alpha+\beta}.$

Taking the derivative in the sense of $(C_{0}^{\infty}(0,T))^{\prime}$ to see

$\displaystyle\frac{d}{dt}\left(\frac{\Gamma(1+\beta)}{\Gamma(2-\alpha+\beta)}t^{1-\alpha+\beta}\right)$
$\displaystyle=$
$\displaystyle\frac{\Gamma(1+\beta)}{\Gamma(2-\alpha+\beta)}(1-\alpha+\beta)t^{\beta-\alpha}=\frac{\Gamma(1+\beta)}{\Gamma(1-\alpha+\beta)}t^{\beta-\alpha}.$

Thus (2.17) is verified.

We cannot define $d_{t}^{\alpha}t^{\beta}$ in general for $-1<\beta\leq 0$,
but we can calculate $D_{t}^{\alpha}t^{\beta}$. We remark that
$\partial_{t}^{\alpha}t^{\beta}$ in the operator sense coincides with the
result calculated by $D_{t}^{\alpha}t^{\beta}$. We here emphasize that our
interest is not only computations of fracitional derivatives, but also
formulate $\partial_{t}^{\alpha}$ as an operator defined on
${}^{-\alpha-\gamma}H(0,T)$ with the isomorphy.

(c) For any constant $t_{0}\in(0,T)$, we consider a function of the Heaviside
type defined by (2.16). By Proposition 2.10 or 2.11, we can see

$\partial_{t}^{\alpha}h_{t_{0}}(t)=\left\\{\begin{array}[]{rl}&0,\quad t\leq
t_{0},\\\ &\frac{(t-t_{0})^{-\alpha}}{\Gamma(1-\alpha)},\quad
t>t_{0}.\end{array}\right.$

We compare with the case $\alpha=1$:

$\frac{d}{dt}h_{t_{0}}(t)=\delta_{t_{0}}(t):\,\mbox{ Dirac delta function at
$t_{0}$}$

in $(C^{\infty}_{0}(0,T))^{\prime}$. On the other hand,
$\partial_{t}^{\alpha}h_{t_{0}}$ for $\alpha<1$ does not generate the
singularity of a Dirac delta function.

Moreover,

$\partial_{t}^{\alpha}h_{t_{0}}\,\longrightarrow\,\delta_{t_{0}}\quad\mbox{as
$\alpha\uparrow 1$ in $(C^{\infty}_{0}(0,T))^{\prime}$},$

which is taken in the distribution sense. More precisely,

$\lim_{\alpha\uparrow
1}\,{}_{(C^{\infty}_{0}(0,T))^{\prime}}<\partial_{t}^{\alpha}h_{t_{0}},\,\psi>_{C^{\infty}_{0}(0,T)}\,=\,_{(C^{\infty}_{0}(0,T))^{\prime}}<\delta_{t_{0}},\,\psi>_{C^{\infty}_{0}(0,T)}=\psi(t_{0})$

for each $\psi\in C^{\infty}_{0}(0,T)$.
Indeed, by integration by parts, we obtain

${}_{(C^{\infty}_{0}(0,T))^{\prime}}<\partial_{t}^{\alpha}h_{t_{0}},\,\psi>_{C^{\infty}_{0}(0,T)}\,=\int^{T}_{t_{0}}\frac{(t-t_{0})^{-\alpha}}{\Gamma(1-\alpha)}\psi(t)dt$
$\displaystyle=$
$\displaystyle\left[\frac{(t-t_{0})^{1-\alpha}}{(1-\alpha)\Gamma(1-\alpha)}\psi(t)\right]^{t=T}_{t=t_{0}}-\frac{1}{(1-\alpha)\Gamma(1-\alpha)}\int^{T}_{t_{0}}(t-t_{0})^{1-\alpha}\psi^{\prime}(t)dt$
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma(2-\alpha)}\int_{T}^{t_{0}}(t-t_{0})^{1-\alpha}\psi^{\prime}(t)dt.$

Letting $\alpha\uparrow 1$ and applying the Lebesgue convergence theorem, we
know that the right-hand side tends to

$\frac{1}{\Gamma(1)}\int_{T}^{t_{0}}\psi^{\prime}(t)dt=\psi(t_{0}).$

$\blacksquare$

Proof of Proposition 2.11.
We prove by approximating $u\in L^{1}(0,T)$ by $u_{n}\in{{}_{0}{C^{1}[0,T]}}$,
$n\in\mathbb{N}$ and using Proposition 2.7. As before, we choose
$\gamma>\frac{1}{2}-\alpha$, so that
$L^{1}(0,T)\subset\,{{}^{-\alpha-\gamma}H}(0,T)$ by the Sobolev embedding:
$\,{}^{\alpha+\gamma}H(0,T)\subset C[0,T]$.

Since we can choose $u_{n}\in{{}_{0}{C^{1}[0,T]}}$, $n\in\mathbb{N}$ such that
$u_{n}\longrightarrow u$ in $L^{1}(0,T)$ as $n\to\infty$, we see that
$u_{n}\longrightarrow u$ in $\,{}^{-\alpha-\gamma}H(0,T)$. Hence, Theorem 2.1
yields that $\partial_{t}^{\alpha}u_{n}\longrightarrow\partial_{t}^{\alpha}u$
in $\,{}^{-2\alpha-\gamma}H(0,T)$. Since
$C^{\infty}_{0}(0,T)\subset\,^{2\alpha+\gamma}H(0,T)$ yields
$\,{}^{-2\alpha-\gamma}H(0,T)\subset(C^{\infty}_{0}(0,T))^{\prime}$, we see
that $\partial_{t}^{\alpha}u_{n}\longrightarrow\partial_{t}^{\alpha}u$ in
$\,{}^{-2\alpha-\gamma}H(0,T)$ implies
$\partial_{t}^{\alpha}u_{n}\longrightarrow\partial_{t}^{\alpha}u$ in
$(C^{\infty}_{0}(0,T))^{\prime}$, that is,

$\lim_{n\to\infty}{{}_{(C^{\infty}_{0}(0,T))^{\prime}}<}\partial_{t}^{\alpha}u_{n},\,\psi>_{C^{\infty}_{0}(0,T)}\,=\,{{}_{(C^{\infty}_{0}(0,T))^{\prime}}<}\partial_{t}^{\alpha}u,\,\psi>_{C^{\infty}_{0}(0,T)}$

for all $\psi\in C^{\infty}_{0}(0,T)$.

On the other hand, by Proposition 2.7, we have
$\partial_{t}^{\alpha}u_{n}=D_{t}^{\alpha}u_{n}$, $n\in\mathbb{N}$. Therefore,

$None$
$\lim_{n\to\infty}{{}_{(C^{\infty}_{0}(0,T))^{\prime}}<}D_{t}^{\alpha}u_{n},\,\psi>_{C^{\infty}_{0}(0,T)}\,=\,{{}_{(C^{\infty}_{0}(0,T))^{\prime}}<}\partial_{t}^{\alpha}u,\,\psi>_{C^{\infty}_{0}(0,T)}$

for all $\psi\in C^{\infty}_{0}(0,T)$.

Then, for any $\psi\in C^{\infty}_{0}(0,T)$, we obtain

${}_{(C^{\infty}_{0}(0,T))^{\prime}}<D_{t}^{\alpha}u_{n},\,\psi>_{C^{\infty}_{0}(0,T)}\,=\,(D_{t}^{\alpha}u_{n}\,\psi)_{L^{2}(0,T)}=\frac{1}{\Gamma(1-\alpha)}\left(\frac{d}{dt}\int^{t}_{0}(t-s)^{-\alpha}u_{n}(s)ds,\,\,\psi\right)_{L^{2}(0,T)}$
$\displaystyle=$
$\displaystyle-\frac{1}{\Gamma(1-\alpha)}\left(\int^{t}_{0}(t-s)^{-\alpha}u_{n}(s)ds,\,\,\frac{d\psi}{dt}\right)_{L^{2}(0,T)}.$

Since the Young inequality on the convolution yields

$\left\|\int^{t}_{0}(t-s)^{-\alpha}u_{n}(s)ds-\int^{t}_{0}(t-s)^{-\alpha}u(s)ds\right\|_{L^{1}(0,T)}\leq\|s^{-\alpha}\|_{L^{1}(0,T)}\|u_{n}-u\|_{L^{1}(0,T)}\longrightarrow
0$

as $n\to\infty$, we have

$\lim_{n\to\infty}(D_{t}^{\alpha}u_{n},\,\psi)_{L^{2}(0,T)}=\left(-\frac{1}{\Gamma(1-\alpha)}\int^{t}_{0}(t-s)^{-\alpha}u(s)ds,\,\,\frac{d\psi}{dt}\right)_{L^{2}(0,T)}.$

Hence, with (2.18), we obtain

${{}_{(C^{\infty}_{0}(0,T))^{\prime}}<}\partial_{t}^{\alpha}u,\,\psi>_{C^{\infty}_{0}(0,T)}\,=\lim_{n\to\infty}{{}_{(C^{\infty}_{0}(0,T))^{\prime}}<}D_{t}^{\alpha}u_{n},\,\psi>_{C^{\infty}_{0}(0,T)}$
$None$
$=\left(-\frac{1}{\Gamma(1-\alpha)}\int^{t}_{0}(t-s)^{-\alpha}u(s)ds,\,\,\frac{d\psi}{dt}\right)_{L^{2}(0,T)}$

for all $\psi\in C^{\infty}_{0}(0,T)$. Consequently, (2.19) means

$\partial_{t}^{\alpha}u(t)=\frac{d}{dt}\left(\frac{1}{\Gamma(1-\alpha)}\int^{t}_{0}(t-s)^{-\alpha}u(s)ds\right)$

in the sense of the derivative of a distribution. Thus the proof of
Proposition 2.11 is complete. $\blacksquare$

##  3\. Basic properties in fractional calculus

In this section, we present fundamental properties of $\partial_{t}^{\alpha}$
in the case of $\mathcal{D}(\partial_{t}^{\alpha})=H_{\alpha}(0,T)$. We can
consider for $\partial_{t}^{\alpha}$ with the domain $L^{2}(0,T)$ but we here
omit.

We set $\partial_{t}^{0}=I$: the identity operator on $L^{2}(0,T)$.
Theorem 3.1.
Let $\alpha,\beta\geq 0$. Then

$\partial_{t}^{\alpha}(\partial_{t}^{\beta}v)=\partial_{t}^{\alpha+\beta}v\quad\mbox{for
all $v\in H_{\alpha+\beta}(0,T)$}.$

This kind of sequential derivatives are more complicated for $d_{t}^{\alpha}$
and $D_{t}^{\alpha}$ when we do not specify the domains. For
$\partial_{t}^{\alpha}$, the domain is already installed in a convenient way.
Proof of Theorem 3.1.
By (2.7) we have $\partial_{t}^{\alpha}=(J^{\alpha})^{-1}$ in
$\mathcal{D}(\partial_{t}^{\alpha})=H_{\alpha}(0,T)$ with $\alpha\geq 0$.
Hence, it suffices to prove

$J^{-\alpha}(J^{-\beta}v)=J^{-(\alpha+\beta)}v,\quad v\in
H_{\alpha+\beta}(0,T).$

Setting $w=J^{-\beta}v$, we have $w\in H_{\alpha}(0,T)$ by Proposition 2.5
(i). Then $J^{-\alpha}J^{-\beta}v=J^{-\alpha}w$ and

$J^{-(\alpha+\beta)}v=J^{-(\alpha+\beta)}J^{\beta}w=J^{-(\alpha+\beta)+\beta}w=J^{-\alpha}w.$

For the second equality to last, we applied Proposition 2.5 (ii) in terms of
$w\in\mathcal{D}(J^{-\alpha})=H_{\alpha}(0,T)$. Hence,
$J^{-\alpha}J^{-\beta}v=J^{-(\alpha+\beta)}v$ for $v\in
H_{\alpha+\beta}(0,T)$. Thus the proof of Theorem 3.1 is complete.
$\blacksquare$

We define the Laplace transform by

$None$ $(Lv)(p)=\widehat{v}(p):=\lim_{T\to\infty}\int^{T}_{0}v(t)e^{-pt}dt$

provided that the limit exists.
Theorem 3.2 (Laplace transform of $\partial_{t}^{\alpha}v$).
Let $u\in H_{\alpha}(0,T)$ with arbitrary $T>0$. If
$(\widehat{|\partial_{t}^{\alpha}u|})(p)$ exists for $p>p_{0}$ which is some
positive constant, then $\widehat{u}(p)$ exists for $p>p_{0}$ and

$\widehat{\partial_{t}^{\alpha}u}(p)=p^{\alpha}\widehat{u}(p),\quad p>p_{0}.$

For initial value problems for fractional ordinary differential equations and
initial boundary value problems for fractional partial differential equations,
it is known that the Laplace transform is useful if one can verify the
existence of the Laplace transform of solutions to these problems. The
existence of the Laplace transfom is concerned with the asymptotic behavior of
unknown solution $u$ as $t\to\infty$, which may not be easy to be verified for
solutions to be determined. At least the method by Laplace transform is
definitely helpful in finding solutions heuristically.

Corollary 3.1.
Let $u\in H_{\alpha}(0,T)$ with arbitrary $T>0$. If $\partial_{t}^{\alpha}u\in
L^{r}(0,\infty)$ with some $r\geq 1$, then

$\widehat{\partial_{t}^{\alpha}u}(p)=p^{\alpha}\widehat{u}(p),\quad p>p_{0}.$

Proof of Theorem 3.2.
First Step. We first prove
Lemma 3.1.
Let $w\in L^{2}(0,T)$ with arbitrary $T>0$. If $\widehat{|w|}(p)$ exists for
$p>p_{0}$. Then

$(\widehat{J^{\alpha}w})(p)=p^{-\alpha}\widehat{w}(p),\quad p>p_{0}.$

Proof of Lemma 3.1.
We recall

$(J^{\alpha}w)(t)=\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}w(s)ds,\quad
t>0\quad\mbox{for $w\in L^{2}(0,T)$}.$

By $w\in L^{2}(0,T)$, we see that $J^{\alpha}w\in L^{2}(0,T)$ for arbitrarily
fixed $T>0$. By the assumption on the existence of $\widehat{w}$, we obtain

$\lim_{T\to\infty}\int^{T}_{0}e^{-pt}w(t)dt=\widehat{w}(p),\quad p>p_{0}.$

Choose $T>0$ arbitrarily. Then $w\in L^{2}(0,T)$, and

$\int^{T}_{0}e^{-pt}\left(\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}w(s)ds\right)dt=\frac{1}{\Gamma(\alpha)}\int^{T}_{0}w(s)\left(\int^{T}_{s}e^{-pt}(t-s)^{\alpha-1}dt\right)ds.$

Changing the variables: $t\longrightarrow\xi$ by $\xi:=(t-s)p$, we have

$\int^{T}_{s}e^{-pt}(t-s)^{\alpha-1}dt=p^{-\alpha}e^{-ps}\int^{(T-s)p}_{0}\xi^{\alpha-1}e^{-\xi}d\xi,$

and so

$\displaystyle\int^{T}_{0}e^{-pt}\left(\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}w(s)ds\right)dt=\frac{1}{\Gamma(\alpha)}p^{-\alpha}\int^{T}_{0}e^{-ps}w(s)\left(\int^{(T-s)p}_{0}\xi^{\alpha-1}e^{-\xi}d\xi\right)ds$
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\alpha)}p^{-\alpha}\int^{T}_{\frac{T}{2}}e^{-ps}w(s)\left(\int^{(T-s)p}_{0}\xi^{\alpha-1}e^{-\xi}d\xi\right)ds$
$\displaystyle+$
$\displaystyle\frac{1}{\Gamma(\alpha)}p^{-\alpha}\int^{\frac{T}{2}}_{0}e^{-ps}w(s)\left(\int^{(T-s)p}_{0}\xi^{\alpha-1}e^{-\xi}d\xi\right)ds=:I_{1}+I_{2}.$

We remark

$\int^{(T-s)p}_{0}\xi^{\alpha-1}e^{-\xi}d\xi\leq\int^{\infty}_{0}\xi^{\alpha-1}e^{-\xi}d\xi=\Gamma(\alpha).$

Then, since $\int^{\infty}_{0}e^{-pt}|w(t)|dt$ exists, we see

$\displaystyle|I_{1}|=\left|\frac{1}{\Gamma(\alpha)}p^{-\alpha}\int^{T}_{\frac{T}{2}}e^{-ps}w(s)\left(\int^{(T-s)p}_{0}\xi^{\alpha-1}e^{-\xi}d\xi\right)ds\right|$
$\displaystyle\leq$ $\displaystyle
Cp^{-\alpha}\int^{T}_{\frac{T}{2}}e^{-ps}|w(s)|ds\quad\longrightarrow
0\quad\mbox{as $T\to\infty$}.$

Since $0<s<\frac{T}{2}$ implies

$\frac{T}{2}p<(T-s)p<Tp,$

we obtain

$None$
$\int^{\frac{Tp}{2}}_{0}\xi^{\alpha-1}e^{-\xi}d\xi\leq\int^{(T-s)p}_{0}\xi^{\alpha-1}e^{-\xi}d\xi\leq\int^{Tp}_{0}\xi^{\alpha-1}e^{-\xi}d\xi\leq\int^{\infty}_{0}\xi^{\alpha-1}e^{-\xi}d\xi=\Gamma(\alpha)$

and so

$None$
$\lim_{T\to\infty}\int^{(T-s)p}_{0}\xi^{\alpha-1}e^{-\xi}d\xi=\int^{\infty}_{0}\xi^{\alpha-1}e^{-\xi}d\xi=\Gamma(\alpha)\quad\mbox{if
$0<s<\frac{T}{2}$}$

for any $p>p_{0}>0$. Since

$\lim_{T\to\infty}\int^{T}_{0}w(s)e^{-ps}ds$

exists for $p>p_{0}$ in view of (3.2) and (3.3), the Lebesgue convergence
theorem yields

$\lim_{T\to\infty}\int^{\frac{T}{2}}_{0}e^{-ps}w(s)\left(\int^{(T-s)p}_{0}\xi^{\alpha-1}e^{-\xi}d\xi\right)ds=\Gamma(\alpha)\int^{\infty}_{0}e^{-ps}w(s)ds,$

and we reach

$\lim_{T\to\infty}I_{2}=\frac{p^{-\alpha}}{\Gamma(\alpha)}\int^{\infty}_{0}w(s)e^{-ps}\Gamma(\alpha)ds=p^{-\alpha}\widehat{w}(p),\quad
p>p_{0}.$

Thus

$\lim_{T\to\infty}\int^{T}_{0}e^{-pt}(J^{\alpha}w)(t)dt=\lim_{T\to\infty}(I_{1}+I_{2})=p^{-\alpha}\widehat{w}(p),\quad
p>p_{0}.$

The proof of Lemma 3.1 is complete. $\blacksquare$
Second Step. We will complete the proof of the theorem. Since $u\in
H_{\alpha}(0,T)$ for any $T>0$, we can find $w_{T}\in L^{2}(0,T)$ such that
$J^{\alpha}w_{T}=u$ in $(0,T)$. For any $t\in(0,T)$, we can define $w$
satisfying $w(t)=w_{T}(t)$ for $0<t<T$. Therefore for all $t>0$, we can define
$w(t)$ and $J^{\alpha}w(t)=u(t)$ for any $t\in(0,T)$, that is, for all $t>0$.
Therefore $\partial_{t}^{\alpha}u(t)=w(t)$ for $t>0$ and $w\in L^{2}(0,T)$
with arbitrary $T>0$.

Since $\partial_{t}^{\alpha}u(t)=w(t)$ for $t>0$ and
$\widehat{|\partial_{t}^{\alpha}u|}(p)$ exists for $p>p_{0}$, we know that
$\widehat{|w|}(p)$ exists for $p>p_{0}$. Therefore Lemma 3.1 yields

$(\widehat{J^{\alpha}w})(p)=p^{-\alpha}\widehat{w}(p),\quad p>p_{0},$

that is,

$\widehat{u}(p)=p^{-\alpha}\widehat{\partial_{t}^{\alpha}u}(p),\quad p>p_{0}.$

Thus the proof of Theorem 3.2 is complete. $\blacksquare$

Moreover $\partial_{t}^{\alpha}$ with
$\mathcal{D}(\partial_{t}^{\alpha})=H_{\alpha}(0,T)$ is consistent also with
the convolution of two functions. We set

$None$ $(u*g)(t)=\int^{t}_{0}u(t-s)g(s)ds,\quad 0<t<T$

for $u\in L^{2}(0,T)$ and $g\in L^{1}(0,T)$. Then, the Young inequality on the
convolution yields

$None$ $\|u*g\|_{L^{2}(0,T)}\leq\|u\|_{L^{2}(0,T)}\|g\|_{L^{1}(0,T)}.$

We prove
Theorem 3.3.
Let $\alpha\geq 0$. Then:

$None$ $J^{\alpha}(u*g)=(J^{\alpha}u)*g\quad\mbox{for $u\in L^{1}(0,T)$ and
$g\in L^{1}(0,T)$}$ $None$ $\|u*g\|_{H_{\alpha}(0,T)}\leq
C\|u\|_{H_{\alpha}(0,T)}\|g\|_{L^{1}(0,T)}\quad\mbox{for $u\in
H_{\alpha}(0,T)$ and $g\in L^{1}(0,T)$.}$ $None$
$\partial_{t}^{\alpha}(u*g)=(\partial_{t}^{\alpha}u)*g\quad\mbox{for $u\in
H_{\alpha}(0,T)$ and $g\in L^{1}(0,T)$.}$

Proof of Theorem 3.3.
Proof of (3.6). By exchange of the order of the integral and change of the
variables $s\to\eta:=s-\xi$, we can derive

$\displaystyle
J^{\alpha}(u*g)(t)=\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}\left(\int^{s}_{0}u(s-\xi)g(\xi)d\xi\right)ds$
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\alpha)}\int^{t}_{0}g(\xi)\left(\int^{t}_{\xi}(t-s)^{\alpha-1}u(s-\xi)ds\right)d\xi=\frac{1}{\Gamma(\alpha)}\int^{t}_{0}g(\xi)\left(\int^{t-\xi}_{0}(t-\xi-\eta)^{\alpha-1}u(\eta)d\eta\right)d\xi$
$\displaystyle=$
$\displaystyle\int^{t}_{0}g(\xi)(J^{\alpha}u)(t-\xi)d\xi=(g*J^{\alpha}u)(t),\quad
0<t<T,$

which completes the proof of (3.6).
Proof of (3.7). Let $u\in H_{\alpha}(0,T)$ and $g\in L^{1}(0,T)$. Then, by
Proposition 2.5, there exists $w\in L^{2}(0,T)$ such that $u=J^{\alpha}w$ and
$\|w\|_{L^{2}(0,T)}\leq C\|u\|_{H_{\alpha}(0,T)}$. By (3.6) we obtain
$u*g=J^{\alpha}w*g=J^{\alpha}(w*g)$. Since $w\in L^{2}(0,T)$ and $g\in
L^{1}(0,T)$ imply $w*g\in L^{2}(0,T)$, by $u*g\in J^{\alpha}L^{2}(0,T)$ we see
that $u*g\in H_{\alpha}(0,T)$. Moreover, by Proposition 2.5 and
$u*g=J^{\alpha}(w*g)$, we have

$\|u*g\|_{H_{\alpha}(0,T)}\leq C\|w*g\|_{L^{2}(0,T)}\leq
C\|w\|_{L^{2}(0,T)}\|g\|_{L^{1}(0,T)}\leq
C\|u\|_{H_{\alpha}(0,T)}\|g\|_{L^{1}(0,T)}.$

which completes the proof of (3.7).
Proof of (3.8). For arbitrary $u\in H_{\alpha}(0,T)$, Proposition 2.5 yields
that there exists $w\in L^{2}(0,T)$ such that $u=J^{\alpha}w$. Applying (3.6),
we have $J^{\alpha}(w*g)=J^{\alpha}w*g$. Therefore,

$\partial_{t}^{\alpha}(J^{\alpha}w*g)=\partial_{t}^{\alpha}J^{\alpha}(w*g)=w*g.$

Since $u=J^{\alpha}w$ is equivalent to $w=\partial_{t}^{\alpha}u$, we reach
$\partial_{t}^{\alpha}(u*g)=(\partial_{t}^{\alpha}u)*g$. Thus the proof of
Theorem 3.3 is complete. $\blacksquare$

We will show a useful variant of Theorem 3.3.
Theorem 3.4.
Let $\gamma>\frac{1}{2}$ and $\beta+\gamma>\frac{1}{2}$. Then for
$\partial^{\beta}_{t}:\,^{-\gamma}H(0,T)\,\longrightarrow\,^{-\beta-\gamma}H(0,T)$,
we have

$\partial_{t}^{\beta}v\,*\,u=\partial_{t}^{\beta}(v\,*\,u)$

for all $u\in L^{1}(0,T)$ and $v\in L^{1}(0,T)$ satisfying
$\partial_{t}^{\beta}v\in L^{1}(0,T)$.

We note that $L^{1}(0,T)\subset\,^{-\gamma}H(0,T)$ by the Sobolev embedding
and $\gamma>\frac{1}{2}$, and so if $v\in
L^{1}(0,T)\subset\mathcal{D}(\partial_{t}^{\beta})=\,^{-\gamma}H(0,T)$, then
so $\partial_{t}^{\beta}v\in\,^{-\beta-\gamma}H(0,t)$ is well-defined. The
assumption of the theorem further requires $\partial_{t}^{\beta}v\in
L^{1}(0,T)$. On the other hand, for $v,u\in L^{1}(0,T)$, the Young inequailty
implies $v\,*\,u\in L^{1}(0,T)$, and so $v\,*\,u\in\,^{-\gamma}H(0,t)$. Hence
in the theorem, $\partial_{t}^{\beta}(v\,*\,u)\in\,^{-\beta-\gamma}H(0,t)$ is
well-defined.
Proof of Theorem 3.4.
The Young inequality yields $\partial_{t}^{\beta}v\,*\,u\in L^{1}(0,T)$.
Consequently, by Proposition 2.10, we see

$J_{\beta}^{\prime}(\partial_{t}^{\beta}v\,*\,u)=J^{\beta}(\partial_{t}^{\beta}v\,*\,u).$

By (3.6) in Theorem 3.3, we have

$J^{\beta}(\partial_{t}^{\beta}v\,*\,u)=(J^{\beta}\partial_{t}^{\beta}v)\,*\,u.$

Since $\partial_{t}^{\beta}v\in L^{1}(0,T)$, by using the definition (2.13),
again Proposition 2.10 implies

$J^{\beta}\partial_{t}^{\beta}v=J_{\beta}^{\prime}\partial_{t}^{\beta}v=J_{\beta}^{\prime}(J_{\beta}^{\prime})^{-1}v=v.$

Hence, $J_{\beta}^{\prime}(\partial_{t}^{\beta}v\,*\,u)=v\,*\,u$. Operating
$(J_{\beta}^{\prime})^{-1}$ to both sides and noting $v\,*\,u\in
L^{1}(0,T)\subset\,^{-\gamma}H(0,T)$, by (2.13), we have

$(J_{\beta}^{\prime})^{-1}(J_{\beta}^{\prime}(\partial_{t}^{\beta}v\,*\,u))=(J_{\beta}^{\prime})^{-1}(v\,*\,u)=\partial_{t}^{\beta}(v\,*\,u),$

that is, $\partial_{t}^{\beta}v\,*\,u=\partial_{t}^{\beta}(v\,*\,u)$. Thus the
proof of Theorem 3.4 is complete. $\blacksquare$

Theorem 3.5 (coercivity).
Let $0<\alpha<1$. Then

$\int^{T}_{0}v(t)\partial_{t}^{\alpha}v(t)dt\geq\frac{1}{2\Gamma(1-\alpha)}T^{-\alpha}\|v\|^{2}_{L^{2}(0,T)}\quad\mbox{for
$v\in H_{\alpha}(0,T)$}$

and

$\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}v(s)\partial_{s}^{\alpha}v(s)ds\geq\frac{1}{2}|v(t)|^{2}\quad\mbox{for
$v\in H_{\alpha}(0,T)$.}$

The proof of Theorem 3.5 can be found in [18]. Theorem 3.5 is not used in this
article, and is useful for proving the well-posedness for initial boundary
value problems for fractional partial differential equations (see e.g., [18],
[19], [37]).

We can generalize Theorem 3.5 to arbitrary $v\in L^{2}(0,T)$, but we omit the
details for the conciseness.

##  4\. Fractional derivatives of the Mittag-Leffler functions

Related to time fractional differential equations, we introduce the Mittag-
Leffler functions:

$None$ $E_{\alpha,\beta}(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(\alpha
k+\beta)},\quad\alpha,\beta>0,\quad z\in\mathbb{C}.$

It is known (e.g., [30]) that $E_{\alpha,\beta}(z)$ is an entire function in
$z\in\mathbb{C}$.

Henceforth let $\lambda\in\mathbb{R}$ be a constant and, $\alpha>0$ and
$\alpha\not\in\mathbb{N}$. We fix an arbitrary constant $\Lambda_{0}>0$ and we
assume that $\lambda>-\Lambda_{0}$.

First we prove
Proposition 4.1.
Let $0<\alpha<2$. We fix constants $T>0$ and $\lambda>-\Lambda_{0}$
arbitrarily. Then
(i) We have $E_{\alpha,1}(-\lambda t^{\alpha})-1\in H_{\alpha}(0,T)$,

$\partial_{t}^{\alpha}(E_{\alpha,1}(-\lambda t^{\alpha})-1)=-\lambda
E_{\alpha,1}(-\lambda t^{\alpha}),\quad t>0$

and there exist constants $C_{1}=C_{1}(\alpha,\Lambda_{0},T)>0$ and
$C_{2}=C_{2}(\alpha)>0$ such that

$None$ $|E_{\alpha,1}(-\lambda
t^{\alpha})|\leq\left\\{\begin{array}[]{rl}&C_{1}\quad\mbox{for $0\leq t\leq
T$ if $-\Lambda_{0}\leq\lambda<0$},\\\ &\frac{C_{2}}{1+\lambda
t^{\alpha}}\quad\mbox{for $t\geq 0$ if $\lambda\geq 0$}.\end{array}\right.$

(ii) We have $tE_{\alpha,2}(-\lambda t^{\alpha})-t\in H_{\alpha}(0,T)$,

$\partial_{t}^{\alpha}(tE_{\alpha,2}(-\lambda t^{\alpha})-t)=-\lambda
tE_{\alpha,2}(-\lambda t^{\alpha}),\quad t>0.$

Furthermore we can find constants $C_{1}=C_{1}(\alpha,\Lambda_{0},T)>0$ and
$C_{2}=C_{2}(\alpha)>0$ such that

$None$ $|E_{\alpha,2}(-\lambda
t^{\alpha})|\leq\left\\{\begin{array}[]{rl}&C_{1}\quad\mbox{for $0\leq t\leq
T$ if $-\Lambda_{0}\leq\lambda<0$},\\\ &\frac{C_{2}}{1+\lambda
t^{\alpha}}\quad\mbox{for $t\geq 0$ if $\lambda\geq 0$}.\end{array}\right.$

A direct proof for the inclusions in $H_{\alpha}(0,T)$ is complicated and
through the operator $J^{\alpha}$, we will provide simpler proofs.
Proof of Proposition 4.1.
(i) Since

$E_{\alpha,1}(-\lambda
t^{\alpha})=\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}t^{\alpha k}}{\Gamma(\alpha
k+1)},$

where the series is uniformly convergent for $0\leq t\leq T$, the termwise
integration yields

$\displaystyle J^{\alpha}(E_{\alpha,1}(-\lambda
t^{\alpha}))=\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{\Gamma(\alpha)}\int^{t}_{0}\frac{(t-s)^{\alpha-1}s^{\alpha
k}}{\Gamma(\alpha k+1)}ds$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{\Gamma(\alpha)}\frac{\Gamma(\alpha)}{\Gamma(\alpha
k+\alpha+1)}t^{\alpha
k+\alpha}=-\frac{1}{\lambda}\sum_{j=1}^{\infty}\frac{(-\lambda
t^{\alpha})^{j}}{\Gamma(\alpha j+1)}.$

Here we set $j=k+1$ to change the indices of the summation. Therefore

$J^{\alpha}(E_{\alpha,1}(-\lambda
t^{\alpha}))=-\frac{1}{\lambda}(E_{\alpha,1}(-\lambda t^{\alpha})-1).$

Since $\partial_{t}^{\alpha}v=(J^{\alpha})^{-1}v$ for $v\in H_{\alpha}(0,T)$
and $E_{\alpha,1}(-\lambda t^{\alpha})\in L^{2}(0,T)$, we obtain

$-\frac{1}{\lambda}(E_{\alpha,1}(-\lambda t^{\alpha})-1)\in H_{\alpha}(0,T)$

and

$(J^{\alpha})^{-1}J^{\alpha}E_{\alpha,1}(-\lambda
t^{\alpha})=-\frac{1}{\lambda}\partial_{t}^{\alpha}(E_{\alpha,1}(-\lambda
t^{\alpha})-1),$

that is, $\partial_{t}^{\alpha}(E_{\alpha,1}(-\lambda t^{\alpha})-1)=-\lambda
E_{\alpha,1}(-\lambda t^{\alpha})$.

On the other hand, for $0<\alpha<2$ and $\beta>0$, by Theorems 1.5 and 1.6
(p.35) in [30], we can find a constant $C_{0}=C_{0}(\alpha,\beta)>0$ such that

$None$ $|E_{\alpha,\beta}(-\lambda
t^{\alpha})|\leq\left\\{\begin{array}[]{rl}&C_{0}(1+|\lambda|t^{\alpha})^{\frac{1-\beta}{\alpha}}\exp(|\lambda|^{\frac{1}{\alpha}}t)\quad\mbox{if
$-\Lambda_{0}\leq\lambda<0$},\\\ &\frac{C_{0}}{1+\lambda
t^{\alpha}}\quad\mbox{if $\lambda\geq 0$}\end{array}\right.$

for all $t\geq 0$. Hence, we can choose constants
$C_{3}=C_{3}(\alpha,\beta,\Lambda_{0},T)>0$ and $C_{4}=C_{4}(\alpha,\beta)>0$
such that

$None$ $|E_{\alpha,\beta}(-\lambda
t^{\alpha})|\leq\left\\{\begin{array}[]{rl}&C_{3}\quad\mbox{for $0\leq t\leq
T$ if $-\Lambda_{0}\leq\lambda<0$},\\\ &\frac{C_{4}}{1+\lambda
t^{\alpha}}\quad\mbox{for $t\geq 0$ if $\lambda\geq 0$}.\end{array}\right.$

In terms of (4.5), we can finish the proof of (4.2). Thus the proof of (i) is
complete.
(ii) Since $tE_{\alpha,2}(-\lambda t^{\alpha})-t\in L^{2}(0,T)$, noting
$\Gamma(2)=1\times\Gamma(1)=1$, we have

$\displaystyle J^{\alpha}(tE_{\alpha,2}(-\lambda
t^{\alpha}))=\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}\sum_{k=0}^{\infty}\frac{s(-\lambda)^{k}s^{\alpha
k}}{\Gamma(\alpha k+2)}ds$ $\displaystyle=$ $\displaystyle
t\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{\Gamma(\alpha k+\alpha+2)}t^{\alpha
k+\alpha}=\frac{t}{-\lambda}\sum_{j=1}^{\infty}\frac{(-\lambda)^{j}}{\Gamma(\alpha
j+2)}t^{\alpha j}$ $\displaystyle=$
$\displaystyle-\frac{t}{\lambda}\left(\sum_{j=0}^{\infty}\frac{(-\lambda)^{j}t^{\alpha
j}}{\Gamma(\alpha j+2)}-\frac{1}{\Gamma(2)}\right),$

that is,

$J^{\alpha}(tE_{\alpha,2}(-\lambda
t^{\alpha}))=-\frac{1}{\lambda}(tE_{\alpha,2}(-\lambda t^{\alpha})-t),\quad
0<t<T.$

Therefore, $tE_{\alpha,2}(-\lambda t^{\alpha})-t\in H_{\alpha}(0,T)$ and

$-\lambda tE_{\alpha,2}(-\lambda
t^{\alpha})=\partial_{t}^{\alpha}(tE_{\alpha,2}(-\lambda t^{\alpha})-t).$

In terms of (4.5), the proof of the estimate is similar to part (i). Thus we
can complete the proof of Proposition 4.1. $\blacksquare$

We set

$None$
$(B_{\lambda}f)(t):=\int^{t}_{0}(t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda(t-s)^{\alpha})f(s)ds.$

Next we show
Proposition 4.2.
Let $f\in L^{2}(0,T)$. Then

$None$ $B_{\lambda}f\in
H_{\alpha}(0,T)\quad\mbox{and}\quad\partial_{t}^{\alpha}(B_{\lambda}f)(t)=-\lambda(B_{\lambda}f)(t)+f(t),\quad
0<t<T.$

Moreover there exist constants $C_{5}=C_{5}(\alpha,\Lambda_{0},T)>0$ and
$C_{6}=C_{6}(\alpha,T)>0$ such that

$None$ $\left\\{\begin{array}[]{rl}&\|B_{\lambda}f\|_{H_{\alpha}(0,T)}\leq
C_{5}(\alpha,\Lambda_{0},T)\|f\|_{L^{2}(0,T)}\quad\mbox{for all $f\in
L^{2}(0,T)$ and $\lambda>-\Lambda_{0}$},\\\
&\|B_{\lambda}f\|_{H_{\alpha}(0,T)}\leq
C_{6}(\alpha,T)\|f\|_{L^{2}(0,T)}\quad\mbox{for all $f\in L^{2}(0,T)$ and
$\lambda\geq 0$}.\end{array}\right.$

The uniformity of estimate (4.8) on $\lambda\geq 0$ plays an important role in
Lemma 6.2 (ii) in Section 6. Such uniformity can be derived from the complete
monotonicity of $E_{\alpha,1}(-\lambda t^{\alpha})$ which is characteristic
only for $0<\alpha<1$ and see (4.9) below.

Here we prove by using $\partial_{t}^{\alpha}=(J^{\alpha})^{-1}$ in
$H_{\alpha}(0,T)$, although other proof by Theorem 3.4 is possible.
Proof.
Since in view of (4.5), we can choose constants
$C_{7}=C_{7}(\alpha,\Lambda_{0},T)>0$ and $C_{8}=C_{8}(\alpha,T)>0$ such that

$|E_{\alpha,\alpha}(-\lambda(t-s)^{\alpha})|\leq\left\\{\begin{array}[]{rl}&C_{7}(\alpha,\Lambda_{0},T)\quad\mbox{for
$0\leq t\leq T$ and $\lambda>-\Lambda_{0}$},\\\ &C_{8}(\alpha,T)\quad\mbox{for
$0\leq t\leq T$ and $\lambda\geq 0$},\end{array}\right.$

we estimate

$\left|\int^{t}_{0}(t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda(t-s)^{\alpha})f(s)ds\right|\leq
C\int^{t}_{0}(t-s)^{\alpha-1}|f(s)|ds,\quad 0<t<T.$

Here and henceforth $C>0$ denotes generic constants which depend on
$\alpha,\Lambda_{0},T$ when we consider $\lambda>-\Lambda_{0}$ and does not
depend on $\Lambda_{0}$ if $\lambda\geq 0$.

Hence, the Young inequality yields that
$\int^{t}_{0}(t-s)^{\alpha-1}|f(s)|ds\in L^{2}(0,T)$ and so $B_{\lambda}f\in
L^{2}(0,T)$.

Now

$\displaystyle
J^{\alpha}(B_{\lambda}f)(t)=\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}(B_{\lambda}f)(s)ds$
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}\left(\int^{s}_{0}(s-\xi)^{\alpha-1}E_{\alpha,\alpha}(-\lambda(s-\xi)^{\alpha})f(\xi)d\xi\right)ds$
$\displaystyle=$
$\displaystyle\int^{t}_{0}f(\xi)\left(\frac{1}{\Gamma(\alpha)}\int^{t}_{\xi}(t-s)^{\alpha-1}(s-\xi)^{\alpha-1}E_{\alpha,\alpha}(-\lambda(s-\xi)^{\alpha})ds\right)d\xi.$

Here

$\displaystyle\frac{1}{\Gamma(\alpha)}\int^{t}_{\xi}(t-s)^{\alpha-1}(s-\xi)^{\alpha-1}E_{\alpha,\alpha}(-\lambda(s-\xi)^{\alpha})ds$
$\displaystyle=$
$\displaystyle\int^{t}_{\xi}\frac{1}{\Gamma(\alpha)}(t-s)^{\alpha-1}(s-\xi)^{\alpha-1}\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}(s-\xi)^{\alpha
k}}{\Gamma(\alpha k+\alpha)}ds$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\alpha)}\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{\Gamma(\alpha
k+\alpha)}\int^{t}_{\xi}(t-s)^{\alpha-1}(s-\xi)^{\alpha k+\alpha-1}ds$
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\alpha)}\sum_{k=0}^{\infty}\frac{(-\lambda)^{k}}{\Gamma(\alpha
k+\alpha)}\frac{\Gamma(\alpha)\Gamma(\alpha k+\alpha)}{\Gamma(\alpha
k+2\alpha)}(t-\xi)^{\alpha k+2\alpha-1}$ $\displaystyle=$
$\displaystyle-\frac{1}{\lambda}(t-\xi)^{\alpha-1}\sum_{j=1}^{\infty}\frac{(-\lambda(t-\xi)^{\alpha})^{j}}{\Gamma(\alpha
j+\alpha)}=-\frac{1}{\lambda}(t-\xi)^{\alpha-1}\left(E_{\alpha,\alpha}(-\lambda(t-\xi)^{\alpha})-\frac{1}{\Gamma(\alpha)}\right).$

Therefore, we have

$\displaystyle
J^{\alpha}(B_{\lambda}f)(t)=-\frac{1}{\lambda}(B_{\lambda}f)(t)+\frac{1}{\lambda}\int^{t}_{0}(t-\xi)^{\alpha-1}\frac{1}{\Gamma(\alpha)}f(\xi)d\xi$
$\displaystyle=$
$\displaystyle-\frac{1}{\lambda}(B_{\lambda}f)(t)+\frac{1}{\lambda}(J^{\alpha}f)(t),\quad
0<t<T,$

that is,

$(B_{\lambda}f)(t)=-\lambda
J^{\alpha}(B_{\lambda}f)(t)+(J^{\alpha}f)(t)=J^{\alpha}(-\lambda
B_{\lambda}f+f)(t),\quad 0<t<T.$

Hence, $B_{\lambda}f\in J^{\alpha}L^{2}(0,T)=H_{\alpha}(0,T)$ by $-\lambda
B_{\lambda}f+f\in L^{2}(0,T)$. Therefore, we apply
$\partial_{t}^{\alpha}=(J^{\alpha})^{-1}$ to reach

$\partial_{t}^{\alpha}(B_{\lambda}f)=-\lambda B_{\lambda}f+f\quad\mbox{in
$(0,T)$}.$

On the other hand, the termwise differentiation of the power series of
$E_{\alpha,1}(z)$ yields

$\frac{d}{dt}E_{\alpha,1}(-\lambda t^{\alpha})=-\lambda
t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha}).$

Furthermore we note the complete monotonicity:

$None$ $E_{\alpha,1}(-\lambda
t^{\alpha})>0,\quad\frac{d}{dt}E_{\alpha,1}(-\lambda_{t}^{\alpha})\leq 0,\quad
t\geq 0$

(e.g., Gorenflo, Kilbas, Mainardi and Rogosin [6]). Therefore, for
$\lambda\geq 0$, we have

$\int^{T}_{0}|\lambda t^{\alpha-1}E_{\alpha,\alpha}(-\lambda
t^{\alpha})|dt=\int^{T}_{0}\lambda t^{\alpha-1}E_{\alpha,\alpha}(-\lambda
t^{\alpha})dt$ $None$ $=-\int^{T}_{0}\frac{d}{dt}E_{\alpha,1}(-\lambda
t^{\alpha})dt=1-E_{\alpha,1}(-\lambda T^{\alpha})\leq 1.$

For $-\Lambda_{0}<\lambda<0$, by (4.5) we have

$\int^{T}_{0}|\lambda t^{\alpha-1}E_{\alpha,\alpha}(-\lambda
t^{\alpha})|dt\leq|\Lambda_{0}|C\int^{T}_{0}t^{\alpha-1}dt,$

where the constant $C>0$ depends on $\alpha,\Lambda_{0},T$. Consequently,

$\lambda\|s^{\alpha-1}E_{\alpha,\alpha}(-\lambda
s^{\alpha})\|_{L^{1}(0,T)}\leq C\quad\mbox{for $\lambda>-\Lambda_{0}$}.$

Applying the Young inequality in (4.6), we obtain

$\lambda\|B_{\lambda}f\|_{L^{2}(0,T)}\leq\lambda\|s^{\alpha-1}E_{\alpha,\alpha}(-\lambda
s^{\alpha})\|_{L^{1}(0,T)}\|f\|_{L^{2}(0,T)}\leq C\|f\|_{L^{2}(0,T)}.$

Therefore,

$\displaystyle\|B_{\lambda}f\|_{H_{\alpha}(0,T)}\leq
C\|\partial_{t}^{\alpha}(B_{\lambda}f)\|_{L^{2}(0,T)}=C\|-\lambda
B_{\lambda}f+f\|_{L^{2}(0,T)}$ $\displaystyle\leq$ $\displaystyle C(\|-\lambda
B_{\lambda}f\|_{L^{2}(0,T)}+\|f\|_{L^{2}(0,T)})\leq
C\|f\|_{L^{2}(0,T)}\quad\mbox{for all $f\in L^{2}(0,T)$}.$

Thus the proof of Proposition 4.2 is complete. $\blacksquare$

We close this section with a lemma which is used in Section 6.
Lemma 4.1.
Let $0<\beta<\alpha$ and $\lambda>0$. Fixing $\gamma>\frac{1}{2}$, we consider
$\partial^{\beta}_{t}:\,^{-\gamma}H(0,T)\,\longrightarrow\,^{-\beta-\gamma}H(0,T)$.
Then
(i)

$t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha})\in
L^{1}(0,T)\subset\,^{-\gamma}H(0,T).$

(ii)

$\partial_{t}^{\beta}(t^{\alpha-1}E_{\alpha,\alpha}(-\lambda
t^{\alpha}))=t^{\alpha-\beta-1}E_{\alpha,\alpha-\beta}(-\lambda t^{\alpha})\in
L^{1}(0,T).$

(iii)

$\partial_{t}^{\beta}(t^{\alpha-1}E_{\alpha,\alpha}(-\lambda
t^{\alpha})\,*\,v)=(t^{\alpha-\beta-1}E_{\alpha,\alpha-\beta}(-\lambda
t^{\alpha})\,*\,v)\quad\mbox{for each $v\in L^{2}(0,T)$.}$

Proof.
(i) By (4.4), we see $|t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha})|\leq
Ct^{\alpha-1}$ for $t>0$ and $t^{\alpha-1}E_{\alpha,\alpha}(-\lambda
t^{\alpha})\in L^{1}(0,T)$. The embedding
$L^{1}(0,T)\subset\,^{-\gamma}H(0,T)$ is derived by the Sobolev embedding.
(ii) By $\alpha-\beta>0$, we can similarly verify that
$t^{\alpha-\beta-1}E_{\alpha,\alpha-\beta}(-\lambda t^{\alpha})\in
L^{1}(0,T)$. Therefore, Proposition 2.10 yields

$J_{\beta}^{\prime}(t^{\alpha-\beta-1}E_{\alpha,\alpha-\beta}(-\lambda
t^{\alpha}))(t)=\frac{1}{\Gamma(\beta)}\int^{t}_{0}(t-s)^{\beta-1}s^{\alpha-\beta-1}E_{\alpha,\alpha-\beta}(-\lambda
s^{\alpha})ds.$

By $\beta>0$ and $\alpha-\beta>0$, we apply the formula (1.100) (p.25) in
[30], which can be also directly verified by the expansion of the power series
of $E_{\alpha,\alpha-\beta}(-\lambda s^{\alpha})$, so that

$\frac{1}{\Gamma(\beta)}\int^{t}_{0}(t-s)^{\beta-1}s^{\alpha-\beta-1}E_{\alpha,\alpha-\beta}(-\lambda
s^{\alpha})ds=t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha}).$

Therefore,
$J_{\beta}^{\prime}(t^{\alpha-\beta-1}E_{\alpha,\alpha-\beta}(-\lambda
t^{\alpha}))(t)=t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha})$. By
$\partial_{t}^{\beta}=(J_{\beta}^{\prime})^{-1}$ in $L^{1}(0,T)$, we see part
(ii).

Since $t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha})$,
$\partial_{t}^{\beta}(t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha})$ are
both in $L^{1}(0,T)$, part (iii) follows from Theorem 3.4. Thus the proof of
Lemma 4.1 is complete. $\blacksquare$

##  5\. Initial value problems for fractional ordinary differential equations

A relevant formulation of initial value problem for time-fractional odinary
differential equations is our main issue in this section, in order to treat
not so smooth data. For keeping the compact descriptions, we are restricted to
a simple linear fractional ordinary differential equation. The treatments are
similar to Chapter 3 in [18]. We formulate an initial value problem as
follows.

$None$ $\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}(u-a)(t)=-\lambda
u(t)+f(t),\quad 0<t<T,\\\ &u-a\in H_{\alpha}(0,T).\end{array}\right.$

As is mentioned in Section 2, if $\alpha>\frac{1}{2}$, then $u-a\in
H_{\alpha}(0,T)\subset{{}_{0}{C^{1}[0,T]}}$ and we know that $u(t)-a$ is
continuous and $u(0)=a$. Thus in the case of $\alpha>\frac{1}{2}$, if $u$
satisfies (5.1), then a usual initial condition $u(0)=a$ is satisfied.

Our formulation (5.1) coincides with a conventional formulation
$D_{t}^{\alpha}(u(t)-a)=f(t)$ of initial value problem, provided that we
suitably specify the regularity of $u$. We emphasize that we always attach
fractional derivative operators with the domains such as $H_{\alpha}(0,T)$ or
${{}^{-\alpha}{H}}(0,T)$ with $\alpha\geq 0$, which means that our approach is
a typical operator theoretic formulation, for example, similarly to that one
is prohibited to consider the Laplacian $-\Delta$ in
$\Omega\subset\mathbb{R}^{d}$ not associated with the domain. In other words,
the operator $-\Delta$ with the domain $\\{u\in H^{2}(\Omega);\,u\in
H^{1}_{0}(\Omega)\\}$, is different from $-\Delta$ with the domain $\\{u\in
H^{2}_{0}(\Omega);\,\nabla u\cdot\nu=0\quad\mbox{on $\partial\Omega$}\\}$.
Here $\nu=\nu(x)$ is the unit outward normal vector to $\partial\Omega$.

In particular, if we consider $\partial_{t}^{\alpha}$ with the domain
$H_{\alpha}(0,T)$ and both sides of (5.1) in $L^{2}(0,T)$, we remark that the
equality
$\partial_{t}^{\alpha}(u-a)=\partial_{t}^{\alpha}u-\partial_{t}^{\alpha}a$
does not make any sense for $\alpha>\frac{1}{2}$, because a constant function
$a$ is not in $H_{\alpha}(0,T)$. On the other hand, if we consider
$\partial_{t}^{\alpha}$ with the domain $L^{2}(0,T)$, then we can justify

$\partial_{t}^{\alpha}(u-a)=\partial_{t}^{\alpha}u-\partial_{t}^{\alpha}a=\partial_{t}^{\alpha}u-\frac{1}{\Gamma(1-\alpha)}t^{-\alpha}$

for any $u\in L^{2}(0,T)$.

Now we can prove
Theorem 5.1.
Let $f\in L^{2}(0,T)$. Then there exists a unique solution $u=u(t)$ to initial
value problem (5.1). Moreover

$None$ $u(t)=aE_{\alpha,1}(-\lambda
t^{\alpha})+\int^{t}_{0}(t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda(t-s)^{\alpha})f(s)ds,\quad
0<t<T.$

Formula (5.2) itself is well-known (e.g., (3.1.34), p.141 in [16]) for $f$ in
some classes. We can refer also to Gorenflo and Mainardi [8], Gorenflo,
Mainardi and Srivastva [9], Gorenflo and Rutman [10], Luchko and Gorenflo
[24].

On the other hand, we should understand that (5.2) holds in the sense that
both sides are in $H_{\alpha}(0,T)$ for each $f\in L^{2}(0,T)$.

Sketch of proof.
We see that (5.1) is equivalent to

$J^{-\alpha}(u-a)=-\lambda u+f\quad\mbox{in $L^{2}(0,T)$}\quad\mbox{and}\quad
u-a\in H_{\alpha}(0,T)$

and also to

$None$ $u=a-\lambda J^{\alpha}u+J^{\alpha}f\quad\mbox{in $H_{\alpha}(0,T)$},$

because $J^{\alpha}J^{-\alpha}(u-a)=J^{\alpha}(-\lambda u+f)$.

We conclude that $J^{\alpha}:L^{2}(0,T)\,\longrightarrow\,L^{2}(0,T)$ is a
compact operator because the embedding $H_{\alpha}(0,T)\subset
H^{\alpha}(0,T)\,\longrightarrow\,L^{2}(0,T)$ is compact (e.g., [1]). On the
other hand, we apply the generalized Gronwall inequality (e.g., Lemma A.2 in
[18]) to $u=-\lambda J^{\alpha}u$ in $(0,T)$, that is,

$u(t)=\frac{-\lambda}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}u(s)ds,\quad
0<t<T.$

Then we obtain $u(t)=0$ for $0<t<T$. Therefore the Fredholm alternative yields
the unique existence of $u$ satifying (5.3).

Finally we define $\widetilde{u}(t)$ by

$\displaystyle\widetilde{u}(t)-a:=a(E_{\alpha,1}(-\lambda
t^{\alpha})-1)+\int^{t}_{0}(t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda(t-s)^{\alpha})f(s)ds$
$\displaystyle=$ $\displaystyle a(E_{\alpha,1}(-\lambda
t^{\alpha})-1)+(B_{\lambda}f)(t),\quad 0<t<T.$

Here $B_{\lambda}$ is defined by (4.6).

By Propositions 4.1 and 4.2, it follows that $\widetilde{u}(t)$ satisfies
(5.1). Thus the proof of Theorem 5.1 is complete. $\blacksquare$

We can discuss an initial value problem for a multi-term time-fractional
ordinary differential equation in the same way:

$None$
$\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}(u-a)+\sum_{k=1}^{N}c_{k}\partial_{t}^{\alpha_{k}}(u-a)=-\lambda
u+f(t),\\\ &u-a\in H_{\alpha}(0,T),\end{array}\right.$

where $c_{1},...,c_{N}\neq 0$, $0<\alpha_{1}<\cdots<\alpha_{N}<\alpha\leq 1$.
Theorem 2.1 implies that
$J^{\alpha}\partial_{t}^{\alpha_{k}}=J^{\alpha}J^{-\alpha_{k}}=J^{\alpha-\alpha_{k}}$.
By Proposition 2.2, we can obtain the equivalent equation:

$u-a=-\sum_{k=1}^{N}c_{k}J^{\alpha-\alpha_{k}}(u-a)-\lambda
J^{\alpha}u+J^{\alpha}f$

and we apply the Fredholm alternative to prove the unique existence of
solution, but we omit the details.

We further consider an initial value problem for $f\in{{}^{-\alpha}{H}}(0,T)$:

$None$ $\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}(u-a)(t)=-\lambda
u(t)+f(t),\quad 0<t<T,\\\ &u-a\in L^{2}(0,T).\end{array}\right.$

Similarly to Theorem 5.1, we prove the well-posedness of (5.5) for
$f\in{{}^{-\alpha}{H}}(0,T)$.
Theorem 5.2.
Let $0<\alpha<1$ and $\alpha\not\in\mathbb{N}$. For
$f\in{{}^{-\alpha}{H}}(0,T)$, there exists a unique solution $u-a\in
L^{2}(0,T)$ to (5.5). Moreover we can choose a constant $C>0$ such that

$\|u-a\|_{L^{2}(0,T)}\leq C(|a|+\|f\|_{{{}^{-\alpha}{H}}(0,T)})$

for all $a\in\mathbb{R}$ and $f\in{{}^{-\alpha}{H}}(0,T)$.
Eaxmple 5.1.
We consider

$None$ $\partial_{t}^{\alpha}(u-a)(t)=f(t),\quad 0<t<T$

with $f(t)=\frac{\Gamma(1+\beta)}{\Gamma(1-\alpha+\beta)}t^{\beta-\alpha}$,
where $\beta>-\frac{1}{2}$. Then, $u(t)=a+t^{\beta}$ satisfies (5.5) with
$\lambda=0$ by (2.17). However, if $-\frac{1}{2}<\beta<0$ and
$-\frac{1}{2}<\beta-\alpha$, then $u,f\in L^{2}(0,T)$, but $u(t)$ does not
satisfy $\lim_{t\downarrow 0}u(t)=a$.

Proof of Theorem 5.2.
Setting $v:=u-a\in L^{2}(0,T)$, we rewrite (5.5) as

$\left\\{\begin{array}[]{rl}&(J_{\alpha}^{\prime})^{-1}v=-\lambda v-\lambda
a+f,\\\ &v\in L^{2}(0,T).\end{array}\right.$

By the definition (2.13) of $\partial_{t}^{\alpha}$, equation (5.5) is
equivalent to

$None$ $v=-\lambda J_{\alpha}^{\prime}v+J_{\alpha}^{\prime}(-\lambda
a+f)\quad\mbox{in $L^{2}(0,T)$}.$

We set $g:=J_{\alpha}^{\prime}(f-\lambda a)\in L^{2}(0,T)$ and $Pv:=-\lambda
J_{\alpha}^{\prime}v$. Then we see that the solution $v=u-a$ is a fixed point
of $P$: $v=Pv+g$.

First the operator $P:L^{2}(0,T)\longrightarrow L^{2}(0,T)$ is a compact
operator. Indeed, Proposition 2.9 (iii) yields
$J_{\alpha}^{\prime}v=J^{\alpha}v$ for $v\in L^{2}(0,T)$, and
$J^{\alpha}:L^{2}(0,T)\longrightarrow H_{\alpha}(0,T)$ is an isomorphism by
Proposition 2.2. Thus $J_{\alpha}^{\prime}$ is a bounded operator from
$L^{2}(0,T)$ to $H_{\alpha}(0,T)$. Since the embedding
$H_{\alpha}(0,T)\longrightarrow L^{2}(0,T)$ is compact, we see that
$J_{\alpha}^{\prime}:L^{2}(0,T)\longrightarrow L^{2}(0,T)$ is a compact
operator (e.g., [1]).

Next we have to prove that $v=0$ in $L^{2}(0,T)$ from assumption that $v=Pv$
in $(0,T)$. Then, since

$J_{\alpha}^{\prime}v(t)=J^{\alpha}v(t)=\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}v(s)ds$

by $v\in L^{2}(0,T)$, we obtain

$v(t)=-\frac{\lambda}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{-\alpha}v(s)ds,\quad
0<t<T.$

Hence,

$|v(t)|\leq C\int^{t}_{0}(t-s)^{\alpha-1}|v(s)|ds\quad\mbox{for almost all
$t\in(0,T)$.}$

The generalized Gronwall inequality (e.g., Lemma A.2 in [18]) implies $v(t)=0$
for almost all $t\in(0,T)$. Therefore, the Fredholm alternative yields the
unique existence of a fixed point of $v=Pv+g$ in $L^{2}(0,T)$. The estimate of
$v$ follows from the application of the generalized Gronwall inequality to
(5.7) and Proposition 2.9 (ii): $\|J_{\alpha}^{\prime}f\|_{L^{2}(0,T)}\leq
C\|f\|_{{{}^{-\alpha}{H}}(0,T)}$. Thus the proof of Theorem 5.2 is complete.
$\blacksquare$

Remark 5.1.
Now we compare formulation (5.1) with (5.5) for $f\in L^{2}(0,T)$.
(a) For $f\in L^{2}(0,T)$, formulations (5.1) and (5.5) are equivalent.

Indeed, we immediately see that (5.1) implies (5.5). Conversely, let $u$
satisfy (5.5). Then the second condition in (5.5) yields $u\in L^{2}(0,T)$,
and the first equation in (5.5) concludes that $\partial_{t}^{\alpha}(u-a)\in
L^{2}(0,T)$. By Proposition 2.5, we see that $u-a\in H_{\alpha}(0,T)$, which
means that $u$ satisfies (5.1).
(b) In formulation (5.1), as we remarked, we should not decompose
$\partial_{t}^{\alpha}(u-a)=\partial_{t}^{\alpha}u-\partial_{t}^{\alpha}a$,
which is wrong for $\alpha\geq\frac{1}{2}$ because $a\not\in H_{\alpha}(0,T)$.
We further consider this issue for $0<\alpha<\frac{1}{2}$. For the clearness,
only here by $\partial_{t0}^{\alpha}$ we denote $\partial_{t}^{\alpha}$ as
operator with the domain $\mathcal{D}(\partial_{t0}^{\alpha})=L^{2}(0,T)$, and
we use the same notation $\partial_{t}^{\alpha}$ with the domain
$H_{\alpha}(0,T)$.

By means of (2.17) with $\beta=0$, for $0<\alpha<\frac{1}{2}$, we see that
$1\in H_{\alpha}(0,T)$ and
$\partial_{t0}^{\alpha}1=\partial_{t}^{\alpha}1=\frac{1}{\Gamma(1-\alpha)}t^{-\alpha}$,
and also that $u-a\in H_{\alpha}(0,T)$ if and only if $u-a\in
H^{\alpha}(0,T)$. Therefore, we see:
If $0<\alpha<\frac{1}{2}$, then (5.1) is equivalent to

$None$ $\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}u=-\lambda
u+\frac{a}{\Gamma(1-\alpha)}t^{-\alpha}+f(t)\quad\mbox{in $L^{2}(0,T)$},\\\
&u\in H_{\alpha}(0,T).\end{array}\right.$

Noting $1\in L^{2}(0,T)$ and using (2.17) with $\beta=0$, for all
$\alpha\in(0,1)$, we can verify that (5.5) is equivalent to

$None$ $\left\\{\begin{array}[]{rl}&\partial_{t0}^{\alpha}u=-\lambda
u+\frac{a}{\Gamma(1-\alpha)}t^{-\alpha}+f(t)\quad\mbox{in
${{}^{-\alpha}{H}}(0,T)$},\\\ &u\in L^{2}(0,T).\end{array}\right.$

Assming that $f\in L^{2}(0,T)$ and $0<\alpha<\frac{1}{2}$, we can conclude
that (5.1)’ is equivalent to (5.5)’. Indeed,
$\frac{at^{-\alpha}}{\Gamma(1-\alpha)}\in L^{2}(0,T)$ and so
$\frac{at^{-\alpha}}{\Gamma(1-\alpha)}+f\in L^{2}(0,T)$. Therefore, by $u\in
L^{2}(0,T)$, the first equation in (5.5)’ yields $\partial_{t0}^{\alpha}u\in
L^{2}(0,T)$, which means that $u\in H_{\alpha}(0,T)$, and
$\partial_{t0}^{\alpha}u=\partial_{t}^{\alpha}u$ in $(0,T)$. Thus (5.1)’ and
(5.5)’ are equivalent provided that $0<\alpha<\frac{1}{2}$.

We emphasize that for $\frac{1}{2}\leq\alpha<1$, we cannot make any
reformulations of (5.1) similar to (5.1)’ by decomposing $u-a$ into $u$ and
$-a$. We can discuss similar reformulations also for initial boundary value
problems for fractional partial differential equations. $\blacksquare$

Now we take the widest domain of $\partial_{t}^{\alpha}$ according to classes
in time of functions under consideration, and we do not distinguish e.g.,
$\partial_{t0}^{\alpha}$ from $\partial_{t}^{\alpha}$, because there is no
fear of confusion.

Example 5.2.
Let $\alpha>\frac{1}{2}$ and let $f(t):=\delta_{t_{0}}(t)$: the Dirac delta
function at $t_{0}\in(0,T)$. In particular, we can prove that
$|\varphi(t_{0})|\leq C\|\varphi\|_{C[0,T]}$ and so
$\delta_{t_{0}}\in(C[0,T])^{\prime}$:

${}_{(C[0,T])^{\prime}}<\delta_{t_{0}},\,\psi>_{C[0,T]}\,=\psi(t_{0})$

for any $\psi\in C[0,T]$. Moreover, by the Sobolev embedding, we can see that
${{}^{\alpha}{H}}(0,T)\subset H^{\alpha}(0,T)\subset C[0,T]$ by
$\alpha>\frac{1}{2}$. Therefore, $\delta_{t_{0}}\in{{}^{-\alpha}{H}}(0,T)$,
and

${}_{{}^{-\alpha}{H}(0,T)}<\delta_{t_{0}}\,\psi>_{{}^{\alpha}{H}(0,T)}\,=\psi(t_{0})$

defines a bounded linear functional on ${{}^{\alpha}{H}}(0,T)$.

This $\delta_{t_{0}}$ describes an impulsive source term in fractional
diffusion. We will search for the representation of the solution to (5.5) with
$f=\delta_{t_{0}}$ and $a=0$:

$None$ $\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}u=-\lambda
u+\delta_{t_{0}}\quad\mbox{in ${{}^{-\alpha}{H}}(0,T)$},\\\ &u\in
L^{2}(0,T).\end{array}\right.$

Simulating a solution formula for

$None$ $d_{t}^{\alpha}u=-\lambda u+f(t),\quad u(0)=0$

(e.g., [16], p.141), we can give a candidate for solution which is formally
written by

$None$
$u(t)=\int^{t}_{0}(t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda(t-s)^{\alpha})\delta_{t_{0}}(s)ds,\quad
0<t<T.$

Our formal calculation suggests

$None$ $u(t)=\left\\{\begin{array}[]{rl}0,&0<t\leq t_{0},\\\
(t-t_{0})^{\alpha-1}E_{\alpha,\alpha}(-\lambda(t-t_{0})^{\alpha}),&t_{0}<t\leq
T.\end{array}\right.$

Now we will verify that $u(t)$ given by (5.11) is the solution to (5.8). First
it is clear that $u\in L^{1}(0,T)$. Then we will verify $u=-\lambda
J_{\alpha}^{\prime}u+J_{\alpha}^{\prime}\delta_{t_{0}}$ in $L^{2}(0,T)$.

Since $u\in L^{2}(0,T)$, we apply Proposition 2.10 to have

$J_{\alpha}^{\prime}u(t)=J^{\alpha}u(t)=\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}u(s)ds.$

For $0<t<t_{0}$, by (5.11), we see $J^{\alpha}u(t)=0$. Next for $t_{0}\leq
t\leq T$, we have

$\displaystyle-\lambda
J^{\alpha}u(t)=\frac{-\lambda}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}(s-t_{0})^{\alpha-1}E_{\alpha,\alpha}(-\lambda(s-t_{0})^{\alpha})ds$
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\alpha)}\int^{t}_{t_{0}}\sum_{k=0}^{\infty}(t-s)^{\alpha-1}(s-t_{0})^{\alpha-1}\frac{(-\lambda)^{k+1}(s-t_{0})^{\alpha
k}}{\Gamma(\alpha k+\alpha)}ds$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\alpha)}\sum_{k=0}^{\infty}\left(\int^{t}_{t_{0}}(t-s)^{\alpha-1}(s-t_{0})^{\alpha
k+\alpha-1}ds\right)\frac{(-\lambda)^{k+1}}{\Gamma(\alpha k+\alpha)}$
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\alpha)}\sum_{k=0}^{\infty}\left(\int^{t-t_{0}}_{0}(t-t_{0}-\eta)^{\alpha-1}\eta^{\alpha
k+\alpha-1}d\eta\right)\frac{(-\lambda)^{k+1}}{\Gamma(\alpha k+\alpha)}$
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\alpha)}\sum_{k=0}^{\infty}\frac{\Gamma(\alpha)\Gamma(\alpha
k+\alpha)}{\Gamma(\alpha+\alpha k+\alpha)}(t-t_{0})^{\alpha
k+2\alpha-1}\frac{(-\lambda)^{k+1}}{\Gamma(\alpha k+\alpha)}$ $None$
$=(t-t_{0})^{\alpha-1}\sum_{k=0}^{\infty}\frac{(-\lambda(t-t_{0})^{\alpha})^{k+1}}{\Gamma(\alpha(k+1)+\alpha)}=(t-t_{0})^{\alpha-1}\sum_{j=1}^{\infty}\frac{(-\lambda(t-t_{0})^{\alpha})^{j}}{\Gamma(\alpha
j+\alpha)}.$

For calculations of $J^{\alpha}u(t)$, we can apply e.g., the formula (1.100)
(p.25) in [30], but we here take a direct way.

On the other hand, by the definition of the dual operator
$J_{\alpha}^{\prime}$, setting $v_{0}:=J_{\alpha}^{\prime}\delta_{t_{0}}$, we
have

$(v_{0},\psi)_{L^{2}(0,T)}=\,\,_{{}^{-\alpha}{H}(0,T)}<\delta_{t_{0}},\,J_{\alpha}\psi>_{{}^{\alpha}{H}(0,T)}\quad\mbox{for
all $\psi\in L^{2}(0,T)$}.$

Since $J_{\alpha}\psi\in{{}^{\alpha}{H}}(0,T)\subset C[0,T]$ by
$\alpha>\frac{1}{2}$ and Proposition 2.8, it follows that $v_{0}$ satisfies

${}_{{}^{-\alpha}{H}(0,T)}<\delta_{t_{0}},\,J_{\alpha}\psi>_{{}^{\alpha}{H}(0,T)}\,=(J_{\alpha}\psi)(t_{0})=\frac{1}{\Gamma(\alpha)}\int^{T}_{t_{0}}(s-t_{0})^{\alpha-1}\psi(s)ds.$

Therefore,

$\frac{1}{\Gamma(\alpha)}\int^{T}_{t_{0}}(s-t_{0})^{\alpha-1}\psi(s)ds=\,(v_{0},\psi)_{L^{2}(0,T)}=\int^{t_{0}}_{0}v_{0}(s)\psi(s)ds+\int^{T}_{t_{0}}v_{0}(s)\psi(s)ds$

for all $\psi\in L^{2}(0,T)$. Choosing $\psi\in L^{2}(0,T)$ satisfying
$\psi=0$ in $(t_{0},T)$, we obtain

$\int^{t_{0}}_{0}v_{0}(s)\psi(s)ds=0,$

and so $v_{0}(s)=0$ for $0<s\leq t_{0}$. Hence,

$None$
$J_{\alpha}^{\prime}\delta_{t_{0}}(s)=v_{0}(s)=\left\\{\begin{array}[]{rl}0,&0<s\leq
t_{0},\\\
\frac{(s-t_{0})^{\alpha-1}}{\Gamma(\alpha)},&t_{0}<s<T.\end{array}\right.$

In other words,

$None$ $\partial_{t}^{\alpha}v_{0}=\delta_{t_{0}},$

where $v_{0}\in L^{2}(0,T)$ is defined by (5.13).

Consequently, since

$\frac{(t-t_{0})^{\alpha-1}}{\Gamma(\alpha)}+(t-t_{0})^{\alpha-1}\sum_{k=1}^{\infty}\frac{(-\lambda(t-t_{0})^{\alpha})^{k}}{\Gamma(\alpha
k+\alpha)}=(t-t_{0})^{\alpha-1}E_{\alpha,\alpha}(-\lambda(t-t_{0})^{\alpha}),$

in terms of (5.12) and (5.13) we reach

$-\lambda
J_{\alpha}^{\prime}u(t)+J_{\alpha}^{\prime}\delta_{t_{0}}(t)=\left\\{\begin{array}[]{rl}0,&0<t\leq
t_{0},\\\
(t-t_{0})^{\alpha-1}E_{\alpha,\alpha}(-\lambda(t-t_{0})^{\alpha}),&t_{0}<t<T.\end{array}\right.$

By (5.11) we verify

$u(t)=-\lambda J_{\alpha}^{\prime}u(t)+J_{\alpha}^{\prime}\delta_{t_{0}}(t).$

Thus we verified that $u(t)$ given by (5.11) is the unique solution to (5.8).
$\blacksquare$

We recall (4.6):

$(B_{\lambda}f)(t):=\int^{t}_{0}(t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda(t-s)^{\alpha})f(s)ds,\quad
0<t<T,\quad\mbox{for $f\in L^{2}(0,T)$}.$

By Proposition 4.2, we know that $B_{\lambda}:L^{2}(0,T)\longrightarrow
H_{\alpha}(0,T)$ is a bounded operator.

We close this section with
Proposition 5.1 (Representation of solution to (5.5) with
$f\in{{}^{-\alpha}{H}}(0,T)$).
Let $\lambda>-\Lambda_{0}$ be fixed. The operator $B_{\lambda}$ can be
extended to $S_{\lambda}:{{}^{-\alpha}{H}}(0,T)\longrightarrow L^{2}(0,T)$ as
follows. For $f\in{{}^{-\alpha}{H}}(0,T)$, there exists a sequence $f_{n}\in
L^{2}(0,T)$, $n\in\mathbb{N}$ such that $f_{n}\longrightarrow f$ in
${{}^{-\alpha}{H}}(0,T)$. Then

$None$
$\lim_{m,n\to\infty}\|B_{\lambda}f_{n}-B_{\lambda}f_{m}\|_{L^{2}(0,T)}=0$

and $lim_{n\to\infty}B_{\lambda}f_{n}$ is unique in $L^{2}(0,T)$ independently
of choices of sequences $f_{n}$, $n\in\mathbb{N}$ such that
$f_{n}\longrightarrow f$ in ${{}^{-\alpha}{H}}(0,T)$. Hence, setting

$S_{\lambda}f:=\lim_{n\to\infty}B_{\lambda}f_{n}\quad\mbox{in $L^{2}(0,T)$},$

we have

$None$ $\partial_{t}^{\alpha}(S_{\lambda}f)=-\lambda
S_{\lambda}f+f\quad\mbox{in ${{}^{-\alpha}{H}}(0,T)$.}$

Proof.
First, since $L^{2}(0,T)$ is dense in ${{}^{-\alpha}{H}}(0,T)$, we can choose
a sequence $f_{n}\in L^{2}(0,T)$, $n\in\mathbb{N}$ such that
$\lim_{n\to\infty}f_{n}=f$ in $L^{2}(0,T)$.
Verification of (5.15).
By Proposition 4.2, we see

$None$ $B_{\lambda}f=-\lambda
J_{\alpha}^{\prime}B_{\lambda}f+J_{\alpha}^{\prime}f$

and

$None$ $\|B_{\lambda}f\|_{L^{2}(0,T)}\leq
C\|J_{\alpha}^{\prime}f\|_{L^{2}(0,T)}\quad\mbox{for $f\in L^{2}(0,T)$}.$

In view of (5.18), we obtain

$\|B_{\lambda}f_{n}-B_{\lambda}f_{m}\|_{L^{2}(0,T)}\leq
C\|J_{\alpha}^{\prime}f_{n}-J_{\alpha}^{\prime}f_{m}\|_{L^{2}(0,T)}.$

Proposition 2.9 (ii) yields

$\|B_{\lambda}f_{n}-B_{\lambda}f_{m}\|_{L^{2}(0,T)}\leq
C\|f_{n}-f_{m}\|_{{{}^{-\alpha}{H}}(0,T)}.$

Since $\lim_{m,n\to\infty}\|f_{n}-f_{m}\|_{{{}^{-\alpha}{H}}(0,T)}=0$, we see
that $B_{\lambda}f_{n}$, $n\in\mathbb{N}$ converge in $L^{2}(0,T)$.

Similarly we can prove that $\lim_{n\to\infty}B_{\lambda}f_{n}$ is determined
independently of choices of sequences $f_{n}$, $n\in\mathbb{N}$ such that
$\lim_{n\to\infty}f_{n}=f$ in ${{}^{-\alpha}{H}}(0,T)$. Therefore,
$S_{\lambda}f$ is well-defined for $f\in L^{2}(0,T)$.
Verification of (5.16).
For an approximating sequence $f_{n}\in L^{2}(0,T)$, $n\in\mathbb{N}$ such
that $f_{n}\longrightarrow f$ in ${{}^{-\alpha}{H}}(0,T)$, we have

$B_{\lambda}f_{n}=-\lambda
J_{\alpha}^{\prime}B_{\lambda}f_{n}+J_{\alpha}^{\prime}f_{n}\quad\mbox{in
$L^{2}(0,T)$ for $n\in\mathbb{N}$}.$

Since $\lim_{n\to\infty}B_{\lambda}f_{n}=S_{\lambda}f$ in $L^{2}(0,T)$,
letting $n\to\infty$, we obtain

$S_{\lambda}f=-\lambda\lim_{n\to\infty}J_{\alpha}^{\prime}B_{\lambda}f_{n}+\lim_{n\to\infty}J_{\alpha}^{\prime}f_{n}\quad\mbox{in
$L^{2}(0,T)$}.$

We apply Proposition 2.9 (ii) to see
$\lim_{n\to\infty}J_{\alpha}^{\prime}f_{n}=J_{\alpha}^{\prime}f$ in
$L^{2}(0,T)$ by $\lim_{n\to\infty}f_{n}=f$ in ${{}^{-\alpha}{H}}(0,T)$.
Finally Proposition 2.9 (iii) implies that
$J_{\alpha}^{\prime}|_{L^{2}(0,T)}:L^{2}(0,T)\longrightarrow L^{2}(0,T)$ is
bounded, so that

$\lim_{n\to\infty}J_{\alpha}^{\prime}B_{\lambda}f_{n}=\lim_{n\to\infty}J^{\alpha}B_{\lambda}f_{n}=J_{\alpha}^{\prime}S_{\lambda}f\quad\mbox{in
$L^{2}(0,T)$}.$

Therefore we reach

$S_{\lambda}f=-\lambda
J_{\alpha}^{\prime}S_{\lambda}f+J_{\alpha}^{\prime}f\quad\mbox{in
$L^{2}(0,T)$}.$

By the definition (2.13) of $\partial_{t}^{\alpha}$, this means

$\partial_{t}^{\alpha}(S_{\lambda}f)=-\lambda S_{\lambda}f+f\quad\mbox{in
${{}^{-\alpha}{H}}(0,T)$.}$

Thus the verification of (5.16) is complete, so that the proof of Proposition
5.1 is finished. $\blacksquare$

We can discuss more about the reprensentation formula of solution to (5.5)
with $f\in{{}^{-\alpha}{H}}(0,T)$ in terms of convolution operators, but we
will postpone to a future work.

##  6\. Initial boundary value problem for fractional partial differential
equations: selected topics

On the basis of $\partial_{t}^{\alpha}$ defined in Section 2, we construct a
feasible framework also for initial boundary value problems. We recall that an
elliptic operator $-\mathcal{A}$ is defined by (1.1), and we assume all the
conditions as described in Section 1 on the coefficients $a_{ij}$, $b_{j}$,
$c\in C^{1}(\overline{\Omega})$.

Here we mostly consider the case $\alpha<1$, but cases $\alpha>1$ can be
formulated and studied similarly.

By $\nu=(\nu_{1},....,\nu_{d})$ we denote the outward unit normal vector to
$\partial\Omega$ at $x$ and set

$\partial_{\nu_{A}}v=\sum_{i,j=1}^{d}a_{ij}(\partial_{j}v)\nu_{i}\quad\mbox{on
$\partial\Omega$}.$

We define an operator $-A$ in $L^{2}(\Omega)$ by

$None$ $\left\\{\begin{array}[]{rl}&Av(x)=\mathcal{A}v(x),\quad x\in\Omega,\\\
&\mathcal{D}(A)=\\{v\in
H^{2}(\Omega);\,v|_{\partial\Omega}=0\\}=H^{2}(\Omega)\cap
H^{1}_{0}(\Omega).\end{array}\right.$

Here $v|_{\partial\Omega}=0$ is understood as the sense of the trace (e.g.,
[1]).

We can similarly discuss other boundary condition, for example,
$\mathcal{D}(A)=\\{v\in
H^{2}(\Omega);\,\partial_{\nu_{A}}v+\sigma(x)v=0\,\,\mbox{on
$\partial\Omega$}\\}$ with fixed function $\sigma(x)$, but we concentrate on
the homogeneous Dirichlet boundary condition $u|_{\partial\Omega}=0$.

We formulate the initial boundary value problem by

$None$ $\partial_{t}^{\alpha}(u(x,t)-a(x))+Au(x,t)=F(x,t)\quad\mbox{in
$L^{2}(0,T;L^{2}(\Omega))$}$

and

$None$ $u-a\in H_{\alpha}(0,T;L^{2}(\Omega)).$

We emphasize that we do not adopt formulation (1.2).

The formulation (6.2) - (6.3) corresponds to (5.1) for an initial value
problem for a time-fractional ordinary differential equation. The term $Au$ in
equation (6.2) means that $u(\cdot,t)\in\mathcal{D}(A)=H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)$ for almost all $t\in(0,T)$, that is,

$u(\cdot,t)|_{\partial\Omega}=0\quad\mbox{for almost all $t\in(0,T)$.}$

In other words, the domain of $A$ describes the boundary condition which is a
conventional way in treating the classical partial differential equations.
Like fractional ordinary differential equations in Section 5, we understand
that (6.3) means the initial condition.

We first present a basic well-posedness result for (6.2) - (6.3):
Theorem 6.1.
Let $0<\alpha<1$. Let $a\in H^{1}_{0}(\Omega)$ and $F\in
L^{2}(0,T;L^{2}(\Omega))$. Then there exists a unique solution $u=u(x,t)$ to
(6.2) - (6.3) such that $u-a\in H_{\alpha}(0,T;L^{2}(\Omega))$ and $u\in
L^{2}(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))$. Moreover there exists a
constant $C>0$ such that

$\|u-a\|_{H_{\alpha}(0,T;L^{2}(\Omega))}+\|u\|_{L^{2}(0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega))}\leq
C(\|a\|_{H^{1}_{0}(\Omega)}+\|F\|_{L^{2}(0,T;L^{2}(\Omega))})$

for all $a\in H^{1}_{0}(\Omega)$ and $F\in L^{2}(0,T;L^{2}(\Omega))$.

Here we remark that

$\|v\|_{H_{\alpha}(0,T;L^{2}(\Omega))}:=\|\partial_{t}^{\alpha}v\|_{L^{2}(0,T;L^{2}(\Omega))}\quad\mbox{for
$v\in H_{\alpha}(0,T;L^{2}(\Omega))$}.$

Unique existence results of solutions are known according to several
formulations of initial boundary value problems. In Sakamoto and Yamamoto
[32], in the case of symmetric $A$ where $b_{j}=0$ for $1\leq j\leq d$ in
(1.1), the unique existence is proved by means of the Fourier method, but the
class of solutions is not the same as here. In the case where $b_{j}$ for
$j=1,...,d$ are not necessarily zero, assuming that the initial value $a$ is
zero, Theorem 6.1 is proved in Gorenflo, Luchko and Yamamoto [7]. In both [7]
and [32], it is assumed that all the coefficients are independent of $t$ and
and $c=c(x)\leq 0$ for $x\in\Omega$. The work Kubica, Ryszewska and Yamamoto
[18] proved Theorem 6.1 in a general case where $a_{ij},b_{j},c$ depends both
on $x$ and $t$ without extra assumpion $c\leq 0$. In other words, Theorem 6.1
is a special case of Theorem 4.2 in [18]. Furthermore, there have been other
works on the well-posedness for initial boundary value problems and we are
restricted to some of them: Bajlekova [3], Kubica and Yamamoto [19], Luchko
[22], [23], Luchko and Yamamoto [26], Zacher [37]. See also Prüss [31] for a
monograph on related integral equations, and a recent book Jin [14] which
mainly studies the symmetric $A$. As for further references up to 2019, the
handbooks [17] edited by A.Kochubei and Y. Luchko are helpful. Most of the
above works discuss the case of $F\in L^{2}(0,T;L^{2}(\Omega))$.

Next we consider less regular $F$ and $a$ in $x$. To this end, we introduce
Sobolev spaces of negative orders in $x$. Similarly to the triple
$\,{}^{\alpha}H(0,T)\subset L^{2}(\Omega)\subset{{}^{-\alpha}{H}}(0,T)$ as is
explained in Section 2, we introduce the dual space
$(H_{1}^{0}(\Omega))^{\prime}$ of $H^{1}_{0}(\Omega)$ by identifying the dual
space $(L^{2}(0,T))^{\prime}$ with $L^{2}(0,T)$:

$H^{1}_{0}(\Omega)\subset
L^{2}(\Omega)\subset(H^{1}_{0}(\Omega))^{\prime}=:H^{-1}(\Omega)$

(e.g., [4]). For less regular $F$ and $a$ in the $x$-variable, we know
Theorem 6.2.
Let $0<\alpha<1$. For the coefficients of $A$, we assume the same conditions
as in Theorem 6.1. Let $a\in L^{2}(\Omega)$ and $F\in
L^{2}(0,T;H^{-1}(\Omega))$. Then there exists a unique solution $u=u(x,t)$ to
(6.2) - (6.3) such that $u-a\in H_{\alpha}(0,T;H^{-1}(\Omega))$ and $u\in
L^{2}(0,T;H^{1}_{0}(\Omega))$. Moreover there exists a constant $C>0$ such
that

$\|u-a\|_{H_{\alpha}(0,T;H^{-1}(\Omega))}+\|u\|_{L^{2}(0,T;H^{1}_{0}(\Omega))}\leq
C(\|a\|_{L^{2}(\Omega)}+\|F\|_{L^{2}(0,T;H^{-1}(\Omega))})$

for all $a\in L^{2}(\Omega)$ and $F\in L^{2}(0,T;H^{-1}(\Omega))$.

In Theorem 6.2, we consider both sides of (6.2) in
$L^{2}(0,T;H^{-1}(\Omega))$. The proof of Theorem 6.2 is found in [18] and see
also [19].

Remark 6.1.
As for general $\alpha>0$, by means of the space defined by (2.6), we can
formulate the initial boundary value problem as follows. For $\alpha>1$, we
set $\alpha=m+\sigma$ with $m\in\mathbb{N}$ and $0<\sigma\leq 1$. Then for
$\alpha>1$ we formulate an initial boundary value problem by

$\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}\left(u-\sum_{k=0}^{m}a_{k}\frac{t^{k}}{k!}\right)=-Au+F(x,t),\quad
x\in\Omega,\,0<t<T,\\\
&\left(u-\sum_{k=0}^{m}a_{k}\frac{t^{k}}{k!}\right)(x,\cdot)\in
H_{\alpha}(0,T)\quad\mbox{for almost all $x\in\Omega$}.\end{array}\right.$

For $\sigma>\frac{1}{2}$, we can interpret the second condition as usual
initial conditions. More precisely,

$\left(u-\sum_{k=0}^{m}a_{k}\frac{t^{k}}{k!}\right)(x,\cdot)\in
H_{\alpha}(0,T)$

if and only if

$\left\\{\begin{array}[]{rl}\frac{\partial^{k}u}{\partial
t^{k}}(\cdot,0)=a_{k},&k=0,1,...,m-1,\\\ \frac{\partial^{m}u}{\partial
t^{m}}(x,\cdot)-a_{m}\in H_{\sigma}(0,T).\end{array}\right.$

In this section, we pick up five topics and apply the results in Sections 2 -
4. We postpone general and complete descriptions to a future work. Moreover,
we are limited to the following $A$:

$None$
$-Av(x)=\sum_{j=1}^{d}\partial_{j}(a_{ij}(x)\partial_{j}v(x))+\sum_{j=1}^{d}b_{j}(x)\partial_{j}v+c(x)v,\quad
x\in\Omega$

for $v\in\mathcal{D}(A):=H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$ with

$None$ $a_{ij}=a_{ji}\in C^{1}(\overline{\Omega}),\quad b_{j}\in
C^{1}(\overline{\Omega})\quad\mbox{for $1\leq i,j\leq d$},\quad c\in
C^{1}(\overline{\Omega}),\,\leq 0\quad\mbox{on $\Omega$}.$

§6.1. Mild solution and strong solution

We define an operator $L$ as a symmetric part of $A$ by

$None$
$-Lv(x)=\sum_{j=1}^{d}\partial_{j}(a_{ij}(x)\partial_{j}v(x))+c(x)v,\quad
x\in\Omega,\quad\mathcal{D}(L)=H^{2}(\Omega)\cap H^{1}_{0}(\Omega).$

Then there exist eigenvalues of $L$ and according to the multiplicities, we
can arrange all the eigenvalues as

$0<\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}\leq\cdots\,\longrightarrow\infty.$

Here by $c\leq 0$ in $\Omega$, we can prove that $\lambda_{1}>0$. Moreover we
can choose eigenfunctions $\varphi_{n}$ for $\lambda_{n}$, $n\in\mathbb{N}$
such that $\\{\varphi_{n}\\}_{n\in\mathbb{N}}$ is an orthonormal basis in
$L^{2}(\Omega)$. Henceforth $(\cdot,\cdot)$ and $\|\cdot\|_{L^{2}(\Omega)}$
denote the scalar product and the norm in $L^{2}(\Omega)$ respectively and we
write $(\cdot,\cdot)_{L^{2}(\Omega)}$ when we like to specify the space. Thus
$L\varphi_{n}=\lambda_{n}\varphi_{n}$, $\|\varphi_{n}\|_{L^{2}(\Omega)}=1$ and
$(\varphi_{n}\,\varphi_{m})=0$ if $n\neq m$.

For $\gamma\in\mathbb{R}$, we can define a fractional power $L^{\gamma}$ of
$L$ by

$None$
$\left\\{\begin{array}[]{rl}&(L^{\gamma}v)(x)=\sum_{n=1}^{\infty}\lambda_{n}^{\gamma}(v,\varphi_{n})\varphi_{n}(x),\\\
&\mathcal{D}(L^{\gamma}):=\left\\{\begin{array}[]{rl}&L^{2}(\Omega)\quad\mbox{if
$\gamma\leq 0$},\\\ &\left\\{v\in
L^{2}(\Omega);\,\sum_{n=1}^{\infty}\lambda_{n}^{2\gamma}|(v,\varphi_{n})|^{2}<\infty\right\\}\quad\mbox{if
$\gamma>0$}.\end{array}\right.\end{array}\right.$

We set

$None$
$\|v\|_{\mathcal{D}(L^{\gamma})}:=\left(\sum_{n=1}^{\infty}\lambda_{n}^{2\gamma}|(v,\varphi_{n})|^{2}\right)^{\frac{1}{2}}\quad\mbox{if
$\gamma>0$}.$

Then it is known that

$\mathcal{D}(L^{\frac{1}{2}})=H^{1}_{0}(\Omega),\quad
C^{-1}\|v\|_{H^{1}_{0}(\Omega)}\leq\|L^{\frac{1}{2}}v\|_{L^{2}(\Omega)}\leq
C\|v\|_{H^{1}_{0}(\Omega)},\quad v\in H^{1}_{0}(\Omega).$

Here $C>0$ is independent of choices of $v\in H^{1}_{0}(\Omega)$.

We further define operator $S(t)$ and $K(t)$ from $L^{2}(\Omega)$ to
$L^{2}(\Omega)$ by

$S(t)a:=\sum_{n=1}^{\infty}E_{\alpha,1}(-\lambda_{n}t^{\alpha})(a,\varphi_{n})\varphi_{n}$

and

$K(t)a:=\sum_{n=1}^{\infty}t^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}t^{\alpha})(a,\varphi_{n})\varphi_{n}$

for all $a\in L^{2}(\Omega)$. Then for $0\leq\gamma\leq 1$, we can find
constants $C_{1}>0$ and $C_{2}=C_{2}(\gamma)>0$ such that

$None$ $\left\\{\begin{array}[]{rl}&\|S(t)a\|_{L^{2}(\Omega)}\leq
C_{1}\|a\|_{L^{(}\Omega)},\\\ &\|L^{\gamma}K(t)a\|_{L^{2}(\Omega)}\leq
C_{2}t^{\alpha(1-\gamma)-1}\|a\|_{L^{2}(\Omega)}\quad\mbox{for $t>0$ and $a\in
L^{2}(\Omega)$.}\end{array}\right.$

The proof of (6.9) is direct by the definition of $S(t)$ and $K(t)$ and can be
found e.g., in [7].

Here and henceforth we write $u(t):=u(\cdot,t)$ as a mapping from $(0,T)$ to
$L^{2}(\Omega)$. The proofs of (6.8) - (6.10) can be found e.g., in [7].

We can show
Proposition 6.1.
Let $a\in H^{1}_{0}(\Omega)$ and $F\in L^{2}(0,T;L^{2}(\Omega))$ and let (6.5)
hold. The following are equivalent:
(i) $u\in L^{2}(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))$ satisfies (6.2) and
(6.3).
(ii) $u\in L^{2}(0,T;L^{2}(\Omega))$ satisfies

$None$
$u(t)=S(t)a+\int^{t}_{0}K(t-s)\sum_{j=1}^{d}b_{j}\partial_{j}u(\cdot,s)ds+\int^{t}_{0}K(t-s)F(s)ds,\quad
0<t<T.$

Equation (6.10) corresponds to formula (5.2) for an initial value problem for
fractional ordinary differential equation.

According to the parabolic equation (e.g., Pazy [29]), the solution guaranteed
by Theorem 6.1 is called a strong solution of the fractional partial
differential equation, while the solution to an integrated equation (6.10) is
called a mild solution. Proposition 6.1 asserts the equivalence between these
two kinds of solutions under assumption that $a\in H^{1}_{0}(\Omega)$ and
$F\in L^{2}(0,T;L^{2}(\Omega))$.

For the proof of Proposition 6.1, we show two lemmata. Henceforth, for $v\in
L^{2}(\Omega)$, we define $\partial_{j}v\in
H^{-1}(\Omega):=(H^{1}_{0}(\Omega))^{\prime}$, $1\leq j\leq d$, by

${}_{H^{-1}(\Omega)}<\partial_{j}v,\,\varphi>_{H^{1}_{0}(\Omega)}\,\,:=-(v,\,\partial_{j}\varphi)_{L^{2}(\Omega)}\quad\mbox{for
all $\varphi\in H^{1}_{0}(\Omega)$}.$

Then by the definition of the norm of the linear functional, we have

$\|\partial_{j}v\|_{H^{-1}(\Omega)}=\sup_{\|\varphi\|_{H^{1}_{0}(\Omega)}=1}|_{H^{-1}(\Omega)}<\partial_{j}v,\,\varphi>_{H^{1}_{0}(\Omega)}|$
$None$
$\leq\sup_{\|\varphi\|_{H^{1}_{0}(\Omega)}=1}|(v,\,\partial_{j}\varphi)_{L^{2}(\Omega)}|\leq\|v\|_{L^{2}(\Omega)}.$

We recall that $L^{-\frac{1}{2}}$ is defined as an operator from
$L^{2}(\Omega)$ to $L^{2}(\Omega)$ by (6.7), while also
$L^{\frac{1}{2}}:H^{1}_{0}(\Omega)\,\longrightarrow\,L^{2}(\Omega)$ is defined
by (6.7).

Then we can state the first lemma which involves the extension of
$L^{-\frac{1}{2}}:L^{2}(\Omega)\,\longrightarrow\,L^{2}(\Omega)$.
Lemma 6.1.
(i) The operator $L^{-\frac{1}{2}}$ can be extended as a bounded operator from
$H^{-1}(\Omega)$ to $L^{2}(\Omega)$. Henceforth, by the same notation, we
denote the extension. There exists a constant $C>0$ such that

$C^{-1}\|v\|_{H^{-1}(\Omega)}\leq\|L^{-\frac{1}{2}}v\|_{L^{2}(\Omega)}\leq
C\|v\|_{H^{-1}(\Omega)}\quad\mbox{for all $v\in H^{-1}(\Omega)$}.$

(ii) $K(t)v=L^{\frac{1}{2}}K(t)L^{-\frac{1}{2}}v$ for $t>0$ and $v\in
H^{-1}(\Omega)$.
(iii)

$\|L^{-\frac{1}{2}}(b_{j}\partial_{j}v)\|_{L^{2}(\Omega)}\leq
C\|v\|_{L^{2}(\Omega)}\quad\mbox{for all $v\in L^{2}(\Omega)$}.$

The second lemma is
Lemma 6.2.
(i) For $a\in L^{2}(\Omega)$, we have $S(t)a\in\mathcal{D}(L)$ for $t>0$ and

$\partial_{t}^{\alpha}(S(t)a-a)+LS(t)a=0,\quad 0<t<T.$

Moreover there exists a constant $C_{0}>0$ such that

$\|S(t)a-a\|_{H_{\alpha}(0,T;L^{2}(\Omega))}\leq
C_{0}\|a\|_{H^{1}_{0}(\Omega)}$

for all $a\in H^{1}_{0}(\Omega)$. Here $C_{0}>0$ is independent of $a\in
H^{1}_{0}(\Omega)$ and $T>0$.
(ii) We have

$\partial_{t}^{\alpha}\left(\int^{t}_{0}K(t-s)F(s)ds\right)+L\left(\int^{t}_{0}K(t-s)F(s)ds\right)=F(t),\quad
0<t<T$

for $F\in L^{2}(0,T;L^{2}(\Omega))$. Moreover, there exists a constant
$C_{1}=C_{1}(T)>0$ such that

$\left\|\int^{t}_{0}K(t-s)F(s)ds\right\|_{H_{\alpha}(0,T;L^{2}(\Omega))}\leq
C_{1}\|F\|_{L^{2}(0,T;L^{2}(\Omega))}$

for each $F\in L^{2}(0,T;L^{2}(\Omega))$.

We note that Lemma 6.2 corresponds to Propositions 4.1 (i) and 4.2.
Let Lemmata 6.1 and 6.2 be proved. Then, the proof of (i) $\longrightarrow$
(ii) of Proposition 6.1 can be done similarly to [32]. The proof of (ii)
$\longrightarrow$ (i) of the proposition can be derived from Lemmata 6.1 and
6.2, and we omit the details. The proofs of the lemmata are provided in
Appendix.

§6.2. Continuity at $t=0$

As is discussed in Sections 1,2 and 5, the continuity of solutions to
fractional differential equations at $t=0$ is delicate, which requires careful
treatments for initial conditions. However, if we assume $F=0$ in (6.2), then
we can prove the following sufficient continuity.
Theorem 6.3.
Under (6.4) and (6.5), for $a\in L^{2}(\Omega)$, the solution $u$ to (6.2) -
(6.3) satisfies

$u\in C([0,T];L^{2}(\Omega)).$

The theorem means that $\lim_{t\to 0}\|u(\cdot,t)-a\|_{L^{2}(\Omega)}=0$, but
the continuity at $t=0$ breaks if a non-homogeneous term $F\neq 0$, which is
already shown in Example 5.1 in Section 5 concerning a fractional ordinary
differential equations.

In the case of $b_{j}=0$, $1\leq j\leq d$, the same result is proved in [32]
and we can refer also to [14]. However, it seems no proofs for a non-symmetric
elliptic operator $A$, although one naturally expect the same continuity. The
proof is typical as arguments by the operator theory applied to the classical
partial differential equations and we carry out similar arguments for
fractional differential equations within our framework.
Proof.
First Step.
From Lemma 6.1, it follows that the solution $u$ to (6.2) - (6.3) with $F=0$
is given by

$None$
$u(\cdot,t)=S(t)a+\int^{t}_{0}K(t-s)\sum_{j=1}^{d}b_{j}\partial_{j}u(\cdot,s)ds=:S(t)a+M(u)(t),\quad
0<t<T.$

The proof of Theorem 6.3 is based on an approximating sequence for the
solution $u(t)$ constructed by

$u_{1}(t):=S(t)a,\quad u_{k+1}(t):=S(t)a+(Mu_{k})(t),\quad
k\in\mathbb{N},\,0<t<T.$

First we can prove

$None$ $S(\cdot)a\in C([0,T];L^{2}(\Omega)).$

Verification of (6.13). For $\ell\leq m$, using (4.4) with $\lambda_{n}>0$,
for $0\leq t\leq T$ we estimate

$\left\|\sum_{n=\ell}^{m}(a,\varphi_{n})E_{\alpha,1}(-\lambda_{n}t^{\alpha})\varphi_{n}\right\|_{L^{2}(\Omega)}^{2}\leq\sum_{n=\ell}^{m}|(a,\varphi_{n})|^{2}|E_{\alpha,1}(-\lambda_{n}t^{\alpha})|^{2}\leq
C\sum_{n=\ell}^{m}|(a,\varphi_{n})|^{2}.$

Therefore,

$\lim_{\ell,m\to\infty}\left\|\sum_{n=\ell}^{m}(a,\varphi_{n})E_{\alpha,1}(-\lambda_{n}t^{\alpha})\varphi_{n}\right\|_{C([0,T];L^{2}(\Omega))}=0,$

which means that

$\sum_{n=1}^{N}(a,\varphi_{n})E_{\alpha,1}(-\lambda_{n}t^{\alpha})\varphi_{n}\in
C([0,T];L^{2}(\Omega))$

converges to $S(t)a$ in $C([0,T];L^{2}(\Omega))$. Thus the verification of
(6.13) is complete. $\blacksquare$
Second Step. We prove

$None$ $\int^{t}_{0}L^{\frac{1}{2}}K(s)w(t-s)ds\in
C([0,T];L^{2}(\Omega))\quad\mbox{for $w\in C([0,T];L^{2}(\Omega))$}.$

Verification of (6.14).
Let $0<t<T$ be arbitrarily fixed. For small $h>0$, we have

$\displaystyle\int^{t+h}_{0}L^{\frac{1}{2}}K(s)w(t+h-s)ds-\int^{t}_{0}L^{\frac{1}{2}}K(s)w(t-s)ds$
$\displaystyle=$
$\displaystyle\int^{t+h}_{t}L^{\frac{1}{2}}K(s)w(t+h-s)ds+\int^{t}_{0}L^{\frac{1}{2}}K(s)(w(t+h-s)-w(t-s))ds$
$\displaystyle=:$ $\displaystyle I_{1}(t,h)+I_{2}(t,h).$

Then by (6.9) we can estimate

$\displaystyle\|I_{1}(t,h)\|\leq
C\left(\int^{t+h}_{t}s^{\frac{1}{2}\alpha-1}ds\right)\|w\|_{C([0,T];L^{2}(\Omega))}$
$\displaystyle=$
$\displaystyle\frac{2C}{\alpha}((t+h)^{\frac{1}{2}\alpha}-t^{\frac{1}{2}\alpha})\|w\|_{C([0,T];L^{2}(\Omega))}\longrightarrow
0\quad\mbox{as $h\to 0$}.$

Next

$\|I_{2}(t,h)\|\leq
C\int^{t}_{0}s^{\frac{1}{2}\alpha-1}\|w(t+h-s)-w(t-s)\|ds.$

We have $\max_{0\leq s\leq T}\|w(t+h-s)-w(t-s)\|\longrightarrow 0$ as $h\to 0$
with fixed $t$ by $w\in C([0,T];L^{2}(\Omega))$. Hence, since
$s^{\frac{1}{2}\alpha-1}\in L^{1}(0,T)$, the Lebesgue convergence theorem
yields that $\lim_{h\to 0}\|I_{2}(t,h)\|=0$. We can argue similarly also for
$h<0$, $t=0$ and $t=T$, and so the verification of (6.14) is complete.
$\blacksquare$

Next we proceed to the proofs of (6.15) and (6.16):

$None$ $u_{k}\in C([0,T];L^{2}(\Omega)),\quad k\in\mathbb{N}$

and

$None$ $u_{k}$ converges in $C([0,T];L^{2}(\Omega))$ as $k\to\infty$.

Third Step: Proof of (6.15). We will prove by induction. By (6.13), we see
that $u_{1}(t)=S(t)a\in C([0,T];L^{2}(\Omega))$. We assume that $u_{k}\in
C([0,T];L^{2}(\Omega))$. Then, by Lemma 6.1 (ii), we have

$M(u_{k})(t)=\int^{t}_{0}K(t-s)\left(\sum_{j=1}^{d}b_{j}\partial_{j}u_{k}(s)\right)ds=\int^{t}_{0}L^{\frac{1}{2}}K(t-s)L^{-\frac{1}{2}}\left(\sum_{j=1}^{d}b_{j}\partial_{j}u_{k}(s)\right)ds.$

Lemma 6.1 (iii) yields

$\|L^{-\frac{1}{2}}(b_{j}\partial_{j}u_{k}(t))-L^{-\frac{1}{2}}(b_{j}\partial_{j}u_{k}(t^{\prime}))\|_{L^{2}(\Omega)}\leq
C\|u_{k}(t)-u_{k}(t^{\prime})\|_{L^{2}(\Omega)},\quad t,t^{\prime}\in[0,T]$

for $1\leq j\leq d$, so that

$L^{-\frac{1}{2}}b_{j}\partial_{j}u_{k}\in C([0,T];L^{2}(\Omega)),\quad 1\leq
j\leq d.$

Therefore, the application of (6.14) to $M(u_{k})$ implies

$M(u_{k})\in C([0,T];L^{2}(\Omega)).$

Consequently,

$u_{k+1}=S(\cdot)a+M(u_{k})\in C([0,T];L^{2}(\Omega)).$

Thus the induction completes the proof of (6.15). $\blacksquare$
Fourth Step: Proof of (6.16).
We have

$\displaystyle(u_{k+2}-u_{k+1})(t)=M(u_{k+1}-u_{k})(t)$ $\displaystyle=$
$\displaystyle\int^{t}_{0}L^{\frac{1}{2}}K(t-s)\sum_{j=1}^{d}L^{-\frac{1}{2}}(b_{j}\partial_{j}(u_{k+1}-u_{k})(s))ds,\quad
k\in\mathbb{N},\,0<t<T.$

We set $v_{k}:=u_{k+1}-u_{k}$. Then Lemma 6.1 (iii) and (6.9) yield

$\|v_{k+1}(t)\|\leq\int^{t}_{0}\left\|L^{\frac{1}{2}}K(t-s)\sum_{j=1}^{d}L^{-\frac{1}{2}}(b_{j}\partial_{j}v_{k}(s))\right\|ds$
$None$ $\leq C\int^{t}_{0}(t-s)^{\frac{1}{2}\alpha-1}\|v_{k}(s)\|ds,\quad
k\in\mathbb{N}.$

Setting $M_{0}:=\|v_{1}\|_{C([0,T];L^{2}(\Omega))}$, we apply (6.17) with
$k=1$ to obtain

$\|v_{2}(t)\|\leq
C\int^{t}_{0}(t-s)^{\frac{1}{2}\alpha-1}dsM_{0}=\frac{CM_{0}}{\Gamma\left(\frac{1}{2}\alpha\right)}t^{\frac{1}{2}\alpha},\quad
0<t<T.$

Therefore, substituting this into (6.17) with $k=2$, we obtain

$\|v_{3}(t)\|\leq\frac{C^{2}M_{0}}{\Gamma\left(\frac{1}{2}\alpha\right)}\int^{t}_{0}(t-s)^{\frac{1}{2}\alpha-1}s^{\frac{1}{2}\alpha}ds=C^{2}M_{0}\frac{\Gamma\left(\frac{1}{2}\alpha+1\right)}{\Gamma(\alpha+1)}t^{\alpha},\quad
0<t<T.$

Continuing this estimate, we can prove

$\|v_{k+1}(t)\|\leq\frac{\frac{1}{2}\alpha
M_{0}C^{k}\left(\Gamma\left(\frac{1}{2}\alpha\right)\right)^{k-1}}{\Gamma\left(\frac{1}{2}k\alpha+1\right)}t^{\frac{1}{2}k\alpha},\quad
0<t<T$

for all $k\in\mathbb{N}$. Consequently,

$\|v_{k+1}\|_{C([0,T];L^{2}(\Omega))}\leq\frac{C_{1}}{\Gamma\left(\frac{1}{2}k\alpha+1\right)}\left(C\Gamma\left(\frac{1}{2}\alpha\right)T^{\frac{1}{2}\alpha}\right)^{k},\quad
k\in\mathbb{N}.$

By the asymptotic behavior of the gamma function, we can verify

$\lim_{k\to\infty}\frac{\left(C\Gamma\left(\frac{1}{2}\alpha\right)T^{\frac{1}{2}\alpha}\right)^{k}}{\Gamma\left(\frac{1}{2}k\alpha+1\right)}=0,$

so that $\sum_{k=0}^{\infty}\|v_{k}\|_{C([0,T];L^{2}(\Omega))}$ converges.
Therefore, $\lim_{k\to\infty}u_{k}$ exists in $C([0,T];L^{2}(\Omega))$. Thus
the proof of (6.16) is complete. $\blacksquare$

From the uniqueness of solution to (6.16), we can derive that its limit is
$u$, the solution to (6.2) - (6.3) with $F=0$. Since $u_{k}\in
C([0,T];L^{2}(\Omega))$ for $k\in\mathbb{N}$, the limits is also in
$C([0,T];L^{2}(\Omega))$. Thus the proof of Theorem 6.3 is complete.
$\blacksquare$

§6.3. Stronger regularity in time of solution

Again we consider (6.2) - (6.3) with $F\neq 0$. Theorem 6.1 provides a basic
result on the unique existence of $u$ for $F\in L^{2}(0,T;L^{2}(\Omega))$ and
$a\in H^{1}_{0}(\Omega)$, while Theorem 6.2 is the well-posedness for $a$ and
$F$ which is less regular in $x$.

Here we consider stronger time-regular $F$ to improve the regularity of
solution $u$. Thanks to the framework of $\partial_{t}^{\alpha}$ deifned by
(2.4), the argument for improving the reglarity of solution is automatic.

Theorem 6.4.
Let $0<\alpha<1$ and $\beta>0$. We assume $a\in H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)$ and

$None$ $F-Aa\in H_{\beta}(0,T;L^{2}(\Omega)).$

Then there exists a unique solution $u$ to (6.2) - (6.3) such that

$None$ $u-a\in H_{\beta}(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))\cap
H_{\alpha+\beta}(0,T;L^{2}(\Omega)),$

and we can find a constant $C>0$ such that

$\|u-a\|_{H_{\beta}(0,T;H^{2}(\Omega))}+\|u-a\|_{H_{\alpha+\beta}(0,T;L^{2}(\Omega))}\leq
C(\|F-Aa\|_{H_{\beta}(0,T;L^{2}(\Omega))}+\|a\|_{H^{2}(\Omega)}).$

If $0<\beta<\frac{1}{2}$, then (6.18) is equivalent to that $F\in
H_{\beta}(0,T;L^{2}(\Omega))$ under condition $a\in H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)$. If $\frac{1}{2}<\beta\leq 1$, then (6.18) is equivalent to

$None$ $F\in H_{\beta}(0,T;L^{2}(\Omega)),\quad Aa=F(\cdot,0)\quad\mbox{in
$\Omega$}$

under condition $a\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$. Condition (6.18)
is a compatibility condition which is necessary for lifting up the regularity
of the solution, and is required also for the parabolic equation for instance
(see Theorem 6 in Chapter 7, Section 1 in Evans [5]).

From Theorem 6.4, we can easily derive
Corollary 6.1.
Let $a=0$ and $0<\alpha<1$. We assume

$F\in H_{\alpha}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)).$

Then

$A^{2}u,\,\partial_{t}^{2\alpha}u\in L^{2}(0,T;L^{2}(\Omega)).$

Proof of Theorem 6.4.
We will gain the regularity of the solution $u$ by means of an equation which
can be expected to be satisfied by $\partial_{t}^{\alpha}u$, although such an
equation is not justified for the moment.

We consider

$None$ $\partial_{t}^{\alpha}v+Av=\partial_{\beta}(F-Aa),\quad v\in
H_{\alpha}(0,T;L^{2}(\Omega)).$

Then, by $\partial_{t}^{\beta}(F-Aa)\in L^{2}(0,T;L^{2}(\Omega))$, Theorem 6.1
yields that $v$ exists and
$v\in H_{\alpha}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega))$.

We set $\widetilde{u}:=J^{\beta}v+a$. Then Proposition 2.5 (i) implies

$None$ $\widetilde{u}-a\in H_{\alpha+\beta}(0,T;L^{2}(\Omega))\cap
H_{\beta}(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega)).$

In view of Proposition 2.5 (ii) and $v\in H_{\alpha}(0,T;L^{2}(\Omega))$, we
have

$J^{-\alpha}(\widetilde{u}-a)=J^{-\alpha}J^{\beta}v=J^{\beta}J^{-\alpha}v,$

and so $J^{-\alpha}(\widetilde{u}-a)=J^{\beta}(J^{-\alpha}v)$. Moreover, by
$v\in L^{2}(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))$, we see

$A\widetilde{u}=A(J^{\beta}v+a)=J^{\beta}Av+Aa.$

Therefore,

$J^{-\alpha}(\widetilde{u}-a)+A\widetilde{u}=J^{\beta}(J^{-\alpha}v+Av)+Aa.$

In terms of (6.21), we obtain

$J^{-\alpha}(\widetilde{u}-a)+A\widetilde{u}=J^{\beta}(\partial^{\beta}_{t}(F-Aa))+Aa=F.$

Therefore, combining (6.22), we can verify that $\widetilde{u}-a$ satisfies
(6.2) - (6.3), and the uniqueness of solution yields $u=\widetilde{u}-a$. Thus
he proof of Theorem 6.4 is complete. $\blacksquare$

Proof of Corollary 6.1.
Theorem 6.4 yields

$u\in H_{\alpha}(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))\cap
H_{2\alpha}(0,T;L^{2}(\Omega)).$

Consequently, applying also Theorem 3.1, we deduce that
$\partial_{t}^{2\alpha}u\in L^{2}(0,T;L^{2}(\Omega))$. Moreover, since
$\partial_{t}^{\alpha}\partial_{t}^{\alpha}=\partial_{t}^{2\alpha}$ by Theorem
3.1, operating $\partial_{t}^{\alpha}$ to (6.2), we obtain

$None$
$\partial_{t}^{\alpha}Au=-\partial_{t}^{2\alpha}u+\partial_{t}^{\alpha}F\in
L^{2}(0,T;L^{2}(\Omega)).$

On the other hand, $A(\partial_{t}^{\alpha}u+Au-F)=0$ in
$L^{2}(0,T;(C_{0}^{\infty}(\Omega))^{\prime})$. Consequently, in terms of
(6.23), we reach

$A^{2}u=-A\partial_{t}^{\alpha}+AF\in L^{2}(0,T;L^{2}(\Omega)).$

Thus the proof of Corollary 6.1 is complete. $\blacksquare$

§6.4. Weaker regularity in time of solution

First we introduce a function space $\,{{}^{-\alpha}{H}}(0,T;X)$ for a Banach
space $X$. Indeed, thanks to Proposition 2.9 or Theorem 2.1, the operator
$J_{\alpha}^{\prime}:\,{{}^{-\alpha}{H}}(0,T)\,\longrightarrow\,L^{2}(0,T)$ is
surjective and isomorhism for $\alpha>0$, so that we define

$None$ $\left\\{\begin{array}[]{rl}&{{}^{-\alpha}{H}}(0,T;X):=\\{v\in
L^{2}(0,T;X);\,J_{\alpha}^{\prime}v\in L^{2}(0,T;X)\\},\\\ &\mbox{with the
norm}\qquad\|v\|_{{{}^{-\alpha}{H}}(0,T;X)}:=\|J_{\alpha}^{\prime}v\|_{L^{2}(0,T;X)}.\end{array}\right.$

By
$\partial_{t}^{\alpha}:=(J_{\alpha}^{\prime})^{-1}:L^{2}(0,T)\,\longrightarrow\,{{}^{-\alpha}{H}}(0,T)$,
we see also that

$None$ $\lim_{n\to\infty}v_{n}=v$ in $L^{2}(0,T;L^{2}(\Omega))$ implies
$\lim_{n\to\infty}\partial_{t}^{\alpha}v_{n}=\partial_{t}^{\alpha}v$ in
${{}^{-\alpha}{H}}(0,T)$.

In this subsection, we consider

$None$ $\partial_{t}^{\alpha}(u-a)+Au=F\quad\mbox{in
${{}^{-\alpha}{H}}(0,T;L^{2}(\Omega))$}$

and

$None$ $u-a\in L^{2}(0,T;L^{2}(\Omega)),\quad
u\in{{}^{-\alpha}{H}}(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega)).$

As is argued in Example 5.2, for $\alpha>\frac{1}{2}$, a source term

$F(x,t)=\delta_{t_{0}}(t)f(x)\in{{}^{-\alpha}{H}}(0,T;L^{2}(\Omega))$

with $f\in L^{2}(\Omega)$, describes an impulsive source at $t=t_{0}$, and it
is not only mathematically but also physically meaningful to treat singular
source $F\in{{}^{-\alpha}{H}}(0,T;L^{2}(\Omega))$ with $\alpha>0$. We here
state one result on the well-posedness for (6.26) - (6.27).
Theorem 6.5.
Let $0<\alpha<1$. We assume

$F\in{{}^{-\alpha}{H}}(0,T;L^{2}(\Omega)),\quad a\in L^{2}(\Omega).$

Then there exists a unique solution $u\in
L^{2}(0,T;L^{2}(\Omega))\cap{{}^{-\alpha}{H}}(0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega))$ to (6.26) - (6.27) and we can find a constant $C>0$ such
that

$None$
$\|u\|_{L^{2}(0,T;L^{2}(\Omega))}+\|u\|_{{{}^{-\alpha}{H}}(0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega))}\leq
C(\|a\|_{L^{2}(\Omega)}+\|F\|_{{{}^{-\alpha}{H}}(0,T;L^{2}(\Omega))})$

for each $a\in L^{2}(\Omega)$ and $F\in{{}^{-\alpha}{H}}(0,T;L^{2}(\Omega))$.
Proof.
We will prove by creating an equation which should be satisfied by
$J_{\alpha}^{\prime}u$, and such an equation can be given by

$\partial_{t}^{\alpha}J_{\alpha}^{\prime}(u-a)+AJ_{\alpha}^{\prime}u=J_{\alpha}^{\prime}F,$

where we have to make justification. Thus we consider the solution $v$ to

$None$ $\partial_{t}^{\alpha}v+Av=J_{\alpha}^{\prime}F+a,\quad v\in
H_{\alpha}(0,T;L^{2}(\Omega)).$

Since $J_{\alpha}^{\prime}F+a\in L^{2}(0,T;L^{2}(\Omega))$, Theorem 6.1
implies the unique existence of solution $v\in
H_{\alpha}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega))$. Therefore, Proposition 2.10 yields
$(J_{\alpha}^{\prime})^{-1}v=J^{-\alpha}v=\partial_{t}^{\alpha}v$ by $v\in
H_{\alpha}(0,T;L^{2}(\Omega))$. Setting

$None$ $\widetilde{u}:=(J_{\alpha}^{\prime})^{-1}v=J^{-\alpha}v,$

we readily verify

$\widetilde{u}\in
L^{2}(0,T;L^{2}(\Omega))\cap{{}^{-\alpha}{H}}(0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega))$

by Propositions 2.2 and 2.9 (ii). Furthermore, since (6.30) implies
$(J_{\alpha}^{\prime})^{-1}v=\partial_{t}^{\alpha}v$, and
$v=J_{\alpha}^{\prime}\widetilde{u}$, we obtain

$\displaystyle(J_{\alpha}^{\prime})^{-1}(\widetilde{u}-a)+A\widetilde{u}-F=(J_{\alpha}^{\prime})^{-1}(\widetilde{u}-a+AJ_{\alpha}^{\prime}\widetilde{u}-J_{\alpha}^{\prime}F)$
$\displaystyle=$
$\displaystyle(J_{\alpha}^{\prime})^{-1}((J_{\alpha}^{\prime})^{-1}v-a+Av-
J_{\alpha}^{\prime}F)=(J_{\alpha}^{\prime})^{-1}(\partial_{t}^{\alpha}v+Av-
J_{\alpha}^{\prime}F-a)=0$

in ${{}^{-\alpha}{H}}(0,T;L^{2}(\Omega))$.

Since $J_{\alpha}^{\prime}$ is injective, the application of (6.29) implies
$\partial_{t}^{\alpha}(\widetilde{u}-a)+A\widetilde{u}=F$ and
$\widetilde{u}-a\in L^{2}(0,T;L^{2}(\Omega))$. Hence, $\widetilde{u}$ is a
solution to (6.26) - (6.27), and (6.28) holds.

We finally prove the uniqueness of solution. Let $w\in
L^{2}(0,T;L^{2}(\Omega))\cap\,{{}^{-\alpha}{H}}(0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega))$ and $\partial_{t}^{\alpha}w+Aw=0$ hold in
$\,{{}^{-\alpha}{H}}(0,T;L^{2}(\Omega))$. Then

$None$
$J_{\alpha}^{\prime}\partial_{t}^{\alpha}w+J_{\alpha}^{\prime}Aw=J_{\alpha}^{\prime}\partial_{t}^{\alpha}w+AJ_{\alpha}^{\prime}w=0\quad\mbox{in
$L^{2}(0,T;L^{2}(\Omega))$.}$

By $w\in L^{2}(0,T;L^{2}(\Omega))$ and (2.13), we see

$J_{\alpha}^{\prime}\partial_{t}^{\alpha}w=J_{\alpha}^{\prime}(J_{\alpha}^{\prime})^{-1}w=w.$

On the other hand,

$w=(J^{\alpha})^{-1}J^{\alpha}w=\partial_{t}^{\alpha}J^{\alpha}w$

by (2.4) and $J^{\alpha}w\in H_{\alpha}(0,T)$. Hence,
${J_{\alpha}}^{\prime}\partial_{t}^{\alpha}w=\partial_{t}^{\alpha}J^{\alpha}w$.
Therefore, in terms of (6.31), we have $J^{\alpha}w\in
H_{\alpha}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega))$ and $\partial_{t}^{\alpha}(J^{\alpha}w)+A(J^{\alpha}w)=0$
in $L^{2}(0,T;L^{2}(\Omega))$. Hence, the uniqueness asserted by Theorem 6.1
implies $J^{\alpha}w=0$ in $(0,T)$. Since
$J^{\alpha}:L^{2}(0,T)\,\longrightarrow\,L^{2}(0,T)$ is injective, we obtain
$w=0$ in $(0,T)$. Thus the proof of Theorem 6.5 is complete. $\blacksquare$

Finally in this subsection, in order to describe the flexibility of our
approach, we show
Proposition 6.2.
Let $0<\alpha<1$. In (6.26) and (6.27), we assume

$None$ $a=0,\quad
F\in{{}^{-\alpha}{H}}(0,T;L^{2}(\Omega))\cap\,^{-2\alpha}{H}(0,T;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)).$

Then the solution $u$ satisfies

$u\in\,^{-2\alpha}{H}(0,T;\mathcal{D}(A^{2}))\subset\,^{-2\alpha}{H}(0,T;H^{4}(\Omega)).$

The proposition means that under (6.32), the solution $u$ can hold the
$H^{4}(\Omega)$-regulairty in $x$ at the expense of weaker regulairty
${}^{-2\alpha}{H}(0,T)$ in $t$.
Proof of Proposition 6.2.
By Theoren 6.5, the solution $u$ to (6.26) - (6.27) with $a=0$ exists
uniquely. More precisely, we have $(J_{\alpha}^{\prime})^{-1}u+Au=F$ in
${{}^{-\alpha}{H}}(0,T;L^{2}(\Omega))$ and

$u\in{{}^{-\alpha}{H}}(0,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))\cap
L^{2}(0,T;L^{2}(\Omega)).$

Therefore
$J_{\alpha}^{\prime}(J_{\alpha}^{\prime})^{-1}=(J_{\alpha}^{\prime})^{-1}J_{\alpha}^{\prime}$
in $L^{2}(0,T;L^{2}(\Omega))$ and
$J_{\alpha}^{\prime}Au=AJ_{\alpha}^{\prime}u$, and so

$(J_{\alpha}^{\prime})^{-1}J_{\alpha}^{\prime}u+AJ_{\alpha}^{\prime}u=J_{\alpha}^{\prime}F\in
L^{2}(0,T;L^{2}(\Omega)).$

Hence, operating $J_{\alpha}^{\prime}$ again, we obtain

$(J_{\alpha}^{\prime})(J_{\alpha}^{\prime})^{-1}J_{\alpha}^{\prime}u+AJ_{\alpha}^{\prime}J_{\alpha}^{\prime}u=J_{\alpha}^{\prime}J_{\alpha}^{\prime}F.$

Consequently, we obtain