# Existence for doubly nonlinear fractional $p$-Laplacian equations

Nobuyuki Kato  Department of Mathematics, Nippon Institute of Technology,
Saitama, Japan<EMAIL_ADDRESS>,  Masashi Misawa  Faculty of Advanced
Science and Technology, Kumamoto University, Kumamoto 860-8555, Japan
<EMAIL_ADDRESS>,  Kenta Nakamura  Headquarters for Admissions and
Education, Kumamoto University, Kumamoto 860-8555, Japan
<EMAIL_ADDRESS>and  Yoshihiko Yamaura  Department of Mathematics,
College of Humanities and Sciences, Nihon University, Tokyo, Japan
<EMAIL_ADDRESS>
(Date: 20210825 ver.)

###### Key words and phrases:

Nonlocal doubly nonlinear equation, Fractional Sobolev space, Space-time
fractional Sobolev inequalities

###### 2010 Mathematics Subject Classification:

Primary: 35B45, 35B65, Secondary: 35D30, 35K61

We prove the existence of a weak solution to a doubly nonlinear parabolic
fractional $p$-Laplacian equation, which has general doubly nonlinearlity
including not only the Sobolev subcritical/critical/supercritical cases but
also the slow/fast diffusion ones. Our proof reveals the weak convergence
method for the doubly nonlinear fractional $p$-Laplace operator.

###### Contents

  1. 1 Introduction and Results
  2. 2 Preliminaries
    1. 2.1 Notation
    2. 2.2 Fundamental tools
    3. 2.3 Weak solution
  3. 3 Approximate solutions
  4. 4 Convergence of approximate solutions
    1. 4.1 Energy estimates for the truncated solution
    2. 4.2 Convergence results for the truncated solution
    3. 4.3 Convergence of approximate solutions I
    4. 4.4 Convergence of approximate solutions II
  5. 5 Proof of Theorem 1.1
  6. A Proof of Lemma 3.1
  7. B Boundedness
    1. B.1 Boundedness
    2. B.2 A regularity of $\partial_{t}\left(|u|^{q-1}u\right)$ in the case $q\geqq 1$
  8. C A space-time fractional Sobolev inequality

##  1\. Introduction and Results

The aim of this paper is the study of existence of a weak solution to doubly
nonlinear parabolic nonlocal equations having the fractional $p$-Laplace
operator as the principal part. The double nonlinearity is treated under the
general settings including not only the Sobolev
subcritical/critical/supercritical cases but also the slow/fast diffusion
ones. Let $\Omega\subset\mathbb{R}^{n}$ ($n\geqq 1$) be a bounded domain with
boundary $\partial\Omega$. We fix indices $s\in(0,1)$, $p>1$ and $q>0$. We are
interested in the doubly nonlinear nonlocal problem:

$\begin{cases}\,\,\partial_{t}(|u|^{q-1}u)+(-\Delta)_{p}^{s}u=0&\textrm{in}\quad\Omega_{\infty}:=\Omega\times(0,\infty),\\\
\,\,u=0&\textrm{on}\quad\\!\\!(\mathbb{R}^{n}\setminus\Omega)\times(0,\infty),\\\
\,\,u(\cdot,0)=u_{0}(\cdot)&\textrm{in}\quad\Omega,\end{cases}$ (1.1)

where the initial datum $u_{0}$ belongs to both the fractional Sobolev space
$W^{s,p}_{0}(\Omega)$ and the Lebesgue space $L^{q+1}(\Omega)$. Here we remark
that $W^{s,p}_{0}(\Omega)$ consists of functions in the fractional Sobolev
space $W^{s,p}(\mathbb{R}^{n})$ vanishing outside $\Omega$. The precise
definition of these function spaces will be given in Section 2. The fractional
$p$-Laplace operator $(-\Delta)_{p}^{s}$ is defined by

$\displaystyle(-\Delta)_{p}^{s}u(x,t)$
$\displaystyle:=\mathrm{P.V.}\int_{\mathbb{R}^{n}}\frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+sp}}\,dy$
$\displaystyle=\lim_{\varepsilon\searrow 0}\int_{\mathbb{R}^{n}\setminus
B_{\varepsilon}(x)}\frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+sp}}\,dy,$
(1.2)

where the symbol $\mathrm{P.V.}$ means “in the principal value sense”.

Let us introduce our interest in (1.1) motivated by an energy structure (refer
to [7, 8, 17]). For this purpose, we shall simply demonstrate a formal
derivation of the equation (1.1)1. Consider the integral functional

$\mathcal{E}(u):=\frac{1}{2p}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dxdy$

for $u\in W^{s,p}_{0}(\Omega)$. The Gâteaux differential of $\mathcal{E}(u)$
in the direction $\varphi\in W^{s,p}_{0}(\Omega)$ turns out to be

$\displaystyle\frac{d}{d\tau}\mathcal{E}(u+\tau\varphi)\Bigg{|}_{\tau=0}$
$\displaystyle=\frac{1}{2p}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{d}{d\tau}\,\frac{|u(x)-u(y)+\tau(\varphi(x)-\varphi(y))|^{p}}{|x-y|^{n+sp}}\,dxdy\,\Bigg{|}_{\tau=0}$
$\displaystyle=\frac{1}{2}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}(\varphi(x)-\varphi(y))\,dxdy$
$\displaystyle=\langle(-\Delta)^{s}_{p}u,\,\varphi\rangle,$

where the symbol $\langle\cdot\,,\,\cdot\rangle$ represents the dual pairing
between $W^{s,p}_{0}(\Omega)$ and its dual space $W^{-s,p^{\prime}}(\Omega)$.
This implies that $(-\Delta)_{p}^{s}u$ is the gradient vector field
$\nabla\mathcal{E}(u)$ on $W^{s,p}_{0}(\Omega)$:

$\nabla\mathcal{E}(u)=(-\Delta)^{s}_{p}u$

and hence, the equation (1.1)1 is interpreted as a nonlinear generalization of
the usual gradient flow equation $\partial_{t}u(t)=-\nabla\mathcal{E}(u(t))$.
We can also refer to [21] about a variety of interesting problems, their
results and the progress of research on the fractional operator.

The main result of this paper is the global existence of a weak solution to
(1.1) as follows:

###### Theorem 1.1

Suppose that the initial datum $u_{0}$ belongs to $W^{s,p}_{0}(\Omega)\cap
L^{q+1}(\Omega)$. Then, there exists a weak solution to (1.1) in the sense of
Definition 2.8. The weak solution $u$ of (1.1) satisfies the regularity:
$\partial_{t}(|u|^{\frac{q-1}{2}}u)\in L^{2}(\Omega_{\infty})$.

The global existence of a weak solution in the fractional Sobolev space as in
Theorem 1.1 is shown under the general double nonlinearity. The concentration
only on the prototype equation makes it possible to describe clearly the
approach for the proof of the existence. The proof is usually based on the
weak compactness method with an appropriate approximation. A space-time
fractional Sobolev inequality is devised and then, induce the weak convergence
for the fractional $p$-Laplace operator with combination of the Vitali
convergence theorem. The approximate equation used here is given by the
difference integro-differential equations of the so-called Rothe type,
associated to (1.1) (see [27]). The approximate solutions are constructed by
an application of the direct method in the calculus of variations, which
enable us to treat the general double nonlinearity with $p>1$, $q>0$ and
$s\in(0,1)$ in the equation. We shall mention the boundary condition in (1.1).
The nonlocal Dirichlet boundary condition (1.1)2, which may look weird at a
first glance, is suitable and consistent with the nonlocal character of the
fractional operator. Here we do _not_ require that $C^{\infty}_{0}(\Omega)$ is
dense in $W^{s,p}_{0}(\Omega)$. In fact, such a density property does not
necessarily hold for a general domain without the smoothness of boundary (see
[11] and [24, Remark 3.5, p. 533]). Here the equation is considered on any
bounded domain $\Omega$ in $\mathbb{R}^{n}$ which of the boundary is not
necessarily smooth.

For the parabolic fractional $p$-Laplace equation of the regular form (Eq.
(1.1) with $q=1$), the existence of weak solutions have been studied under
weaker conditions for initial data (see [26, 22, 28, 1]). Here the doubly
nonlinear parabolic fractional $p$-Laplace equation (1.1) of fairly general
form is considered and the existence of a solution is studied in the energy
class of the fractional Sobolev space, as explained before. Our proof of the
existence in the energy class develops a weak convergence method, based on
certain space-time fractional Sobolev inequality and Vitali’s convergence
theorem. To the best of our knowledge, that includes a new content for
evolutionary fractional differential equations. In fact, a truncation
technique and a space-time fractional Sobolev inequality are devised and
combined with Vitali’s convergence theorem to derive the space-time Lebesgue
strong convergence. See Lemmata 4.1, 4.2, 4.3 and 4.4 and, Lemma C.1 in
Appendix C. Our truncation argument may be somehow technical and, however,
clearly overcomes some difficulties stemming from the power nonlinearity in
the time derivative. The Lebesgue strong convergence is naturally applied to
take the limit in the nonlinear integral quantities (see the proof of Lemma
4.6 for details). It is also shown that the initial datum is continuously
attained in the fractional Sobolev space, whose proof can be seen in the proof
of (D3).

In our future work we shall consider the gradient flow associated with the
best constant of fractional Sobolev inequality, where our approach requires
the prototype equation as in (1.1) as a fundamental tool. See [4] for the
topics of the best constant and its extremal function in the fractional
Sobolev inequality. In the usual local setting we have studied the $p$-Sobolev
flow, which is the gradient flow concerning the usual Sobolev inequality and
based on the prototype doubly nonlinear parabolic equation (refer to [23, 19,
20]). The related results for PDEs can be seen in [2, 5, 9, 16]. A certain
variational inequality method is well adopted to the doubly nonlinear PDEs in
[3] with its precise description and comprehensive reference on related
equations. Here we shall explain another motive from the geometric
perspective: Given a compact Riemannian manifold $(M^{n},g_{0})$ of dimension
$n\geqq 2$ and a constant $s\in(0,\frac{n}{2})$, Graham and Zworski ([14])
originally introduced, in their study of a family of conformally covariant
operators, the nonlocal evolutionary equation for a metric
$g(t)=u^{\frac{4}{n-2s}}(t)g_{0}$ on $M$, called a _fractional Yamabe flow_.
When $M=\mathbb{S}^{n}$ in particular, through the stereographic projection
$\pi:\mathbb{S}^{n}\setminus\\{\mathrm{northpole}\\}\longrightarrow\mathbb{R}^{n}$,
the fractional Yamabe flow is written in the form

$\partial_{t}\left(u^{\frac{n+2s}{n-2s}}(t)\right)=-(-\Delta)^{s}u(t)+r^{g}_{s}u^{\frac{n+2s}{n-2s}}(t)\quad\textrm{in}\,\,\,\mathbb{R}^{n},$
(1.3)

where $(-\Delta)^{s}u$ denotes the usual fractional Laplacian defined by (1)
with $p=2$. The quantity $r^{g}_{s}:=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int\limits_{M}R^{g}_{s}\,dvol_{g}$ in (1.3) is the
average of the fractional order curvature $R^{g}_{s}$ that coincides with the
classical scalar curvature in the case $s=1$. If $s=1$ specifically, then the
fractional Yamabe flow (1.3) becomes exactly the classical Yamabe flow
introduced by Hamilton ([15]). The stationary problem for the fractional
Yamabe flow is the fractional Yamabe problem, for which the existence of a
solution is shown in [13] under the assumption that $s\in(0,1)$ and that the
domain manifold $M$ is locally conformally flat with positive Yamabe constant.
Jing and Xiong ([18]) succeeded in obtaining the global existence for the
fractional Yamabe flow (1.3) and the asymptotic convergence of (1.3) to the
stationary solution. Recently, when the domain manifold is locally conformally
flat with positive Yamabe constant, Daskalopoulous, Sire and Vázquez proved
the global existence of a smooth solution to the fractional Yamabe flow and
its convergence as $t\longrightarrow\infty$ to a metric of constant scalar
curvature metric (see [6]). In the Euclidean space, the metric is flat and the
above curvature conditions are not verified and thus, there may occur some
difficulties from the viewpoint of PDEs. In the outstanding result of [10],
the existence and uniqueness for more general fractional porous medium
equations is proved via the Fourier analysis and $L^{1}$-semigroup theory. In
this paper, we are forced to consider on the bounded domain
$\Omega\subset\mathbb{R}^{n}$ and take a direct approach dictated by the
structure of fractional $p$-Laplacian term inspired by the energy
$\mathcal{E}$, as mentioned above. Our results presented here can cover those
of the fractional Yamabe flow in the Euclidean case.

The paper is organized as follows. In Section 2, we fix the notation and
summarize some fundamental facts involved with the fractional Sobolev space.
Section 3 is devoted to constructing approximate solutions of (1.1) and some
uniform energy estimates for them. The approximate solutions are given by the
time-difference integro-differential equations (3.1) associated with (1.1). In
Section 4, we derive the convergence of approximate solutions on the basis of
the energy estimates in the previous section. In Section 5, we show Theorem
1.1 by the weak convergence method. Appendix A is devoted to the proof of the
existence of approximate solutions to (3.1) via the direct method in the
calculus of variations. In Appendix B, we show the boundedness of our
approximate solutions under the assumption of that of the initial datum. As an
application, we deduce a regularity of $\partial_{t}\left(|u|^{q-1}u\right)$
in the case $q\geqq 1$, as seen in Theorem B.3. Here we stress that our main
result, Theorem 1.1 does not require any pointwise boundedness of approximate
solutions. In the final Appendix C we report our key idea, the space-time
fractional Sobolev inequality.

Acknowledgements

Masashi Misawa acknowledges the partial support by the Grant-in-Aid for
Scientific Research (C) Grant No.21K03330 (2021) at JSPS. The work by Kenta
Nakamura was also partially supported by Grant-in-Aid for Young Scientists
Grant No.21K13824 (2021) at JSPS. Yoshihiko Yamaura was also partially
supported by Grant-in-Aid for Scientific Research (C) Grant No.18K03381 (2018)
at JSPS, and by Individual Research Expense of College of Humanities and
Sciences at Nihon University for 2020.

##  2\. Preliminaries

We shall split this section in three parts: first, we display our notation,
then we collect some inequalities used in the course of the paper, and finally
we introduce the definition of weak solutions to our problem (1.1).

###  2.1. Notation

In this paper, let $\Omega\subset\mathbb{R}^{n}\,\,(n\geqq 1)$ be a bounded
domain with boundary $\partial\Omega$ and denote for simplicity by
$\Omega^{c}:=\mathbb{R}^{n}\setminus\Omega$ the outside of $\Omega$. For
$T\in(0,\infty]$, let $\Omega_{T}:=\Omega\times(0,T)$ be a space-time
cylinder. We shall denote in a standard way
$B_{\varrho}(x_{0}):=\left\\{x\in\mathbb{R}^{n}:|x-x_{0}|<\varrho\right\\}$,
the open ball with radius $\varrho>0$ and center $x_{0}\in\mathbb{R}^{n}$. We
adopt the convention of writing $B_{\varrho}$ instead of $B_{\varrho}(x_{0})$,
when the center is origin $0$ of $\mathbb{R}^{n}$, or when the center is clear
from the context. For a measurable set $A\subset\mathbb{R}^{k}$,
$k\in\mathbb{N}$, we designate by $|A|$ the usual $k$-dimensional Lebesgue
measure of $A$.

In what follows, we denote by $C$, $C_{1}$, $C_{2},\cdots$ different positive
constants in a given context. Relevant dependencies on parameters will be
emphasized using parentheses, e.g., $C\equiv C(n,p,\Omega)$ means that $C$
depends on $n,p$ and $\Omega$. For the sake of readability, the dependencies
of the constants will be often omitted within the chains of estimates.
Furthermore, the equation number $(\,\cdot\,)_{\ell}$ denotes the $\ell$-th
line of the Eq. $(\,\cdot\,)$.

Now we define some fractional Sobolev spaces (refer to [24]).

For any $p\in[1,\infty)$ and $s\in(0,1)$, the fractional Sobolev space is
defined as

$W^{s,p}(\mathbb{R}^{n}):=\left\\{v\in
L^{p}(\mathbb{R}^{n})\,:\,\frac{|v(x)-v(y)|}{|x-y|^{\frac{n}{p}+s}}\in
L^{p}(\mathbb{R}^{n}\times\mathbb{R}^{n})\right\\}$

with the finite norm

$\|v\|_{W^{s,p}(\mathbb{R}^{n})}:=\|v\|_{L^{p}(\mathbb{R}^{n})}+[v]_{W^{s,p}(\mathbb{R}^{n})},$
(2.1)

the usual $L^{p}$-norm $\|v\|_{L^{p}(\mathbb{R}^{n})}$ in $\mathbb{R}^{n}$
together with the so-called _Gagliardo semi-norm_ in $\mathbb{R}^{n}$

$[v]_{W^{s,p}(\mathbb{R}^{n})}:=\left(\,\,\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\,dxdy\right)^{\frac{1}{p}}.$
(2.2)

Analogously, the fractional Sobolev space on a domain $K\subset\mathbb{R}^{n}$
is defined as

$W^{s,p}(K):=\left\\{v\in
L^{p}(K)\,:\,\frac{|v(x)-v(y)|}{|x-y|^{\frac{n}{p}+s}}\in L^{p}(K\times
K)\right\\}$

with the finite norm

$\|v\|_{W^{s,p}(K)}:=\|v\|_{L^{p}(K)}+[v]_{W^{s,p}(K)}$ (2.3)

and

$[v]_{W^{s,p}(K)}:=\left(\,\,\iint_{K\times
K}\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\,dxdy\right)^{\frac{1}{p}}.$ (2.4)

We also define

$W_{0}^{s,p}(K):=\left\\{u\in
W^{s,p}(\mathbb{R}^{n})\,:\,u=0\,\,\,\textrm{a.e.\,\,in}\,\,\mathbb{R}^{n}\setminus
K\right\\}.$ (2.5)

In the context on the Dirichlet boundary condition for a nonlocal operator
$(-\Delta)^{s}_{p}$ in $K$, we say that a function $u$ in
$W^{s,p}(\mathbb{R}^{n})$ takes zero value on the boundary of $K$ if $u$
belongs to $W^{s,p}_{0}(K)$. By the definition (2.5) with $K=\mathbb{R}^{n}$,
it can be naturally understood that
$W_{0}^{s,p}(\mathbb{R}^{n})=W^{s,p}(\mathbb{R}^{n})$.

For $s\in(0,1)$ and $p\in[1,\infty)$, the fractional spaces
$W^{s,p}(\mathbb{R}^{n})$, $W^{s,p}(K)$ and $W_{0}^{s,p}(K)$ are the Banach
spaces with finite norms (2.1), (2.3) and
$\|\cdot\|_{W_{0}^{s,p}(K)}:=\|\cdot\|_{L^{p}(K)}+[\,\cdot\,]_{W^{s,p}(\mathbb{R}^{n})}$,
respectively.The dual space of $W^{s,p}(\mathbb{R}^{n})$ is denoted as
$(W^{s,p}(\mathbb{R}^{n}))^{\ast}=W^{-s,p^{\prime}}(\mathbb{R}^{n})$.
Similarly, $(W^{s,p}_{0}(K))^{\ast}=W^{-s,p^{\prime}}(K)$ denotes the dual
space of $W^{s,p}_{0}(K)$. Here $p^{\prime}$ denotes the Hölder conjugate of
the exponent $p$.

We also address the space-time function spaces. Let $K$ be a domain in
$\mathbb{R}^{n}$ and $0\leqq t_{1},t_{2}<\infty$. Let $1\leqq p,q\leqq\infty$.
Let $\mathcal{X}$ be a Banach space consists of functions defined on $K$. We
use the space of Bochner $L^{q}(t_{1},t_{2})$-integrable functions
$v:(t_{1},t_{2})\ni t\mapsto v(t)\in\mathcal{X}$, denoted by
$L^{q}(t_{1},t_{2};\mathcal{X})$. Letting $\mathcal{X}$ be the Lebesgue space
$L^{p}(K)$, we have $L^{q}(t_{1},t_{2}\,;\,L^{p}(K))$ with a finite norm

$\|v\|_{L^{q}(t_{1},t_{2}\,;\,L^{p}(K))}:=\begin{cases}\displaystyle\left(\int_{t_{1}}^{t_{2}}\|v(t)\|_{p}^{q}\,dt\right)^{1/q}\quad&(1\leqq
q<\infty)\\\
\displaystyle\sup_{t_{1}<t<t_{2}}\|v(t)\|_{p}\quad&(q=\infty),\end{cases}$

where $\|v(t)\|_{L^{p}(K)}$ is abbreviated to $\|v(t)\|_{p}$ for $1\leqq
p\leqq\infty$ and
$\operatorname*{ess\,sup}\limits_{t_{1}<t<t_{2}}\|v(t)\|_{p}$ to
$\sup\limits_{t_{1}<t<t_{2}}\|v(t)\|_{p}$. If $p=q$ then we have the
identification as
$L^{p}(K\times(t_{1},t_{2}))=L^{p}(t_{1},t_{2}\,;\,L^{p}(K))$.

Let $s\in(0,1)$. The choice of $\mathcal{X}=W_{0}^{s,p}(K)$ implies that
$L^{q}(t_{1},t_{2}\,;\,W_{0}^{s,p}(K))$ is Banach space with a finite norm

$\|v\|_{L^{q}(t_{1},t_{2}\,;\,W_{0}^{s,p}(K))}:=\left(\int_{t_{1}}^{t_{2}}\|v(t)\|_{W^{s,p}(K)}^{q}\,dt\right)^{1/q}$

whenever $q<\infty$, and

$\|v\|_{L^{\infty}(t_{1},t_{2}\,;\,W_{0}^{s,p}(K))}:=\operatorname*{ess\,sup}_{t_{1}<t<t_{2}}\|v(t)\|_{W^{s,p}(K)}.$

The dual spaces of Bochner spaces $L^{q}(t_{1},t_{2};W^{s,p}(\mathbb{R}^{n}))$
and $L^{q}(t_{1},t_{2};W^{s,p}_{0}(K))$ are denoted by
$L^{q}(t_{1},t_{2};W^{s,p}(\mathbb{R}^{n}))^{\ast}$ and
$L^{q}(t_{1},t_{2};W^{s,p}_{0}(K))^{\ast}$, respectively. Here
$L^{q}(t_{1},t_{2};W^{s,p}(\mathbb{R}^{n}))^{\ast}$ is identified with
$L^{q^{\prime}}(t_{1},t_{2};W^{-s,p^{\prime}}(\mathbb{R}^{n}))$, where
$p^{\prime}$ and $q^{\prime}$ are the Hölder conjugate of $p$ and $q$,
respectively. Similarly, we identify
$L^{q}(t_{1},t_{2};W^{s,p}(K))^{\ast}=L^{q^{\prime}}(t_{1},t_{2};W^{-s,p^{\prime}}(K))$.

For studying the parabolic nonlocal equation in (1.1), we shall exploit the
space-time fractional Sobolev space. Let $p\in[1,\infty)$ and $s\in(0,1)$. For
a domain $K$ in $\mathbb{R}^{n}$ and $0<T\leqq\infty$, put
$K_{T}:=K\times(0,T)$. The space-time fractional Sobolev space, denoted by
$W^{s,p}(K_{T})$, consists of functions $v$ in $L^{p}(K_{T})$ having a finite
semi-norm

$[v]_{W^{s,p}(K_{T})}:=\left(\,\,\iint_{K_{T}\times
K_{T}}\frac{|v(x,t)-v(x^{\prime},t^{\prime})|^{p}}{\left(\sqrt{|x-x^{\prime}|^{2}+(t-t^{\prime})^{2}}\right)^{n+1+sp}}\,dxdtdx^{\prime}dt^{\prime}\right)^{\frac{1}{p}},$
(2.6)

which is also the Banach space with respect to the norm

$\|v\|_{W^{s,p}(K_{T})}:=\|v\|_{L^{p}(K_{T})}+[v]_{W^{s,p}(K_{T})}.$ (2.7)

###  2.2. Fundamental tools

We gather fundamental tools and facts. First, the three inequalities listed in
the following lemma are needed throughout this paper and, in particular, used
to evaluate the fractional $p$-Laplacian term.

###### Lemma 2.1 (Algebraic inequality)

For all $\alpha\in(1,\infty)$ there are positive constants $C_{j}(\alpha)$,
$j=1,2$, such that, for all $\xi,\,\eta\in\mathbb{R}$,

$\displaystyle||\xi|^{\alpha-2}\xi-|\eta|^{\alpha-2}\eta|\leqq
C_{1}(|\xi|+|\eta|)^{\alpha-2}|\xi-\eta|,$ (2.8)
$\displaystyle(|\xi|^{\alpha-2}\xi-|\eta|^{\alpha-2}\eta)(\xi-\eta)\geqq
C_{2}(|\xi|+|\eta|)^{\alpha-2}|\xi-\eta|^{2}$ (2.9)

and, in particular, when $\alpha\geqq 2$

$(|\xi|^{\alpha-2}\xi-|\eta|^{\alpha-2}\eta)(\xi-\eta)\geqq
C_{2}|\xi-\eta|^{\alpha}.$ (2.10)

###### Proof.

The proof of (2.8) and (2.9) can be derived from the proof of [12, Lemma 8.3,
p.266]. Inequality (2.10) is simply obtained from the Minkowski inequality and
(2.9) in [9, Lemma 4.4 in Chapter I, p.13]. ∎

The following lemma is often used later and the key in some convergences. The
proof is due to Vitali’s convergence theorem.

###### Lemma 2.2

Let $a\in(1,\infty)$ and $a^{\prime}$ satisfy
$\frac{1}{a}+\frac{1}{a^{\prime}}=1$. Assume $\\{f_{h}\\}_{h>0}$ is bounded in
$L^{a}(K_{T})$ and $f_{h}$ converges to $f$ almost everywhere in $K_{T}$. Then

$\iint_{K_{T}}f_{h}\psi\,dxdt\longrightarrow\iint_{K_{T}}f\psi\,dxdt\quad\textrm{for}\,\,\,\psi\in
L^{a^{\prime}}(K_{T}).$

In particular, if $\\{f_{h}\\}_{h>0}$ is bounded in $L^{a}(K_{T})$ and $f_{h}$
converges to 0 almost everywhere in $K_{T}$, then for any $b\in[1,a)$,

$\int_{K_{T}}|f_{h}|^{b}\,dxdt\longrightarrow 0\quad\textrm{as}\,\,\,h\searrow
0.$

###### Proof.

We first verify that $\left\\{f_{h}\psi\right\\}_{h>0}$ is uniformly
integrable on $K_{T}$. Let $\Lambda\subset K_{T}$ be any measurable set. By
the boundedness of $f_{h}$ in $L^{a}(K_{T})$ and Hölder’s inequality, we
observe that

$\displaystyle\iint_{\Lambda}\left|f_{h}\psi\right|\,dxdt$
$\displaystyle\leqq\left(\iint_{K_{T}}|f_{h}|^{a}\,dxdt\right)^{\frac{1}{a}}\left(\iint_{\Lambda}|\psi|^{a^{\prime}}\,dxdt\right)^{\frac{1}{a^{\prime}}}$
$\displaystyle\leqq
C\left(\iint_{\Lambda}|\psi|^{a^{\prime}}\,dxdt\right)^{\frac{1}{a^{\prime}}}.$

Due to the absolute continuity of the Lebesgue integral on the right-hand
side, for every $\varepsilon>0$, there exists $\delta>0$ such that

$\iint_{\Lambda}|\psi|^{a^{\prime}}\,dxdt<\varepsilon\quad\textrm{whenever}\quad|\Lambda|<\delta$

and thus, $\left\\{f_{h}\psi\right\\}_{h>0}$ is uniformly integrable on
$K_{T}$. Furthermore, it follows from the assumption that

$f_{h}\psi\longrightarrow f\psi\quad\textrm{a.e.\,\,in}\,\,K_{T}.$

Therefore Vitali’s convergence theorem gives that

$\iint_{K_{T}}f_{h}\psi\,dxdt\longrightarrow\iint_{K_{T}}f\psi\,dxdt$

as $h\searrow 0$. The validity of second statement follows from choosing
$\frac{a}{b}$ as $a$, $|f_{h}|^{b}$ as $f_{h}$, 0 as $f$ and $\psi\equiv 1$.
The proof is concluded. ∎

We now report the fractional Sobolev embedding theorem, which is proved in
[24, Theorem 6.7, p.557], [25].

###### Lemma 2.3 (Fractional Sobolev continuous embedding)

Let $s\in(0,1)$ and $p\in[1,\infty)$ be numbers satisfying $sp<n$. Let
$p^{\star}_{s}:=\frac{np}{n-sp}$ be the Sobolev exponent. Then
$W^{s,p}(\mathbb{R}^{n})$ is embedded into
$L^{p^{\star}_{s}}(\mathbb{R}^{n})$:

$\|v\|_{L^{p^{\star}_{s}}(\mathbb{R}^{n})}\leqq
C[v]_{W^{s,p}(\mathbb{R}^{n})}$ (2.11)

for $v\in W^{s,p}(\mathbb{R}^{n})$, where $C$ is the positive constant
depending only on $n,s$ and $p$. For any bounded domain $K$, $W^{s,p}(K)$ is
embedded into $L^{p^{\star}_{s}}(K)$:

$\|v\|_{L^{r}(K)}\leqq C\|v\|_{W^{s,p}(K)}$

for $v\in W^{s,p}(K)$ and all $r\in[1,p^{\star}_{s}]$, where the positive
constant $C$ depends on $K$ as well as $n,s,p$ and $r$.

The next lemma refers to the fractional Sobolev compact embedding, whose proof
is in [24, Corollary 7.2, p.561].

###### Lemma 2.4 (Fractional Sobolev compact embedding)

Let $s\in(0,1)$ and $p\in[1,\infty)$ be numbers satisfying $sp<n$ and
$p_{s}^{\star}$ be as in Lemma 2.3. Let $K$ be a Lipschitz bounded domain in
$\mathbb{R}^{n}$. Then, for any $r\in[1,p^{\star}_{s})$, the Sobolev embedding
$W^{s,p}(K)\hookrightarrow L^{r}(K)$ given in Lemma 2.3 is compact. In other
words, if a sequence $\\{v_{h}\\}_{h>0}\subset W^{s,p}(K)$ is bounded, then
$\\{v_{h}\\}_{h>0}$ is pre-compact in $L^{r}(K)$ for any
$r\in[1,p^{\star}_{s})$.

We also need the space-time fractional Sobolev continuous and compact
embedding. The proof is done similarly as in Lemma 2.4, on the space-time
domain with dimension $n+1$.

###### Lemma 2.5 (Space-time fractional Sobolev embedding)

Let $s\in(0,1)$ and $p\in[1,\infty)$ be numbers with $sp<n+1$. Let $K$ be a
bounded domain in $\mathbb{R}^{n}$ and $0<T<\infty$. For any $v\in
W^{s,p}(K_{T})$, there holds that, for any
$\gamma\in[1,\overline{p^{\star}_{s}}\,]$ with
$\overline{p^{\star}_{s}}:=\frac{(n+1)p}{n+1-sp}$,

$\|v\|_{L^{\gamma}(K_{T})}\leqq C\|v\|_{W^{s,p}(K_{T})},$ (2.12)

where the positive constant $C$ depends only on $K$, $T$, $s$ $p$ and $n$.
Furthermore, if $K$ is a Lipschitz bounded domain, then the space-time Sobolev
embedding $W^{s,p}(K_{T})\hookrightarrow L^{\gamma}(K_{T})$ is compact for any
$\gamma\in[1,\overline{p^{\star}_{s}}\,)$. In other words, if a sequence
$\\{v_{h}\\}_{h>0}\subset W^{s,p}(K_{T})$ is bounded, then $\\{v_{h}\\}_{h>0}$
is pre-compact in $L^{\gamma}(K_{T})$ for any
$\gamma\in[1,\overline{p^{\star}_{s}}\,)$.

The following is the fractional Poincaré inequality available for functions in
the fractional Sobolev space $W^{s,p}_{0}(K)$. We shall present the proof for
completeness (refer to [24, Lemma 8.1, Theorem 8.2, pp.562–565]).

###### Lemma 2.6 (Fractional Poincaré inequality)

Let $s\in(0,1)$, $p>1$ and $K$ be a bounded domain in $\mathbb{R}^{n}$. Then
for any $u\in W^{s,p}_{0}(K)$ there holds

$\|u\|_{L^{p}(K)}\leqq
C(n,s,p)(\operatorname*{diam}{K})^{sp}[u]_{W^{s,p}(\mathbb{R}^{n})}.$ (2.13)

###### Proof.

First we notice that the assertion follows from the Sobolev inequality (2.11)
in Lemma 2.3 and Hölder’s inequality. Here the direct proof is given, which is
the modification of the Campanato semi-norm estimate in the proof of [24,
Lemma 8.1, pp.562–565].

Since $u(x)=0$ for $x\in K^{c}$, letting $d:=\operatorname*{diam}K$, one can
estimate as

$\displaystyle[u]^{p}_{W^{s,p}(\mathbb{R}^{n})}$
$\displaystyle=\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dxdy\geqq\iint_{K\times
K^{c}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dxdy$
$\displaystyle\geqq\int_{K}\left(\,\int_{K^{c}\,\cap\,\\{x\in\mathbb{R}^{n}\,:\,|x-y|\,\geqq\,2d\\}}\dfrac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dx\right)\,dy$
$\displaystyle=\int_{K}\left(\,\int_{K^{c}\,\cap\,\\{x\in\mathbb{R}^{n}\,:\,|x-y|\,\geqq\,2d\\}}\dfrac{|u(y)|^{p}}{|x-y|^{n+sp}}\,dx\right)\,dy.$

Since for $y\in K$ the fact that $K\subset\\{x\in\mathbb{R}^{n}:|x-y|<2d\\}$
shows the region of the integral in the above display is
$K^{c}\,\cap\,\\{x\in\mathbb{R}^{n}:|x-y|\geqq
2d\\}=\\{x\in\mathbb{R}^{n}:|x-y|\geqq 2d\\}$ and so, the change of variable
$\varrho=|x-y|$ gives that

$\displaystyle[u]^{p}_{W^{s,p}(\mathbb{R}^{n})}$
$\displaystyle\geqq\int_{K}\left(\int_{\\{x\in\mathbb{R}^{n}:|x-y|\geqq
2d\\}}\dfrac{|u(y)|^{p}}{|x-y|^{n+sp}}\,dx\right)\,dy$
$\displaystyle=\int_{K}\left(\,\int_{2d}^{\infty}\dfrac{n\alpha_{n}}{\varrho^{1+sp}}\,d\varrho\right)|u(y)|^{p}\,dy$
$\displaystyle=n\alpha_{n}\dfrac{(2d)^{-sp}}{sp}\int_{K}|u(y)|^{p}\,dy=Cd^{-sp}\|u\|_{L^{p}(K)}^{p},$

where $\alpha_{n}$ denotes the volume of the unit ball in $\mathbb{R}^{n}$ and
$C\equiv C(n,s,p)$. Thus the proof is complete. ∎

We conclude this subsection by listing the result retrieved from [24,
Proposition 2.1, p.524].

###### Lemma 2.7

Let $p\geqq 1$ and $0<s\leqq s^{\prime}<1$, and $K$ be an open set in
$\mathbb{R}^{n}$. Then there holds

$\|u\|_{W^{s,p}(K)}\leqq C\|u\|_{W^{s^{\prime},p}(K)}$

for a measurable function $u$ on $K$ and suitable positive constant $C$
depending only on $n$,$s$ and $p$.

###  2.3. Weak solution

Finally, we introduce the notion of weak solution in the following. We
hereafter abbreviate

$\displaystyle U(x,y)$ $\displaystyle:=|u(x)-u(y)|^{p-2}(u(x)-u(y)),$ (2.14)
$\displaystyle U(x,y,t)$ $\displaystyle:=|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))$
(2.15)

for any measurable function $u$ defined on $\mathbb{R}^{n}$ and
$\mathbb{R}^{n}\times\mathbb{R}$, respectively. Such a short-hand notation is
used in the fractional operator because of space limitation.

###### Definition 2.8 (Weak solution)

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain and $T\in(0,\infty]$.
Suppose that the initial datum $u_{0}$ is in $W^{s,p}_{0}(\Omega)\cap
L^{q+1}(\Omega)$. Let $\mathscr{T}$ be the class of test functions defined by

$\mathscr{T}:=\left\\{\varphi\in
L^{p}(0,T\,;W^{s,p}_{0}(\Omega)):\left.\begin{array}[]{c}\partial_{t}\varphi\in
L^{q+1}(\Omega_{T}),\\\\[2.84526pt]
\varphi(x,0)=\varphi(x,T)=0\quad\textrm{a.e.}\,\,x\in\Omega\end{array}\right.\right\\}.$

A measurable function $u=u(x,t)$ defined on a space-time region
$\mathbb{R}^{n}_{T}:=\mathbb{R}^{n}\times(0,T)$ is said to be a _weak
solution_ to (1.1) provided that the following conditions are satisfied:

  1. (D1)

$u\in L^{\infty}(0,T\,;W^{s,p}(\mathbb{R}^{n})\cap L^{q+1}(\mathbb{R}^{n}))$.

  2. (D2)

There holds

$-\iint_{\Omega_{T}}|u|^{q-1}u\cdot\partial_{t}\varphi\,dxdt+\frac{1}{2}\int_{0}^{T}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{U(x,y,t)}{|x-y|^{n+sp}}(\varphi(x,t)-\varphi(y,t))\,dxdydt=0$

for every $\varphi\in\mathscr{T}$.

  3. (D3)

$u$ satisfies the Dirichlet boundary condition in the sense that

$u(t)\in W^{s,p}_{0}(\Omega)\quad\textrm{for a.e.}\,\,\,t\in(0,T),$

and attains the initial datum $u_{0}$ continuously in the fractional Sobolev
space:

$\lim_{t\searrow 0}\|u(t)-u_{0}\|_{W^{s,p}(\mathbb{R}^{n})}=0.$

##  3\. Approximate solutions

This section is devoted to constructing approximate solutions of (1.1) and
deriving the uniform estimate with respect to $h$. Following [23, Section 3],
we introduce a family of nonlocal elliptic equations (3.1) of Rothe type,
characterized by the difference quotient in time variable (refer to [27]). Let
$h$ be a fixed positive number, sent to zero later. Starting from the initial
datum $u_{0}$ in $W_{0}^{s,p}(\Omega)\cap L^{q+1}(\Omega)$, we shall
inductively define $u_{m}\in W_{0}^{s,p}(\Omega)\cap L^{q+1}(\Omega)$ for
$m\in\mathbb{N}$ as a weak solution in the sense of (3.2) below to the
following nonlocal integro equation

$\displaystyle\begin{cases}\displaystyle\frac{|u_{m}|^{q-1}u_{m}-|u_{m-1}|^{q-1}u_{m-1}}{h}+(-\Delta)_{p}^{s}u_{m}=0\quad&\hbox{in}\,\,\mathbb{R}^{n}\\\
u_{m}=0\quad&\hbox{in}\,\,\Omega^{c}.\end{cases}$ (3.1)

Here, $C^{\infty}_{0}(\Omega)$ is a subspace in $W^{s,p}_{0}(\Omega)$ and
$L^{q+1}(\Omega)$, respectively, and thus, $W^{s,p}_{0}(\Omega)\cap
L^{q+1}(\Omega)$ is nonempty.

The existence of the weak solution $u_{m}$ to Eq. (3.1) is guaranteed by the
following lemma, whose proof is given in Appendix A.

###### Lemma 3.1 (Existence of solutions for the nonlocal difference equation)

Suppose that (3.1) has solutions $u_{k}$ for $k=1,2,\ldots,m-1$. Then there
exists a weak solution $u_{m}\in W_{0}^{s,p}(\Omega)\cap L^{q+1}(\Omega)$ to
(3.1).

Let $\\{u_{m}\\}_{m\in\mathbb{N}}\subset W^{s,p}_{0}(\Omega)\cap
L^{q+1}(\Omega)$ be a sequence of weak solutions to (3.1) constructed by Lemma
3.1. Note that the weak solutions satisfy (3.1) in the following weak sense

$\displaystyle\int_{\Omega}\dfrac{|u_{m}|^{q-1}u_{m}-|u_{m-1}|^{q-1}u_{m-1}}{h}\,\xi\,dx+\frac{1}{2}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{U_{m}(x,y)(\xi(x)-\xi(y))}{|x-y|^{n+sp}}\,dxdy=0$
(3.2)

for any $\xi\in W_{0}^{s,p}(\Omega)\cap L^{q+1}(\Omega)$, where $U_{m}(x,y)$
is defined along the rule in (2.14):

$U_{m}(x,y):=|u_{m}(x)-u_{m}(y)|^{p-2}(u_{m}(x)-u_{m}(y)).$

Let $h>0$ be an approximation parameter, finally sent to zero. In terms of the
family $\\{u_{m}\\}_{m\in\mathbb{N}}$ of solutions to (3.1), we define the six
functions defined on $\Omega_{\infty}$:
$\bar{u}_{h},\,u_{h},\,\bar{v}_{h},\,v_{h},\,\bar{w}_{h}$ and $w_{h}$. Put
$t_{m}:=mh,\,m=0,1,2,\ldots$. For $m=1,2,\ldots$, let

$\begin{cases}\bar{u}_{h}(x,t):=\begin{cases}u_{0}(x),&\quad(x,t)\in\Omega\times[-h,0]\\\\[2.84526pt]
u_{m}(x),&\quad(x,t)\in\Omega\times(t_{m-1},t_{m}]\end{cases}\\\\[28.45274pt]
\bar{v}_{h}(x,t):=|\bar{u}_{h}(x,t)|^{q-1}\bar{u}_{h}(x,t),\qquad(x,t)\times\Omega\in[-h,\infty)\\\\[8.53581pt]
\bar{w}_{h}(x,t):=|\bar{u}_{h}(x,t)|^{\frac{q-1}{2}}\bar{u}_{h}(x,t),\qquad(x,t)\in\Omega\times[-h,\infty)\end{cases}$
(3.3)

and for $(x,t)\in\Omega\times[t_{m-1},t_{m}]$, $m=1,2,\ldots$,

$\begin{cases}u_{h}(x,t):=\dfrac{t-t_{m-1}}{h}u_{m}(x)+\dfrac{t_{m}-t}{h}u_{m-1}(x),\vspace{3mm}\\\
v_{h}(x,t):=\dfrac{t-t_{m-1}}{h}|u_{m}(x)|^{q-1}u_{m}(x)+\dfrac{t_{m}-t}{h}|u_{m-1}(x)|^{q-1}u_{m-1}(x),\vspace{3mm}\\\
w_{h}(x,t):=\dfrac{t-t_{m-1}}{h}|u_{m}(x)|^{\frac{q-1}{2}}u_{m}(x)+\dfrac{t_{m}-t}{h}|u_{m-1}(x)|^{\frac{q-1}{2}}u_{m-1}(x).\end{cases}$
(3.4)

We refer to all of six functions $\bar{u}_{h}$, $\bar{v}_{h}$, $\bar{w}_{h}$,
$u_{h}$ , $v_{h}$ and $w_{h}$ as _approximate solutions_ of (1.1). By use of
these notation, (3.1) is formally rewritten as

$\partial_{t}v_{h}+(-\Delta)_{p}^{s}\bar{u}_{h}=0\quad\textrm{in}\,\,\,\Omega_{\infty},$

which holds in the sense of distribution as follows:

$\displaystyle\iint_{\Omega_{T}}\partial_{t}v_{h}(x,t)\,\varphi(x,t)\,dxdt+\frac{1}{2}\int_{0}^{T}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{\overline{U}_{h}(x,y,t)}{|x-y|^{n+sp}}(\varphi(x,t)-\varphi(y,t))\,dxdydt=0$
(3.5)

for any positive number $T<\infty$ and all test-functions $\varphi\in
L^{1}\left(0,T\,;W^{s,p}_{0}(\Omega)\cap L^{q+1}(\Omega)\right)$, where

$\overline{U}_{h}(x,y,t):=|\bar{u}_{h}(x,t)-\bar{u}_{h}(y,t)|^{p-2}(\bar{u}_{h}(x,t)-\bar{u}_{h}(y,t)).$

We first gather the elementary estimates for the step on time functions and
the linear-interpolated on time ones. These are often used in the proof of
convergence later. The proof simply follows from the definition of (3.3) and
(3.4).

###### Lemma 3.2

Let $\bar{u}_{h},u_{h},\bar{v}_{h},v_{h},{\bar{w}}_{h}$ and $w_{h}$ defined as
in (3.3) and (3.4) by use of the family of functions $\\{u_{m}\\}$,
$m=0,1,\ldots$. Let us denote by $f_{m}\,\,(m=0,1,\cdots)$ each $u_{m}$,
$|u_{m}|^{\frac{q-1}{2}}u_{m}$ or $|u_{m}|^{q-1}u_{m}$. Further, denote by
$\bar{f}_{h}$ each of $\bar{u}_{h}$, $\bar{w}_{h}$ or $\bar{v}_{h}$, and by
$f_{h}$ each of $u_{h}$, $w_{h}$ or $v_{h}$. Then there holds, for any
$(x,t)\in\mathbb{R}^{n}\times[t_{m-1},t_{m}]\,\,(m=1,2,\ldots)$,

$\left|f_{h}(x,t)\right|\leqq\frac{t-t_{m-1}}{h}|f_{m}(x)|+\frac{t_{m}-t}{h}|f_{m-1}(x)|;$
(3.6)
$\left|\bar{f}_{h}(x,t)-f_{h}(x,t)\right|\leqq\frac{t_{m}-t}{h}\left|f_{m}(x)-f_{m-1}(x)\right|.$
(3.7)

In particular, for any $(x,t)\in\mathbb{R}^{n}\times[0,\infty)$

$\left|f_{h}(x,t)\right|\leqq\left|\bar{f}_{h}(x,t)\right|+\left|\bar{f}_{h}(x,t-h)\right|;$
(3.8)
$\left|\bar{f}_{h}(x,t)-f_{h}(x,t)\right|\leqq\left|\bar{f}_{h}(x,t)-\bar{f}_{h}(x,t-h)\right|.$
(3.9)

We next derive the energy estimates as follows.

###### Lemma 3.3 (Energy estimates)

Let $\bar{u}_{h}$ and $u_{h}$ be the approximate solutions of (1.1) defined as
in (3.3) and (3.4). Then it holds that, for any positive number $T<\infty$,

$\sup_{0<t<T}\int_{\Omega}|\bar{u}_{h}(t)|^{q+1}\,dx\leqq\int_{\Omega}|u_{0}|^{q+1}\,dx,$
(3.10)
$\int_{0}^{T}[\bar{u}_{h}(t)]_{W^{s,p}(\mathbb{R}^{n})}^{p}\,dt\leqq\frac{2q}{q+1}\int_{\Omega}|u_{0}|^{q+1}\,dx,$
(3.11)
$\iint_{\Omega_{T}}\bigg{(}|\bar{u}_{h}(x,t)|+|\bar{u}_{h}(x,t-h)|\bigg{)}^{q-1}|\partial_{t}u_{h}(x,t)|^{2}\,dxdt\leqq
C[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p},$ (3.12)

where the positive constant $C$ depends only on $p$ and $q$, and

$\sup_{0<t<T}[\bar{u}_{h}(t)]_{W^{s,p}(\mathbb{R}^{n})}\leqq[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}.$
(3.13)

###### Proof.

Throughout this proof, we choose an integer $N$ to satisfsy $t_{N-1}<T\leqq
t_{N}$. Testing (3.2) by $\xi=hu_{m}$ and summing up from $m=1$ to $k$
($k=1,2,\dotsc,N$), we have

$\displaystyle\sum\limits_{m=1}^{k}\int_{\Omega}\left(|u_{m}|^{q-1}u_{m}-|u_{m-1}|^{q-1}u_{m-1}\right)u_{m}\,dx+\frac{1}{2}\sum_{m=1}^{k}h\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{|u_{m}(x)-u_{m}(y)|^{p}}{|x-y|^{n+sp}}\,dxdy=0$

and thus,

$\sum\limits_{m=1}^{k}\int_{\Omega}\left(|u_{m}|^{q-1}u_{m}-|u_{m-1}|^{q-1}u_{m-1}\right)u_{m}\,dx+\frac{1}{2}\sum_{m=1}^{k}h[u_{m}]_{W^{s,p}(\mathbb{R}^{n})}^{p}=0.$
(3.14)

By Young’s inequality, the first term on the left-hand side of (3.14) is
estimated as

$\displaystyle\sum\limits_{m=1}^{k}\int_{\Omega}\left(|u_{m}|^{q-1}u_{m}-|u_{m-1}|^{q-1}u_{m-1}\right)u_{m}\,dx$
$\displaystyle\geqq\sum_{m=1}^{k}\frac{q}{q+1}\int_{\Omega}\left(|u_{m}|^{q+1}-|u_{m-1}|^{q+1}\right)\,dx$
$\displaystyle=\frac{q}{q+1}\int_{\Omega}\left(|u_{k}|^{q+1}-|u_{0}|^{q+1}\right)\,dx.$

On substitution into (3.14), we arrive at

$\frac{q}{q+1}\int_{\Omega}|u_{k}|^{q+1}\,dx+\frac{1}{2}\sum\limits_{m=1}^{k}h[u_{m}]_{W^{s,p}(\mathbb{R}^{n})}^{p}\leqq\frac{q}{q+1}\int_{\Omega}|u_{0}|^{q+1}\,dx.$
(3.15)

Then we neglect the second term and the first one in the left-hand side of
(3.15) to get

$\sup_{1\leqq k\leqq
N}\int_{\Omega}|u_{k}|^{q+1}\,dx\leqq\int_{\Omega}|u_{0}|^{q+1}\,dx,$

and

$\int_{0}^{t_{N}}[\bar{u}_{h}(t)]_{W^{s,p}(\mathbb{R}^{n})}^{p}\,dt\leqq\frac{2q}{q+1}\int_{\Omega}|u_{0}|^{q+1}\,dx,$

respectively. Those inequalities easily lead to

$\sup_{0<t<T}\int_{\Omega}|\bar{u}_{h}(t)|^{q+1}\,dx\leqq\int_{\Omega}|u_{0}|^{q+1}\,dx$

and

$\int_{0}^{T}[\bar{u}_{h}(t)]_{W^{s,p}(\mathbb{R}^{n})}^{p}\,dt\leqq\frac{2q}{q+1}\int_{\Omega}|u_{0}|^{q+1}\,dx.$

Thus estimates (3.10) and (3.11) are claimed.

On the other hand, we take $\displaystyle\xi=u_{m}-u_{m-1}$ in (3.2) and sum
up the resultant equality from $m=1$ to $k$, where $k=1,2,\cdots,N$. Then we
have

$\displaystyle\sum\limits_{m=1}^{k}\dfrac{1}{h}\int_{\Omega}(|u_{m}|^{q-1}u_{m}-|u_{m-1}|^{q-1}u_{m-1})(u_{m}-u_{m-1})\,dx$
$\displaystyle+\frac{1}{2}\sum\limits_{m=1}^{k}\,\,\,\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{U_{m}(x,y)\big{[}(u_{m}(x)-u_{m-1}(x))-(u_{m}(y)-u_{m-1}(y))\big{]}}{|x-y|^{n+sp}}\,dxdy=0.$
(3.16)

By (2.9) in Lemma 2.1 with $\alpha=q+1$, the integrand of the 1st term on the
left-hand side of (3) is estimated as

$\displaystyle\quad\left(|u_{m}|^{q-1}u_{m}-|u_{m-1}|^{q-1}u_{m-1}\right)(u_{m}-u_{m-1})$
$\displaystyle\geqq
C(q)\big{(}|u_{m}|+|u_{m-1}|\big{)}^{q-1}|u_{m}-u_{m-1}|^{2}.$ (3.17)

Again, by Young’s inequality, we observe that

$\displaystyle\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{U_{m}(x,y)\big{[}(u_{m}(x)-u_{m-1}(x))-(u_{m}(y)-u_{m-1}(y))\big{]}}{|x-y|^{n+sp}}\,dxdy$
$\displaystyle\geqq\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{|u_{m}(x)-u_{m}(y)|^{p}}{|x-y|^{n+sp}}\,dxdy$
$\displaystyle\quad\quad\quad-\dfrac{1}{p}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{|u_{m-1}(x)-u_{m-1}(y)|^{p}}{|x-y|^{n+sp}}\,dxdy-\dfrac{p-1}{p}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{|u_{m}(x)-u_{m}(y)|^{p}}{|x-y|^{n+sp}}\,dxdy$
$\displaystyle=\frac{1}{p}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{|u_{m}(x)-u_{m}(y)|^{p}}{|x-y|^{n+sp}}\,dxdy-\frac{1}{p}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{|u_{m-1}(x)-u_{m-1}(y)|^{p}}{|x-y|^{n+sp}}\,dxdy.$
(3.18)

Gathering (3) and (3) in (3) yields

$\displaystyle\sum\limits_{m=1}^{k}h\int_{\Omega}C\big{(}|u_{m}|+|u_{m-1}|\big{)}^{q-1}\bigg{|}\frac{u_{m}-u_{m-1}}{h}\bigg{|}^{2}\,dx$
$\displaystyle\quad\quad\quad+\frac{1}{2p}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{|u_{k}(x)-u_{k}(y)|^{p}}{|x-y|^{n+sp}}\,dxdy\leqq\dfrac{1}{2p}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{|u_{0}(x)-u_{0}(y)|^{p}}{|x-y|^{n+sp}}\,dxdy.$
(3.19)

Neglecting either term in the left-hand side, we obtain

$C(q)\int_{0}^{t_{k}}\int_{\Omega}\bigg{(}|\bar{u}_{h}(x,t)|+|\bar{u}_{h}(x,t-h)|\bigg{)}^{q-1}|\partial_{t}u_{h}(x,t)|^{2}\,dxdt\leqq\frac{1}{2p}[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p}$

and

$\max_{1\leqq k\leqq
N}[\bar{u}_{h}(t)]_{W^{s,p}(\mathbb{R}^{n})}^{p}\leqq[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p},$

respectively. Therefore the proof is complete. ∎

The statements of Lemma 3.3 naturally also holds true for the linear-
interpolated solutions by the definition. The proof is clear by (3.6)–(3.8) in
Lemma 3.2:

###### Lemma 3.4 (Energy estimates)

Let $u_{h}$ be the approximate solution of (1.1) defined in (3.4). Then it
holds true that, for any positive number $T<\infty$,

$\sup_{0<t<T}\int_{\Omega}|u_{h}(t)|^{q+1}\,dx\leqq\int_{\Omega}|u_{0}|^{q+1}\,dx,$
(3.20) $\int_{0}^{T}[u_{h}(t)]_{W^{s,p}(\mathbb{R}^{n})}^{p}\,dt\leqq
2^{p-1}\left(\frac{4q}{q+1}\int_{\Omega}|u_{0}|^{q+1}\,dx+h[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p}\right),$
(3.21)

where the positive constant $C$ depends only on $p$ and $q$, and

$\sup_{0<t<T}[u_{h}(t)]_{W^{s,p}(\mathbb{R}^{n})}\leqq[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}.$
(3.22)

###### Proof.

The inequalities (3.20) and (3.22) follows from the Minkowski inequality,
(3.6) in Lemma 3.2 and (3.10), (3.13) in Lemma 3.3, respectively. For the
inequality (3.21), by (3.8) in Lemma 3.2 and the Minkowski inequality we have

$\int_{0}^{T}[u_{h}(t)]_{W^{s,p}(\mathbb{R}^{n})}^{p}\,dt\leqq
2^{p-1}\left(\int_{0}^{T}[{\bar{u}}_{h}(t)]_{W^{s,p}(\mathbb{R}^{n})}^{p}\,dt+\int_{0}^{T}[{\bar{u}}_{h}(t-h)]_{W^{s,p}(\mathbb{R}^{n})}^{p}\,dt\right),$

of which the last terms are applied by (3.11) in Lemma 3.3 and thus, (3.21)
follows. ∎

###### Lemma 3.5 (Time-derivative estimate)

Let $v_{h}$ and $w_{h}$ be the approximate solutions of (1.1) defined by
(3.4). Then the time derivatives of approximating solutions
$\\{\partial_{t}w_{h}\\}_{h>0}$ are bounded in $L^{2}(\Omega_{T})$ for any
positive $T<\infty$, namely,

$\iint_{\Omega_{T}}|\partial_{t}w_{h}|^{2}\,dxdt\leqq
C[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p}$ (3.23)

and, specifically, if $q\geqq 1$, the time derivatives of approximating
solutions $\\{\partial_{t}v_{h}\\}$ are bounded in $L^{1}(\Omega_{T})$ for any
positive $T<\infty$,

$\iint_{\Omega_{T}}|\partial_{t}v_{h}|\,dxdt\leqq
C|\Omega_{T}|^{\frac{1}{q+1}}\|u_{0}\|_{L^{q+1}(\Omega)}^{\frac{q-1}{2}}[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{\frac{p}{2}}$
(3.24)

with a positive constant $C=C(p,q)$.

###### Proof.

Throughout the proof, we choose an integer $N$ satisfying $t_{N-1}<T\leqq
t_{N}$. We now begin the proof of (3.23). By (2.8) in Lemma 2.1 with
$\alpha=\frac{q+3}{2}$

$\left||u_{m}|^{\frac{q-1}{2}}u_{m}-|u_{m-1}|^{\frac{q-1}{2}}u_{m-1}\right|^{2}\leqq
C\big{(}|u_{m}|+|u_{m-1}|\big{)}^{q-1}|u_{m}-u_{m-1}|^{2}.$ (3.25)

From (3.25) and (3.12) in Lemma 3.3 it follows that, for any $N=1,2,\ldots$

$\displaystyle\sum_{m=1}^{N}h\int_{\Omega}|\partial_{t}w_{h}|^{2}\,dx$
$\displaystyle=\sum_{m=1}^{N}h\int_{\Omega}\left|\frac{|u_{m}|^{\frac{q-1}{2}}u_{m}-|u_{m-1}|^{\frac{q-1}{2}}u_{m-1}}{h}\right|^{2}\,dx$
$\displaystyle\\!\\!\\!\stackrel{{\scriptstyle\eqref{wh
eq.1}}}{{\leqq}}C\sum_{m=1}^{N}h\int_{\Omega}\big{(}|u_{m}|+|u_{m-1}|\big{)}^{q-1}\left|\frac{u_{m}-u_{m-1}}{h}\right|^{2}\,dx$
$\displaystyle=C\int_{0}^{t_{N}}\int_{\Omega}\big{(}|\bar{u}_{h}(x,t)|+|\bar{u}_{h}(x,t-h)|\big{)}^{q-1}|\partial_{t}u_{h}(x,t)|^{2}\,dxdt$
$\displaystyle\\!\\!\stackrel{{\scriptstyle\eqref{energy est.
NE3}}}{{\leqq}}C[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p},$

which implies (3.23).

We shall show the estimation (3.24). Suppose that $q\geqq 1$. We use Hölder’s
inequality twice with pair of exponents $\left(2,2\right)$ and
$\left(\frac{q+1}{q-1},\,\frac{q+1}{2}\right)$, the energy inequalities (3.10)
and (3.12) in Lemma 3.3 to have

$\displaystyle\|\partial_{t}v_{h}\|_{L^{1}(\Omega_{T})}$
$\displaystyle\leqq\iint_{\Omega_{T}}\Big{(}|\bar{u}_{h}(t)|+|\bar{u}_{h}(t-h)|\Big{)}^{q-1}|\partial_{t}u_{h}|\,dxdt$
$\displaystyle\leqq\left(\iint_{\Omega_{T}}\Big{(}|\bar{u}_{h}(t)|+|\bar{u}_{h}(t-h)|\Big{)}^{q-1}\,dxdt\right)^{\frac{1}{2}}$
$\displaystyle\quad\quad\quad\quad\quad\times\left(\iint_{\Omega_{T}}\Big{(}|\bar{u}_{h}(t)|+|\bar{u}_{h}(t-h)|\Big{)}^{q-1}|\partial_{t}u_{h}|^{2}\,dxdt\right)^{\frac{1}{2}}$
$\displaystyle\leqq|\Omega_{T}|^{\frac{1}{q+1}}\left(\iint_{\Omega_{T}}\Big{(}|\bar{u}_{h}(t)|+|\bar{u}_{h}(t-h)|\Big{)}^{q+1}\,dxdt\right)^{\frac{q-1}{2(q+1)}}$
$\displaystyle\quad\quad\quad\quad\quad\times\left(\iint_{\Omega_{T}}\Big{(}|\bar{u}_{h}(t)|+|\bar{u}_{h}(t-h)|\Big{)}^{q-1}|\partial_{t}u_{h}|^{2}\,dxdt\right)^{\frac{1}{2}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\stackrel{{\scriptstyle\eqref{energy
est. NE1},~{}\eqref{energy est.
NE3}}}{{\leqq}}C(p,q)|\Omega_{T}|^{\frac{1}{q+1}}\|u_{0}\|_{L^{q+1}(\Omega)}^{\frac{q-1}{2}}[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{\frac{p}{2}},$

which finishes the proof of (3.24). ∎

We conclude this section by deriving the integral estimate for $\bar{u}_{h}$
and $u_{h}$, used later in Section 4.

###### Lemma 3.6

Let $\bar{u}_{h}$ and $u_{h}$ be the approximate solutions to (1.1) defined by
(3.3) and (3.4). Then, for any bounded domain $K$ in $\mathbb{R}^{n}$ and any
positive $T<\infty$

$\begin{split}\|\bar{u}_{h}\|_{L^{p}(K_{T})}^{p}&\leqq
C(\mathrm{diam}\,K)^{sp}\int_{0}^{T}[\bar{u}_{h}(t)]_{W^{s,p}(\mathbb{R}^{n})}^{p}dt\leqq
C(\mathrm{diam}\,K)^{sp}\|u_{0}\|_{L^{q+1}(\Omega)}^{q+1};\\\\[5.69054pt]
\|u_{h}\|_{L^{p}(K_{T})}^{p}&\leqq
C(\mathrm{diam}\,K)^{sp}\left(\|u_{0}\|_{L^{q+1}(\Omega)}^{q+1}+h[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p}\right)\end{split}$
(3.26)

and

$\begin{split}\sup_{0<t<T}\|\bar{u}_{h}(t)\|_{L^{p_{s}^{\star}}(K)}&\leqq\widetilde{C}[u_{0}]_{W^{s,p}(\mathbb{R}^{n})};\\\\[5.69054pt]
\sup_{0<t<T}\|u_{h}(t)\|_{L^{p_{s}^{\star}}(K)}&\leqq\widetilde{C}[u_{0}]_{W^{s,p}(\mathbb{R}^{n})},\end{split}$
(3.27)

where the $C=C(n,p,q)$ and $\tilde{C}=\tilde{C}(n,s,p)$ are positive
constants.

###### Proof.

From (2.13) in Lemma 2.6 and (3.11) in Lemma 3.3 we obtain (3.26)1. Similarly,
(3.21) in Lemma 3.4 yields (3.26)2. By (2.11) in Lemma 2.3 and (3.13) in Lemma
3.3 one has, for any $t>0$

$\displaystyle\|\bar{u}_{h}(t)\|_{L^{p_{s}^{\star}}(K)}$
$\displaystyle\\!\\!\\!\stackrel{{\scriptstyle\eqref{embedRn}}}{{\leqq}}\widetilde{C}(n,s,p)[\bar{u}_{h}(t)]_{W^{s,p}(\mathbb{R}^{n})}$
$\displaystyle\\!\\!\\!\stackrel{{\scriptstyle\eqref{energy est.
NE4}}}{{\leqq}}\widetilde{C}[u_{0}]_{W^{s,p}(\mathbb{R}^{n})},$

which together with (3.6) in Lemma 3.2 shows (3.27)1. The estimate (3.27)2 is
shown similarly. ∎

##  4\. Convergence of approximate solutions

This section is devoted to deriving the convergence of approximate solution of
(1.1) defined by (3.3) and (3.4). The initial value $u_{0}$ is assumed to
belong to $W_{0}^{s,p}(\Omega)\cap L^{q+1}(\Omega)$.

To achieve the convergence of approximate solution of (1.1), as stated in
Lemma 4.6 below, we need to introduce the following truncated function: For
the approximate solution $\\{u_{m}(x)\\}$ to (3.1) and
$2\leqq\ell\in\mathbb{N}$, let us define as

$(u_{m}(x))_{\pm}^{(\ell)}:=\min\left\\{\max\left\\{(u_{m}(x))_{\pm},\,\dfrac{1}{\ell}\right\\},\,\ell\right\\},$

where $(u_{m}(x))_{+}:=\max\\{u_{m}(x),0\\}$ and
$(u_{m}(x))_{-}:=(-u_{m}(x))_{+}$.

$u_{m}$$y$$y=(u_{m})_{+}^{(\ell)}$$1/\ell$$\ell$ Figure 1. Graph of
$(u_{m}(x))_{+}^{(\ell)}$

Further, for $(x,t)\in\mathbb{R}^{n}\times(t_{m-1},t_{m}]\,\,(m=1,2,\cdots)$,
we define $\\{(u_{h})_{\pm}^{(\ell)}\\}$ and
$\\{(\bar{u}_{h})_{\pm}^{(\ell)}\\}$ by

$(u_{h})_{\pm}^{(\ell)}(x,t):=\dfrac{t-t_{m-1}}{h}(u_{m}(x))_{\pm}^{(\ell)}+\dfrac{t_{m}-t}{h}(u_{m-1}(x))_{\pm}^{(\ell)}$
(4.1)

and

$(\bar{u}_{h})_{\pm}^{(\ell)}(x,t):=(u_{m}(x))_{\pm}^{(\ell)},$

respectively. By $u_{m}=u_{m-1}=0$ outside $\Omega\,\,(m=1,2,\cdots)$, for any
$t\in[0,\infty)$,

$(u_{h})_{\pm}^{(\ell)}(x,t)\equiv\dfrac{1}{\ell}\quad\mathrm{outside}\,\,\Omega.$
(4.2)

Sending $\ell\longrightarrow\infty$, for
$(x,t)\in\mathbb{R}^{n}\times[t_{m-1},t_{m}]\,\,(m=1,2,\ldots)$,

$(u_{h})_{\pm}^{(\ell)}(x,t)\longrightarrow\dfrac{t-t_{m-1}}{h}(u_{m}(x))_{\pm}+\dfrac{t_{m}-t}{h}(u_{m-1}(x))_{\pm}.$

Also, the time-derivative of $(u_{h})_{\pm}^{(\ell)}$ is

$\partial_{t}(u_{h})_{\pm}^{(\ell)}=\dfrac{(u_{m}(x))_{\pm}^{(\ell)}-(u_{m-1}(x))_{\pm}^{(\ell)}}{h}$

for $(x,t)\in\mathbb{R}^{n}\times[t_{m-1},t_{m}]\,\,(m=1,2,\ldots)$.

###  4.1. Energy estimates for the truncated solution

First we deduce certain energy estimate of the truncated solution defined in
(4.1) above.

###### Lemma 4.1

Let $(u_{h})_{\pm}^{(\ell)}$ be the function defined by (4.1). Then there
holds

$\iint_{\Omega_{T}}|\partial_{t}(u_{h})_{\pm}^{(\ell)}|^{2}\,dxdt\leqq
C\ell^{q-1}[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p}$ (4.3)

whenever $q\geqq 1$ and,

$\min\left\\{\iint_{\Omega_{T}}|\partial_{t}(u_{h})_{\pm}^{(\ell)}|^{2}\,dxdt,\,\iint_{\Omega_{T}}|\partial_{t}(u_{h})_{\pm}^{(\ell)}|^{q+1}\,dxdt\right\\}\leqq
C\ell^{1-q}[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p}$ (4.4)

whenever $0<q<1$, where the positive constant $C$ depends only on $p$ and $q$.

###### Proof.

The proof is based on a sequence of elementary algebraic integral estimates,
those are separately performed in several cases. We prove the statement for
$\\{(u_{h})_{+}^{(\ell)}\\}$ only, as the other case is similarly treated. To
begin, we employ the energy estimate (3) in the proof of Lemma 3.3

$\sum\limits_{m=1}^{N}h\int_{\Omega}\big{(}|u_{m}|+|u_{m-1}|\big{)}^{q-1}\bigg{|}\frac{u_{m}-u_{m-1}}{h}\bigg{|}^{2}\,dx\leqq
C(p,q)[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p}.$ (4.5)

We use a short-hand notation $[u_{m}\geqq L]\equiv\Omega\cap\\{u_{m}\geqq
L\\}$ in the sequel. Other symbols are also abbreviated similarly. Now, we
distinguish between the cases $q\geqq 1$ and $0<q<1$.

In the first case $q\geqq 1$, for any $m=1,\ldots,N$,

$\displaystyle\int_{\Omega}\big{(}|u_{m}|+|u_{m-1}|\big{)}^{q-1}\bigg{|}\frac{u_{m}-u_{m-1}}{h}\bigg{|}^{2}\,dx$
$\displaystyle\geqq\int_{\Omega}\big{(}|u_{m}|+|u_{m-1}|\big{)}^{q-1}\bigg{|}\frac{(u_{m})_{+}^{(\ell)}-(u_{m-1})_{+}^{(\ell)}}{h}\bigg{|}^{2}\,dx$
$\displaystyle=\int_{[u_{m}>\frac{1}{\ell}]\,\cup\,[u_{m-1}>\frac{1}{\ell}]}\ldots\,dx+\int_{[u_{m}\leqq\frac{1}{\ell}]\,\cap\,[u_{m-1}\leqq\frac{1}{\ell}]}\ldots\,dx$
$\displaystyle\geqq\int_{[u_{m}>\frac{1}{\ell}]\,\cup\,[u_{m-1}>\frac{1}{\ell}]}\left(\frac{1}{\ell}\right)^{q-1}\bigg{|}\frac{(u_{m})_{+}^{(\ell)}-(u_{m-1})_{+}^{(\ell)}}{h}\bigg{|}^{2}$
$\displaystyle=\int_{[u_{m}>\frac{1}{\ell}]\,\cup\,[u_{m-1}>\frac{1}{\ell}]}\left(\frac{1}{\ell}\right)^{q-1}\bigg{|}\frac{(u_{m})_{+}^{(\ell)}-(u_{m-1})_{+}^{(\ell)}}{h}\bigg{|}^{2}\,dx$
$\displaystyle\quad\quad+\int_{[u_{m}\leqq\frac{1}{\ell}]\,\cap\,[u_{m-1}\leqq\frac{1}{\ell}]}\left(\frac{1}{\ell}\right)^{q-1}\bigg{|}\frac{(u_{m})_{+}^{(\ell)}-(u_{m-1})_{+}^{(\ell)}}{h}\bigg{|}^{2}\,dx$
$\displaystyle=\left(\frac{1}{\ell}\right)^{q-1}\int_{\Omega}\bigg{|}\frac{(u_{m})_{+}^{(\ell)}-(u_{m-1})_{+}^{(\ell)}}{h}\bigg{|}^{2}\,dx,$
(4.6)

where, in the previous line of (4.1), we used the fact that
$\frac{(u_{m})_{+}^{(\ell)}-(u_{m-1})_{+}^{(\ell)}}{h}=0$ on the set
$[u_{m}\leqq\frac{1}{\ell}]\,\cap\,[u_{m-1}\leqq\frac{1}{\ell}]$. Hence, by
(4.5) and (4.1), we arrive at the first desired estimate (4.3).

In the latter case $0<q<1$, we need to show

$\big{(}|u_{m}|+|u_{m-1}|\big{)}^{q-1}\Big{|}\tfrac{u_{m}-u_{m-1}}{h}\Big{|}^{2}\geqq
3^{q-1}\min\left\\{\ell^{q-1}\Big{|}\tfrac{(u_{m})_{+}^{(\ell)}-(u_{m-1})_{+}^{(\ell)}}{h}\Big{|}^{2},\,\Big{|}\tfrac{(u_{m})_{+}^{(\ell)}-(u_{m-1})_{+}^{(\ell)}}{h}\Big{|}^{q+1}\right\\}.$
(4.7)

For the proof of (4.7), a distinction must be made among the cases in the
following: Firstly, on the set
$\Omega_{1}:=[u_{m}\leqq\frac{1}{\ell}]\cap[u_{m-1}\leqq\frac{1}{\ell}]$ or
$\Omega_{2}:=[u_{m}>\ell]\cap[u_{m-1}>\ell]$, it holds
$\frac{(u_{m})_{+}^{(\ell)}-(u_{m-1})_{+}^{(\ell)}}{h}=0$ and thus, (4.7) is
clearly valid. On the set $\Omega\setminus(\Omega_{1}\cup\Omega_{2})$, we
distinguish among the four cases (I)–(IV).

Case (I) $[u_{m}\leqq\frac{1}{\ell}]\cap[\frac{1}{\ell}<u_{m-1}\leqq\ell]$: In
this region, we further consider the two cases separately: In the case
$u_{m}>0$, noting carefully that $u_{m}-u_{m-1}<0$, we have

$\displaystyle|u_{m}|+|u_{m-1}|=u_{m}+u_{m-1}$
$\displaystyle=(u_{m}-u_{m-1})+2u_{m-1}$ $\displaystyle\leqq 2u_{m-1}\leqq
2\ell<3\ell.$

In the remaining case $u_{m}\leqq 0$, we plainly get

$|u_{m}|+|u_{m-1}|=-u_{m}+u_{m-1}\leqq|u_{m}-u_{m-1}|.$

Case (II) $[u_{m}\leqq\frac{1}{\ell}]\cap[u_{m-1}>\ell]$: Also, we distinguish
between the two cases $u_{m}>0$ and $u_{m}\leqq 0$. In the first case
$u_{m}>0$, noting carefully that $u_{m-1}-u_{m}\geqq\ell-\frac{1}{\ell}\geqq
2-\frac{1}{2}\geqq\frac{1}{\ell}$, we find

$\displaystyle|u_{m}|+|u_{m-1}|=(u_{m-1}-u_{m})+2u_{m}$
$\displaystyle\leqq(u_{m-1}-u_{m})+\frac{2}{\ell}$
$\displaystyle\leqq(u_{m-1}-u_{m})+2(u_{m-1}-u_{m})$
$\displaystyle=3(u_{m-1}-u_{m})$ $\displaystyle\leqq 3|u_{m}-u_{m-1}|.$

In the latter case $u_{m}\leqq 0$, we have again,

$|u_{m}|+|u_{m-1}|=-u_{m}+u_{m-1}\leqq|u_{m}-u_{m-1}|.$

Case (III)
$[\frac{1}{\ell}<u_{m}\leqq\ell]\cap[\frac{1}{\ell}<u_{m-1}\leqq\ell]$: On
this set, it clearly holds that

$|u_{m}|+|u_{m-1}|=u_{m}+u_{m-1}\leqq 2\ell.$

Case (IV) $[\frac{1}{\ell}<u_{m}\leqq\ell]\cap[u_{m-1}>\ell]$: In this region,
we further distinguish between the two cases $u_{m-1}-u_{m}\leqq\ell$ and
$u_{m-1}-u_{m}>\ell$. In the first case $u_{m-1}-u_{m}\leqq\ell$, we in turn
see that

$|u_{m}|+|u_{m-1}|=(u_{m-1}-u_{m})+2u_{m}\leqq\ell+2\ell=3\ell.$

In the latter case $u_{m-1}-u_{m}>\ell$, we have

$\displaystyle|u_{m}|+|u_{m-1}|=(u_{m-1}-u_{m})+2u_{m}$
$\displaystyle\leqq(u_{m-1}-u_{m})+2\ell$
$\displaystyle\leqq(u_{m-1}-u_{m})+2(u_{m-1}-u_{m})$
$\displaystyle=3(u_{m-1}-u_{m})$ $\displaystyle=3|u_{m}-u_{m-1}|.$

By exchanging of the role of $u_{m}(x)$ and $u_{m-1}(x)$, we can get the
estimates same as Cases (I), (II) and (IV) on the sets
$[\frac{1}{\ell}<u_{m}\leqq\ell]\cap[u_{m-1}\leqq\frac{1}{\ell}]$,
$[u_{m}>\ell]\cap[u_{m-1}\leqq\frac{1}{\ell}]$ and
$[u_{m}>\ell]\cap[\frac{1}{\ell}<u_{m-1}\leqq\ell]$, respectively. Joining all
cases leads to

$|u_{m}|+|u_{m-1}|\leqq\max\left\\{3\ell,\,3|u_{m}-u_{m-1}|\right\\}\quad\mathrm{in}\quad\Omega\setminus(\Omega_{1}\cup\Omega_{2}),$

which, by noticing $q-1<0$, in turn implies that

$\big{(}|u_{m}|+|u_{m-1}|\big{)}^{q-1}\geqq
3^{q-1}\min\left\\{\ell^{q-1},\,|u_{m}-u_{m-1}|^{q-1}\right\\}\quad\mathrm{in}\quad\Omega\setminus(\Omega_{1}\cup\Omega_{2}).$
(4.8)

Using $h^{q-1}\geqq 1$ and (4.8), we obtain on the set
$\Omega\setminus(\Omega_{1}\cup\Omega_{2})$,

$\displaystyle\big{(}|u_{m}|+|u_{m-1}|\big{)}^{q-1}\Big{|}\tfrac{u_{m}-u_{m-1}}{h}\Big{|}^{2}$
$\displaystyle\geqq
3^{q-1}\min\left\\{\ell^{q-1}\left|\tfrac{u_{m}-u_{m-1}}{h}\right|^{2},\,h^{q-1}\left|\tfrac{u_{m}-u_{m-1}}{h}\right|^{q+1}\right\\}$
$\displaystyle\geqq
3^{q-1}\min\left\\{\ell^{q-1}\Big{|}\tfrac{(u_{m})_{+}^{(\ell)}-(u_{m-1})_{+}^{(\ell)}}{h}\Big{|}^{2},\,\Big{|}\tfrac{(u_{m})_{+}^{(\ell)}-(u_{m-1})_{+}^{(\ell)}}{h}\Big{|}^{q+1}\right\\}.$

In the sequel, we actually have (4.7) in the whole region $\Omega$ and thus,
by use of (4.5) and (4.7), we in turn achieve (4.4). The proof is concluded. ∎

###  4.2. Convergence results for the truncated solution

We list here the convergence results, Lemmata 4.2 and 4.4 for the truncated
solution in (4.1), used later in Lemma 4.6.

###### Lemma 4.2

Let $u_{h}$ be the approximate solutions of (1.1) defined by (3.3). For a
given $s\in(0,1)$, let $s^{\prime}<s$ be a positive number. Let $\ell\geqq 2$
be a fixed natural number. Let $K$ be a Lipschitz bounded domain in
$\mathbb{R}^{n}$. Then there exists a subsequence
$\\{(u_{h})_{\pm}^{(\ell)}\\}$, denoted by the same notation, and limit
functions $\omega_{\ell}^{\pm}$ such that for every power $\gamma$ with
$1\leqq\gamma<\frac{n+1}{n+1-s^{\prime}}$,

$(u_{h})_{+}^{(\ell)}\longrightarrow\omega_{\ell}^{\pm}\quad\mathrm{strongly\,\,in}\quad
L^{\gamma}(K_{T}).$ (4.9)

###### Proof.

We prove the statement for $\\{(u_{h})^{(\ell)}_{+}\\}$ only, as the other
case is similarly shown. Let $2\leqq\ell\in\mathbb{N}$ and $s\in(0,1)$ be
arbitrarily given and fixed. Firstly, by (C.1) in Lemma C.1, for all
$s^{\prime},\bar{s}$ satisfying $0<s^{\prime}<\bar{s}<s<1$ there holds

$\left[(u_{h})_{+}^{(\ell)}\right]_{W^{s^{\prime},1}(K_{T})}\leqq
C\left(\left\|\partial_{t}(u_{h})_{+}^{(\ell)}\right\|_{L^{1}(K_{T})}+\int_{0}^{T}\left[(u_{h})_{+}^{(\ell)}(\cdot,t)\right]_{W^{\bar{s},1}(K)}\,dt\right).$
(4.10)

By the energy estimates (4.3) and (4.4) in Lemma 4.1, the first term on the
right-hand side of (4.10) is bounded uniformly in $h$. The Gagliardo semi-norm
appearing on the right-hand side of (4.10) is estimated as

$\displaystyle\left[(u_{h})_{+}^{(\ell)}(\cdot,t)\right]_{W^{\bar{s},1}(K)}$
$\displaystyle\leqq\iint_{K\times
K}\dfrac{\left|(u_{h})_{+}(x,t)-(u_{h})_{+}(y,t)\right|}{|x-y|^{(n+sp)\frac{1}{p}}}\cdot\dfrac{1}{|x-y|^{n+\bar{s}-(n+sp)\frac{1}{p}}}\,dxdy$
$\displaystyle\leqq\left(\,\,\,\iint_{K\times
K}\dfrac{\left|(u_{h})_{+}(x,t)-(u_{h})_{+}(y,t)\right|^{p}}{|x-y|^{n+sp}}\,dxdy\right)^{\frac{1}{p}}$
$\displaystyle\quad\quad\quad\times\left(\,\,\,\iint_{K\times
K}\dfrac{1}{|x-y|^{\frac{p}{p-1}\left(n+\bar{s}-(n+sp)\frac{1}{p}\right)}}\,dxdy\right)^{\frac{p-1}{p}}.$
(4.11)

Noticing that
$\frac{p}{p-1}\left(n+\bar{s}-(n+sp)\frac{1}{p}\right)=n+\frac{p}{p-1}(\bar{s}-s)<n$
and letting $R:=\mathrm{diam}\,K$ we compute that

$\displaystyle\iint_{K\times
K}\dfrac{1}{|x-y|^{\frac{p}{p-1}\left(n+\bar{s}-(n+sp)\frac{1}{p}\right)}}\,dxdy$
$\displaystyle\leqq\int_{K}\left(\,\int_{B_{R}(x)}\dfrac{dy}{|x-y|^{n+\frac{p}{p-1}(\bar{s}-s)}}\right)\,dx$
$\displaystyle=C(n)|K|\int_{0}^{R}\dfrac{d\varrho}{\varrho^{1+\frac{p}{p-1}(\bar{s}-s)}}$
$\displaystyle=C(n)|K|\dfrac{p-1}{p(s-\bar{s})}R^{\frac{p}{p-1}(s-\bar{s})}$

and thus, the second integral on the right-hand side of (4.2) is finite. We
therefore obtain from the energy estimate (3.11) in Lemma 3.3 that

$\displaystyle\left[(u_{h})_{+}^{(\ell)}(\cdot,t)\right]_{W^{\bar{s},1}(K)}$
$\displaystyle\leqq
C(n,p,\bar{s},s)\left[(u_{h})_{+}(\cdot,t)\right]_{W^{\bar{s},p}(\mathbb{R}^{n})}$
$\displaystyle\leqq
C(n,p,\bar{s},s)\left[u_{0}\right]_{W^{\bar{s},p}(\mathbb{R}^{n})},$

which, together this with (4.10) and the fact that
$\left\|(u_{h})_{+}^{(\ell)}(\cdot,t)\right\|_{L^{1}(K)}\leqq\ell|K_{T}|$, in
turn yields that $\left\\{(u_{h})_{+}^{(\ell)}\right\\}_{h>0}$ is bounded in
$W^{s^{\prime},1}(K_{T})$. Hence, by Lemma 2.5, for all $\gamma$ with
$1\leqq\gamma<\frac{n+1}{n+1-s^{\prime}}$, there exist a subsequence
$\\{(u_{h})_{+}^{(\ell)}\\}_{h>0}$ (also labelled with $h$) and a limit
function $\omega_{\ell}^{+}$ fo each $\ell$ such that

$(u_{h})_{+}^{(\ell)}\longrightarrow\omega_{\ell}^{+}\quad\mathrm{strongly\,\,in}\quad
L^{\gamma}(K_{T})$

as $h\searrow 0$. Therefore the proof of Lemma 4.2 is concluded. ∎

We shall make a Chebyshev type estimate for the truncated solutions:

###### Lemma 4.3

Let $\\{u_{m}\\}$ be the approximate solution to (3.1). Let $\ell\geqq 2$ and
$\gamma\in[1,p_{s}^{\star})$ be taken arbitrarily. Then there holds that

$\left|K\cap\\{(u_{m})_{+}\geqq\ell\\}\right|\leqq\dfrac{C}{\ell^{p_{s}^{\star}}}[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p_{s}^{\star}}$
(4.12)

and

$\left\|(u_{m})_{+}\right\|_{L^{\gamma}(K\,\cap\,\\{(u_{m})_{+}\geqq\,\ell\\})}\leqq\dfrac{C}{\ell^{\frac{p_{s}^{\star}}{\gamma}-1}}[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{\frac{p_{s}^{\star}}{\gamma}}.$
(4.13)

###### Proof.

We report the short proof. By (3.27)1 in Lemma 3.6, one has

$\displaystyle\ell^{p_{s}^{\star}}\left|K\cap\\{(u_{m})_{+}\geqq\ell\\}\right|\leqq\int_{K}(u_{m})_{+}^{p_{s}^{\star}}\,dx$
$\displaystyle\leqq\|u_{m}\|_{L^{p_{s}^{\star}}(K)}^{p_{s}^{\star}}\leqq
C[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p_{s}^{\star}},$

which in turn implies (4.12). By Hölder’s inequality, (4.12) and (3.27) in
Lemma 3.6, one estimates that

$\displaystyle\left\|(u_{m})_{+}\right\|_{L^{\gamma}(K\,\cap\,\\{(u_{m})_{+}\geqq\,\ell\\})}^{\gamma}$
$\displaystyle\leqq\left(\,\int_{K\,\cap\,\\{(u_{m})_{+}\geqq\,\ell\\}}|(u_{m})_{+}|^{p_{s}^{\star}}\,dx\right)^{\frac{\gamma}{p_{s}^{\star}}}\left|K\cap\\{(u_{m})_{+}\geqq\ell\\}\right|^{1-\frac{\gamma}{p_{s}^{\star}}}$
$\displaystyle\leqq
C\dfrac{[u_{0}]_{W^{s,p}(K)}^{p_{s}^{\star}}}{\ell^{p_{s}^{\star}-\gamma}},$

which is exactly (4.13). ∎

We now will show that $\\{(u_{h})_{\pm}\\}_{h>0}$ is a Cauchy sequence in
$L^{\gamma}(K_{T})$ in the next lemma.

###### Lemma 4.4

$\\{(u_{h})_{\pm}\\}_{h>0}$ is a Cauchy sequence in $L^{\gamma}(K_{T})$ for
$1\leqq\gamma<\min\left\\{p_{s}^{\star},\frac{n+1}{n+1-s^{\prime}}\right\\}$
and $0<s^{\prime}<s(<1)$.

###### Proof.

We prove the statement for $\\{(u_{h})^{(\ell)}_{+}\\}$ only, as the other
case is similarly treated. We start with dividing into three terms:

$\displaystyle\left\|(u_{h})_{+}-(u_{h^{\prime}})_{+}\right\|_{L^{\gamma}(K_{T})}$
$\displaystyle\leqq\left\|(u_{h})_{+}-(u_{h})_{+}^{(\ell)}\right\|_{L^{\gamma}(K_{T})}+\left\|(u_{h})_{+}^{(\ell)}-(u_{h^{\prime}})_{+}^{(\ell)}\right\|_{L^{\gamma}(K_{T})}+\left\|(u_{h^{\prime}})_{+}^{(\ell)}-(u_{h^{\prime}})_{+}\right\|_{L^{\gamma}(K_{T})}$
$\displaystyle:=\mathbf{I}_{h,\ell}+\mathbf{I}_{h,h^{\prime},\ell}+\mathbf{I}_{h^{\prime},\ell}.$
(4.14)

From now on, we shall estimate each term
$\mathbf{I}_{h,\ell},\,\mathbf{I}_{h,h^{\prime},\ell}$ and
$\mathbf{I}_{h^{\prime},\ell}$ in (4.2). Since for $(x,t)\in
K\times(t_{m-1},t_{m})\,\,(m=1,2,\cdots)$

$\displaystyle(u_{h})_{+}(x,t)-(u_{h})_{+}^{(\ell)}(x,t)$
$\displaystyle=\dfrac{t-t_{m-1}}{h}\left((u_{m})_{+}(x)-(u_{m})_{+}^{(\ell)}(x)\right)+\dfrac{t_{m}-t}{h}\left((u_{m-1})_{+}(x)-(u_{m-1})_{+}^{(\ell)}(x)\right)$

By the Minkowski inequality, $\mathbf{I}_{h,\ell}$ is estimated as

$\displaystyle\mathbf{I}_{h,\ell}$
$\displaystyle\leqq\left(h\sum_{m=1}^{N}\left\|(u_{m})_{+}-(u_{m})_{+}^{(\ell)}\right\|_{L^{\gamma}(K)}^{\gamma}\right)^{\frac{1}{\gamma}}+\left(h\sum_{m=1}^{N}\left\|(u_{m-1})_{+}-(u_{m-1})_{+}^{(\ell)}\right\|_{L^{\gamma}(K)}^{\gamma}\right)^{\frac{1}{\gamma}}$
$\displaystyle=:I_{1}+I_{2}.$

$I_{2}$ is estimated similarly as for $I_{1}$, so we will estimate $I_{1}$.
Since $(u_{m})_{+}-(u_{m})_{+}^{(\ell)}=0$ in
$K\cap\left\\{\frac{1}{\ell}<(u_{m})_{+}<\ell\right\\}\,\,(m=1,2,\ldots)$ from
(4.12) and (4.13) we observe that

$\displaystyle\left\|(u_{m})_{+}-(u_{m})_{+}^{(\ell)}\right\|_{L^{\gamma}(K)}$
$\displaystyle\leqq\left\|(u_{m})_{+}-(u_{m})_{+}^{(\ell)}\right\|_{L^{\gamma}(K\,\cap\,\\{(u_{m})_{+}\leqq\frac{1}{\ell}\\})}+\left\|(u_{m})_{+}-(u_{m})_{+}^{(\ell)}\right\|_{L^{\gamma}(K\,\cap\,\\{(u_{m})_{+}\geqq\ell\\})}$
$\displaystyle\leqq\left\|(u_{m})_{+}\right\|_{L^{\gamma}(K\,\cap\,\\{(u_{m})_{+}\leqq\frac{1}{\ell}\\})}+\Big{\|}\underbrace{(u_{m})_{+}^{(\ell)}}_{\equiv\,\frac{1}{\ell}}\Big{\|}_{L^{\gamma}(K\,\cap\,\\{(u_{m})_{+}\leqq\frac{1}{\ell}\\})}$
$\displaystyle\quad\quad\quad+\left\|(u_{m})_{+}\right\|_{L^{\gamma}(K\,\cap\,\\{(u_{m})_{+}\geqq\ell\\})}+\Big{\|}\underbrace{(u_{m})_{+}^{(\ell)}}_{\equiv\,\ell}\Big{\|}_{L^{\gamma}(K\,\cap\,\\{(u_{m})_{+}\geqq\ell\\})}$
$\displaystyle\leqq\dfrac{1}{\ell}\left|K\cap\left\\{(u_{m})_{+}\leqq\frac{1}{\ell}\right\\}\right|^{\frac{1}{\gamma}}+\dfrac{1}{\ell}\left|K\cap\left\\{(u_{m})_{+}\leqq\frac{1}{\ell}\right\\}\right|^{\frac{1}{\gamma}}$
$\displaystyle\quad\quad\quad+\left\|(u_{m})_{+}\right\|_{L^{\gamma}(K\,\cap\,\\{(u_{m})_{+}\geqq\ell\\})}+\ell\left|K\cap\left\\{(u_{m})_{+}\geqq\ell\right\\}\right|^{\frac{1}{\gamma}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\stackrel{{\scriptstyle\eqref{lemma
4.3 eq.3},~{}\eqref{lemma 4.3
eq.4}}}{{\leqq}}\dfrac{2}{\ell}\left|K\right|^{\frac{1}{\gamma}}+\frac{C}{\ell^{\frac{p_{s}^{\star}}{\gamma}-1}}[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{\frac{p_{s}^{\star}}{\gamma}}+\ell\left(\frac{C}{\ell^{p_{s}^{\star}}}[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p_{s}^{\star}}\right)^{\frac{1}{\gamma}}$
$\displaystyle=C\left(\frac{1}{\ell}\left|K\right|^{\frac{1}{\gamma}}+\frac{[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{\frac{p_{s}^{\star}}{\gamma}}}{\ell^{\frac{p_{s}^{\star}}{\gamma}-1}}\right).$

This leads to

$I_{1}\leqq
C(T+1)\left(\frac{1}{\ell^{\gamma}}\left|K\right|+\frac{[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p_{s}^{\star}}}{\ell^{p_{s}^{\star}-\gamma}}\right),$

which in turn gives

$\mathbf{I}_{h,\ell}\leqq
C(T+1)\left(\frac{1}{\ell^{\gamma}}\left|K\right|+\frac{[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p_{s}^{\star}}}{\ell^{p_{s}^{\star}-\gamma}}\right)=:\zeta(\ell).$
(4.15)

Clearly, $\mathbf{I}_{h^{\prime},\ell}$ is estimated similarly as
$\mathbf{I}_{h,\ell}$. Joining (4.15) and (4.2) with Lemma 4.2, we come up
with

$\displaystyle\limsup_{h,h^{\prime}\searrow
0}\left\|(u_{h})_{+}-(u_{h^{\prime}})_{+}\right\|_{L^{\gamma}(K_{T})}$
$\displaystyle\leqq\limsup_{h\searrow
0}\mathbf{I}_{h,\ell}+\limsup_{h,h^{\prime}\searrow
0}\mathbf{I}_{h,h^{\prime},\ell}+\limsup_{h^{\prime}\searrow
0}\mathbf{I}_{h^{\prime},\ell}$ $\displaystyle\leqq 2\zeta(\ell).$

Noticing by $p_{s}^{\star}>\gamma$ that
$\lim\limits_{\ell\longrightarrow\infty}\zeta(\ell)=0$ and sending
$\ell\longrightarrow\infty$ in the above display, we finally arrive at

$\lim\limits_{h,h^{\prime}\searrow
0}\left\|(u_{h})_{+}-(u_{h^{\prime}})_{+}\right\|_{L^{\gamma}(K_{T})}=0.$

Similarly as above we also have,

$\lim\limits_{h,h^{\prime}\searrow
0}\left\|(u_{h})_{-}-(u_{h^{\prime}})_{-}\right\|_{L^{\gamma}(K_{T})}=0.$

∎

###  4.3. Convergence of approximate solutions I

Here we shall present the convergence of the approximate solutions $u_{h}$ and
$\bar{u}_{h}$.

###### Lemma 4.5 (Convergence of approximate solutions I)

For any $\gamma\in[1,\max\\{p_{s}^{\star},q+1\\})$, there exists a limit
function $u\in L^{\gamma}(\Omega_{T})$ such that, for any a Lipschitz bounded
domain $K\subset\mathbb{R}^{n}$ and every positive number $T<\infty$,

$\displaystyle u_{h},\,\bar{u}_{h}\longrightarrow
u\quad\textrm{a.e.\,\,in}\,\,K_{T},$ (4.16) $\displaystyle
u_{h},\,\bar{u}_{h}\longrightarrow
u\quad\textrm{strongly\,\,in}\,\,L^{\gamma}(K_{T}).$ (4.17)

###### Proof.

Noticing that $u_{h}=(u_{h})_{+}-(u_{h})_{-}$, by Lemma 4.4, for all $\gamma$,
$1\leqq\gamma<\min\\{p_{s}^{\star},\frac{n+1}{n+1-s^{\prime}}\\}$, there
exists a limit function $u\in L^{\gamma}(K_{T})$ such that

$u_{h}\longrightarrow u\quad\mathrm{strongly\,\,in}\,\,L^{\gamma}(K_{T})$
(4.18)

as $h\searrow 0$. Thus, we have that a subsequence $\\{u_{h}\\}_{h>0}$ denoted
by the same notation, converges to $u$ almost everywhere in $K_{T}$, that is
the first statement of (4.16). This fact is combined with (3.20) in Lemma 3.4,
(3.27)2 in Lemma 3.6 to yield through Fatou’s lemma that, for
$\gamma=p_{s}^{\star}$ and $\gamma=q+1$,

$\sup_{0<t<T}\|u(t)\|_{L^{\gamma}(K)}\leqq
C\max\Big{\\{}\|u_{0}\|_{L^{q+1}(\Omega)},\,[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}\Big{\\}}.$
(4.19)

For the validity of second statement of (4.16), we employ Lemma 2.2 and the
truncated argument, as discussed in Section 4.2. For this, we shall handle
$\\{(\bar{u}_{h})_{+}\\}_{h>0}$ only, as the other case is similarly treated.
By the Minkowski inequality we have

$\displaystyle\left\|(\bar{u}_{h})_{+}-(u_{h})_{+}\right\|_{L^{1}(K_{T})}$
$\displaystyle\leqq\left\|(\bar{u}_{h})_{+}-(\bar{u}_{h})_{+}^{(\ell)}\right\|_{L^{1}(K_{T})}+\left\|(\bar{u}_{h})_{+}^{(\ell)}-(u_{h})_{+}^{(\ell)}\right\|_{L^{1}(K_{T})}+\left\|(u_{h})_{+}^{(\ell)}-(u_{h})_{+}\right\|_{L^{1}(K_{T})}$
$\displaystyle=:\mathbf{I}_{h,\ell}+\mathbf{II}_{h,\ell}+\mathbf{III}_{h,\ell}.$
(4.20)

Arguing similarly as (4.15), we infer that

$\mathbf{I}_{h,\ell},\,\,\mathbf{III}_{h,\ell}\leqq
C(T+1)\left(\frac{1}{\ell}\left|K\right|+\frac{[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p_{s}^{\star}}}{\ell^{p_{s}^{\star}-1}}\right)=:\zeta(\ell),$
(4.21)

where $\zeta(\ell)\longrightarrow 0$ as $\ell\longrightarrow\infty$. Similarly
as in (3.7) in Lemma 3.2, we have

$\displaystyle\left|(\bar{u}_{h})_{+}^{(\ell)}-(u_{h})_{+}^{(\ell)}\right|\leqq
h\left|\partial_{t}(u_{h})_{+}^{(\ell)}\right|$

and thus,

$\mathbf{II}_{h,\ell}=\left\|(\bar{u}_{h})_{+}^{(\ell)}-(u_{h})_{+}^{(\ell)}\right\|_{L^{1}(K_{T})}\leqq
h\left\|\partial_{t}(u_{h})_{+}^{(\ell)}\right\|_{L^{1}(K_{T})}.$

Noting that $q+1>1$, from (4.3) and (4.4) in Lemma 4.1 we find

$\lim_{h\searrow 0}\mathbf{II}_{h,\ell}=0.$ (4.22)

From (4.3), (4.21) and (4.22), we obtain

$\displaystyle\limsup_{h\searrow
0}\left\|(\bar{u}_{h})_{+}-(u_{h})_{+}\right\|_{L^{1}(K_{T})}$
$\displaystyle\leqq\limsup_{h\searrow 0}\mathbf{I}_{h,\ell}+\limsup_{h\searrow
0}\mathbf{II}_{h}+\limsup_{h\searrow 0}\mathbf{II}_{h}$ $\displaystyle\leqq
2\zeta(\ell)$

and, taking the limit as $\ell\longrightarrow\infty$ we have

$\lim_{h\searrow
0}\left\|(\bar{u}_{h})_{+}-(u_{h})_{+}\right\|_{L^{1}(K_{T})}=0.$ (4.23)

Furthermore, we make the inequality

$\|(\bar{u}_{h})_{+}-u_{+}\|_{L^{1}(K_{T})}\leqq\|(\bar{u}_{h})_{+}-(u_{h})_{+}\|_{L^{1}(K_{T})}+\|(u_{h})_{+}-u_{+}\|_{L^{1}(K_{T})},$

which converge to zero as $h\searrow 0$, by the convergences (4.18) and
(4.23). Thus, a subsequence $\\{(\bar{u}_{h})_{+}\\}_{h>0}$ converges to
$u_{+}$ almost everywhere in $K_{T}$ as $h\searrow 0$. Analogously, the
convergence $(\bar{u}_{h})_{-}\longrightarrow u_{-}$ a.e. in $K_{T}$ is
verified. Hence the second assertion of (4.16) is concluded.

For the proof (4.17), we verify that
$\left\\{(\bar{u}_{h})_{+}-u_{+}\right\\}_{h>0}$ is bounded in
$L^{\gamma}(K_{T})$ for $\gamma=p_{s}^{\star}$ and $\gamma=q+1$. In fact, by
(3.10)1 in Lemma 3.3, (3.27)1 in Lemma 3.6 and (4.19)

$\|(\bar{u}_{h})_{+}-u_{+}\|_{L^{\gamma}(K_{T})}\leqq
C\max\Big{\\{}\|u_{0}\|_{L^{q+1}(\Omega)},\,[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}\Big{\\}}.$

Therefore, from Lemma 2.2 it follows that, for any
$\gamma\in[1,\max\\{p_{s}^{\ast},q+1\\})$,

$\lim_{h\searrow 0}\|({\bar{u}}_{h})_{+}-u_{+}\|_{L^{\gamma}(K_{T})}=0.$

Similarly as above we also have

$\lim_{h\searrow
0}\left\|(\bar{u}_{h})_{-}-u_{-}\right\|_{L^{\gamma}(K_{T})}=0.$

Joining two displays above, we actually have the second statement of (4.17).
The first statement of (4.17) is shown in the same way as for the second one
above. The proof is concluded. ∎

###  4.4. Convergence of approximate solutions II

The following convergence results actually holds true:

###### Lemma 4.6 (Convergence of approximate solutions II)

Let $u_{h}$, $\bar{u}_{h}$, $\bar{v}_{h}$, $v_{h}$, $w_{h}$ and $\bar{w}_{h}$
be the approximate solutions of (1.1) defined by (3.3)–(3.4). For $s\in(0,1)$,
let $s^{\prime}<s$ be a positive number. Then there exist subsequences of
$\\{u_{h}\\}_{h>0}$, $\\{\bar{u}_{h}\\}_{h>0}$, $\\{v_{h}\\}_{h>0}$,
$\\{\bar{v}_{h}\\}_{h>0}$, $\\{w_{h}\\}_{h>0}$ and $\\{\bar{w}_{h}\\}_{h>0}$
(denoted by the same symbol unless otherwise stated) and a limit function
$u\in L^{\infty}(0,T\,;L^{q+1}(\Omega))$ such that, for any bounded domain
$K\subset\mathbb{R}^{n}$, for every positive number $T<\infty$, the following
convergences hold true:

$\displaystyle\bar{u}_{h}\longrightarrow u\quad$
$\displaystyle\ast\textrm{-weakly\,\,in}\,\,L^{\infty}(0,\infty\,;L^{q+1}(\Omega)),$
(4.24) $\displaystyle
w_{h},\bar{w}_{h}\longrightarrow|u|^{\frac{q-1}{2}}u\quad$
$\displaystyle\textrm{strongly\,\,in}\,\,L^{\beta}(K_{T}),\quad\forall\beta\in[1,2),$
(4.25) $\displaystyle v_{h},\,\bar{v}_{h}\longrightarrow|u|^{q-1}u\quad$
$\displaystyle\textrm{strongly\,\,in}\,\,L^{\alpha}(K_{T}),\quad\forall\alpha\in[1,\tfrac{q+1}{q})$
(4.26)

as $h\searrow 0$.

###### Proof of Lemma 4.6.

To show each of the convergence in the statement, we shall use the energy
estimates in Lemmata 3.3 and 3.5. Let us prove Lemma 4.6 step by step, as
displayed below.

Step 1: The proof of (4.24). By (3.10) in Lemma 3.3, we have

$\\{\bar{u}_{h}\\}_{h>0}:\quad\textrm{bounded\,\,in}\,\,L^{\infty}(0,\infty;L^{q+1}(\Omega)).$
(4.27)

Then there exist a subsequence $\\{\bar{u}_{h}\\}$ (written by the same
symbol) and the limit function $u\in L^{\infty}(0,\infty;L^{q+1}(\Omega))$
such that (4.24) holds true. Note that this limit function $u$ is equal to
that obtained in Lemma 4.5, by the convergence in Lemma 4.5 of $u_{h}$ and
$\bar{u}_{h}$.

Step 2: The proof of (4.25). We further divide this step into two steps:

Step 2.1. We shall prove the strong convergence $\bar{w}_{h}$ to
$|u|^{\frac{q-1}{2}}u$ in $L^{\beta}(K_{T})$ for $\beta\in[1,2)$. By the
energy estimate (3.10) in Lemma 3.3, we have

$\displaystyle\iint_{K_{T}}\big{|}|\bar{u}_{h}|^{\frac{q-1}{2}}\bar{u}_{h}-|u|^{\frac{q-1}{2}}u\big{|}^{2}\,dxdt\leqq
2\iint_{\Omega_{T}}\left(|u_{0}|^{q+1}+|u|^{q+1}\right)\,dx$

and thus, $\\{\bar{w}_{h}-|u|^{\frac{q-1}{2}}u\\}_{h>0}$ is bounded in
$L^{2}(K_{T})$. By (4.16) $\bar{w}_{h}$ converges to $|u|^{\frac{q-1}{2}}u$
almost everywhere in $K_{T}$. From Lemma 2.2 with $\psi\equiv 1$ it follows
that
$\displaystyle\iint_{K_{T}}\big{|}|\bar{u}_{h}|^{\frac{q-1}{2}}\bar{u}_{h}-|u|^{\frac{q-1}{2}}u\big{|}^{\beta}\,dxdt\longrightarrow
0$, that is,

$\bar{w}_{h}\longrightarrow|u|^{\frac{q-1}{2}}u\quad\mathrm{strongly\,\,in}\,\,L^{\beta}(K_{T}).$
(4.28)

This proves the first statement of (4.25).

Step 2.2. We now observe the difference of $\bar{w}_{h}$ and $w$. By (3.7) in
Lemma 3.2 we have, for $(x,t)\in\mathbb{R}^{n}\times[0,\infty)$,

$\displaystyle\left|w_{h}-\bar{w}_{h}\right|\leqq
h\left|\partial_{t}w_{h}\right|$

and thus, by the energy estimate (3.23) in Lemma 3.5,

$\displaystyle\left\|w_{h}-\bar{w}_{h}\right\|_{L^{2}(K_{T})}$
$\displaystyle\leqq h\left\|\partial_{t}w_{h}\right\|_{L^{2}(K_{T})}\leqq
Ch[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{\frac{p}{2}}$

which converges to zero as $h\searrow 0$. This together with (4.28) finishes
the proof of (4.25).

Step 3: The proof of (4.26). Similarly as in Step 3, we proceed the proof.

Step 3.1. Let $\alpha\in[1,\tfrac{q+1}{q})$ be any number. Using the energy
estimate (3.10) in Lemma 3.3, we have

$\displaystyle\iint_{K_{T}}\left||\bar{u}_{h}|^{q-1}\bar{u}_{h}-|u|^{q-1}u\right|^{\frac{q+1}{q}}\,dxdt\leqq
2^{\frac{1}{q}}\iint_{\Omega_{T}}\left(|u_{0}|^{q+1}+|u|^{q+1}\right)\,dx$
(4.29)

and thus, $\\{\bar{v}_{h}-|u|^{q-1}u\\}_{h>0}$ is bounded in
$L^{\frac{q+1}{q}}(K_{T})$. By (4.17) $\bar{v}_{h}$ converges to $|u|^{q-1}u$
almost everywhere in $K_{T}$. From Lemma 2.2 with $\psi\equiv 1$ we obtain
that, for $\alpha\in[1,\frac{q}{q+1})$,

${\bar{v}}_{h}\longrightarrow|u|^{q-1}u\quad\textrm{strongly\,\,in}\,\,\,L^{\alpha}(K_{T}).$

Step 3.2. Finally, we shall verify the convergence of $v_{h}$ to $|u|^{q-1}u$
in $L^{\alpha}(K_{T})$ for $\alpha\in[1,\frac{q+1}{q})$. To conclude the proof
we now distinguish two cases. Whenever $q\geqq 1$, there holds that

$\left||\bar{u}_{h}|^{q-1}\bar{u}_{h}-v_{h}\right|=\left|\bar{v}_{h}-v_{h}\right|\leqq
h\left|\partial_{t}v_{h}\right|$

and thus, by the energy estimate (3.24) in Lemma 3.5

$\displaystyle\left\|\bar{v}_{h}-v_{h}\right\|_{L^{1}(K_{T})}$
$\displaystyle\leqq h\left\|\partial_{t}v_{h}\right\|_{L^{1}(K_{T})}$
$\displaystyle\leqq
Ch\left|K_{T}\right|^{\frac{1}{q+1}}\|u_{0}\|_{L^{q+1}(\Omega)}^{\frac{q-1}{2}}[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{\frac{p}{2}}.$
(4.30)

When $0<q<1$, by (3.8) in Lemma 3.2 and (2.8) in Lemma 2.1 we estimate for any
$(x,t)\in\mathbb{R}^{n}\times[0,\infty)$,

$\displaystyle|v_{h}-{\bar{v}}_{h}|$ $\displaystyle\leqq
C\left(|\bar{u}_{h}(t)|+|\bar{u}_{h}(t-h)|\right)^{q-1}\left|\bar{u}_{h}(t)-\bar{u}_{h}(t-h)\right|$
$\displaystyle\leqq C\left|\bar{u}_{h}(t)-\bar{u}_{h}(t-h)\right|^{q}$
$\displaystyle=Ch^{q}\left(|\bar{u}_{h}(t)|+|\bar{u}_{h}(t-h)|\right)^{\frac{q(1-q)}{2}}\left(|\bar{u}_{h}(t)|+|\bar{u}_{h}(t-h)|\right)^{\frac{q(q-1)}{2}}|\partial_{t}u_{h}|^{q}$

and thus, we subsequently use Hölder’s inequality twice with pair of exponents
$\left(\frac{2}{q},\frac{2}{2-q}\right)$ and
$\left(\frac{(2-q)(q+1)}{q(1-q)},\,\frac{(2-q)(q+1)}{2}\right)$, the energy
inequalities (3.10) and (3.12) in Lemma 3.3 to have

$\displaystyle\iint_{K_{T}}|v_{h}-\bar{v}_{h}|\,dxdt$ $\displaystyle\leqq
h^{q}\left(\,\iint_{K_{T}}\left(|\bar{u}_{h}(t)|+|\bar{u}_{h}(t-h)|\right)^{q-1}|\partial_{t}u_{h}|^{2}\,dxdt\right)^{\frac{q}{2}}$
$\displaystyle\quad\quad\quad\quad\quad\times\left(\,\iint_{K_{T}}\left(|\bar{u}_{h}(t)|+|\bar{u}_{h}(t-h)|\right)^{\frac{q(1-q)}{2-q}}\,dxdt\right)^{\frac{2-q}{2}}$
$\displaystyle\leqq
h^{q}\left(\,\iint_{K_{T}}\left(|\bar{u}_{h}(t)|+|\bar{u}_{h}(t-h)|\right)^{q-1}|\partial_{t}u_{h}|^{2}\,dxdt\right)^{\frac{q}{2}}$
$\displaystyle\quad\quad\quad\quad\quad\times\left(\,\iint_{K_{T}}\left(|\bar{u}_{h}(t)|+|\bar{u}_{h}(t-h)|\right)^{q+1}\,dxdt\right)^{\frac{q(1-q)}{2(q+1)}}\left|K_{T}\right|^{\frac{1}{q+1}}$
$\displaystyle\leqq
Ch^{q}\left|K_{T}\right|^{\frac{1}{q+1}}\|u_{0}\|_{L^{q+1}(\Omega)}^{\frac{q(1-q)}{2}}[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{\frac{pq}{2}}.$
(4.31)

Passing to the limit $h\searrow 0$ in (4.4) and (4.4) from the convergence in
Step 3.2, we deduce the convergence in $L^{1}(K_{T})$ of $v_{h}$ and thus,
convergence almost everywhere in $K_{T}$ of a subsequence of $v_{h}$ to
$|u|^{q-1}u$.

Similarly as (4.29) above we also see that
$\left\\{v_{h}-|u|^{q-1}u\right\\}_{h>0}$ is bounded in
$L^{\frac{q+1}{q}}(K_{T})$. Thus, by Lemma 2.2, this together with the almost
convergence show the validity of the convergence of $v_{h}$ to $|u|^{q-1}u$ in
$L^{\alpha}(K_{T})$ for $\alpha\in[1,\frac{q+1}{q})$.

Finally, the proof of Lemma 4.6 is complete. ∎

From Lemma 2.2 we easily deduce the following convergence, used later in
Section 5.

###### Lemma 4.7

For every test function $\varphi\in\mathscr{T}$, there holds that

$-\iint_{\Omega_{T}}w_{h}\,\partial_{t}\varphi\,dxdt\longrightarrow-\iint_{\Omega_{T}}|u|^{\frac{q-1}{2}}u\,\partial_{t}\varphi\,dxdt$
(4.32)

and

$-\iint_{\Omega_{T}}v_{h}\,\partial_{t}\varphi\,dxdt\longrightarrow-\iint_{\Omega_{T}}|u|^{q-1}u\,\partial_{t}\varphi\,dxdt$
(4.33)

as $h\searrow 0$.

###### Proof.

We shall show (4.33) only, as the other case (4.32) is similarly treated. By
(3.20) in Lemma 3.4

$\displaystyle\iint_{\Omega_{T}}|v_{h}|^{\frac{q+1}{q}}\,dxdt\leqq
C(T+1)\|u_{0}\|_{L^{q+1}}^{q+1}$

and thus, $\\{v_{h}\\}_{h>0}$ is bounded in $L^{\frac{q+1}{q}}(\Omega_{T})$.
From (4.26) in Lemma 4.6

$v_{h}\,\partial_{t}\varphi\longrightarrow|u|^{q-1}u\,\partial_{t}\varphi\quad\textrm{a.e.\,\,in}\,\,\Omega_{T}.$

Noticing that $\partial_{t}\varphi\in L^{q+1}(\Omega_{T})$ since
$\varphi\in\mathscr{T}$ and choosing $\psi=\partial_{t}\varphi$ in Lemma 2.2
lead to

$\iint_{\Omega_{T}}v_{h}\,\partial_{t}\varphi\,dxdt\longrightarrow\iint_{\Omega_{T}}|u|^{q-1}u\,\partial_{t}\varphi\,dxdt$

as $h\searrow 0$, finishing the proof. ∎

##  5\. Proof of Theorem 1.1

Now we are in the position to prove the main Theorem 1.1 since we have all
prerequisites at hand.

###### Proof of Theorem 1.1.

We shall show that the limit function $u$ obtained in Lemma 4.6 is a weak
solution of (3.1). For this we will verify that $u$ obeys the conditions in
Definition 2.8. Firstly, we ensure the condition (D1) in Definition 2.8. By
the convergence (4.16) in Lemma 4.6, applied to (3.13) in Lemma 3.3 and
Fatou’s lemma we have that, for any positive $t<\infty$,

$[u(t)]_{W^{s,p}(\mathbb{R}^{n})}\leqq\liminf_{h\searrow
0}\,[\bar{u}_{h}(t)]_{W^{s,p}(\mathbb{R}^{n})}\leqq[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}.$
(5.1)

Furthermore, by (2.13) in Lemma 2.6 and (5.1) we have, for any positive
$t<\infty$,

$\|u(t)\|_{L^{p}(\Omega)}\leqq C[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}.$ (5.2)

Thus, it follows from (5.1) and (5.2) that the limit function $u$ is in
$L^{\infty}(0,\infty;W^{s,p}_{0}(\Omega))$. Similarly as (5.1), by (3.20) in
Lemma 3.4, we also have, for any positive $t<\infty$,

$\|u(t)\|_{L^{q+1}(\Omega)}\leqq\|u_{0}\|_{L^{q+1}(\Omega)}$ (5.3)

which implies $u\in L^{\infty}(0,\infty;L^{q+1}(\Omega))$ and thus, the proof
of (D1) is accomplished.

By (3.23) in Lemma 3.5, $\\{\partial_{t}w_{h}\\}_{h>0}$ is bounded in
$L^{2}(\Omega_{\infty})$ and thus, there are a subsequence
$\\{\partial_{t}w_{h}\\}_{h>0}$ (denoted by the same notation) and a limit
function $\omega\in L^{2}(\Omega_{\infty})$ such that, as $h\searrow 0$,

$\partial_{t}w_{h}\longrightarrow\omega\quad\textrm{weakly\,\,in}\,\,\,L^{2}(\Omega_{\infty}).$
(5.4)

By use of the convergence (4.32) in Lemma 4.7 we pass to the limit as
$h\searrow 0$ in the identity

$\iint_{\Omega_{\infty}}\partial_{t}w_{h}\cdot\varphi\,dxdt=-\iint_{\Omega_{\infty}}w_{h}\,\partial_{t}\varphi\,dxdt$
(5.5)

for every $\varphi\in C^{\infty}_{0}(\Omega_{\infty})$ to have

$\omega=\partial_{t}(|u|^{\frac{q-1}{2}}u)\quad\textrm{in}\,\,L^{2}(\Omega_{\infty}).$
(5.6)

As a result, it follows from (5.4) and (5.6) that

$\partial_{t}w_{h}\longrightarrow\partial_{t}(|u|^{\frac{q-1}{2}}u)\quad\textrm{weakly\,\,in}\,\,\,L^{2}(\Omega_{\infty})$
(5.7)

and thus, from the convergence (5.7) and (3.23) in Lemma 3.5, for any
$T<\infty$,

$\left\|\partial_{t}(|u|^{\frac{q-1}{2}}u)\right\|_{L^{2}(\Omega_{T})}^{2}\leqq
C[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p}.$ (5.8)

Therefore the regularity part of Theorem 1.1 is concluded.

We now turn to prove the condition (D2) in Definition 2.8. Let
$\varphi\in\mathscr{T}$, whose the definition is given in Definition 2.8.
According to the notation in (2.15), we first set

$\displaystyle\overline{U}_{h}(x,y,t)$
$\displaystyle:=|\bar{u}_{h}(x,t)-\bar{u}_{h}(y,t)|^{p-2}(\bar{u}_{h}(x,t)-\bar{u}_{h}(y,t)),$
$\displaystyle U(x,y,t)$
$\displaystyle:=|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t)).$

For the estimation below let $R$ be any positive number such that the open
ball $B_{R}$ with center of origin and radius $R$ compactly contains $\Omega$.
In order to verify (D2) in Definition 2.8, we shall show the convergence, as
$h\searrow 0$,

$\int_{0}^{T}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{\overline{U}_{h}(x,y,t)\left(\varphi(x,t)-\varphi(y,t)\right)}{|x-y|^{n+sp}}\,dxdydt\longrightarrow\int_{0}^{T}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{U(x,y,t)\left(\varphi(x,t)-\varphi(y,t)\right)}{|x-y|^{n+sp}}\,dxdydt.$
(5.9)

We now set $B\equiv B_{R}$, $B^{c}:=\mathbb{R}^{n}\setminus B$ and
$D_{B}:=(\mathbb{R}^{n}\times\mathbb{R}^{n})\setminus(B^{c}\times B^{c})$. For
the validity of (5.9), we shall consider the difference of two fractional
integrations on $D_{B}$, subsequently, we divide it into three terms as
follows:

$\displaystyle I_{h}$
$\displaystyle:=\left|\int_{0}^{T}\iint_{D_{B}}\Bigg{[}\dfrac{\overline{U}_{h}(x,y,t)\left(\varphi(x,t)-\varphi(y,t)\right)}{|x-y|^{n+sp}}-\dfrac{U(x,y,t)\left(\varphi(x,t)-\varphi(y,t)\right)}{|x-y|^{n+sp}}\Bigg{]}\,dxdydt\right|$
$\displaystyle\leqq\int_{0}^{T}\iint_{B\times
B}\ldots\,dxdydt+\int_{0}^{T}\iint_{B\times
B^{c}}\ldots\,dxdydt+\int_{0}^{T}\iint_{B^{c}\times B}\ldots\,dxdydt$
$\displaystyle=:I_{h}^{(1)}+I_{h}^{(2)}+I_{h}^{(3)}.$ (5.10)

We now proceed estimating the terms $I_{h}^{(1)},I_{h}^{(2)}$ and
$I_{h}^{(3)}$. For the estimation of $I_{h}^{(1)}$, we shall verify that
$\left\\{\dfrac{\overline{U}_{h}(x,y,t)\left(\varphi(x,t)-\varphi(y,t)\right)}{|x-y|^{n+sp}}\right\\}_{h>0}$
is uniformly integrable on $B\times B\times(0,T)$. Let $\Lambda\subset B\times
B\times(0,T)$ be any measurable set. By Hölder’s inequality and energy
estimate (3.11) in Lemma 3.3, we have

$\displaystyle\iiint_{\Lambda}\left|\dfrac{\overline{U}_{h}(x,y,t)\left(\varphi(x,t)-\varphi(y,t)\right)}{|x-y|^{n+sp}}\,\right|\,dxdydt$
$\displaystyle\leqq\left(\iiint_{\Lambda}\dfrac{|\varphi(x,t)-\varphi(y,t)|^{p}}{|x-y|^{n+sp}}\,dxdydt\right)^{\frac{1}{p}}\left(\int_{0}^{T}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{|\bar{u}_{h}(x,t)-\bar{u}_{h}(y,t)|^{p}}{|x-y|^{n+sp}}\,dxdydt\right)^{\frac{p-1}{p}}$
$\displaystyle\\!\\!\stackrel{{\scriptstyle\eqref{energy est.
NE2}}}{{\leqq}}C\left(\iiint_{\Lambda}\dfrac{|\varphi(x,t)-\varphi(y,t)|^{p}}{|x-y|^{n+sp}}\,dxdydt\right)^{\frac{1}{p}}\left(\int_{\Omega}|u_{0}|^{q+1}\,dx\right)^{\frac{p-1}{p}}.$

Due to the absolute continuity of the first Lebesgue integral on the right-
hand side, for every positive number $\varepsilon$, there exists a positive
number $\delta$ such that

$\iiint_{\Lambda}\left|\dfrac{\overline{U}_{h}(x,y,t)\left(\varphi(x,t)-\varphi(y,t)\right)}{|x-y|^{n+sp}}\right|\,dxdydt<\varepsilon\quad\textrm{whenever}\quad|\Lambda|<\delta$

and thus,
$\left\\{\dfrac{\overline{U}_{h}(x,y,t)\left(\varphi(x,t)-\varphi(y,t)\right)}{|x-y|^{n+sp}}\right\\}_{h>0}$
is uniformly integrable on $B\times B\times(0,T)$. From the almost everywhere
convergence (4.16) in Lemma 4.6, it follows that

$\displaystyle\dfrac{\overline{U}_{h}(x,y,t)\left(\varphi(x,t)-\varphi(y,t)\right)}{|x-y|^{n+sp}}\longrightarrow\dfrac{U(x,y,t)\left(\varphi(x,t)-\varphi(y,t)\right)}{|x-y|^{n+sp}}$

almost everywhere $(x,y,t)\in B\times B\times(0,T)$, where we notice that the
set

$\left\\{(x,y,t)\in B\times B\times(0,T):x=y\right\\}$

is negligible with respect to the $(2n+1)$-dimensional Lebesgue measure on
$\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}$. Therefore, Vitalli’s
convergence theorem yields that

$\displaystyle\int_{0}^{T}\iint_{B\times
B}\dfrac{\overline{U}_{h}(x,y,t)\left(\varphi(x,t)-\varphi(y,t)\right)}{|x-y|^{n+sp}}\,dxdydt\longrightarrow\int_{0}^{T}\iint_{B\times
B}\dfrac{U(x,y,t)\left(\varphi(x,t)-\varphi(y,t)\right)}{|x-y|^{n+sp}}\,dxdydt,$

as $h\searrow 0$, that is,

$\lim_{h\searrow 0}I_{h}^{(1)}=0.$ (5.11)

We are going to estimate $I_{h}^{(2)}$ and $I_{h}^{(3)}$. Since these two
integrals are symmetric with respect to $x$ and $y$, $I_{h}^{(3)}$ can be
estimated similarly to $I_{h}^{(2)}$ and thus, we give an estimate of
$I_{h}^{(2)}$. The boundary condition (3.1)2 implies that $\bar{u}_{h}=u=0$ in
$\Omega^{c}\times(0,T)$ and, by $B^{c}\subset\Omega^{c}$,
$\bar{u}_{h}(y,t)=u(y,t)=\varphi(y,t)=0$ for almost all $(y,t)\in
B^{c}\times(0,T)$ because $\Omega$ is compactly contained in $B$ and
$u_{h}=u=0$ almost everywhere in $\Omega^{c}\times(0,T)$. Using Hölder’s
inequality, $I^{(2)}_{h}$ is estimated as

$\displaystyle I_{h}^{(2)}$ $\displaystyle\leqq\int_{0}^{T}\iint_{\Omega\times
B^{c}}\dfrac{\Big{|}|\bar{u}_{h}(x,t)|^{p-2}\bar{u}_{h}(x,t)-|u(x,t)|^{p-2}u(x,t)\Big{|}}{|x-y|^{n+sp}}|\varphi(x,t)|\,dxdydt$
$\displaystyle\leqq\int_{0}^{T}\int_{\Omega}\left(\,\,\int_{B^{c}}\dfrac{dy}{|x-y|^{n+sp}}\right)\Big{|}|\bar{u}_{h}(x,t)|^{p-2}\bar{u}_{h}(x,t)-|u(x,t)|^{p-2}u(x,t)\Big{|}|\varphi(x,t)|\,dxdt$
$\displaystyle\leqq
C(n,s,p)d^{-sp}\iint_{\Omega_{T}}\Big{|}|\bar{u}_{h}(x,t)|^{p-2}\bar{u}_{h}(x,t)-|u(x,t)|^{p-2}u(x,t)\Big{|}|\varphi(x,t)|\,dxdt,$
(5.12)

where, in the second line, we note that $|x-y|\geqq
d:=\mathrm{dist}(\Omega,\partial B)$ for any $(x,y)\in\Omega\times B^{c}$, and
estimate for $(x,y)\in\Omega\times B^{c}$

$\displaystyle\int_{B^{c}}\dfrac{dy}{|x-y|^{n+sp}}$
$\displaystyle\leqq\int_{B_{d}(x)^{c}}\dfrac{dy}{|x-y|^{n+sp}}$
$\displaystyle=C(n)\int_{d}^{\infty}\dfrac{d\varrho}{\varrho^{1+sp}}=C(n)\dfrac{d^{-sp}}{sp}.$

To further estimate the integral of the right hand side of (5) we shall verify
that
$\left\\{\Big{|}|\bar{u}_{h}|^{p-2}\bar{u}_{h}-|u|^{p-2}u\Big{|}\right\\}_{h>0}$
is bounded in $L^{\frac{p}{p-1}}(\Omega_{T})$. By (3.26) in Lemma 3.6

$\iint_{\Omega_{T}}\left|\bar{u}_{h}(x,t)\right|^{p}\,dxdt\leqq
C\int_{0}^{T}[\bar{u}_{h}(t)]_{W^{s,p}(\mathbb{R}^{n})}^{p}\,dt\leqq
C\|u_{0}\|_{L^{q+1}(\Omega)}^{q+1}.$ (5.13)

By the almost convergence (4.16) in Lemma 4.6, (5.13) and Fatou’s lemma, we
have

$\iint_{\Omega_{T}}\left|u(x,t)\right|^{p}\,dxdt\leqq\liminf_{h\searrow
0}\iint_{\Omega_{T}}\left|\bar{u}_{h}(x,t)\right|^{p}\,dxdt\leqq
C\|u_{0}\|_{L^{q+1}(\Omega)}^{q+1}.$

This together with (5.13) yields

$\displaystyle\iint_{K_{T}}\Big{|}|\bar{u}_{h}(x,t)|^{p-2}\bar{u}_{h}(x,t)-|u(x,t)|^{p-2}u(x,t)\Big{|}^{\frac{p}{p-1}}\,dxdt$
$\displaystyle\leqq
2^{\frac{1}{p-1}}\iint_{\Omega_{T}}\left(|\bar{u}_{h}(x,t)|^{p}+|u(x,t)|^{p}\right)\,dxdt$
$\displaystyle\leqq C\|u_{0}\|_{L^{q+1}(\Omega)}^{q+1}$

and thus,
$\left\\{\Big{|}|\bar{u}_{h}|^{p-2}\bar{u}_{h}-|u|^{p-2}u\Big{|}\right\\}_{h>0}$
is bounded in $L^{\frac{p}{p-1}}(\Omega_{T})$. This, (5) and (4.16) in Lemma
4.6 through Lemma 2.2 yields

$\lim_{h\searrow 0}I_{h}^{(2)}=0.$ (5.14)

The preceding estimates (5.11) and (5.14) can be merged to give the following
final estimate for $I_{h}$:

$\displaystyle\limsup_{h\searrow 0}I_{h}\leqq\limsup_{h\searrow
0}I_{h}^{(1)}+2\limsup_{h\searrow 0}I^{(2)}_{h}$ $\displaystyle=0,$

Therefore, by (5.9) and (4.33) in Lemma 4.7, we pass to the limit as
$h\searrow 0$ in (3.5) to have

$-\iint_{\Omega_{T}}|u|^{q-1}u\,\partial_{t}\varphi\,dxdt+\dfrac{1}{2}\int_{0}^{T}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{U(x,y,t)}{|x-y|^{n+sp}}\left(\varphi(x,t)-\varphi(y,t)\right)\,dxdydt=0$

for all $\varphi\in\mathscr{T}$, which conclude the condition (D2). Here we
remark that the class $\mathscr{T}$ of test functions in Definition 2.8 is a
subspace of that in (3.5).

Finally, we shall verify the condition (D3). As noted before, we have that
$\bar{u}_{h}(t)=0$ almost everywhere in $\mathbb{R}^{n}\setminus\Omega$ for
any $t\geqq 0$. Thus, by the convergence (4.16) in Lemma 4.6, $u=0$ almost
everywhere in $(\mathbb{R}^{n}\setminus\Omega)\times(0,T)$ for any $T>0$,
showing the validity of boundary condition in (D3).

We test the approximating equation (3.5) on
$\mathbb{R}^{n}\times(t_{1},t_{2})$ for any $t_{2}>t_{1}\geqq 0$, with
$\varphi(x)\sigma_{(t_{1},t_{2})}(t)$, where $\varphi\in
C^{\infty}_{0}(\Omega)$ and $\sigma_{(t_{1},t_{2})}(t)$ is the usual Lipschitz
approximation of characteristic function of time-interval $(t_{1},t_{2})$. By
use of the Hölder inequality and (3.13) in Lemma 3.3, we have that, for any
$t_{2}>t_{1}\geqq 0$,

$\displaystyle\left|\,\,\int_{\Omega}\big{(}v_{h}(t_{2})-v_{h}(t_{1})\big{)}\varphi\,dx\right|$
$\displaystyle=\left|-\frac{1}{2}\int_{t_{1}}^{t_{2}}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{\overline{U}_{h}(x,y,t)}{|x-y|^{n+sp}}\big{(}\varphi(x)-\varphi(y)\big{)}\,dxdydt\right|$
$\displaystyle\leqq\frac{1}{2}\int_{t_{1}}^{t_{2}}[\bar{u}_{h}(t)]_{W^{s,p}(\mathbb{R}^{n})}^{p-1}\,dt\cdot[\varphi]_{W^{s,p}(\mathbb{R}^{n})}$
$\displaystyle\\!\\!\stackrel{{\scriptstyle\eqref{energy est.
NE4}}}{{\leqq}}\frac{1}{2}[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p-1}[\varphi]_{W^{s,p}(\mathbb{R}^{n})}(t_{2}-t_{1}),$
(5.15)

implying by the density of $C^{\infty}_{0}(\Omega)$ in $L^{q+1}(\Omega)$ that
$v_{h}(t)$ is weakly continuous on $t\in[0,\infty)$ in
$L^{\frac{q+1}{q}}(\Omega)$, uniformly on approximation parameter $h$. By
(3.8) in Lemma 3.2 and (3.10) in Lemma 3.3, $\\{v_{h}(t)\\}_{h>0}$ is bounded
in $L^{\frac{q+1}{q}}(\Omega)$. Thus, the Arzelà-Ascoli theorem gives that, as
$h\searrow 0$,

$v_{h}\longrightarrow|u|^{q-1}u$

uniformly locally on $[0,\infty)$ and weakly in $L^{\frac{q+1}{q}}(\Omega)$.
This together with (5.9) applied to the first line of (5) implies that, for
$t_{2}>t_{1}=0$,

$\left|\,\,\int_{\Omega}\big{(}|u|^{q-1}u(t_{2})-|u_{0}|^{q-1}u_{0}\big{)}\varphi\,dx\right|=\left|-\frac{1}{2}\int_{0}^{t_{2}}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{U(x,y,t)}{|x-y|^{n+sp}}\big{(}\varphi(x)-\varphi(y)\big{)}\,dxdydt\right|$

yielding that $|u(t)|^{q-1}u(t)$ converges to $|u_{0}|^{q-1}u_{0}$ weakly in
$L^{\frac{q+1}{q}}(\Omega)$, as $t\searrow 0$. Similarly as above, we use the
algebraic inequality (2.8) in Lemma 2.1 to get

$u(t)\longrightarrow u_{0}\quad\textrm{weakly\,\,in}\quad L^{q+1}(\Omega)$
(5.16)

as $t\searrow 0$. Furthermore, the inequality (5.3) gives

$\limsup_{t\searrow
0}\|u(t)\|_{L^{q+1}(\Omega)}\leqq\|u_{0}\|_{L^{q+1}(\Omega)}.$ (5.17)

From (5.16) and (5.17) it follows that

$u(t)\longrightarrow
u_{0}\quad\textrm{strongly\,\,in}\,\,L^{q+1}(\Omega)\,\,\,\textrm{as}\,\,t\searrow
0.$ (5.18)

We now have prerequisites at hand to verify the initial condition (D3). In
view of the convergence (5.18), we have only to show
$[u(t)-u_{0}]_{W^{s,p}(\mathbb{R}^{n})}\longrightarrow 0$ as $t\searrow 0$. We
first observe from (5.18) and Fatou’s lemma that

$[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}\leqq\liminf_{t\searrow
0}[u(t)]_{W^{s,p}(\mathbb{R}^{n})},$

which, together with (5.1) yields

$\lim_{t\searrow
0}[u(t)]_{W^{s,p}(\mathbb{R}^{n})}=[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}.$ (5.19)

From the convergence (5.18) and the boundedness (5.1) we can apply Vitali’s
convergence theorem to have

$\dfrac{u(x,t)-u(y,t)}{|x-y|^{s+\frac{n}{p}}}\longrightarrow\dfrac{u_{0}(x)-u_{0}(y)}{|x-y|^{s+\frac{n}{p}}}\quad\textrm{strongly\,\,in}\,\,L^{p}(B_{R}\times
B_{R})$

for any ball $B_{R}$. We shall consider the integral on
$(x,y)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$ with a test function of the
difference of two terms above, which corresponds to (5) for (5.9). Arguing
similarly as the proof of (5) we obtain that

$\dfrac{u(x,t)-u(y,t)}{|x-y|^{s+\frac{n}{p}}}\longrightarrow\dfrac{u_{0}(x)-u_{0}(y)}{|x-y|^{s+\frac{n}{p}}}\quad\textrm{weakly\,\,in}\,\,L^{p}(\mathbb{R}^{n}\times\mathbb{R}^{n}).$
(5.20)

Gluing (5.19) with (5.20) and adopting the Radon-Riesz theorem, we are led to
the conclusion $[u(t)-u_{0}]_{W^{s,p}(\mathbb{R}^{n})}\longrightarrow 0$ as
$t\searrow 0$ and thus, the initial condition (D3) is actually verified.

Therefore the proof is complete. ∎

##  Appendix A Proof of Lemma 3.1

In this appendix, we prove the existence of solutions of (3.1), Lemma 3.1. The
proof is based on the direct method in the calculus of variations. In the
course of this appendix, let $s\in(0,1)$, $p>1$ and $q>0$ be given and set
$\mathcal{X}:=W_{0}^{s,p}(\Omega)\cap L^{q+1}(\Omega)$.

###### Proof of Lemma 3.1.

Given $u_{m-1}\in\mathcal{X}$, we define the functional on $\mathcal{X}$:

$\mathcal{X}\ni
w\longmapsto\mathcal{F}(w):=\dfrac{1}{h}\int_{\Omega}\left(\dfrac{1}{q+1}|w|^{q+1}-|u_{m-1}|^{q-1}u_{m-1}w\right)\,dx+\dfrac{1}{2p}[w]_{W^{s,p}(\mathbb{R}^{n})}^{p}.$

The proof is twofold: first, we shall show the existence of a minimizer of the
functional $\mathcal{F}(w)$, then we shall demonstrate the first variation of
$\mathcal{F}(w)$, both of them show the validity of Lemma 3.1.

By Young’s inequality, there holds for any $w\in\mathcal{X}$

$\displaystyle\mathcal{F}(w)$
$\displaystyle=\dfrac{1}{h}\int_{\Omega}\left(\dfrac{1}{q+1}|w|^{q+1}-|u_{m-1}|^{q-1}u_{m-1}w\right)\,dx+\dfrac{1}{2p}[w]_{W^{s,p}(\mathbb{R}^{n})}^{p}$
$\displaystyle\geqq\dfrac{1}{h}\int_{\Omega}\left(\dfrac{1}{2(q+1)}|w|^{q+1}-C(q)|u_{m-1}|^{q+1}\right)\,dx+\dfrac{1}{2p}[w]_{W^{s,p}(\mathbb{R}^{n})}^{p}$
(A.1) $\displaystyle>-\infty.$

Let $\\{u_{k}\\}_{k=1}^{\infty}$ be a minimizing sequence, such that

$\inf\mathcal{F}\leqq\mathcal{F}(u_{k})<\inf\mathcal{F}+\dfrac{1}{k}$

for all $k\in\mathbb{N}$, where the infimum is taken for
$w\in\mathcal{X}:=W^{s,p}_{0}(\Omega)\cap L^{q+1}(\Omega)$. Note that, for all
$w\in\mathcal{X}$,

$-C\|u_{m-1}\|_{L^{q+1}(\Omega)}^{q+1}\leqq\inf\mathcal{F}\leqq\mathcal{F}(v)<\infty$

with some $v\in\mathcal{X}$ not empty, because
$C^{\infty}_{0}(\Omega)\subset\mathcal{X}$. We then observe that
$\sup\limits_{k\in\mathbb{N}}\mathcal{F}(u_{k})\leqq\inf\mathcal{F}+1=:M<\infty$
and, by (A),

$\dfrac{1}{h}\int_{\Omega}\dfrac{1}{2(q+1)}|u_{k}|^{q+1}\,dx+\dfrac{1}{2p}[u_{k}]_{W^{s,p}(\mathbb{R}^{n})}^{p}\leqq
M+C\int_{\Omega}|u_{m-1}|^{q+1}\,dx,$ (A.2)

implying

$\sup\limits_{k\in\mathbb{N}}\|u_{k}\|_{L^{q+1}(\Omega)}<\infty\quad;\quad\sup\limits_{k\in\mathbb{N}}\,[u_{k}]_{W^{s,p}(\mathbb{R}^{n})}<\infty.$
(A.3)

Up to not relabelled subsequences, there exists a weak limit $u\in
L^{q+1}(\Omega)$ such that

$u_{k}\longrightarrow u\quad\textrm{weakly\,\,in}\quad L^{q+1}(\Omega)$ (A.4)

and thus, by the Banach-Steinhaus theorem and (A.3),

$\|u\|_{L^{q+1}(\Omega)}\leqq\liminf_{k\longrightarrow\infty}\|u_{k}\|_{L^{q+1}(\Omega)}<\infty.$
(A.5)

Since $u_{k}\in W^{s,p}_{0}(\Omega)$, by the fractional Poincaré inequality
(2.13) in Lemma 2.6 we have,
$\sup\limits_{k\in\mathbb{N}}\|u_{k}\|_{L^{p}(\mathbb{R}^{n})}<\infty$, in
turn yielding with (A.3)

$\sup\limits_{k\in\mathbb{N}}\|u_{k}\|_{W^{s,p}(\mathbb{R}^{n})}<\infty.$
(A.6)

Letting $s^{\prime}\equiv\min\\{\frac{n}{2p},s\\}$ impiles $0<s^{\prime}\leqq
s<1$. By Lemma 2.7,

$\sup_{k\in\mathbb{N}}\|u_{k}\|_{W^{s^{\prime},p}(\mathbb{R}^{n})}<\infty.$

and, for any ball $B_{R}$ satisfying $\Omega\subset B_{R}$

$\sup_{k\in\mathbb{N}}\|u_{k}\|_{W^{s^{\prime},p}(B_{R})}<\infty.$

Since $s^{\prime}p\leqq\frac{n}{2}<n$, we can apply Lemma 2.4 to get (after
passing to a subsequence, if necessary)

$u_{k}\longrightarrow u\quad\textrm{strongly\,\,in}\quad L^{r}(B_{R})$

for all $r$ with $1\leqq r<\frac{np}{n-s^{\prime}p}$ and

$u_{k}\longrightarrow u\quad\textrm{a.e.\,\, in}\quad\Omega,$ (A.7)

which, in particular, yields $u=0$ in $\Omega^{c}$. From Fatou’s Lemma with
(A.7) and (A.6), it follows that

$\|u\|^{p}_{W^{s,p}(\mathbb{R}^{n})}\leqq\liminf_{k\longrightarrow\infty}\|u_{k}\|^{p}_{W^{s,p}(\mathbb{R}^{n})}<\infty.$
(A.8)

Consequently, by (A.5) and (A.8), it verifies that $u\in
W^{s,p}_{0}(\Omega)\cap L^{q+1}(\Omega)=\mathcal{X}$.

Finally, in view of (A.5) and (A.8) again, we have

$\displaystyle\inf_{w\in\mathcal{X}}\mathcal{F}(w)$
$\displaystyle=\liminf_{k\longrightarrow\infty}\mathcal{F}(u_{k})$
$\displaystyle=\liminf_{k\longrightarrow\infty}\Bigg{[}\dfrac{1}{h}\int_{\Omega}\left(\dfrac{1}{q+1}|u_{k}|^{q+1}-|u_{m-1}|^{q-1}u_{m-1}u_{k}\right)\,dx+\dfrac{1}{2p}[u_{k}]^{p}_{W^{s,p}(\mathbb{R}^{n})}\Bigg{]}$
$\displaystyle\geqq\dfrac{1}{h}\int_{\Omega}\left(\dfrac{1}{q+1}|u|^{q+1}-|u_{m-1}|^{q-1}u_{m-1}u\right)\,dx+\dfrac{1}{2p}[u]^{p}_{W^{s,p}(\mathbb{R}^{n})}$
$\displaystyle=\mathcal{F}(u),$

that implies that $u\in\mathcal{X}$ is actually a minimizer of $\mathcal{F}$.

We now turn to compute the first variation of $\mathcal{F}$. Let
$u_{m}\in\mathcal{F}$ be a minimizer. For $\varepsilon\in\mathbb{R}$ and
$\varphi\in\mathcal{X}$, we compute as

$\displaystyle\dfrac{\mathcal{F}(u_{m}+\varepsilon\varphi)-\mathcal{F}(u_{m})}{\varepsilon}$
$\displaystyle=\dfrac{1}{h}\dfrac{1}{q+1}\int_{\Omega}\dfrac{|u_{m}+\varepsilon\varphi|^{q+1}-|u_{m}|^{q+1}}{\varepsilon}\,dx$
$\displaystyle-\dfrac{1}{h}\int_{\Omega}\dfrac{|u_{m-1}|^{q-1}u_{m-1}(u_{m}+\varepsilon\varphi)-|u_{m-1}|^{q-1}u_{m-1}u_{m}}{\varepsilon}\,dx$
$\displaystyle+\dfrac{1}{2p}\dfrac{[u_{m}+\varepsilon\varphi]_{W^{s,p}(\mathbb{R}^{n})}^{p}-[u_{m}]^{p}_{W^{s,p}(\mathbb{R}^{n})}}{\varepsilon}$
$\displaystyle=:I_{1}+I_{2}+I_{3}.$ (A.9)

We clearly have

$I_{2}=-\dfrac{1}{h}\int_{\Omega}|u_{m-1}|^{q-1}u_{m-1}\varphi\,dx$ (A.10)

while, using the fundamental theorem of calculus, we find from Hölder’s
inequality with $u_{m}\in\mathcal{X}$ that, as $\varepsilon\longrightarrow 0$,

$\displaystyle I_{1}$
$\displaystyle=\dfrac{1}{h}\dfrac{1}{q+1}\int_{\Omega}\dfrac{1}{\varepsilon}\left(\,\int_{0}^{1}\dfrac{d}{d\theta}\left|u_{m}+\theta\varepsilon\varphi\right|^{q+1}\,d\theta\right)\,dx$
$\displaystyle=\dfrac{1}{h}\int_{\Omega}\left(\,\int_{0}^{1}\left|u_{m}+\theta\varepsilon\varphi\right|^{q-1}(u_{m}+\theta\varepsilon\varphi)\,d\theta\right)\varphi\,dx\longrightarrow\dfrac{1}{h}\int_{\Omega}|u_{m}|^{q-1}u_{m}\varphi\,dx.$
(A.11)

Due to the fundamental theorem of calculus again,

$\displaystyle I_{3}$
$\displaystyle=\dfrac{1}{2p}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{1}{\varepsilon}\dfrac{\left|(u_{m}+\varepsilon\varphi)(x)-(u_{m}+\varepsilon\varphi)(y)\right|^{p}-\left|u_{m}(x)-u_{m}(y)\right|^{p}}{|x-y|^{n+sp}}\,dxdy$
$\displaystyle=\dfrac{1}{2p}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{1}{\varepsilon}\dfrac{1}{|x-y|^{n+sp}}\left(\int_{0}^{1}\dfrac{d}{d\theta}\Big{|}(u_{m}+\theta\varepsilon\varphi)(x)-(u_{m}+\theta\varepsilon\varphi)(y)\Big{|}^{p}\,d\theta\right)\,dxdy$
$\displaystyle=\dfrac{1}{2}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{1}{|x-y|^{n+sp}}\left(\int_{0}^{1}U_{m}^{(\theta)}(x,y)\left(\varphi(x)-\varphi(y)\right)\,d\theta\right)\,dxdy$
(A.12)

with short-hand notation as well as (2.14)

$U_{m}^{(\theta)}(x,y):=\Big{|}(u_{m}+\theta\varepsilon\varphi)(x)-(u_{m}+\theta\varepsilon\varphi)(y)\Big{|}^{p-2}\Big{(}(u_{m}+\theta\varepsilon\varphi)(x)-(u_{m}+\theta\varepsilon\varphi)(y)\Big{)}.$

The integrand of the right-hand side of (A) is estimated as

$\displaystyle\dfrac{1}{|x-y|^{n+sp}}\left|\int_{0}^{1}U_{m}^{(\theta)}(x,y)\left(\varphi(x)-\varphi(y)\right)\,d\theta\right|$
$\displaystyle\quad\quad\quad\quad\quad\leqq
2^{p-1}\left(\dfrac{\left|u_{m}(x)-u_{m}(y)\right|^{p-1}|\varphi(x)-\varphi(y)|}{|x-y|^{n+sp}}+\dfrac{\left|\varphi(x)-\varphi(y)\right|^{p}}{|x-y|^{n+sp}}\right)$

and, from the Hölder inequality,

$\displaystyle\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}C\left(\dfrac{\left|u_{m}(x)-u_{m}(y)\right|^{p-1}|\varphi(x)-\varphi(y)|}{|x-y|^{n+sp}}+\dfrac{\left|\varphi(x)-\varphi(y)\right|^{p}}{|x-y|^{n+sp}}\right)\,dxdy$
$\displaystyle\quad\quad\quad\quad\quad\leqq
C\left([u_{m}]_{W^{s,p}(\mathbb{R}^{n})}^{p-1}[\varphi]_{W^{s,p}(\mathbb{R}^{n})}+[\varphi]_{W^{s,p}(\mathbb{R}^{n})}^{p}\right)<\infty.$

By the Lebesgue dominated convergence theorem, we pass to the limit
$\varepsilon\longrightarrow 0$ in (A) to conclude that

$\lim_{\varepsilon\longrightarrow
0}I_{3}=\dfrac{1}{2}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{U_{m}(x,y)}{|x-y|^{n+sp}}\left(\varphi(x)-\varphi(y)\right)\,dxdy,$
(A.13)

where
$U_{m}(x,y):=\left|u_{m}(x)-u_{m}(y)\right|^{p-2}\left(u_{m}(x)-u_{m}(y)\right)$
as in (2.14). Merging (A.10), (A) and (A.13) in (A), we therefore obtain from
the minimality of $u_{m}$ of $\mathcal{F}$ in $\mathcal{X}$ that, for all
$\varphi\in\mathcal{X}$,

$\displaystyle 0$ $\displaystyle=\lim_{\varepsilon\longrightarrow
0}\dfrac{\mathcal{F}(u_{m}+\varepsilon\varphi)-\mathcal{F}(u_{m})}{\varepsilon}$
$\displaystyle=\int_{\Omega}\dfrac{|u_{m}|^{q-1}u_{m}-|u_{m-1}|^{q-1}u_{m-1}}{h}\,\varphi\,dx+\dfrac{1}{2}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\dfrac{U_{m}(x,y)}{|x-y|^{n+sp}}\left(\varphi(x)-\varphi(y)\right)\,dxdy,$

which finishes the proof of Lemma 3.1. ∎

##  Appendix B Boundedness

In this Appendix B, by the additional stronger assumption that the boundedness
of the initial datum $u_{0}$, we conclude slightly stronger assertions
compared to ones in the previous section (see Lemmata B.1 and B.2 and Theorem
B.3).

###  B.1. Boundedness

In this Appendix B.1, we show the boundedness for approximate solutions of
(1.1). The test function here is retrieved from [2].

###### Lemma B.1 (Boundedness)

Suppose that $u_{0}\in W^{s,p}_{0}(\Omega)\cap L^{\infty}(\Omega)$. Let
$\bar{u}_{h}$, $u_{h}$, $\bar{w}_{h}$, $w_{h}$, $v_{h}$ and $\bar{v}_{h}$ be a
approximate solution of (1.1) defined in (3.3) and (3.4). Then, these
solutions are bounded in $\Omega_{\infty}$:

$\begin{split}\sup_{t>0}\|\bar{u}_{h}(t)\|_{L^{\infty}(\Omega)}&\leqq\|u_{0}\|_{L^{\infty}(\Omega)},\\\
\sup_{t>0}\|u_{h}(t)\|_{L^{\infty}(\Omega)}&\leqq\|u_{0}\|_{L^{\infty}(\Omega)};\end{split}$
(B.1)
$\sup_{t>0}\|\bar{w}_{h}(t)\|_{L^{\infty}(\Omega)},\,\,\sup_{t>0}\|w_{h}(t)\|_{L^{\infty}(\Omega)}\leqq\|u_{0}\|_{L^{\infty}(\Omega)}^{\frac{q+1}{2}};$
(B.2)
$\sup_{t>0}\|\bar{v}_{h}(t)\|_{L^{\infty}(\Omega)},\,\,\sup_{t>0}\|v_{h}(t)\|_{L^{\infty}(\Omega)}\leqq\|u_{0}\|_{L^{\infty}(\Omega)}^{q}.$
(B.3)

###### Proof.

The estimations (B.1)2, (B.2) and (B.3) simply follow from (B.1)1 and Lemma
3.2. Following the argument developed in [23, Lemma 3.3, p.164], we shall
prove (B.1)1. Put $M:=\|u_{0}\|_{L^{\infty}(\Omega)}$ for brevity. In (3.2),
choose a testing function $\xi_{\delta}(u_{m})$ with

$\xi_{\delta}(\sigma)=\dfrac{\sigma}{|\sigma|}\min\left\\{1,\,\frac{(|\sigma|-M)_{+}}{\delta}\right\\}\quad\textrm{for}\quad\delta>0.$

Note that $\xi_{\delta}(\sigma)$ is clearly bounded and Lipschitz function for
$\sigma\in\mathbb{R}$ and thus, $\xi_{\delta}(u_{m})\in
W^{s,p}_{0}(\Omega)\cap L^{\infty}(\Omega)$.

$\sigma$$y$$y=\xi_{\delta}(\sigma)$$M$$M+\delta$$1$$-M$$-(M+\delta)$$-1$
Figure B.1. Graph of $y=\xi_{\delta}(\sigma)$

Then, for $m=1,2,\ldots$, we have

$\displaystyle\frac{1}{h}\int_{\Omega}$
$\displaystyle\left(|u_{m}|^{q-1}u_{m}-|u_{m-1}|^{q-1}u_{m-1}\right)\xi_{\delta}(u_{m})\,dx$
$\displaystyle+\frac{1}{2}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{U_{m}(x,y)}{|x-y|^{n+sp}}\big{(}\xi_{\delta}(u_{m}(x))-\xi_{\delta}(u_{m}(y))\big{)}\,dxdy=0.$
(B.4)

The integrand in the fractional integral in (B.1) is evaluated as follows:
Since $\xi_{\delta}(u_{m}(x))-\xi_{\delta}(u_{m}(y))$ has the same sign as
$u_{m}(x)-u_{m}(y)$ via the monotonicity of $\xi_{\delta}(\sigma)$, we observe

$\displaystyle
U_{m}(x,y)\bigl{(}\xi_{\delta}(u_{m}(x))-\xi_{\delta}(u_{m}(y))\bigr{)}$
$\displaystyle=|u_{m}(x)-u_{m}(y)|^{p-2}\bigl{(}u_{m}(x)-u_{m}(y)\bigr{)}\bigl{(}\xi_{\delta}(u_{m}(x))-\xi_{\delta}(u_{m}(y))\bigr{)}\geqq
0$

on $\mathbb{R}^{n}\times\mathbb{R}^{n}$.

Dropping the fractional term of (B.1) implies

$\displaystyle\int_{\Omega}\frac{|u_{m}|^{q}-|u_{m-1}|^{q}}{h}\min\left\\{1,\,\frac{(|u_{m}|-M)_{+}}{\delta}\right\\}\,dx\leqq
0.$ (B.5)

By Lebesgue’s dominated convergence theorem and passage to the limit
$\delta\searrow 0$ in (B.5) we have

$\int_{\Omega\,\cap\,\\{|u_{m}|>M\\}}\frac{|u_{m}|^{q}-|u_{m-1}|^{q}}{h}\,dx\leqq
0,$ (B.6)

which is equivalent to

$\int_{\Omega}(|u_{m}|^{q}-M^{q})_{+}\,dx\leqq\int_{\Omega\cap\\{|u_{m}|>M\\}}(|u_{m-1}|^{q}-M^{q})\,dx.$

By neglecting the region such that $|u_{m-1}|\leqq M\leqq|u_{m}|$, the right-
hand side is further estimated as

$\displaystyle\int_{\Omega\cap\\{|u_{m}|>M\\}}(|u_{m-1}|^{q}-M^{q})\,dx$
$\displaystyle\leqq\int_{\Omega\cap\\{|u_{m}|>M\\}\cap\\{|u_{m-1}|>M\\}}(|u_{m-1}|^{q}-M^{q})\,dx$
$\displaystyle\leqq\int_{\Omega}(|u_{m-1}|^{q}-M^{q})_{+}\,dx.$

Summarizing, we have

$\int_{\Omega}(|u_{m}|^{q}-M^{q})_{+}\,dx\leqq\int_{\Omega}(|u_{m-1}|^{q}-M^{q})_{+}\,dx.$

Iterating the above display with respect to $m$, we obtain

$\int_{\varOmega}(|u_{m}|^{q}-M^{q})_{+}\,dx\leqq\int_{\varOmega}(|u_{0}|^{q}-M^{q})_{+}\,dx=0$

for all $m=1,2,\ldots$, which in turn implies $|u_{m}|\leqq M$ in $\Omega$ for
any $m=1,2,\ldots$, finishing the proof. ∎

By the boundedness Lemma B.1, the convergence result in Lemma 4.6 becomes
better as the following.

###### Lemma B.2 (Convergence of approximate solutions)

Let $\bar{u}_{h}$, $u_{h}$, $\bar{w}_{h}$, $w_{h}$, $\bar{v}_{h}$ and $v_{h}$
be the approximate solutions of (1.1) defined by (3.3) and (3.4), satisfying
the convergence in Lemma 4.6. For all finite exponent $\gamma\geqq 1$, any
bounded domain $K\subset\mathbb{R}^{n}$ and every positive number $T<\infty$,

$\displaystyle u_{h},\,\bar{u}_{h}\longrightarrow u\quad$
$\displaystyle\textrm{strongly\,\,in}\,\,L^{\gamma}(K_{T}),$ (B.7)
$\displaystyle w_{h},{\bar{w}}_{h}\longrightarrow|u|^{\frac{q-1}{2}}u\quad$
$\displaystyle\textrm{strongly\,\,in}\,\,L^{\gamma}(K_{T}),$ (B.8)
$\displaystyle v_{h},\,\bar{v}_{h}\longrightarrow|u|^{q-1}u\quad$
$\displaystyle\textrm{strongly\,\,in}\,\,L^{\gamma}(K_{T})$ (B.9)

as $h\searrow 0$.

###### Proof.

From the boundedness (B.1) and (4.16) it follows that

$\sup_{0<t<T}\|u(t)\|_{L^{\infty}(K)}\leqq\|u_{0}\|_{L^{\infty}(\Omega)}.$
(B.10)

The boundedness (B.1), (B.10) and the convergence (4.17) with the Hölder
inequality gives (B.7)1. The validity of other statements (B.7)2, (B.8) and
(B.9) is shown by use of the boundedness Lemma B.1, (4.25) and (4.26). ∎

###  B.2. A regularity of $\partial_{t}\left(|u|^{q-1}u\right)$ in the case
$q\geqq 1$

Here we state the regularity on the time-derivative, obtained from the
boundedness Lemma B.1.

###### Theorem B.3

Let $q\geqq 1$. Suppose that the initial datum $u_{0}\in
W^{s,p}_{0}(\Omega)\cap L^{\infty}(\Omega)$. Then, the weak solution $u$ of
(1.1) obtained in Theorem 1.1 satisfies $\partial_{t}(|u|^{q-1}u)\in
L^{2}(\Omega_{\infty})$ and the solution $u$ itself satisfies (1.1) almost
everywhere in $\Omega_{\infty}$, as an $L^{2}$-function.

We first derive the $L^{2}$-estimate of the time-derivative:

###### Lemma B.4

Let $q\geqq 1$ and $v_{h}$ be the approximate solutions of (1.1) defined by
(3.4). Assume that $u_{0}\in W^{s,p}_{0}(\Omega)\cap L^{\infty}(\Omega)$. Then
the time derivatives of approximate solutions $\\{\partial_{t}v_{h}\\}$ are
bounded in $L^{2}(\Omega_{T})$ for any positive $T<\infty$,

$\iint_{\Omega_{T}}|\partial_{t}v_{h}|^{2}\,dxdt\leqq
C\|u_{0}\|_{L^{\infty}(\Omega)}^{q-1}[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p}$
(B.11)

with a positive constant $C=C(p,q)$.

###### Proof of Lemma B.4.

From the energy estimate (3.12) in Lemma 3.3 and (B.1) in Lemma B.1, we obtain

$\displaystyle\sum_{m=1}^{N}h\int_{\Omega}|\partial_{t}v_{h}|^{2}\,dx$
$\displaystyle\leqq
C\sum_{m=1}^{N}h\int_{\Omega}\left(|u_{m}|+|u_{m-1}|\right)^{2(q-1)}\left|\frac{u_{m}-u_{m-1}}{h}\right|^{2}\,dx$
$\displaystyle\leqq
C\sum_{m=1}^{N}h\left(\|u_{m}\|_{L^{\infty}(\Omega)}^{q-1}+\|u_{m-1}\|_{L^{\infty}(\Omega)}^{q-1}\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\times\int_{\Omega}\left(|u_{m}|+|u_{m-1}|\right)^{q-1}\left|\frac{u_{m}-u_{m-1}}{h}\right|^{2}\,dx$
$\displaystyle\\!\\!\\!\stackrel{{\scriptstyle\eqref{MPeq.}}}{{\leqq}}C\|u_{0}\|_{L^{\infty}(\Omega)}^{q-1}\int_{0}^{t_{N}}\int_{\Omega}\left(|{\bar{u}}_{h}(x,t)|+|{\bar{u}}_{h}(x,t-h)|\right)^{q-1}|\partial_{t}u_{h}(x,t)|^{2}\,dxdt$
$\displaystyle\\!\\!\\!\\!\stackrel{{\scriptstyle\eqref{energy est.
NE3}}}{{\leqq}}C\|u_{0}\|_{L^{\infty}(\Omega)}^{q-1}[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p},$

which finishes the proof of (B.11). ∎

We report the proof of Theorem B.3.

###### Proof of Theorem B.3.

Let $q\geqq 1$. By (B.11) in Lemma B.4, $\\{\partial_{t}v_{h}\\}_{h>0}$ is
bounded in $L^{2}(\Omega_{\infty})$ and thus, there are a subsequence
$\\{\partial_{t}v_{h}\\}_{h>0}$ (also labelled with $h$) and a limit function
$\vartheta\in L^{2}(\Omega_{\infty})$ such that, as $h\searrow 0$,

$\partial_{t}v_{h}\longrightarrow\vartheta\quad\textrm{weakly\,\,in}\,\,\,L^{2}(\Omega_{\infty}).$
(B.12)

By the convergence (B.9) in Lemma B.2 we pass to the limit as $h\searrow 0$ in
the identity

$\iint_{\Omega_{\infty}}\partial_{t}v_{h}\cdot\varphi\,dxdt=-\iint_{\Omega_{\infty}}v_{h}\,\partial_{t}\varphi\,dxdt$

for every $\varphi\in C^{\infty}_{0}(\Omega_{\infty})$ to obtain from (4.33)
that

$\vartheta=\partial_{t}(|u|^{q-1}u)\quad\textrm{in}\,\,L^{2}(\Omega_{\infty}).$
(B.13)

As a result, by the convergence (B.9) in Lemma B.2 with (B.11) in Lemma B.4,
there holds that

$\left\|\partial_{t}(|u|^{q-1}u)\right\|_{L^{2}(\Omega_{T})}^{2}\leqq
C\|u_{0}\|_{L^{\infty}(\Omega)}^{q-1}[u_{0}]_{W^{s,p}(\mathbb{R}^{n})}^{p},$

which finishes the proof of Theorem B.3.

∎

##  Appendix C A space-time fractional Sobolev inequality

In this appendix, we accomplish in Lemma C.1 a general inequality
interpolating by the Gagliardo seminorm in time-space domain in the special
case where the function is weakly differentiable in time direction. The
inequality in Lemma C.1 is used essentially as the key ingredient for Lemma
4.6

The following inequality gives an interpolation between fractional Sobolev
semi-norms and the integral of time derivative.

###### Lemma C.1 (Space-time fractional Sobolev inequality)

Let $K\subset\mathbb{R}^{n}$ be a bounded domain and let $I:=(0,T)$ with
$0<T<\infty$. Let $s^{\prime},\bar{s}$ be two exponents obeying
$0<s^{\prime}<\bar{s}<1$. Then it holds

$[f]_{W^{s^{\prime},1}(K_{T})}\leqq
C(n,n,s^{\prime},\bar{s})\left(\|\partial_{t}f\|_{L^{1}(K_{T})}+\int_{I}[f(\cdot,t)]_{W^{\bar{s},1}(K)}\,dt\right),$
(C.1)

where the integrals of a function $f$ in the right-hand side is assumed to be
finite.

###### Proof.

To begin, we divide the Gagliardo semi-norm into two terms:

$\displaystyle[f]_{W^{s^{\prime},1}(K_{T})}$ $\displaystyle\leqq\iint_{I\times
I}\iint_{K\times
K}\dfrac{|f(x,t)-f(x,t^{\prime})|}{\left(\sqrt{|x-x^{\prime}|^{2}+(t-t^{\prime})^{2}}\right)^{(n+1)+s^{\prime}}}\,dxdx^{\prime}dtdt^{\prime}$
$\displaystyle+\iint_{I\times I}\iint_{K\times
K}\dfrac{|f(x,t^{\prime})-f(x^{\prime},t^{\prime})|}{\left(\sqrt{|x-x^{\prime}|^{2}+(t-t^{\prime})^{2}}\right)^{(n+1)+s^{\prime}}}\,dxdx^{\prime}dtdt^{\prime}$
$\displaystyle=:I+II.$ (C.2)

By the assumption it plainly holds $1+(s^{\prime}-\bar{s})>0$. Under this
condition, the integration $I$ is simply estimated as

$\displaystyle I$ $\displaystyle=\iint_{I\times I}\iint_{K\times
K}\dfrac{|f(x,t)-f(x,t^{\prime})|}{\left(\sqrt{|x-x^{\prime}|^{2}+(t-t^{\prime})^{2}}\right)^{1+\bar{s}}}\times\frac{dxdx^{\prime}dtdt^{\prime}}{\left(\sqrt{|x-x^{\prime}|^{2}+(t-t^{\prime})^{2}}\right)^{n+(s^{\prime}-\bar{s})}}$
$\displaystyle\leqq\int_{K}\left(\,\,\iint_{I\times
I}\dfrac{|f(x,t^{\prime})-f(x,t^{\prime})|}{|t-t^{\prime}|^{1+\bar{s}}}\,dtdt^{\prime}\right)\left(\,\int_{K_{x^{\prime}}}\frac{dx^{\prime}}{|x-x^{\prime}|^{n+(s^{\prime}-\bar{s})}}\right)\,dx$
$\displaystyle=\int_{K}\left[\,\,\int_{I}\left(\,\,\int_{\\{\tau\in(-T,T)\,:\,t+\tau\in
I\\}}\dfrac{|f(x,t^{\prime})-f(x,\tau+t)|}{|t-t^{\prime}|^{1+\bar{s}}}\,d\tau\right)dt^{\prime}\right]\left(\,\int_{K_{x^{\prime}}}\frac{dx^{\prime}}{|x-x^{\prime}|^{n+(s^{\prime}-\bar{s})}}\right)\,dx.$
(C.3)

Due to the fundamental theorem of calculus,

$\displaystyle\big{|}f(x,t)-f(x,\tau+t)\big{|}=\left|\int_{0}^{1}\dfrac{d}{d\sigma}f(x,t+\sigma\tau)\,d\sigma\right|$
$\displaystyle\leqq|\tau|\int_{0}^{1}\Big{|}\partial_{t}f(x,t+\sigma\tau)\Big{|}\,d\tau,$

which, together with (C) and Fubini’s theorem, in turn implies that

$\displaystyle I$
$\displaystyle\leqq\int_{K}\int_{0}^{1}\left[\,\,\int_{-T}^{T}\dfrac{1}{|\tau|^{\bar{s}}}\left(\,\,\int_{\\{t\in
I\,:\,t+\tau\in
I\\}}\Big{|}\partial_{t}f(x,t+\sigma\tau)\Big{|}\,dt\right)d\tau\right]\left(\,\int_{K_{x^{\prime}}}\frac{dx^{\prime}}{|x-x^{\prime}|^{n+(s^{\prime}-\bar{s})}}\right)\,d\sigma
dx$
$\displaystyle\leqq\left(\,\,\int_{-T}^{T}\dfrac{1}{|\tau|^{\bar{s}}}d\tau\right)\int_{K}\|\partial_{t}f(x,\cdot)\|_{L^{1}(I)}\left(\,\int_{K_{x^{\prime}}}\frac{dx^{\prime}}{|x-x^{\prime}|^{n+(s^{\prime}-\bar{s})}}\right)\,dx.$
(C.4)

Via change of variable $\varrho=|x-x^{\prime}|$ we have

$\displaystyle\int_{K_{x^{\prime}}}\frac{dx^{\prime}}{|x-x^{\prime}|^{n+(s^{\prime}-\bar{s})}}$
$\displaystyle\leqq
C(n)\int_{0}^{\operatorname*{diam}K}\frac{d\varrho}{\varrho^{1+(s^{\prime}-\bar{s})}}$
$\displaystyle=\frac{(\operatorname*{diam}K)^{\bar{s}-s^{\prime}}}{\bar{s}-s^{\prime}}<\infty.$
(C.5)

Analogously, by the condition $0<\bar{s}<1$, the integral
$\int_{-T}^{T}\frac{1}{|\tau|^{\bar{s}}}d\tau$ appearing in (C) takes a finite
value depending $T$ only. Combining this with (C) and (C), we get

$\displaystyle I$ $\displaystyle\leqq
C(n,s^{\prime},\bar{s},T)\int_{K}\|\partial_{t}f(x,\cdot)\|_{L^{1}(I)}\,dx$
$\displaystyle=C(n,s^{\prime},\bar{s},T)\|\partial_{t}f\|_{L^{1}(K\times I)}.$
(C.6)

We now turn to evaluate $II$. The integration $II$ is estimated as

$\displaystyle II$ $\displaystyle\leqq\iint_{I\times I}\iint_{K\times
K}\dfrac{|f(x,t^{\prime})-f(x^{\prime},t^{\prime})|}{\left(\sqrt{|x-x^{\prime}|^{2}+(t-t^{\prime})^{2}}\right)^{n+\bar{s}}}\times\frac{dxdx^{\prime}dtdt^{\prime}}{\left(\sqrt{|x-x^{\prime}|^{2}+(t-t^{\prime})^{2}}\right)^{1+s^{\prime}-\bar{s}}}$
$\displaystyle\leqq\iint_{I\times I}\iint_{K\times
K}\dfrac{|f(x,t^{\prime})-f(x^{\prime},t^{\prime})|}{|x-x^{\prime}|^{n+\bar{s}}}\times\frac{dxdx^{\prime}dtdt^{\prime}}{|t-t^{\prime}|^{1+(s^{\prime}-\bar{s})}}$
$\displaystyle=\int_{I}\left(\,\int_{I_{t}}\dfrac{dt}{|t-t^{\prime}|^{1+(s^{\prime}-\bar{s})}}\right)[f(\cdot,t^{\prime})]_{W^{\bar{s},1}(K)}\,dt^{\prime}$
$\displaystyle\leqq
C\int_{I}\left(\,\,\int_{-T}^{T}\dfrac{d\tau}{|\tau|^{1+(s^{\prime}-\bar{s})}}\right)\,[f(\cdot,t^{\prime})]_{W^{\bar{s},1}(\mathbb{R}^{n})}\,dt^{\prime}$
$\displaystyle=C(n,s,\bar{s},T)\int_{I}[f(\cdot,t^{\prime})]_{W^{\bar{s},1}(K)}\,dt^{\prime},$
(C.7)

where, in the fourth line,
$\displaystyle\int_{-T}^{T}\dfrac{d\tau}{|\tau|^{1+(s^{\prime}-\bar{s})}}=\frac{2T^{\bar{s}-s^{\prime}}}{\bar{s}-s^{\prime}}$
is used.

Merging (C), (C) in (C), we end up with the conclusion (C.1). ∎

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