# A final value problem with a non-local and a source term: regularization by
truncation

Subhankar Mondal  TIFR Centre for Applicable Mathematics, Bangalore-560065,
India<EMAIL_ADDRESS>
###### Abstract.

This paper is concerned with recovering the solution of a final value problem
associated with a parabolic equation involving a non linear source and a non-
local term, which to the best of our knowledge has not been studied earlier.
It is shown that the considered problem is ill-posed, and thus, some
regularization method has to be employed in order to obtain stable
approximations. In this regard, we obtain regularized approximations by
solving some non linear integral equations which is derived by considering a
truncated version of the Fourier expansion of the sought solution. Under
different Gevrey smoothness assumptions on the exact solution, we provide
parameter choice strategies and obtain the error estimates. A key tool in
deriving such estimates is a version of Grönwalls’ inequality for iterated
integrals, which perhaps, is proposed and analysed for the first time.

Keywords: backward problem, non-local term, ill-posed problem, regularization,
parameter choice, fixed point

MSC 2020: 47A52, 35R25, 35R30, 45K05

##  1\. Introduction

Let $\Omega\subset{\mathbb{R}}^{d},\,d\geq 1,$ be a bounded domain with smooth
boundary $\partial\Omega$ and $\tau>0$ be fixed. For $F:[0,\tau]\times
L^{2}(\Omega)\to L^{2}(\Omega)$ and $g\in L^{2}(\Omega)$ we consider the
problem of recovering $u:[0,\tau]\to L^{2}(\Omega)$ that satisfies the
following final value problem (FVP) with a non-local term

(1.1) $\begin{cases}u_{t}(x,t)-\Delta
u(x,t)=F(t,u(x,t))+\int_{t}^{\tau}u(x,s)\,ds,&\quad\text{in}\quad\Omega\times(0,\tau),\\\
u(x,t)=0,&\quad\text{on}\quad\partial\Omega\times(0,\tau),\\\
u(x,\tau)=g(x),&\quad\text{in}\quad\Omega.\end{cases}$

This type of PDE appears in various context of engineering such as modelling
the heat conduction for materials with memory [16], analysis of space-time
dependent nuclear dynamics [17], epidemic phenomena in biology [12]. For
further details about applications of some parabolic equations with non-local
term, the interested reader may refer to [11, 12] and the references therein.
Additionally, it is worth to note that these type of PDE with non-local terms
have attracted a lot of attention in control theory in the recent past (see
for e.g., [6, 9, 21]).

As mentioned earlier, in this paper we are interested in recovering
$u:[0,\tau]\to L^{2}(\Omega)$ from the knowledge of the final value $g\in
L^{2}(\Omega)$ that satisfies (1.1) for a general source function
$F:[0,\tau]\times L^{2}(\Omega)\to L^{2}(\Omega).$ Throughout, we assume that
the possibly non linear source function $F$ satisfies the following
hypothesis:

  1. $(\bf F{1})$

There exists $\kappa>0$ such that
$\left\|F(s,\psi_{1})-F(s,\psi_{2})\right\|\leq\kappa\left\|\psi_{1}-\psi_{2}\right\|$
for all $s\in[0,\tau]$ and $\psi_{1},\psi_{2}\in L^{2}(\Omega).$

  2. $(\bf F{2})$

There exists $\phi\in L^{2}(\Omega)$ such that $s\mapsto F(s,\phi)\in
L^{1}(0,\tau;L^{2}(\Omega)).$

Note that the assumptions $(\bf F{1})$ and $(\bf F{2})$ are quite standard.
Indeed, $(\bf F{1})$ requires that $F$ is Lipschitz with respect to the second
variable, and the condition $(\bf F{2})$ is assumed just to ensure some
integrability of $F$ (see Lemma 3.4 (i)).

Moreover, we assume that the considered FVP (1.1) has a unique solution $u\in
C([0,\tau];L^{2}(\Omega))$ corresponding to the exact final value $g\in
L^{2}(\Omega).$ (In the next section we shall provide an example (see Example
2.1) of a FVP of the form (1.1) for which we obtain explicitly the analytical
representation of the solution of the corresponding FVP.) Once we assume that
a solution exists, a natural question one would like to ask is whether the
solution is stable with respect to the perturbation in the data. We shall show
that solving the FVP (1.1) is an ill-posed problem (see Example 2.2) in the
sense that a small perturbation in the final value $g$ may lead to a large
deviation in the solution $u(t)$ for each $0\leq t<\tau.$ Thus, the considered
FVP possesses the ill-posedness behaviour similar to that of the classical
backward heat conduction equation [8].

In real world scenario, it is obvious that the final value $g$ will be
available from some measurements, and hence noise in the data is inevitable.
Thus, we assume that for $\delta>0,$ $g^{\delta}\in L^{2}(\Omega)$ is the
noisy data corresponding to the exact final value $g$ satisfying the
deterministic noise model

(1.2) $\left\|g^{\delta}-g\right\|_{L^{2}(\Omega)}\leq\delta.$

Thus, the problem at hand is to recover $u(t),\,0\leq t<\tau,$ where $u$ is
the unique solution of the FVP (1.1) for the exact final value $g\in
L^{2}(\Omega),$ from the knowledge of $g^{\delta}$ satisfying (1.2). Since the
considered FVP is ill-posed, we have to employ some regularization method (see
for e.g.[4, 14]) in order to obtain some stable approximations. In this
regard, we shall obtain regularized solution by solving some non linear
integral equation derived using a truncated version of the Fourier expansion
of the sought solution. The level at which the Fourier expansion is truncated
plays the role of regularization parameter.

The literature on the regularization aspect for the classical backward heat
conduction problem is quite rich and several regularization methods (see for
example, quasi reversibility method [10, 20]; quasi boundary value method [13,
7]; truncated regularization method [3, 24, 15]) were studied extensively over
the last few decades. In comparison to that, there seems to be a very limited
work devoted to the study of regularization aspect for the ill-posed FVP with
non-local terms. To the best of our knowledge, a recent work of Luan and Khanh
[12] is perhaps the first work, where the authors investigated the
FVP/backward problem of identification of $v(\cdot,t)$ for $0\leq t<\tau$ from
the knowledge of the final value $h\in L^{2}(\Omega),$ where $v$ is the
solution of

(1.3) $\begin{cases}v_{t}(x,t)-\Delta
v(x,t)=\beta\int_{0}^{t}v(x,s)\,ds,&\quad\text{in}\quad\Omega\times(0,\tau),\\\
v(x,t)=0,&\quad\text{on}\quad\partial\Omega\times(0,\tau),\\\
v(x,\tau)=h(x),&\quad\text{in}\quad\Omega,\end{cases}$

for a fixed $\beta>0.$ They have shown that for $0<t<\tau,$ the backward
problem is well-posed in the sense that it has a unique solution and the
solution depends continuously on the final value $h$, whereas for $t=0,$ the
backward problem is ill-posed (see [12, Theorem 3.2]). That is, the problem of
identification of $v_{0}:=v(\cdot,0)$ from the knowledge of the final value
$h$ is ill-posed. In order to obtain stable approximations they have employed
the truncated spectral regularization method and obtained error estimates of
Hölder type for different type of parameter choice strategies.

It is clear from (1.1) and (1.3) that the non-local term associated with these
equations are different. Also, it is worth noting that with the type of non-
local term as considered in this paper, the ill-posedness behaviour of the
backward problem changes completely, as discussed earlier. Thus, in view of
this, studying the problem associated with (1.1) is indeed an interesting
problem.

We now point out the important contributions of this paper.

  * •

To the best of our knowledge, the FVP (1.1) with a general source term that
allows some non-linearity and involving a different version of the non-local
term is studied for the first time from the regularization point of view.

  * •

We establish that the considered FVP is ill-posed for every $0\leq t<\tau$ and
thus obtain stable approximations by employing a regularization method using
the truncated Fourier expansion.

  * •

Under different Gevrey smoothness assumption, we obtain the corresponding
error estimates. One of the key tool in deriving such rates is a version of
Grönwalls’ inequality for iterated integrals, which perhaps is proposed and
analysed for the first time in this paper.

The rest of the paper is organised as follows: In Section 2, we recall some
well known results associated to the Dirichlet eigenvalue problem for a
Laplace operator. Using the eigenvalues of the Dirichlet eigenvalue problem,
we obtain an integral representation of the exact solution of the FVP (1.1),
which plays the key role in defining the stable approximations. We also recall
the definition of Gevrey spaces, that will play the role of source sets. In
Section 3, we state and prove (wherever required) various necessary results
that plays the essential role in deriving error estimates and defining the
stable approximations. Moreover, in this section we provide a new version of
Grönwalls’ type inequality for iterated integrals. Section 4 contains the main
results of this paper. In this section we shall deal with regularization and
obtain various error estimates. Moreover, we shall provide a parameter choice
strategy and obtain the corresponding estimates.

##  2\. Preliminaries

It is well known that for $\lambda\in{\mathbb{R}}$ the Dirichlet eigenvalue
problem

(2.1) $\begin{cases}-\Delta v=\lambda v,&\quad\text{in}\quad\Omega,\\\
v=0,&\quad\text{on}\quad\partial\Omega,\end{cases}$

admits a sequence of eigenvalues
$0<\lambda_{1}\leq\lambda_{2}\leq\ldots\to\infty$ and the corresponding
eigenfunctions $\varphi_{n}\in H^{1}_{0}(\Omega)\cap H^{2}(\Omega)$ are such
that $\left\\{\varphi_{n}:\,n\in{\mathbb{N}}\right\\}$ forms an orthonormal
basis of $L^{2}(\Omega)$ (see for e.g., [5]). It is also known that (see for
e.g., [2]) there exist $e_{1},e_{2}>0$ such that

(2.2) $e_{1}n^{2/d}\leq\lambda_{n}\leq e_{2}n^{2/d},\quad n\in{\mathbb{N}}.$

We first give an example of a FVP that is of the form (1.1) and show that it
has a unique solution.

######  Example 2.1.

Let $d=1$ and $\Omega=(0,1).$ Let $\lambda_{j}$ be the the eigenvalues for the
Dirichlet problem (2.1) and $\left\\{\varphi_{j}:\,j\in{\mathbb{N}}\right\\}$
be the corresponding eigenfunctions. Then it is known that
$\lambda_{j}=j^{2}\pi^{2},\,j\in{\mathbb{N}}.$ Now consider the FVP

(2.3)
$\begin{cases}v_{t}(x,t)-v_{xx}(x,t)=v(x,t)+\int_{t}^{\tau}v(x,s)\,ds,&\quad\text{in}\quad\Omega\times(0,\tau),\\\
v(x,t)=0,&\quad\text{on}\quad\partial\Omega\times(0,\tau),\\\
v(x,\tau)=\varphi_{n}(x),&\quad\text{in}\quad\Omega.\end{cases}$

Let

$\alpha_{j}=\frac{1}{2}\left(-\left(\lambda_{j}-1\right)+\sqrt{\left(\lambda_{j}-1\right)^{2}-4}\right)\,\,\text{and}\,\,\beta_{j}=\frac{1}{2}\left(-\left(\lambda_{j}-1\right)-\sqrt{\left(\lambda_{j}-1\right)^{2}-4}\right).$

Then it can be verified easily that

$v(x,t)=\frac{\alpha_{n}e^{-\alpha_{n}(\tau-t)}-\beta_{n}e^{-\beta_{n}(\tau-t)}}{\alpha_{n}-\beta_{n}}\varphi_{n}(x).$

is the solution of (2.3). $\lozenge$

We now give an example to show that the considered FVP (1.1) is ill-posed in
the sense that a small perturbation in the final value $g$ may lead to a large
deviation in the solution $u$ of the FVP (1.1).

######  Example 2.2.

Let $\Omega,\lambda_{j},\varphi_{j}\,\alpha_{j}\,\beta_{j}$ be as considered
in Example 2.1. Consider the FVP of recovering the solution $v(\cdot,t)$ for
$0\leq t<\tau$ such that

(2.4)
$\begin{cases}v_{t}-v_{xx}=v+\int_{t}^{\tau}v(s)\,ds,&\quad\text{in}\quad\Omega\times(0,\tau),\\\
v=0,&\quad\text{on}\quad\partial\Omega\times(0,\tau),\\\
v(\cdot,\tau)=h^{n},&\quad\text{in}\quad\Omega,\end{cases}$

where $h^{n}=\frac{1}{\left|\beta_{n}\right|}\varphi_{n}.$ It can be verified
easily that for every $n\in{\mathbb{N}},$

$v^{n}(x,t)=\frac{\alpha_{n}e^{-\alpha_{n}(\tau-t)}-\beta_{n}e^{-\beta_{n}(\tau-t)}}{\left(\alpha_{n}-\beta_{n}\right)\left|\beta_{n}\right|}\varphi_{n}(x)$

is the solution of (2.4). Recall that
$\lambda_{n}=n^{2}\pi^{2},\,\,\left|\alpha_{n}\right|\leq\left|\beta_{n}\right|,\,\,\text{for
all}\,\,n\in{\mathbb{N}}$ and

$\left|\beta_{n}\right|=\frac{\left(\lambda_{n}-1\right)+\sqrt{\left(\lambda_{n}-1\right)^{2}-4}}{2}=\frac{n^{2}\pi^{2}-1+\sqrt{\left(n^{2}\pi^{2}-1\right)^{2}-4}}{2}\to\infty\,\,\text{as}\,\,n\to\infty.$

Now, we observe that

$\left\|h^{n}\right\|_{L^{2}(\Omega)}=\frac{1}{\left|\beta_{n}\right|}\to
0\quad\text{as}\,\,n\to\infty.$

Moreover, for every $0\leq t<\tau,$ we have

$\displaystyle\left\|v^{n}(t)\right\|_{L^{2}(\Omega)}$ $\displaystyle=$
$\displaystyle\left|\frac{\alpha_{n}e^{-\alpha_{n}(\tau-t)}-\beta_{n}e^{-\beta_{n}(\tau-t)}}{\left(\alpha_{n}-\beta_{n}\right)\left|\beta_{n}\right|}\right|=\frac{\left|\beta_{n}\right|e^{\left|\beta_{n}\right|(\tau-t)}-\left|\alpha_{n}\right|e^{\left|\alpha_{n}\right|(\tau-t)}}{\left(\left|\beta_{n}\right|-\left|\alpha_{n}\right|\right)\left|\beta_{n}\right|}$
$\displaystyle\geq$
$\displaystyle\frac{\left|\beta_{n}\right|e^{\left|\beta_{n}\right|(\tau-t)}-\left|\alpha_{n}\right|e^{\left|\beta_{n}\right|(\tau-t)}}{\left(\left|\beta_{n}\right|-\left|\alpha_{n}\right|\right)\left|\beta_{n}\right|}=\frac{e^{\left|\beta_{n}\right|(\tau-t)}}{\left|\beta_{n}\right|},$

and hence,

$\left\|v^{n}(t)\right\|_{L^{2}(\Omega)}\to\infty\,\,\text{as}\,\,n\to\infty,\,\,0\leq
t<\tau.$

This, shows that a small perturbation in the final value may lead to a large
deviation in the corresponding solution of the FVP for every $0\leq t<\tau.$
Thus, the problem of solving the FVP (2.4) is ill-posed. $\lozenge$

Before proceeding further, let us fix some notations that we shall follow in
the rest of the paper.

  * •

We shall denote the inner-product on $L^{2}(\Omega)$ by
$\langle\cdot,\cdot\rangle$ and the norm by $\left\|\cdot\right\|.$

  * •

For a Banach space $\mathcal{X},$ $\mathcal{B}(\mathcal{X})$ shall denote the
Banach space of all bounded linear operators from $\mathcal{X}$ to
$\mathcal{X}.$ The norm on $\mathcal{B}\left(\mathcal{X}\right)$ will be
denoted by $\left\|\cdot\right\|_{\mathcal{X}\to\mathcal{X}}.$

  * •

For a Banach space $\mathcal{X},$ $C([0,\tau];\mathcal{X})$ denotes the Banach
space of all continuous functions $v:[0,\tau]\to\mathcal{X}$ with the norm
$\displaystyle\left\|v\right\|_{C([0,\tau];\mathcal{X})}:=\max_{0\leq
t\leq\tau}\left\|v(t)\right\|_{\mathcal{X}}.$

  * •

$L^{1}(0,\tau;L^{2}(\Omega))$ denotes the Banach space of all measurable
functions $v:[0,\tau]\to L^{2}(\Omega)$ such that
$\int_{0}^{\tau}\left\|v(t)\right\|\,dt<\infty$ and the norm is given by
$\left\|v\right\|_{L^{1}(0,\tau;L^{2}(\Omega))}:=\int_{0}^{\tau}\left\|v(t)\right\|\,dt.$

  * •

$L^{\infty}(0,\tau;L^{2}(\Omega))$ denotes the Banach space of all measurable
functions $v:[0,\tau]\to L^{2}(\Omega)$ such that there exists $M>0$
satisfying $\left\|v(t)\right\|\leq M$ for almost all $t\in[0,\tau],$ and the
norm is given by
$\left\|v\right\|_{L^{\infty}(0,\tau;L^{2}(\Omega))}:=\inf\left\\{M:\,\left\|v(t)\right\|\leq
M\,\,\text{for almost all}\,\,t\in[0,\tau]\right\\}.$

  * •

For $w\in L^{1}(0,\tau;L^{2}(\Omega))$ we shall use the standard notation
$w(t)$ to denote the function $w(\cdot,t).$ Moreover, we may denote $\langle
w(t),\varphi_{j}\rangle$ by $w_{j}(t)$ and $\langle
F(t,w((t)),\varphi_{j}\rangle$ by $F_{j}(t,w(t)).$

  * •

For $f\in L^{2}(\Omega)$ we shall denote $\langle f,\varphi_{j}\rangle$ by
$f_{j}$.

Recall that we have assumed that the FVP (1.1) has a unique solution $u\in
C([0,\tau];L^{2}(\Omega))$ for the exact final value $g\in L^{2}(\Omega)$.
Thus, taking the $L^{2}(\Omega)$ inner product on both sides of the governing
equation of (1.1) with $\varphi_{j}$, we have for all $j\in{\mathbb{N}},$

(2.5)
$\begin{cases}\frac{d}{dt}\left(u_{j}(t)\right)+\lambda_{j}u_{j}(t)=F_{j}(t,u(t))+\int_{t}^{\tau}u_{j}(s)\,ds,&\quad
t\in(0,\tau)\\\ u_{j}(\tau)=g_{j}.\end{cases}$

Now, it can be verified easily that

(2.6)
$u_{j}(t)=e^{\lambda_{j}(\tau-t)}g_{j}-\int_{t}^{\tau}e^{\lambda_{j}(s-t)}F_{j}(s,u(s))\,ds-\int_{t}^{\tau}e^{\lambda_{j}(s-t)}\int_{s}^{\tau}u_{j}(\xi)\,d\xi\,ds.$

Thus, the solution $u$ to the FVP (1.1) is

$u(t)=\sum_{j=1}^{\infty}u_{j}(t)\varphi_{j}$

where $u_{j}$ is as given in (2.6).

Moreover, for $\psi\in L^{2}(\Omega)$ if we define
$\mathcal{S}(t)\psi:=\sum_{j=1}^{\infty}e^{\lambda_{j}t}\langle\psi,\varphi_{j}\rangle\varphi_{j}$,
then we have

(2.7)
$u(t)=\mathcal{S}(\tau-t)g-\int_{t}^{\tau}\mathcal{S}(s-t)F(s,u(s))\,ds-\int_{t}^{\tau}\mathcal{S}(s-t)\int_{s}^{\tau}u(\xi)\,d\xi\,ds.$

The above integral equation form of the exact solution will be helpful in our
analysis in the upcoming sections. In fact, we obtain the approximations for
the exact solution $u$ by replacing $\mathcal{S}$ with some finite rank
operators from $L^{2}(\Omega)$ to $L^{2}(\Omega).$

We end this section by recalling the definition of Gevrey space (see for e.g.,
[1, 23]) which shall play the role of source sets in deriving error estimates.

For $p,q\geq 0,$ the Gevrey space $\mathbb{G}_{p,q}$ is defined as

$\mathbb{G}_{p,q}:=\left\\{\psi\in
L^{2}(\Omega):\sum_{j=1}^{\infty}\lambda_{j}^{2p}e^{2q\lambda_{j}}\left|\langle\psi,\varphi_{j}\rangle\right|^{2}<\infty\right\\}.$

It is well known that $\mathbb{G}_{p,q}$ is a Hilbert space with respect to
the inner product

$\langle\chi,\psi\rangle_{\mathbb{G}_{p,q}}:=\sum_{j=1}^{\infty}\lambda_{j}^{2p}e^{2q\lambda_{j}}\langle\chi,\varphi_{j}\rangle\langle\varphi_{j},\psi\rangle,\,\,\chi,\psi\in\mathbb{G}_{p,q},$

so that the norm is given by

$\left\|\psi\right\|_{\mathbb{G}_{p,q}}^{2}=\sum_{j=1}^{\infty}\lambda_{j}^{2p}e^{2q\lambda_{j}}\left|\langle\psi,\varphi_{j}\rangle\right|^{2}.$

##  3\. Preparatory results

We begin by proving a version of Grönwalls’ type inequality for iterated
integrals that will play the crucial role in obtaining stability estimates in
later part of the paper. This version, as mentioned in the introduction, is
perhaps proposed for the first time. We should mention that this is motivated
from a different version of Grönwalls’ inequality for iterated integrals
established in [18, Theorem 1.4.1].

######  LEMMA 3.1.

Let $U:[0,\tau]\to{\mathbb{R}}$ be a non-negative continuous function and
$c_{0},c_{1}>0$ be fixed. Suppose that

$U(t)\leq
c_{0}+c_{1}\int_{t}^{\tau}\left(U(s)+\int_{s}^{\tau}U(\xi)\,d\xi\right)ds,\quad\,0\leq
t\leq\tau.$

Then

$U(t)\leq c_{0}e^{(1+c_{1})(\tau-t)},\,\,0\leq t\leq\tau.$

###### Proof.

Let

$X(t):=c_{0}+c_{1}\int_{t}^{\tau}\left(U(s)+\int_{s}^{\tau}U(\xi)\,d\xi\right)ds,\,\,0\leq
t\leq\tau.$

Then it follows that $U(t)\leq X(t)$ for all $t\in[0,\tau]$ and
$X(\tau)=c_{0}.$ Thus,

$\displaystyle X^{\prime}(t)$ $\displaystyle=$ $\displaystyle-
c_{1}U(t)-c_{1}\int_{t}^{\tau}U(s)\,ds\geq-
c_{1}\left(X(t)+\int_{t}^{\tau}X(s)\,ds\right).$

Let

$V(t):=X(t)+\int_{t}^{\tau}X(s)\,ds,\,\,t\in[0,\tau].$

Clearly, $V(\tau)=X(\tau)=c_{0},\,0\leq X(t)\leq V(t)$ and $X^{\prime}(t)\geq-
c_{1}V(t).$ Thus,

$\displaystyle V^{\prime}(t)=X^{\prime}(t)-X(t)\geq-
c_{1}V(t)-V(t)=-(1+c_{1})V(t).$

That is,

$V^{\prime}(t)+(1+c_{1})V(t)\geq 0,\,\,t\in(0,\tau),\,\,V(\tau)=c_{0}.$

Thus, it follows that

$e^{(1+c_{1})\tau}V(\tau)-e^{(1+c_{1})t}V(t)\geq 0$

and hence

$V(t)\leq c_{0}e^{(1+c_{1})(\tau-t)}.$

Now using the relation $U(t)\leq X(t)\leq V(t),$ we have

$U(t)\leq c_{0}e^{(1+c_{1})(\tau-t)}.$

This completes the proof. ∎

######  Remark 3.2.

In Lemma 3.1 if we replace $c_{0}$ and $c_{1}$ by two non-negative continuous
functions, say $\alpha(t)$ and $\beta(t),$ respectively, then also one can
infer similar conclusion in terms of integrals of $\alpha(t)$ and $\beta(t).$
Since those generalizations are not needed for our purpose, we are not
including that. $\lozenge$

Recall that $g\in L^{2}(\Omega)$ is the exact final value and $u\in
C([0,\tau];L^{2}(\Omega))$ is the corresponding unique solution of the FVP
(1.1). We have seen in Example 2.2 that the problem of recovering $u(t)$ for
(1.1) from the knowledge of final value $g$ is an ill-posed problem for all
$0\leq t<\tau.$ That is, a small perturbation in the final value may lead to a
large deviation in the corresponding solution of the FVP. Since the accessible
final value will be typically a measured data, noise in the final value is
inevitable. Thus, it is reasonable to obtain some approximations for the exact
solution from the knowledge of the measured final value/noisy data. Because of
the ill-posedness of the problem, we have to employ a regularization method in
order to obtain stable approximations. In this regard, we consider a Fourier
truncation method for the regularization purpose. More specifically, we shall
replace $\mathcal{S}$ in (2.7) by some finite rank operator described below,
and then try to solve some non linear integral equation. The obtained solution
will serve as the regularized approximations. Keeping this in mind, in this
section we shall define those finite rank operators and study various
important properties associated with it.

For $N\in{\mathbb{N}}$ and $t\in[0,\tau],$ we define
$\mathcal{S}_{N}(t):L^{2}(\Omega)\to L^{2}(\Omega)$ by

(3.1)
$\mathcal{S}_{N}(t)\psi:=\sum_{j=1}^{N}e^{\lambda_{j}t}\langle\psi,\varphi_{j}\rangle\varphi_{j}.$

From the definition of $\mathcal{S}_{N}$ it is easy to verify the following
result.

######  LEMMA 3.3.

For $N\in{\mathbb{N}}$ and $t\in[0,\tau],$ let $\mathcal{S}_{N}(t)$ be as
defined in (3.1). Then the following holds:

  1. (i)

$\mathcal{S}_{N}(t)$ is a bounded linear operator with
$\left\|S_{N}(t)\right\|_{L^{2}\to L^{2}}\leq e^{t\lambda_{N}}$ for every
$t\in[0,\tau].$

  2. (ii)

$t\mapsto\mathcal{S}_{N}(t)\in C([0,\tau];\mathcal{B}(L^{2}(\Omega)))$

For $v\in C([0,\tau];L^{2}(\Omega)),$ we define

(3.2)
$\mathfrak{T}v=\mathcal{S}_{N}(\tau-t)g-\int_{t}^{\tau}\mathcal{S}_{N}(s-t)F(s,v(s))\,ds-\int_{t}^{\tau}\mathcal{S}_{N}(s-t)\int_{s}^{\tau}v(\xi)\,d\xi\,ds.$

######  LEMMA 3.4.

Let $F:[0,\tau]\times L^{2}(\Omega)\to L^{2}(\Omega)$ satsfies $(\bf F{1})$
and $(\bf F{2})$. Let $v\in L^{1}(0,\tau;L^{2}(\Omega)).$ Then the following
hold:

  1. (i)

$t\mapsto F(t,v(t))\in L^{1}(0,\tau;L^{2}(\Omega)).$

  2. (ii)

$t\mapsto\int_{t}^{\tau}\mathcal{S}_{N}(s-t)\int_{s}^{\tau}v(\xi)\,d\xi\,ds\in
C([0,\tau];L^{2}(\Omega)).$

  3. (iii)

$t\mapsto\int_{t}^{\tau}\mathcal{S}_{N}(s-t)F(s,v(s))\,ds\,\in
C([0,\tau];L^{2}(\Omega)).$

###### Proof.

Let $\kappa>0,\phi\in L^{2}(\Omega)$ be as in $(\bf F{1})$ and $(\bf F{2})$,
respectively.

(i). The proof follows by observing that

$\displaystyle\int_{0}^{\tau}\left\|F(s,v(s))\right\|\,ds$ $\displaystyle\leq$
$\displaystyle\int_{0}^{\tau}\left(\left\|F(s,\phi)\right\|+\left\|F(s,\phi)-F(s,v(s))\right\|\right)ds$
$\displaystyle\leq$
$\displaystyle\int_{0}^{\tau}\left\|F(s,\phi)\right\|\,ds+\kappa\int_{0}^{\tau}\left\|\phi-v(s)\right\|\,ds.$

(ii). Let
$\Psi(t):=\int_{t}^{\tau}\mathcal{S}_{N}(s-t)\int_{s}^{\tau}v(\xi)\,d\xi\,ds,\,0\leq
t\leq\tau.$ Then for each $t\in[0,\tau]$ it follows that $\Psi(t)\in
L^{2}(\Omega).$ Indeed, by Lemma 3.3 (i), we have

$\left\|\Psi(t)\right\|\leq\int_{t}^{\tau}\left\|\mathcal{S}_{N}(s-t)\int_{s}^{\tau}v(\xi)\,d\xi\right\|\,ds\leq\int_{t}^{\tau}e^{\lambda_{N}(s-t)}\int_{s}^{\tau}\left\|v(\xi)\right\|\,d\xi\,ds.$

We now show that $t\mapsto\Psi(t),\,0\leq t\leq\tau,$ is continuous.

Let $t_{0}\in[0,\tau].$ Then for $t\leq t_{0},$ we have

$\displaystyle\Psi(t)-\Psi(t_{0})$ $\displaystyle=$
$\displaystyle\int_{t}^{\tau}\mathcal{S}_{N}(s-t)\int_{s}^{\tau}v(\xi)\,d\xi\,ds-\int_{t_{0}}^{\tau}\mathcal{S}_{N}(s-t_{0})\int_{s}^{\tau}v(\xi)\,d\xi\,ds$
$\displaystyle=$
$\displaystyle\int_{t}^{\tau}\left(\mathcal{S}_{N}(s-t)-\mathcal{S}_{N}(s-t_{0})\right)\int_{s}^{\tau}v(\xi)\,d\xi\,ds+\int_{t}^{\tau}\mathcal{S}_{N}(s-t_{0})\int_{s}^{\tau}v(\xi)\,d\xi\,ds$
$\displaystyle\quad\quad-\int_{t_{0}}^{\tau}\mathcal{S}_{N}(s-t_{0})\int_{s}^{\tau}v(\xi)\,d\xi\,ds$
$\displaystyle=$
$\displaystyle\int_{t}^{\tau}\left(\mathcal{S}_{N}(s-t)-\mathcal{S}_{N}(s-t_{0})\right)\int_{s}^{\tau}v(\xi)\,d\xi\,ds+\int_{t}^{t_{0}}\mathcal{S}_{N}(s-t_{0})\int_{s}^{\tau}v(\xi)\,d\xi\,ds.$

Therefore,

$\displaystyle\left\|\Psi(t)-\Psi(t_{0})\right\|$ $\displaystyle\leq$
$\displaystyle\int_{t}^{\tau}\left\|\mathcal{S}_{N}(s-t)-\mathcal{S}_{N}(s-t_{0})\right\|_{L^{2}\to
L^{2}}\int_{s}^{\tau}\left\|v(\xi)\right\|\,d\xi\,ds$
$\displaystyle\quad\quad+\int_{t}^{t_{0}}\left\|\mathcal{S}_{N}(s-t_{0})\right\|_{L^{2}\to
L^{2}}\int_{s}^{\tau}\left\|v(\xi)\right\|\,d\xi\,ds$ $\displaystyle\leq$
$\displaystyle\left(\int_{0}^{\tau}\left\|\mathcal{S}_{N}(s-t)-\mathcal{S}_{N}(s-t_{0})\right\|_{L^{2}\to
L^{2}}\,ds+\int_{t}^{t_{0}}\left\|\mathcal{S}_{N}(s-t_{0})\right\|_{L^{2}\to
L^{2}}\,ds\right)\left\|v\right\|_{L^{1}(0,\tau;L^{2}(\Omega))}.$

For $t_{0}\leq t,$ similarly we obtain

$\displaystyle\left\|\Psi(t)-\Psi(t_{0})\right\|$ $\displaystyle\leq$
$\displaystyle\left(\int_{0}^{\tau}\left\|\mathcal{S}_{N}(s-t)-\mathcal{S}_{N}(s-t_{0})\right\|_{L^{2}\to
L^{2}}\,ds+\int_{t}^{t_{0}}\left\|\mathcal{S}_{N}(s-t_{0})\right\|_{L^{2}\to
L^{2}}\,ds\right)\left\|v\right\|_{L^{1}(0,\tau;L^{2}(\Omega))}.$

Now, from Lemma 3.3 (ii) it follows that
$\left\|\mathcal{S}_{N}(s-t)-\mathcal{S}_{N}(s-t_{0})\right\|\to 0$ as $t\to
t_{0},$ therefore by dominated convergence theorem it follows that
$\int_{0}^{\tau}\left\|\mathcal{S}_{N}(s-t)-\mathcal{S}_{N}(s-t_{0})\right\|\,ds\to
0$ as $t\to t_{0}.$ Also,
$\left|\int_{t}^{t_{0}}\left\|\mathcal{S}_{N}(s-t_{0})\right\|\,ds\right|\to
0$ as $t\to t_{0}.$ Thus, it follows that for each $t_{0}\in[0,\tau],$
$\left\|\Psi(t)-\Psi(t_{0})\right\|\to 0$ as $t\to t_{0}.$

(iii). Let
$\Phi(t):=\int_{t}^{\tau}\mathcal{S}_{N}(s-t)F(s,v(s))\,ds,\,\,0\leq
t\leq\tau.$ By Lemma 3.3 (i), it follows that

$\left\|\Phi(t)\right\|\leq\int_{t}^{\tau}e^{\lambda_{N}(s-t)}\left\|F(s,v(s))\right\|\,ds.$

Thus, by Lemma 3.4 (i) it follows that $\Phi(t)\in L^{2}(\Omega)$ for all
$t\in[0,\tau].$ We now show that $t\mapsto\Phi(t)$ is continuous. Let
$t_{0}\in[0,\tau].$ Then similar to the proof of Lemma 3.4 (ii), we obtain

$\displaystyle\left\|\Phi(t)-\Phi(t_{0})\right\|$ $\displaystyle\leq$
$\displaystyle\int_{0}^{\tau}\left\|\mathcal{S}_{N}(s-t)-\mathcal{S}_{N}(s-t_{0})\right\|_{L^{2}\to
L^{2}}\left\|F(s,v(s))\right\|\,ds$
$\displaystyle\quad\quad+\left|\int_{t}^{t_{0}}\left\|\mathcal{S}_{N}(s-t_{0})\right\|_{L^{2}\to
L^{2}}\left\|F(s,v(s))\right\|\,ds\right|.$

Thus, we conclude that $\left\|\Phi(t)-\Phi(t_{0})\right\|\to 0$ as $t\to
t_{0}.$ ∎

######  Theorem 3.5.

Let $F:[0,\tau]\times L^{2}(\Omega)\to L^{2}(\Omega)$ satisfies $(\bf F{1})$
and $(\bf F{2})$, and $\mathfrak{T}$ be as in (3.2). If $v\in
C([0,\tau];L^{2}(\Omega)),$ then $\mathfrak{T}v\in C([0,\tau];L^{2}(\Omega)).$

###### Proof.

By Lemma 3.3 (ii) we observe that $t\mapsto\mathcal{S}_{N}(\tau-t)g\in
C([0,\tau];L^{2}(\Omega)).$ Now the proof follows by Lemma 3.4. ∎

######  Theorem 3.6.

Let $F$ and $\mathfrak{T}:C([0,\tau];L^{2}(\Omega))\to
C([0,\tau];L^{2}(\Omega))$ be as in Theorem 3.5. Then there exists a unique
$v_{0}\in C([0,\tau];L^{2}(\Omega))$ such that $\mathfrak{T}v_{0}=v_{0}.$

###### Proof.

For $0\leq t\leq\tau,$ we have

$\displaystyle\left\|\mathfrak{T}v_{1}(t)-\mathfrak{T}v_{2}(t)\right\|$
$\displaystyle\leq$
$\displaystyle\int_{t}^{\tau}\left\|\mathcal{S}_{N}(s-t)\left(F(s,v_{1}(s))-F(s,v_{2}(s))\right)\right\|\,ds$
$\displaystyle\quad+\int_{t}^{\tau}\left\|\mathcal{S}_{N}(s-t)\left(\int_{s}^{\tau}\left(v_{1}(\xi)-v_{2}(\xi)\right)\,d\xi\right)\right\|\,ds$
$\displaystyle\leq$
$\displaystyle\int_{t}^{\tau}e^{\lambda_{N}(s-t)}\left(\left\|F(s,v_{1}(s))-F(s,v_{2}(s))\right\|+\int_{s}^{\tau}\left\|v_{1}(\xi)-v_{2}(\xi)\right\|\,d\xi\right)\,ds$
$\displaystyle\leq$
$\displaystyle\kappa\int_{t}^{\tau}e^{\lambda_{N}(s-t)}\left\|v_{1}(s)-v_{2}(s)\right\|\,ds+\left\|v_{1}-v_{2}\right\|_{C([0,\tau];L^{2}(\Omega))}\int_{t}^{\tau}e^{\lambda_{N}(s-t)}\int_{s}^{\tau}\,d\xi\,ds$
$\displaystyle\leq$ $\displaystyle
e^{\lambda_{N}\tau}\left\|v_{1}-v_{2}\right\|_{C([0,\tau];L^{2}(\Omega))}\left(\kappa\left(\tau-t\right)+\frac{\left(\tau-t\right)^{2}}{2}\right)$
$\displaystyle\leq$ $\displaystyle
e^{\lambda_{N}\tau}\left\|v_{1}-v_{2}\right\|_{C([0,\tau];L^{2}(\Omega))}\left(\kappa\left(\tau-t\right)+\tau\left(\tau-t\right)\right)$

and hence

$\left\|\mathfrak{T}v_{1}(t)-\mathfrak{T}v_{2}(t)\right\|\leq\kappa_{0}e^{\lambda_{N}\tau}\left(1+\tau\right)\left(\tau-t\right)\left\|v_{1}-v_{2}\right\|_{C([0,\tau];L^{2}(\Omega))},$

where $\kappa_{0}=\max\\{\kappa,1\\}.$ Using this, we obtain

$\displaystyle\left\|\mathfrak{T}^{2}v_{1}(t)-\mathfrak{T}^{2}v_{2}(t)\right\|$
$\displaystyle\leq$
$\displaystyle\int_{t}^{\tau}\left\|\mathcal{S}_{N}(s-t)\left(F(s,\mathfrak{T}v_{1}(s))-F(s,\mathfrak{T}v_{2}(s))\right)\right\|\,ds$
$\displaystyle\quad+\int_{t}^{\tau}\left\|\mathcal{S}_{N}(s-t)\left(\int_{s}^{\tau}\left(\mathfrak{T}v_{1}(\xi)-\mathfrak{T}v_{2}(\xi)\right)\,d\xi\right)\right\|\,ds$
$\displaystyle\leq$
$\displaystyle\int_{t}^{\tau}e^{\lambda_{N}(s-t)}\left(\left\|F(s,\mathfrak{T}v_{1}(s))-F(s,\mathfrak{T}v_{2}(s))\right\|+\int_{s}^{\tau}\left\|\mathfrak{T}v_{1}(\xi)-\mathfrak{T}v_{2}(\xi)\right\|\,d\xi\right)ds$
$\displaystyle\leq$
$\displaystyle\kappa\int_{t}^{\tau}e^{\lambda_{N}(s-t)}\left\|\mathfrak{T}v_{1}(s)-\mathfrak{T}v_{2}(s)\right\|\,ds+\int_{t}^{\tau}e^{\lambda_{N}(s-t)}\int_{s}^{\tau}\left\|\mathfrak{T}v_{1}(\xi)-\mathfrak{T}v_{2}(\xi)\right\|\,d\xi\,ds$
$\displaystyle\leq$
$\displaystyle\kappa_{0}e^{\lambda_{N}\tau}\left(\int_{t}^{\tau}\left\|\mathfrak{T}v_{1}(s)-\mathfrak{T}v_{2}(s)\right\|\,ds+\tau\int_{t}^{\tau}\left\|\mathfrak{T}v_{1}(\xi)-\mathfrak{T}v_{2}(\xi)\right\|\,d\xi\right)$
$\displaystyle=$
$\displaystyle\kappa_{0}e^{\lambda_{N}\tau}\left(1+\tau\right)\int_{t}^{\tau}\left\|\mathfrak{T}v_{1}(s)-\mathfrak{T}v_{2}(s)\right\|\,ds$
$\displaystyle\leq$
$\displaystyle\left(\kappa_{0}e^{\lambda_{N}\tau}\left(1+\tau\right)\right)^{2}\left\|v_{1}-v_{2}\right\|_{C([0,\tau];L^{2}(\Omega))}\int_{t}^{\tau}\left(\tau-s\right)\,ds$
$\displaystyle=$
$\displaystyle\left(\kappa_{0}e^{\lambda_{N}\tau}\left(1+\tau\right)\right)^{2}\frac{\left(\tau-t\right)^{2}}{2!}\left\|v_{1}-v_{2}\right\|_{C([0,\tau];L^{2}(\Omega))}.$

Thus, inductively, for all $m\in{\mathbb{N}}$ we have for $0\leq t\leq\tau,$

$\left\|\mathfrak{T}^{m}v_{1}(t)-\mathfrak{T}^{m}v_{2}(t)\right\|\leq\frac{\left(\kappa_{0}e^{\lambda_{N}\tau}\left(1+\tau\right)\left(\tau-t\right)\right)^{m}}{m!}\left\|v_{1}-v_{2}\right\|_{C([0,\tau];L^{2}(\Omega))},$

and hence

$\left\|\mathfrak{T}^{m}v_{1}-\mathfrak{T}^{m}v_{2}\right\|_{C([0,\tau];L^{2}(\Omega))}\leq\frac{\left(\kappa_{0}e^{\lambda_{N}\tau}\left(1+\tau\right)\tau\right)^{m}}{m!}\left\|v_{1}-v_{2}\right\|_{C([0,\tau];L^{2}(\Omega))}.$

Since
$\frac{\left(\kappa_{0}e^{\lambda_{N}\tau}\left(1+\tau\right)\tau\right)^{m}}{m!}\to
0$ as $m\to\infty,$ there exists $m_{0}\in{\mathbb{N}}$ such that
$\mathfrak{T}^{m_{0}}$ is a contraction. Therefore, by Banach fixed point
theorem, there exists a unique $v_{0}\in C([0,\tau];L^{2}(\Omega))$ such that
$\mathfrak{T}^{m_{0}}v_{0}=v_{0}.$ This shows that
$\mathfrak{T}\left(\mathfrak{T}^{m_{0}}v_{0}\right)=\mathfrak{T}v_{0},$ that
is, $\mathfrak{T}v_{0}$ is a fixed point of $\mathfrak{T}^{m_{0}}.$ Therefore,
by uniqueness, we have $\mathfrak{T}v_{0}=v_{0}.$ Clearly, this $v_{0}$ is the
unique fixed point of $\mathfrak{T}.$ ∎

######  Remark 3.7.

Note that the unique $v_{0}\in C([0,\tau];L^{2}(\Omega))$ as obtained in
Theorem 3.6 depends on $N$. Thus, in order to visualize that dependence, from
now onwards we shall denote that $v_{0}$ by $u^{N}.$ That is, $u^{N}\in
C([0,\tau];L^{2}(\Omega))$ is the unique element such that
$\mathfrak{T}u^{N}=u^{N}.$ $\lozenge$

##  4\. Regularization and error analysis

Let $u^{N}\in C([0,\tau];L^{2}(\Omega))$ be as described in Remark 3.7. In
this section, we shall see that $u^{N}$ are indeed regularized approximations.
Moreover, we shall obtain estimates for the quantity
$\left\|u(t)-u^{N}(t)\right\|$ by assuming that $u$ belongs to some Gevrey
spaces. Note that, in order to obtain some estimate of
$\left\|u(t)-u^{N}(t)\right\|$ in terms of the regularization parameter $N$ so
that the estimate tends to zero as $N\to\infty,$ it is necessary to assume
some apriori condition on the solution $u$ otherwise no such estimate can be
derived (see for e.g., [19]). Finally, we provide a parameter choice strategy,
that is a procedure to choose the regularization parameter $N$, and obtain the
corresponding error estimates.

######  Theorem 4.1.

Let $F:[0,\tau]\times L^{2}(\Omega)\to L^{2}(\Omega)$ satisfies $(\bf F{1})$
and $(\bf F{2})$. Let $p,q>0,$ and $\kappa_{0}=\max\\{\kappa,1\\},$ where
$\kappa$ is as in $(\bf F{1})$. For $N\in{\mathbb{N}},$ let $u^{N}\in
C([0,\tau];L^{2}(\Omega))$ be as mentioned in Remark 3.7. Then the following
hold:

  1. (i)

If $u\in L^{\infty}(0,\tau;\mathbb{G}_{p,\tau})$ then

$\left\|u(t)-u^{N}(t)\right\|\leq
e^{\left(1+\kappa_{0}\right)\left(\tau-t\right)}\left\|u\right\|_{L^{\infty}(0,\tau;\mathbb{G}_{p,\tau})}\lambda_{N}^{-p}e^{-\lambda_{N}t},\,\,0\leq
t\leq\tau.$

  2. (ii)

If $u\in L^{\infty}(0,\tau;\mathbb{G}_{0,q+\tau})$ then

$\left\|u(t)-u^{N}(t)\right\|\leq
e^{\left(1+\kappa_{0}\right)\left(\tau-t\right)}\left\|u\right\|_{L^{\infty}(0,\tau;\mathbb{G}_{0,q+\tau})}e^{-(q+t)\lambda_{N}},\,\,0\leq
t\leq\tau.$

###### Proof.

Recall that the exact solution $u$ of (1.1) satisfies

$u(t)=\mathcal{S}(\tau-t)g-\int_{t}^{\tau}\mathcal{S}(s-t)F(s,u(s))\,ds-\int_{t}^{\tau}\mathcal{S}(s-t)\int_{s}^{\tau}u(\xi)\,d\xi\,ds$

and hence

$\sum_{j=1}^{N}\langle
u(t),\varphi_{j}\rangle\varphi_{j}=\mathcal{S}_{N}(\tau-t)g-\int_{t}^{\tau}\mathcal{S}_{N}(s-t)F(s,u(s))\,ds-\int_{t}^{\tau}\mathcal{S}_{N}(s-t)\int_{s}^{\tau}u(\xi)\,d\xi\,ds.$

Thus,

$\displaystyle\left\|u(t)-u^{N}(t)\right\|$ $\displaystyle\leq$
$\displaystyle\left\|u(t)-\sum_{j=1}^{N}u_{j}(t)\varphi_{j}\right\|+\left\|\sum_{j=1}^{N}u_{j}(t)\varphi_{j}-u^{N}(t)\right\|$
$\displaystyle\leq$
$\displaystyle\sqrt{\sum_{j=N+1}^{\infty}\left|u_{j}(t)\right|^{2}}+\int_{t}^{\tau}\left\|\mathcal{S}_{N}(s-t)\left(F(s,u^{N}(s))-F(s,u(s))\right)\right\|\,ds$
$\displaystyle\quad\quad+\int_{t}^{\tau}\left\|\mathcal{S}_{N}(s-t)\left(\int_{s}^{\tau}\left(u^{N}(\xi)-u(\xi)\right)d\xi\right)\right\|\,ds$

(i). Let $u\in L^{\infty}(0,\tau;\mathbb{G}_{p,\tau}).$ Then we have

$\displaystyle\left\|u(t)-u^{N}(t)\right\|$ $\displaystyle\leq$
$\displaystyle\sqrt{\sum_{j=N+1}^{\infty}\lambda_{j}^{2p}e^{2t\lambda_{j}}\lambda_{j}^{-2p}e^{-2t\lambda_{j}}\left|u_{j}(t)\right|^{2}}+\kappa\int_{t}^{\tau}e^{\lambda_{N}(s-t)}\left\|u^{N}(s)-u(s)\right\|\,ds$
$\displaystyle\quad\quad+\int_{t}^{\tau}e^{\lambda_{N}(s-t)}\int_{s}^{\tau}\left\|u^{N}(\xi)-u(\xi)\right\|\,d\xi\,ds$
$\displaystyle\leq$
$\displaystyle\lambda_{N}^{-p}e^{-t\lambda_{N}}\left\|u(t)\right\|_{\mathbb{G}_{p,\tau}}$
$\displaystyle\quad\quad+\kappa_{0}\left(\int_{t}^{\tau}e^{\lambda_{N}(s-t)}\left(\left\|u^{N}(s)-u(s)\right\|+\int_{s}^{\tau}\left\|u^{N}(\xi)-u(\xi)\right\|\,d\xi\right)\,ds\right).$

Thus,

$\displaystyle e^{t\lambda_{N}}\left\|u(t)-u^{N}(t)\right\|$
$\displaystyle\leq$
$\displaystyle\lambda_{N}^{-p}\left\|u\right\|_{L^{\infty}(0,\tau;\mathbb{G}_{p,\tau})}$
$\displaystyle\quad+\kappa_{0}\left(\int_{t}^{\tau}e^{s\lambda_{N}}\left(\left\|u^{N}(s)-u(s)\right\|+\int_{s}^{\tau}\left\|u^{N}(\xi)-u(\xi)\right\|\,d\xi\right)\,ds\right).$

Now, note that

$\displaystyle\int_{t}^{\tau}e^{s\lambda_{N}}\int_{s}^{\tau}\left\|u^{N}(\xi)-u(\xi)\right\|\,d\xi\,ds$
$\displaystyle=$
$\displaystyle\int_{t}^{\tau}e^{s\lambda_{N}}\int_{s}^{\tau}e^{\xi\lambda_{N}}e^{-\xi\lambda_{N}}\left\|u^{N}(\xi)-u(\xi)\right\|\,d\xi\,ds$
$\displaystyle\leq$
$\displaystyle\int_{t}^{\tau}e^{s\lambda_{N}}\int_{s}^{\tau}e^{-s\lambda_{N}}e^{\xi\lambda_{N}}\left\|u^{N}(\xi)-u(\xi)\right\|\,d\xi\,ds$
$\displaystyle=$
$\displaystyle\int_{t}^{\tau}\int_{s}^{\tau}e^{\xi\lambda_{N}}\left\|u^{N}(\xi)-u(\xi)\right\|\,d\xi\,ds.$

Thus, we obtain

$\displaystyle e^{t\lambda_{N}}\left\|u(t)-u^{N}(t)\right\|$
$\displaystyle\leq$
$\displaystyle\lambda_{N}^{-p}\left\|u\right\|_{L^{\infty}(0,\tau;\mathbb{G}_{p,\tau})}$
$\displaystyle\quad\quad+\kappa_{0}\left(\int_{t}^{\tau}e^{s\lambda_{N}}\left\|u(s)-u^{N}(s)\right\|\,ds+\int_{t}^{\tau}\int_{s}^{\tau}e^{\xi\lambda_{N}}\left\|u(\xi)-u^{N}(\xi)\right\|\,d\xi\,ds\right).$

Therefore, by Lemma 3.1, we obtain

$e^{t\lambda_{N}}\left\|u(t)-u^{N}(t)\right\|\leq\lambda_{N}^{-p}\left\|u\right\|_{L^{\infty}(0,\tau;\mathbb{G}_{p,\tau})}e^{\left(1+\kappa_{0}\right)\left(\tau-t\right)}.$

Thus,

$\left\|u(t)-u^{N}(t)\right\|\leq
e^{\left(1+\kappa_{0}\right)\left(\tau-t\right)}\left\|u\right\|_{L^{\infty}(0,\tau;\mathbb{G}_{p,\tau})}\lambda_{N}^{-p}e^{-\lambda_{N}t}.$

This completes the proof of (i).

(ii). Let $u\in L^{\infty}(0,\tau;\mathbb{G}_{0,q+\tau}).$ Then similar to the
above argument we obtain

$\displaystyle\left\|u(t)-u^{N}(t)\right\|$ $\displaystyle\leq$
$\displaystyle\sqrt{\sum_{j=N+1}^{\infty}e^{2(q+t)\lambda_{j}}e^{-2(q+t)\lambda_{j}}\left|u_{j}(t)\right|^{2}}+\kappa\int_{t}^{\tau}e^{\lambda_{N}(s-t)}\left\|u^{N}(s)-u(s)\right\|\,ds$
$\displaystyle\quad\quad+\int_{t}^{\tau}e^{\lambda_{N}(s-t)}\int_{s}^{\tau}\left\|u^{N}(\xi)-u(\xi)\right\|\,d\xi\,ds$
$\displaystyle\leq$ $\displaystyle
e^{-(q+t)\lambda_{N}}\left\|u(t)\right\|_{\mathbb{G}_{0,q+\tau}}$
$\displaystyle\quad\quad+\kappa_{0}\left(\int_{t}^{\tau}e^{\lambda_{N}(s-t)}\left(\left\|u^{N}(s)-u(s)\right\|+\int_{s}^{\tau}\left\|u^{N}(\xi)-u(\xi)\right\|\,d\xi\right)\,ds\right),$

and hence

$\displaystyle e^{t\lambda_{N}}\left\|u(t)-u^{N}(t)\right\|$
$\displaystyle\leq$ $\displaystyle
e^{-q\lambda_{N}}\left\|u\right\|_{L^{\infty}(0,\tau;\mathbb{G}_{0,q+\tau})}$
$\displaystyle\quad\quad+\kappa_{0}\left(\int_{t}^{\tau}e^{s\lambda_{N}}\left\|u(s)-u^{N}(s)\right\|\,ds+\int_{t}^{\tau}\int_{s}^{\tau}e^{\xi\lambda_{N}}\left\|u(\xi)-u^{N}(\xi)\right\|\,d\xi\,ds\right).$

Therefore, by Lemma 3.1, we have

$\left\|u(t)-u^{N}(t)\right\|\leq
e^{\left(1+\kappa_{0}\right)\left(\tau-t\right)}\left\|u\right\|_{L^{\infty}(0,\tau;\mathbb{G}_{0,q+\tau})}e^{-(q+t)\lambda_{N}}.$

∎

We now analyse the case when we have a noisy final value in place of the exact
final value. For $\delta>0,$ let $g^{\delta}\in L^{2}(\Omega)$ be the noisy
final value satisfying (1.2).

For $v\in C([0,\tau];L^{2}(\Omega))$, let

(4.1)
$\mathfrak{T}_{\delta}v=\mathcal{S}_{N}(\tau-t)g^{\delta}-\int_{t}^{\tau}\mathcal{S}_{N}(s-t)F(s,v(s))\,ds-\int_{t}^{\tau}\mathcal{S}_{N}(s-t)\int_{s}^{\tau}v(\xi)\,d\xi\,ds.$

Then following the arguments of Theorems 3.5 and 3.6 it is obvious that
$\mathfrak{T}_{\delta}:C([0,\tau];L^{2}(\Omega))\to C([0,\tau];L^{2}(\Omega))$
is well-defined and there exists a unique $u_{\delta}^{N}\in
C([0,\tau];L^{2}(\Omega))$ such that

(4.2) $\mathfrak{T}_{\delta}u^{N}_{\delta}=u^{N}_{\delta},$

that is,

$u^{N}_{\delta}(t)=\mathcal{S}_{N}(\tau-t)g^{\delta}-\int_{t}^{\tau}\mathcal{S}_{N}(s-t)F(s,u^{N}_{\delta}(s))\,ds-\int_{t}^{\tau}\mathcal{S}_{N}(s-t)\int_{s}^{\tau}u^{N}_{\delta}(\xi)\,d\xi\,ds.$

Having obtained $u^{N}_{\delta},$ we now obtain estimate for
$\left\|u^{N}(t)-u^{N}_{\delta}(t)\right\|.$

######  Theorem 4.2.

Let $F:[0,\tau]\times L^{2}(\Omega)\to L^{2}(\Omega)$ satisfies $(\bf F{1})$
and $(\bf F{2})$. Let $\kappa$ be as in $(\bf F{1})$ and
$\kappa_{0}=\max\\{\kappa,1\\}.$ Let $u^{N},u^{N}_{\delta}$ be as mentioned in
Remark 3.7 and (4.2), respectively. Then

$\left\|u^{N}(t)-u^{N}_{\delta}(t)\right\|\leq\delta
e^{\lambda_{N}(\tau-t)}e^{(1+\kappa_{0})(\tau-t)},\quad 0\leq t\leq\tau.$

###### Proof.

We have

$\displaystyle\left\|u^{N}(t)-u^{N}_{\delta}(t)\right\|$ $\displaystyle\leq$
$\displaystyle\left\|\mathcal{S}_{N}(\tau-t)\left(g-g^{\delta}\right)\right\|+\int_{t}^{\tau}\left\|\mathcal{S}_{N}(s-t)\left(F(s,u^{N}(s))-F(s,u^{N}_{\delta}(s))\right)\right\|\,ds$
$\displaystyle\quad\quad+\int_{t}^{\tau}\left\|\mathcal{S}_{N}(s-t)\int_{s}^{\tau}\left(u^{N}(\xi)-u^{N}_{\delta}(\xi)\right)d\xi\right\|\,ds$
$\displaystyle\leq$ $\displaystyle
e^{\lambda_{N}\left(\tau-t\right)}\delta+\int_{t}^{\tau}\kappa
e^{\lambda_{N}(s-t)}\left\|u^{N}(s)-u^{N}_{\delta}(s)\right\|\,ds+\int_{t}^{\tau}e^{\lambda_{N}(s-t)}\int_{s}^{\tau}\left\|u^{N}(\xi)-u^{N}_{\delta}(\xi)\right\|\,d\xi\,ds$
$\displaystyle\leq$ $\displaystyle e^{\lambda_{N}\left(\tau-t\right)}\delta$
$\displaystyle\quad\quad+\kappa_{0}\left(\int_{t}^{\tau}e^{\lambda_{N}(s-t)}\left\|u^{N}(s)-u^{N}_{\delta}(s)\right\|\,ds+\int_{t}^{\tau}e^{\lambda_{N}(s-t)}\int_{s}^{\tau}\left\|u^{N}(\xi)-u^{N}_{\delta}(\xi)\right\|\,d\xi\,ds\right).$

Thus,

$\displaystyle e^{\lambda_{N}t}\left\|u^{N}(t)-u^{N}_{\delta}(t)\right\|$
$\displaystyle\leq$ $\displaystyle
e^{\lambda_{N}\tau}\delta+\kappa_{0}\left(\int_{t}^{\tau}e^{\lambda_{N}s}\left\|u^{N}(s)-u^{N}_{\delta}(s)\right\|\,ds+\int_{t}^{\tau}e^{\lambda_{N}s}\int_{s}^{\tau}\left\|u^{N}(\xi)-u^{N}_{\delta}(\xi)\right\|\,d\xi\,ds\right)$
$\displaystyle\leq$ $\displaystyle
e^{\lambda_{N}\tau}\delta+\kappa_{0}\left(\int_{t}^{\tau}e^{\lambda_{N}s}\left\|u^{N}(s)-u^{N}_{\delta}(s)\right\|\,ds+\int_{t}^{\tau}\int_{s}^{\tau}e^{\lambda_{N}\xi}\left\|u^{N}(\xi)-u^{N}_{\delta}(\xi)\right\|\,d\xi\,ds\right).$

Therefore, by Lemma 3.1, we obtain

$e^{\lambda_{N}t}\left\|u^{N}(t)-u^{N}_{\delta}(t)\right\|\leq\delta
e^{\lambda_{N}\tau}e^{(1+\kappa_{0})(\tau-t)},$

that is,

$\left\|u^{N}(t)-u^{N}_{\delta}(t)\right\|\leq\delta
e^{\lambda_{N}(\tau-t)}e^{(1+\kappa_{0})(\tau-t)}.$

∎

Therefore combining Theorems 4.1 and 4.2, we have the following result.

######  Theorem 4.3.

Let $p,q>0,$ and $\kappa_{0}=\max\\{\kappa,1\\},$ where $\kappa$ is as in
$(\bf F{1})$. For $N\in{\mathbb{N}},$ let $u^{N},u^{N}_{\delta}$ be as in
Theorem 4.2. Then the following hold:

  1. (i)

If $u\in L^{\infty}(0,\tau;\mathbb{G}_{p,\tau})$ then

$\left\|u(t)-u^{N}_{\delta}(t)\right\|\leq
e^{\left(1+\kappa_{0}\right)\left(\tau-t\right)}\left\|u\right\|_{L^{\infty}(0,\tau;\mathbb{G}_{p,\tau})}\lambda_{N}^{-p}e^{-\lambda_{N}t}+\delta
e^{\lambda_{N}(\tau-t)}e^{(1+\kappa_{0})(\tau-t)},\quad 0\leq t\leq\tau.$

  2. (ii)

If $u\in L^{\infty}(0,\tau;\mathbb{G}_{0,q+\tau})$ then

$\left\|u(t)-u^{N}_{\delta}(t)\right\|\leq
e^{\left(1+\kappa_{0}\right)\left(\tau-t\right)}\left\|u\right\|_{L^{\infty}(0,\tau;\mathbb{G}_{0,q+\tau})}e^{-(q+t)\lambda_{N}}+\delta
e^{\lambda_{N}(\tau-t)}e^{(1+\kappa_{0})(\tau-t)},\quad 0\leq t\leq\tau.$

We now provide parameter choice strategies, that is, a strategy to choose the
regularization parameter $N$ depending on $\delta$ in such a way that
$N\to\infty$ and $\left\|u(t)-u^{N}_{\delta}(t)\right\|\to 0$ as $\delta\to
0.$ We first recall an important result from [22] that will be useful in the
upcoming analysis.

######  LEMMA 4.4.

[22, Lemma 3.3] Let $0<a<1,b,c,d$ be positive constants. Let
$\zeta:(0,a]\to{\mathbb{R}}$ be given by
$\zeta(s):=s^{b}\left(d\ln\frac{1}{s}\right)^{-c}.$ Then for the inverse
function $\zeta^{-1}(s),$ we have

$\zeta^{-1}(s)=s^{1/b}\left(\frac{d}{b}\ln\frac{1}{s}\right)^{c/b}\left(1+o(1)\right)\quad\text{for}\,\,s\to
0.$

For $p,q,\varrho>0$, let

(4.3) $\mathcal{K}_{p,\tau}=\left\\{v\in
C([0,\tau];L^{2}(\Omega)):\,\left\|v\right\|_{L^{\infty}(0,\tau;\mathbb{G}_{p,\tau})}\leq\varrho\right\\}$

and

(4.4) $\mathcal{M}_{0,q}=\left\\{v\in
C([0,\tau];L^{2}(\Omega)):\,\left\|v\right\|_{L^{\infty}(0,\tau;\mathbb{G}_{0,q+\tau})}\leq\varrho\right\\}.$

Also, let $\kappa_{1}=1+\kappa_{0},$ where $\kappa_{0}$ is as in Theorem 4.3.
Let $u\in\mathcal{K}_{p,\tau}.$ Let $e_{1},e_{2}>0$ be as in (2.2). Then from
Theorem 4.3, we obtain

$\displaystyle\left\|u(t)-u^{N}_{\delta}(t)\right\|$ $\displaystyle\leq$
$\displaystyle
e^{\kappa_{1}(\tau-t)}\left(\frac{\varrho}{\lambda_{N}^{p}e^{\lambda_{N}t}}+\delta
e^{\lambda_{N}(\tau-t)}\right)$ $\displaystyle\leq$ $\displaystyle
E_{1}e^{\kappa_{1}(\tau-t)}\left(\frac{\varrho}{N^{2p/d}\left(e^{N^{2/d}}\right)^{e_{1}t}}+\delta\left(e^{N^{2/d}}\right)^{e_{2}(\tau-t)}\right),$

where $E_{1}=\max\\{e_{1}^{-p},1\\}.$ For $r>0,$ let
$A(r)=\frac{\varrho}{r^{2p/d}}+\delta\left(e^{r^{2/d}}\right)^{\eta(t)},$
where $\eta(t)=e_{1}t+e_{2}(\tau-t),\,\,0\leq t\leq\tau.$ Note that $A(r)$
will attain minimum at $r_{\delta}$ if
$\frac{\varrho}{{r_{\delta}}^{2p/d}}=\delta\left(e^{{r_{\delta}}^{2/d}}\right)^{\eta(t)}$
that is, if
$\frac{\delta}{\varrho}=\left(\frac{1}{r_{\delta}}\right)^{2p/d}e^{-\eta(t){r_{\delta}}^{2/d}}.$
Let $s_{\delta}=e^{-{r_{\delta}}^{2/d}}.$ Then it follows that

$\frac{\delta}{\varrho}={s_{\delta}}^{\eta(t)}\left(\ln\frac{1}{s_{\delta}}\right)^{-p}.$

Thus, by Lemma 4.4, we have

$s_{\delta}=\left(\frac{\delta}{\varrho}\right)^{1/\eta(t)}\left(\frac{1}{\eta(t)}\ln\frac{\varrho}{\delta}\right)^{p/\eta(t)}\left(1+o(1)\right)\quad\text{for}\,\,\delta\to
0.$

Thus,

${r_{\delta}}^{2/d}=\ln\left(\left(\frac{\delta}{\varrho}\right)^{-1/\eta(t)}\left(\frac{1}{\eta(t)}\ln\frac{\varrho}{\delta}\right)^{-p/\eta(t)}\right)\left(1+o(1)\right)\quad\text{for}\,\,\delta\to
0$

and hence

$r_{\delta}=\left(\ln\left(\left(\frac{\delta}{\varrho}\right)^{-1/\eta(t)}\left(\frac{1}{\eta(t)}\ln\frac{\varrho}{\delta}\right)^{-p/\eta(t)}\right)\right)^{d/2}\left(1+o(1)\right)\quad\text{for}\,\,\delta\to
0.$

For $r\in{\mathbb{R}}$, let $\left[\left[r\right]\right]$ denotes the greatest
integer not exceeding $r.$ We define
$N_{\delta}=\left[\left[r_{\delta}\right]\right].$ Then for this choice of the
regularization parameter, from Theorem 4.3, it follows that for each
$t\in[0,\tau]$ there exists $\delta_{t}>0$ such that

$\displaystyle\left\|u(t)-u^{N_{\delta}}_{\delta}(t)\right\|$
$\displaystyle\leq$ $\displaystyle
2E_{1}e^{\kappa_{1}(\tau-t)}\,\delta\left(\frac{\delta}{\varrho}\right)^{-e_{2}\left(\tau-t\right)/\eta(t)}\left(\frac{1}{\eta(t)}\ln\frac{\varrho}{\delta}\right)^{-pe_{2}\left(\tau-t\right)/\eta(t)},\quad\text{for}\,\delta\leq\delta_{t}$
$\displaystyle=$ $\displaystyle
2E_{1}e^{\kappa_{1}(\tau-t)}\,\varrho^{e_{2}\left(\tau-t\right)/\eta(t)}\,\delta^{e_{1}t/\eta(t)}\left(\frac{1}{\eta(t)}\ln\frac{\varrho}{\delta}\right)^{-pe_{2}\left(\tau-t\right)/\eta(t)},\quad\text{for}\,\delta\leq\delta_{t}$

Now suppose that $u\in\mathcal{M}_{0,q}.$ Let $e_{1},e_{2}>0$ be as in (2.2).
Then from Theorem 4.3 (ii), we have

$\displaystyle\left\|u(t)-u^{N}_{\delta}(t)\right\|$ $\displaystyle\leq$
$\displaystyle
e^{\kappa_{1}(\tau-t)}\left(\frac{\varrho}{e^{(q+t)\lambda_{N}}}+\delta
e^{\lambda_{N}(\tau-t)}\right)$ $\displaystyle\leq$ $\displaystyle
e^{\kappa_{1}(\tau-t)}\left(\frac{\varrho}{e^{(q+t)e_{1}N^{2/d}}}+\delta
e^{(\tau-t)e_{2}N^{2/d}}\right).$

For $r>0,$ let $A(r)=\frac{\varrho}{e^{e_{1}(q+t)r^{2/d}}}+\delta
e^{e_{2}(\tau-t)r^{2/d}}.$ Clearly $A(r)$ attains it minimum at $r_{\delta}$
if

$\frac{\varrho}{\delta}=\left(e^{{r_{\delta}}^{2/d}}\right)^{e_{1}(q+t)+e_{2}(\tau-t)},$

that is, if

$r_{\delta}=\left(\frac{1}{e_{1}(q+t)+e_{2}(\tau-t)}\ln\frac{\varrho}{\delta}\right)^{d/2}.$

Let $N_{\delta}=\left[\left[r_{\delta}\right]\right].$ Then from Theorem 4.3
(ii), it follows that for each $t\in[0,\tau]$

$\displaystyle\left\|u(t)-u^{N_{\delta}}_{\delta}(t)\right\|$
$\displaystyle\leq$ $\displaystyle
2e^{\kappa_{1}(\tau-t)}\,\delta\left(\frac{\varrho}{\delta}\right)^{e_{2}(\tau-t)/\left(e_{1}(q+t)+e_{2}(\tau-t)\right)}$
$\displaystyle=$ $\displaystyle
2e^{\kappa_{1}(\tau-t)}\,\varrho^{\frac{e_{2}(\tau-t)}{e_{1}(q+t)+e_{2}(\tau-t)}}\,\delta^{\frac{e_{1}(q+t)}{e_{1}(q+t)+e_{2}(\tau-t)}}.$

Thus, we have proved the following result.

######  Theorem 4.5.

For $p,q,\varrho>0,$ let $\mathcal{K}_{p,\tau}$ and $\mathcal{M}_{0,q}$ be as
defined in (4.3) and (4.4), respectively. Let $e_{1},e_{2}$ be as in (2.2).
Let $\kappa_{0}=\max\\{\kappa,1\\},\,\,\kappa_{1}=1+\kappa_{0},$ where
$\kappa$ is as in $(\bf F{1})$, and $\eta(t)=e_{1}t+e_{2}(\tau-t),\,\,0\leq
t\leq\tau.$ For $r\in{\mathbb{R}},$ let $\left[\left[r\right]\right]$ denotes
the greatest integer not exceeding $r.$ Then the following holds:

  1. (i)

If $u\in\mathcal{K}_{p,\tau}$ then for each $t\in[0,\tau]$ there exists
$\delta_{t}>0$ such that for the choice

$N_{\delta}=\left[\left[\left(\ln\left(\left(\frac{\delta}{\varrho}\right)^{-1/\eta(t)}\left(\frac{1}{\eta(t)}\ln\frac{\varrho}{\delta}\right)^{-p/\eta(t)}\right)\right)^{d/2}\right]\right],$

we have

$\left\|u(t)-u^{N_{\delta}}_{\delta}(t)\right\|=O\left(e^{\kappa_{1}(\tau-t)}\varrho^{e_{2}(\tau-t)/\eta(t)}\,\delta^{e_{1}t/\eta(t)}\left(\frac{1}{\eta(t)}\ln\frac{\varrho}{\delta}\right)^{-pe_{2}\left(\tau-t\right)/\eta(t)}\right),\quad\delta\to
0.$

  2. (ii)

If $u\in\mathcal{M}_{0,q}$ then for each $t\in[0,\tau]$ and

$N_{\delta}=\left[\left[\left(\frac{1}{e_{1}(q+t)+e_{2}(\tau-t)}\ln\frac{\varrho}{\delta}\right)^{d/2}\right]\right],$

we have

$\left\|u(t)-u^{N_{\delta}}_{\delta}(t)\right\|=O\left(e^{\kappa_{1}(\tau-t)}\,\varrho^{\frac{e_{2}(\tau-t)}{e_{1}(q+t)+e_{2}(\tau-t)}}\,\delta^{\frac{e_{1}(q+t)}{e_{1}(q+t)+e_{2}(\tau-t)}}\right),\quad\delta\to
0.$

######  Remark 4.6.

From the above Theorem it follows that if $u\in\mathcal{K}_{p,\tau}$ then for
$t=0$ we obtain logarithmic rate of convergence and for $0<t<\tau$ we obtain a
logarithmic-type of convergence. Moreover, if $u\in\mathcal{M}_{0,q}$ then we
obtain a Hölder rate of convergence for every $t\in[0,\tau].$ $\lozenge$

##  5\. Conclusion

We have considered the problem of recovering the solution of a final value
problem for a parabolic equation with a non-linear source and a non-local
term, from the knowledge of the final value. We have shown by an example that
the considered problem is ill-posed, that is, a small perturbation in the data
may lead to a large deviation in the sought solution. Since the problem is
ill-posed, we have obtained regularized solutions by solving a non-linear
integral equation which is derived by considering the truncated version of
Fourier representation of the solution. Under appropriate Gevrey smoothness
assumption (i.e., source condition), we have established a logarithmic-type
convergence rate for the regularized approximation and with some comparatively
higher Gevrey smoothness assumption, Hölder rate of convergence was
established for the regularized approximations. We have proposed and proved a
new version of Grönwalls’ inequality for iterated integrals, which plays the
key role in obtaining the above mentioned rates.

Acknowledgements. The author is supported by the post doctoral fellowship of
TIFR Centre for Applicable Mathematics, Bangalore.

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