\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 109, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/109\hfil A quasi-boundary value method]
{A quasi-boundary value method for regularizing
 nonlinear ill-posed problems}

\author[ D. D. Trong, P. H. Quan, N. H. Tuan\hfil EJDE-2009/109\hfilneg]
{Dang Duc Trong, Pham Hoang Quan, Nguyen Huy Tuan}  % in alphabetical order


\address{Dang Duc Trong \newline
Department of Mathematics, Ho Chi Minh City National University \\
227 Nguyen Van Cu, Q. 5, HoChiMinh City, Vietnam}
\email{ddtrong@mathdep.hcmuns.edu.vn}

\address{Pham Hoang Quan \newline
Department of Mathematics, Sai Gon University\\
 273 An Duong Vuong , Ho Chi Minh city, Vietnam}
\email{tquan@pmail.vnn.vn}

\address{Nguyen Huy Tuan \newline
Department of Mathematics and Informatics,
Ton Duc Thang University \\
98, Ngo Tat To, Binh Thanh district, Ho Chi Minh city, Vietnam}
\email{tuanhuy\_bs@yahoo.com}

\thanks{Submitted June 19, 2009. Published September 10, 2009.}
\subjclass[2000]{35K05, 35K99, 47J06, 47H10}
\keywords{Backward heat problem; nonlinearly Ill-posed problem,
\hfill\break\indent
quasi-boundary value methods; quasi-reversibility methods,
contraction principle}

\begin{abstract}
 In this article, a modified quasi-boudary regularization method
 for solving nonlinear backward heat equation is given.
 Sharp error estimates for the approximate solutions,
 and numerical examples to illustrate the effectiveness
 our method are provided.  This work extends to the nonlinear
 case earlier results by  the authors \cite{t2,t3} and by
 Clark and Oppenheimer \cite{c1}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

For $T$ be a positive number, we consider the problem of finding
a function $u(x,t)$, the temperature,  such that
\begin{gather}
u_t-u_{xx} = f(x,t,u(x,t)),\quad (x,t)\in (0,\pi)\times (0,T),
\label{eq1}\\
u(0,t)= u(\pi,t)=0,\quad t\in (0,T), \label{eq2}\\
u(x,T)= g(x),\quad  x\in (0,\pi), \label{eq3}
\end{gather}
where $g(x), f(x,t,z)$ are given functions. This problem is called
backward heat problem, backward Cauchy problem, and
final value problem.

As is known, the nonlinear problem is severely ill-posed;
 i.e., solutions do not always exist, and in the case of existence,
 these do not depend continuously on the given data.
In fact, from small noise contaminated physical measurements,
the corresponding solutions have large errors.
It makes difficult to numerical calculations. Hence,
a regularization is in order. In the mathematical literature
various methods have been proposed for solving backward Cauchy problems.
We can notably mention the method of quasi-solution
(QS-method) by Tikhonov, the method of quasi-reversibility
(QR method) by Lattes and Lions, the quasi boundary value method
(Q.B.V method) and the C-regularized semigroups technique.

In the method of quasi-reversibility, the main idea consists
in replacing operator $A$ by $A_\epsilon=g_\epsilon(A)$, where $A[u]$
is the left-hand side of \eqref{eq1}.
In the original method, Lattes and Lions \cite{l1} proposed
$g_\epsilon(A)=A-\epsilon A^2$, to obtain well-posed problem
in the backward direction. Then, using the information from
the solution of the perturbed problem and solving the original
problem, we get another well-posed problem and this solution
sometimes can be taken to be the approximate solution
of the ill-posed problem.

Difficulties may arise when using the method quasi-reversibility
discussed above. The essential difficulty is that the order
of the operator is replaced by an operator of second order,
which produces serious difficulties on the numerical implementation,
in addition, the error $c(\epsilon)$ introduced by small change in
the final value $g$ is of the order $e^{T/\epsilon}$.

In 1983, Showalter \cite{s1} presented a method called
the quasi-boundary value (QBV) method to regularize that linear
homogeneous problem which gave a stability estimate better
than the one in the previous method. The main idea of the method is
of adding an appropriate ``corrector'' into the final data.
Using this method, Clark and Oppenheimer \cite{c1}, and Denche-Bessila,
\cite{d1}, regularized the backward problem by
replacing the final condition by
\[ %4
u(T)+\epsilon u(0)=g
\]
and
\[ %5
u(T)-\epsilon u'(0)=g
\]
respectively.

 To the author's knowledge, so far there are many papers on the
linear homogeneous case of the backward problem, but  we only
find a few papers on the nonhomogeneous case, and especially,
the nonlinear  case of their is very scarce.
In \cite{t1}, we used the Quasi-reversibility method to regularize
a 1-D linear nonhomogeneous backward problem. Very recently,
in \cite{q1}, the methods of integral equations and of Fourier
transform have been used to solved a 1-D problem in an
unbounded region.

For  recent  articles considering the nonlinear backward-parabolic heat,
we refer the reader to \cite{t3,t4}.
In \cite{t2}, the authors used the QBV method to regularize the latter
problem. However, in \cite{t2}, the authors showed that  the error
 between the approximate problem and the exact solution  is
\[
\|u(.,t)-u^\epsilon(.,t)\| \le \sqrt{M}\exp
\big( \frac{3k^2T(T-t)}{2}\big) \epsilon^{t/T}.
\]
In \cite{t4}, the error is also of similar form,
$$
\|u(t) - u^\epsilon  (t)\|  \leq  M \beta(\epsilon) ^{t/T}.
$$
It is easy to see that two errors above
are  not near to zero, if $\epsilon$ fixed and $t$ tend to zero.
Hence, the convergence of the approximate solution is very
slow when $t$ is in a neighborhood of zero. Moreover,
the regularization error in  $t=0$ is not given.

In the present paper, we shall regularize  \eqref{eq1}-\eqref{eq3}
using a modified quasi-boundary method given in \cite{t3}.
This regularization method is rather simple and convenient
for dealing with some ill-posed problems. The nonlinear backward
problem is approximated by the following one dimensional problem
\begin{gather}
u_t^{\epsilon}-u_{xx}^{\epsilon}=\sum_{k=1}^\infty
  \frac{e^{-T k^2}}{\epsilon k^2+e^{-Tk^2}}  f_k(u^{\epsilon})(t)
  \sin (kx), \quad (x,t)\in(0,\pi)\times (0,T), \label{eq6}\\
u^{\epsilon}(0,t)=u^{\epsilon}(\pi,t)=0, \quad t \in[0,T], \label{eq7}\\
u^{\epsilon}(x,T)=\sum_{k=1}^\infty
\frac{e^{-T k^2}}{\epsilon k^2+e^{-Tk^2}} g_k \sin (kx),\quad
 x\in[0,\pi], \label{eq8}
\end{gather}
where $\epsilon \in (0,eT)$,
\begin{gather*}
g_k=\frac{2}{\pi}\langle g(x),\sin kx\rangle  = \frac{2}{\pi}\int_0^{\pi}g(x)
 \sin (kx)dx, \\
f_k(u)(t)=\frac{2}{\pi}\langle f(x,t,u(x,t)),\sin kx \rangle
=\frac{2}{\pi}\int_0^{\pi}f(x,t,u(x,t))\sin kx dx
\end{gather*}
and $\langle \cdot,\cdot\rangle$ is the inner product in $L^2(0,\pi)$.

The paper is organized as follows.
In Theorem \ref{thm2.1} and \ref{thm2.2}, we shall show that
\eqref{eq6}-\eqref{eq8} is
 well-posed and that the unique  solution  $u^{\epsilon}(x,t)$ of it
satisfies the  equality
\begin{equation} \label{eq9}
u^{\epsilon}(x,t)=\sum_{k=1}^\infty
\big({\epsilon k^2+e^{-T k^2}}\big)^{-1}
\Big({e^{-t k^2}}g_k -\int_t^T {e^{(s-t-T) k^2}}f_k(u^{\epsilon})
(s)ds\Big)\sin k x.
\end{equation}
Then, in theorem  \ref{thm2.3} and \ref{thm2.4}, we estimate the
error between an exact solution $u$ of  \eqref{eq1}-\eqref{eq3} and
the approximation solution $u^\epsilon$ of \eqref{eq6}-\eqref{eq8}.
 In fact, we shall prove that
\begin{equation} \label{eq10}
\| u^{\epsilon}(.,t)-u(.,t)\|\le
H   \epsilon^{t/T-1}\big(\ln(T/\epsilon)\big)^{\frac{t}{T}-1}
\end{equation}
where $\|\cdot\|$ is the norm of $L^2(0,\pi)$ and $H$ is the
term depend on $u$.
Note that the above  results are improvements of
some  results in \cite{q1,t1,t2,t3,t4}. In fact,
in most of the previous results, the errors often have the form
 $C\epsilon^{t/T}$. This is one of their disadvantages in which
$t$ is zero. It is easy to see that from \eqref{eq10},
the convergence of the approximate solution at $t=0$ is also proved.
The notation about the usefulness and advantages of this method
can be founded in Remark 1 and Remark 2.
Finally, a numerical experiment will be given in Section 4,
which proves  the efficiency of our method.


\section{Main results}

 For clarity of notation, we denote the solution of
\eqref{eq1}-\eqref{eq3} by $u(x,t)$,  and the solution of the
problem \eqref{eq6}-\eqref{eq8} by $u^\epsilon(x,t)$.
Let $\epsilon $ be a positive number such that $0<\epsilon <eT$.

A function $f$ is called a  global Lipchitz function if
$f \in L^\infty ([0,\pi]\times [0,T] \times R)$ and satisfies
\begin{equation} \label{H1}
|f(x,y,w) - f(x,y,v)| \leq L |w - v|
\end{equation}
for a positive constant $L$ independent of $x, y, w, v$.
Throughout this paper, we denote $T_1=\max\{1,T\}$.
The existence and uniqueness of the regularized solution is stated
as follows.

\begin{theorem} \label{thm2.1}
Assume $0<\epsilon< eT$ and  \eqref{H1}.
Then  \eqref{eq6}-\eqref{eq8} has a unique  weak solution
$u^\epsilon \in W=C([0,T];L^2(0,\pi)) \cap L^2(0,T;H_0^1(0,\pi))
\cap C^1(0,T; H_0^1(0,\pi))$ satisfying \eqref{eq9}.
\end{theorem}

Regarding the stability of the regularized solution we have the following
result.

\begin{theorem} \label{thm2.2}
 Let $u$ and $v$ be two solutions of \eqref{eq6}-\eqref{eq8}
corresponding  to the final values $g$ and $h$ in $L^2(0,\pi)$.
Then
\[
\|u(.,t)-v(.,t)\|\le  T_1\exp(L^2T_1^2(T-t)^2)
\big(\epsilon \ln(T/\epsilon)\big)^{\frac{t-T}{T}}\| g-h\|.
\]
\end{theorem}

We remark that in \cite{a1,e1,g1,l3,t1},
the magnitude of stability inequality is $e^{T/\epsilon}$.
While in \cite{c1,m1,s1}, it is $ \epsilon^{-1}$.
In \cite[p. 5]{q1}, in \cite[p. 238]{t2}, and in \cite{t4},
the stability
estimate is of order $ \epsilon^{\frac{t}{T}-1}$, which is better
than the some previous results.

Since Theorem \ref{thm2.2}  gives a   estimate of the stability of order
 \begin{equation}
C\epsilon^{\frac{t}{T}-1}\big({ \ln(T/\epsilon)}\big)
^{\frac{t}{T}-1}. \label{e11}
\end{equation}
It is clear that this order of stability is less than the orders
given in \cite{q1,t2,t4}, which is one advantage of our method.
Despite the uniqueness, Problem \eqref{eq1}-\eqref{eq3} is
still ill-posed. Hence,  we have to resort to a regularization.
We have the following result.

\begin{theorem} \label{thm2.3}
Assume \eqref{H1}.
(a) If $u(x,t) \in W$ is a solution of  \eqref{eq1}-\eqref{eq3}
such that
\begin{equation}
\int_0^T \sum_{k=1}^\infty k^4 e^{2s k^2}f_k^2(u)(s)ds
<\infty \label{eq12}
\end{equation}
 and $\|u_{xx}(.,0)\| <\infty$.
Then
\[ %13
\| u(.,t)- u^{\epsilon}(.,t)\| \leq C_M \epsilon^{t/T}
\big(\ln(T/\epsilon)\big)^{\frac{t}{T}-1}.
\]
(b) If $u(x,t) $ satisfies
\begin{equation}
Q=\sup_{0 \le t \le T}\Big(\sum_{k=1}^\infty k^4 e^{2t k^2}
|\langle u(x,t),\sin kx\rangle |^2 \Big)< \infty \label{pppp}
\end{equation}
then
\[
\| u(.,t)- u^{\epsilon}(.,t)\| \leq
 C_Q \epsilon^{t/T}\big(\ln(T/\epsilon)\big)^{\frac{t}{T}-1}
\]
for every $t\in[0,T]$, where
\begin{gather*} %15,16,17
M=3\|u_{xx}(0)\|^2+ \frac{3\pi}{2}T \int_0^T
\sum_{k=1}^\infty k^4 e^{2s k^2}f^2_{k}(u)(s))ds,  \\
C_M=\sqrt{ MT_1^2e^{3L^2TT_1^2(T-t)} },\quad
C_Q=\sqrt{ Q T_1^2 e^{2L^2TT_1^2(T-t)}}.
\end{gather*}
\end{theorem}

\subsection*{Remarks}
 1. In \cite[p. 241]{t2} and in \cite{q1},  the error estimates between the
exact solution and the approximate solution is
$ U(\epsilon,t)= C\epsilon^{t/T}$. So,   if  the time $t$ is near
to the original time $t=0$, the converges rate is very slowly.
Thus,some  methods studied in \cite{q1,t2} are not useful to derive
the error estimations in the case $t$ is in a neighbourhood of zero.
To improve this, the convergence rate in the present theorem
is in slightly different form than given in
\cite{q1,t2}, defined by
$ V(\epsilon,t)= D \epsilon^{t/T}\big(\ln(T/\epsilon)\big)^{\frac{t}{T}-1}$.
 We note that
$\lim _{\epsilon \to 0} \frac{V(\epsilon,t)}{U(\epsilon,t)}=0$.
Hence, this error is the optimal error estimates which we know.
Moreover, we also have   $ \lim _{\epsilon \to 0}
\big( \lim  _{t \to 0} U(\epsilon,t)\big)= C $ and
$\lim  _{\epsilon \to 0} \big(\lim  _{t \to 0} V (\epsilon,t)\big)
= \lim_{\epsilon \to 0} \big(D \frac{1}{ \ln(T/\epsilon)}\big)=0$.
This also proves that our method give a better approximation than
the previous case which we know. Comparing  \eqref{eq12} with the
results obtained in \cite{t2,t4}, we realize this  estimate  is sharp
and the best known estimate. This is  generalization of  many previous
results in \cite{a1,a2,c1,d1,e1,g1,l2,l3,m1,q1,s1,s2,s3,t2,t4}.

2. One superficial advantage of this method is that there is an error
estimation in the  time $t=0$, which does not appear in many recently
known results in \cite{q1,t2,t4}. We have the following estimate
\[
\| u(.,0)- u^{\epsilon}(.,0)\|
\leq \frac{H}{\ln(T/\epsilon)}.
\]
where $H$ is a term  depending only on $u$.
These estimates, as noted above, are very seldom in the
theory of ill-posed problems.

 3. In the linear  nonhomogeneous case  $f(x,t,u)=f(x,t)$,
the error estimates were  given in \cite{t3}.
And  the assumption of $f$ in  \eqref{eq12} is not used.
It is  only in $L^2(0,T;L^2(0,\pi))$.

4. In  theorem \ref{thm2.3}(a),  we  ask for the condition on the expansion
coefficient $f_k$. We note that the solution $u$ depend on
the nonlinear term $f$ and therefore $f_k, f_k(u)$ is
very difficult to be valued. Such a obscurity makes this Theorem
hard to be used for numerical computations.
To improve this, in Theorem \ref{thm2.3}(b), we require the assumption
of  $u$, not to depend on the function  $f(u)$.  In fact,
we note that in the simple case  of the right-hand side
$f(u)=0$, the term $Q$ becomes
\[
\sum_{k=1}^\infty k^4 e^{2t k^2} |\langle u(x,t),\sin kx\rangle |^2
=\|u_{xx}(.,0) \|.
\]
So, the condition \eqref{pppp} is  acceptable.


In the case of non-exact data, one has the following result.

\begin{theorem} \label{thm2.4}
 Let the exact solution $u$ of \eqref{eq1}-\eqref{eq3} corresponding
to $g$. Let $g_\epsilon$ be a measured data such that
$\| g_\epsilon-g\| \le \epsilon$.
Then there exists a function $ w^{\epsilon}$ satisfying:
(a) for every $t \in [0,T]$,
\[
\| w^{\epsilon}(.,t)-u(.,t)\|
\le T_1(1+\sqrt{M})
\exp(\frac{3L^2TT_1^2(T-t)}{2})\epsilon^{t/T}\big(\ln(T/\epsilon)
\big)^{\frac{t}{T}-1} ,
\]
where $u$ is  defined in Theorem \ref{thm2.3}(a).

(b) for every $t \in [0,T]$,
\[
\| w^{\epsilon}(.,t)-u(.,t)\|
\le T_1(1+\sqrt{Q})\exp({L^2TT_1^2(T-t)})\epsilon^{t/T}
\big(\ln(T/\epsilon)\big)^{\frac{t}{T}-1} ,
\]
where $u$ is  defined in Theorem \ref{thm2.3}(b),
and  $M,Q$ is defined in Theorem \ref{thm2.3}.
\end{theorem}

\section{Proof of the Main Theorems}

First we give some assumptions and lemmas
which will be useful in proving the main Theorems.

\begin{lemma} \label{lem1}
For $0<\epsilon < eT$, denote $h(x)=\frac{1}{\epsilon x+e^{-x T}}$.
Then it follows that
\[
h(x)\le \frac{T}{\epsilon \big(1+\ln(T/\epsilon)\big)}
\le \frac{T}{\epsilon \ln(T/\epsilon)}.
\]
\end{lemma}

 The proof of the above lemma can be found in \cite{t3}.
For $0 \le t \le s \le T$, denote
\begin{equation} \label{16}
G_\epsilon(s,t,k) = \frac{e^{(s-t-T)k^2}}{\epsilon k^2 +e^{-Tk^2}},\quad
G_\epsilon(T,t,k) = \frac{e^{-tk^2}}{\epsilon k^2 +e^{-Tk^2}},
\end{equation}
and $T_1=\max\{1,T \}$.


 \begin{lemma} \label{lem2}
 \[ %22
G_\epsilon(s,t,k) \le T_1
\big(\epsilon \ln(T/\epsilon)\big)^{\frac{t-s}{T}}.
\]
\end{lemma}

\begin{proof}
We have
\begin{align*} %23
G_\epsilon(s,t,k)
&= \frac{e^{(s-t-T)k^2}}{\epsilon k^2 +e^{-Tk^2}} = \frac{  e^{(s-t-T)k^2}}  { {(\epsilon k^2 +e^{-Tk^2})^{\frac{s-t}{T}}}(\epsilon k^2 +e^{-Tk^2})^{\frac{T+t-s}{T}}}
 \\
&\leq  \frac{e^{(s-t-T)k^2}} {(e^{-Tk^2})^{\frac{T+t-s}{T}} }
\frac{1}{(\epsilon k^2 +e^{-Tk^2})^{\frac{s}{T}-\frac{t}{T}}}
 \\
&\leq  \Big(\frac{T}{\epsilon \ln(T/\epsilon)}
 \Big)^{\frac{s}{T}-\frac{t}{T}}\\
&= T^{ \frac{s-t}{T}   }\epsilon^{\frac{t-s}{T}}
\big(\ln(T/\epsilon)\big)^{\frac{t-s}{T}} \\
&\leq    \max\{1,T  \}
 \big(\epsilon \ln(T/\epsilon)\big)^{\frac{t-s}{T}}.
\end{align*}
\end{proof}

\begin{lemma} \label{lem3}
Let $s=T$ in Lemma \ref{lem2}, to obtain
\begin{equation} \label{21}
G_\epsilon(T,t,k)  \le  T_1
\Big(\epsilon \ln(T/\epsilon)\Big)^{\frac{t-T}{T}}.
\end{equation}
\end{lemma}


\begin{proof}[Proof of Theorem \ref{thm2.1}]
\textbf{Step 1.} Existence and  uniqueness of a solution
of the integral equation \eqref{eq9}.
Put
\[
F(w)(x,t) = P(x,t) - \sum_{k=1}^\infty\int_t^T
G_\epsilon(s,t,k)f_k(w)(s)\,ds \sin (kx)
\]
for $w \in C([0,T];L^2(0,\pi))$, where
$$
P(x,t) = \sum_{k=1}^\infty G_\epsilon(T,t,k)\langle g(x),\sin kx\rangle
 \sin kx.
$$
Note that by Lemma \ref{lem2}, we have
\begin{equation} \label{eq25}
\begin{aligned}
G_\epsilon(s,t,k)& \leq   \max\{1,T  \}
\big(\epsilon \ln(T/\epsilon)\big)^{\frac{t-s}{T}}  \\
&\leq   \max\{1,T  \} \frac{1}{ \epsilon}  \\
&= \max\{ \frac{1}{ \epsilon} , \frac{T}{ \epsilon}   \}=B_\epsilon.
\end{aligned}
\end{equation}
We claim that, for every $w, v \in C([0,T];L^2(0,\pi)), p \geq 1$,
 we have
\begin{equation}
\|F^p (w)(.,t)-F^p(v)(.,t)\|^2 \leq (LB_\epsilon )^{2p}
\frac{(T-t)^p C^p}{p!}|||w -v|||^2, \label{pt1}
\end{equation}
where $C = \max \{T,1\}$ and $|||.|||$ is supremum norm
in $C([0,T];L^2(0,\pi))$.
We shall prove this inequality by induction.
For $p =1$, and using Lemma \ref{lem2}, we have
\begin{align*}
&\|F(w)(.,t)-F(v)(.,t)\|^2  \\
&=\frac{\pi}{2} \sum_{k=1}^\infty
\Big[\int_t^T G_\epsilon(s,t,k)\left(f_k(w)(s)-f_k(v)(s)\right)ds\Big]^2
 \\
&\leq \frac{\pi}{2} \sum_{k=1}^\infty \int_t^T
 \Big(\frac{e^{(s-t-T)k^2}}{\epsilon k^2 +e^{-Tk^2}}\Big)^2ds
 \int_t^T\left(f_k(w)(s)-f_k(v)(s)\right)^2 ds
 \\
&\leq\frac{\pi}{2} \sum_{k=1}^\infty B_\epsilon^{2}(T-t)  \int_t^T\left(f_k(w)(s)-f_k(v)(s)\right)^2 ds
 \\
&=\frac{\pi}{2}B_\epsilon^{2}(T-t)  \int_t^T\sum_{k=1}^\infty\left(f_k(w)(s)-f_k(v)(s)\right)^2 ds
 \\
&= B_\epsilon^{2}(T-t)  \int_t^T \int_0^\pi \left(f(x,s,w(x,s))-f(x,s,v(x,s))\right)^2 dx ds
 \\
&\leq L^2B_\epsilon^{2}(T-t)  \int_t^T \int_0^\pi |w(x,s)-v(x,s)|^2 dx ds
 \\
&= C L^2 B_\epsilon^{2}(T-t)  |||w-v|||^2.
\end{align*}
Thus \eqref{pt1} holds.
Suppose that \eqref{pt1} holds for $p =m$. We prove that \eqref{pt1}
 holds for $p = m + 1$. We have
\begin{align*}
&\|F^{m+1}(w)(.,t) - F^{m+1}(v)(.,t)\|^2  \\
&= \frac{\pi}{2} \sum_{k = 1}^\infty
\Big[\int_t^T G_\epsilon(s,t,k) \left(f_k(F^m(w))(s)
- f_k(F^m (v))(s)\right)ds \Big]^2 \\
&\leq \frac{\pi}{2} B_\epsilon^{2}\sum_{k = 1}^\infty
\Big[\int_t^T |f_k(F^m(w))(s) - f_k(F^m (v))(s)|ds \Big]^2 \\
&\leq  \frac{\pi}{2}B_\epsilon^{2} (T - t)\int_t^T
\sum_{k = 1}^\infty |f_k(F^m(w))(s) - f_k(F^m (v))(s)|^2 ds  \\
&\leq  B_\epsilon^{2} (T - t)\int_t^T \|f(.,s,F^m(w)(.,s))
- f(.,s,F^m (v)(.,s))\|^2 ds  \\
&\leq B_\epsilon^{2} (T - t)L^2 \int_t^T \|F^m(w)(.,s)
 - F^m (v)(.,s)\|^2 ds \\
&\leq  B_\epsilon^{2} (T - t)L^{2m+2}B_\epsilon^{2m}
 \int_t^T \frac{(T-s)^m}{m!} ds C^m |||w-v|||^2 \\
&\leq  (LB_\epsilon)^{2m+2}\frac{(T-t)^{m+1}}{(m+1)!}
C^{m+1} |||w-v|||^2.
\end{align*}
Therefore, by the induction principle, we have
\[
|||F^p (w) -F^p(v)||| \leq
(LB_\epsilon)^{p}  \frac{T^{p/2}}{\sqrt{p!}}C^{p/2} |||w-v|||
\]
for all $w, v \in C([0,T];L^2(0,\pi))$.

We consider $F: C([0,T];L^2(0,\pi))\to C([0,T];L^2(0,\pi))$.
Since
$$
\lim  _{p \to \infty} (LB_\epsilon)^{p} \frac{T^{p/2}C^{p/2}}{\sqrt{p!}}=0,
$$
 there exists a positive integer number $p_0$ such that
$$
(LB_\epsilon)^{p_0}    \frac{T^{p_0/2}C^{p_0/2}}{\sqrt{(p_0)!}}<1,
$$
and $F^{p_0}$ is a contraction. It follows that the equation
$F^{p_0} (w) = w$ has a unique solution
$u^\epsilon \in C([0,T];L^2(0,\pi))$.

We claim that $F(u^\epsilon) = u^\epsilon$. In fact, one has
$F(F^{p_0} (u^\epsilon)) = F(u^\epsilon)$.
Hence $F^{p_0} (F(u^\epsilon)) = F(u^\epsilon)$. By the uniqueness
of the fixed point of $F^{p_0}$, one has $F(u^\epsilon) = u^\epsilon$;
i.e., the equation $F(w) = w$ has a unique solution
$u^\epsilon \in C([0,T];L^2(0,\pi))$.

\textbf{Step 2.} If $ u^{\epsilon} \in W$ satisfies  \eqref{eq9}
then $ u^{\epsilon}$ is solution of \eqref{eq6}-\eqref{eq8}.
For $0\le t \le T$, we have
\[ %27
u^{\epsilon}(x,t)=\sum_{k=1}^\infty
 \left({\epsilon k^2+e^{-T k^2}}\right)^{-1}
\Big({e^{-t k^2}}g_k -\int_t^T {e^{(s-t-T) k^2}}f_k(u^{\epsilon})(s)ds
\Big)\sin k x,
\]
We can verify directly that
\[
u^\epsilon \in C([0,T];L^2(0,\pi) \cap C^1 ((0,T);H_0^1(0,\pi))
\cap L^2 (0,T; H_0^1(0,\pi))).
\]
In fact, $u^\epsilon \in C^\infty ((0,T];H_0^1(0,\pi)))$.
Moreover, by direct computation, one has
\begin{align*}
&u^{\epsilon}_t(x,t)\\
&= \sum_{k=1}^\infty  -k^2\big({\epsilon k^2+e^{-T k^2}}\big)^{-1}
\Big({e^{-t k^2}}g_k -\int_t^T {e^{(s-t-T) k^2}}f_k(u^{\epsilon})(s)ds
\Big)\sin k x  \\
&\quad + \sum_{k=1}^\infty {e^{-T k^2}}({\epsilon k^2+e^{-Tk^2}})^{-1}
f_k( u^{\epsilon})(t) \sin kx
 \\
&= -\frac{2}{\pi}\sum_{k=1}^\infty k^2\langle  u^{\epsilon}(x,t),\sin kx\rangle
 \sin kx+\sum_{k=1}^\infty  {e^{-T k^2}}({\epsilon k^2+e^{-Tk^2}})^{-1}
 f_k( u^{\epsilon})(t) \sin kx
 \\
&=  u^{\epsilon}_{xx}(x,t)+\sum_{k=1}^\infty {e^{-T k^2}}
({\epsilon k^2+e^{-Tk^2}})^{-1} f_k( u^{\epsilon})(t) \sin kx
\end{align*}
and
\begin{gather}
 u^{\epsilon}(x,T)=\sum_{k=1}^\infty
{e^{-T k^2}}({\epsilon k^2+e^{-Tk^2}})^{-1} g_k \sin (kx).
\end{gather}
So $ u^{\epsilon}$ is the solution of \eqref{eq6}-\eqref{eq8}.

\textbf{Step 3.} The problem \eqref{eq6}-\eqref{eq8} has at most
one (weak) solution $ u^{\epsilon} \in W$.
In fact, let $ u^\epsilon $ and $ v^\epsilon$ be two solutions
of  \eqref{eq6}-\eqref{eq8} such that
$ u^{\epsilon}, v^\epsilon \in W$.
Putting $w^\epsilon(x,t)=u^{\epsilon}(x,t)-v^{\epsilon}(x,t)$,
then $w^\epsilon$ satisfies
\[ %30
w_t^{\epsilon}-w_{xx}^{\epsilon}=\sum_{k=1}^\infty
{e^{-T k^2}}({\epsilon k^2+e^{-Tk^2}})^{-1}
\left(f_k(u^{\epsilon})(t) -f_k(v^{\epsilon})(t) \right)\sin (kx).
\]
It  follows that
\begin{align*}
\|w_t^{\epsilon}-w_{xx}^{\epsilon}\|^2
&\leq  \frac{1}{\epsilon^2}  \sum_{k=1}^\infty
 \left(f_k(u^{\epsilon})(t) -f_k(v^{\epsilon})(t)\right )^2 \\
&\leq \frac{1}{\epsilon^2} \|f(.,t,u^\epsilon(.,t)
 -f(.,t,v^\epsilon(.,t))\|^2\\
&\leq  \frac{L^2}{\epsilon^2} \|u^\epsilon(.,t)-v^\epsilon(.,t)\|^2\\
&=\frac{L^2}{\epsilon^2} \|w^\epsilon(.,t)\|^2.
\end{align*}
Using a result in Lees-Protter \cite{l2},  we get
$w^\epsilon(.,t)=0$. This completes proof of Step 3.
Combining three Step 1,2,3, we complete the proof of
Theorem \ref{thm2.1}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2.2}]
 From \eqref{eq9} one has in view of the inequality
 $(a + b)^2 \leq 2(a^2 + b^2)$,
\begin{equation}
\begin{aligned}
 & \| u(.,t)-v(.,t)\|^2 \\
&= \frac{\pi}{2}\sum_{k=1}^\infty \big| G_\epsilon(T,t,k)(g_k-h_k)
-\int_t^T G_\epsilon(s,t,k)(f_k(u)(s)-f_k(v)(s)ds)\big|^2
 \\
&\leq \pi \sum_{k=1}^\infty(  G_\epsilon(T,t,k) |g_k-h_k|)^2
+\pi \sum _{k=1}^\infty(\int_t^T G_\epsilon(s,t,k)|f_k(u)(s)
-f_k(v)(s)|ds)^2.
\end{aligned} \label{pt2}
\end{equation}
Combining Lemma \ref{lem2}, Lemma \ref{lem3} and \eqref{pt2}, we get
\begin{equation} \label{eq32}
\begin{aligned}
&\|u(.,t) - v(.,t)\|^2 \\
&\leq \max\{1,T^2  \}
\big(\epsilon \ln(T/\epsilon)\big)^{\frac{2t-2T}{T}} \| g-h\|^2\\
&\quad +2 L^2(T-t)\max\{1,T^2  \}
\big(\epsilon \ln(T/\epsilon)\big)^{\frac{2t}{T}}
\int_t^T \big(\epsilon \ln(T/\epsilon)\big)^{\frac{-2s}{T}}
\|u(.,s) -v(.,s)\|^2ds.
\end{aligned}
\end{equation}
It follows that
\begin{align*}
&\big(\epsilon \ln(T/\epsilon)\big)^{\frac{-2t}{T}} \|u(.,t) - v(.,t)\|^2\\
&\leq  \max\{1,T^2 \} \big(\epsilon \ln(T/\epsilon)\big)^{-2}
 \| g-h\|^2 \\
&\quad + 2\max\{1,T^2  \}  L^2(T-t) \int_t^T
\big(\epsilon \ln(T/\epsilon)\big)^{\frac{-2s}{T}} \|u(.,s)-v(.,s)\|^2ds.
\end{align*}
Using Gronwall's inequality we have
\[
\|u(.,t)-v(.,t)\|\le T_1\exp(L^2T_1^2(T-t)^2)
\big(\epsilon \ln(T/\epsilon)\big)^{\frac{t-T}{T}}\| g-h\|.
\]
This completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2.3}]
Part (a): Suppose the Problem \eqref{eq1}-\eqref{eq3} has an
exact solution $u$, then $u$ can be rewritten as
\begin{equation}
 u(x,t)=\sum_{k=1}^\infty (e^{-(t-T) k^2}g_k
-\int_t^T e^{-(t-s) k^2}f_k(u)(s)ds)\sin kx. \label {pt3}
\end{equation}
Since
\[
u_k (0)=e^{Tk^2}g_k-\int_0^T e^{sk^2}f_k(u)(s)ds,
\]
implies
\[
g_k=e^{-Tk^2}u_k(0)+\int_0^T e^{(s-T)k^2}f_k(u)(s)ds,
\]
we get
\begin{align*}
 u(x,T)&= \sum_{k=1}^\infty g_k \sin kx\\
&= \sum_{k=1}^\infty (e^{-T k^2} u_k(0)
+\int_0^T e^{-(T-s)k^2}f_k(u)(s)ds)\sin kx.
\end{align*}
 From \eqref{eq9} and \eqref{pt3}, we have
\begin{align}
u^{\epsilon}_k(t)= \big({\epsilon k^2+e^{-T k^2}}\big)^{-1}
\Big({e^{-t k^2}}g_k
-\int_t^T {e^{(s-t-T) k^2}}f_k(u^{\epsilon})(s)ds\Big) \label{pt4}\\
u_k(t)= e^{T k^2}\Big({e^{-t k^2}}g_k
-\int_t^T {e^{(s-t-T) k^2}}f_k(u)(s)ds\Big).\label{pt5}
\end{align}
 From \eqref{16}, \eqref{pt4} and \eqref{pt5}, we have
\begin{align*}
u_k(t)- u^{\epsilon}_k (t)
&= \Big(e^{T k^2}-\frac{1}{\epsilon k^2+e^{-T k^2}}\Big)
\Big({e^{-t k^2}}g_k -\int_t^T {e^{(s-t-T) k^2}}f_k(u)(s)ds\Big)
 \\
&\quad +\int_t^T  G_\epsilon(s,t,k)\left(f_k(u^\epsilon)(s)-f_k( u)(s)\right)ds
 \\
&= \frac{\epsilon k^2 e^{-tk^2}}{\epsilon k^2+e^{-T k^2}}
\Big({e^{T k^2}}g_k -\int_t^T {e^{s k^2}}f_k(u)(s)ds\Big)
 \\
&\quad +\int_t^T  G_\epsilon(s,t,k)(f_k(u^\epsilon)(s)-f_k( u)(s)ds.
%\label{32}
\end{align*}
 From \eqref{21} and
$$
 T_1 \big(\epsilon \ln(T/\epsilon)\big)^{\frac{t-T}{T}} .
\big(\epsilon \ln(T/\epsilon)\big)^{1-\frac{s}{T}}  = T_1
     \big(\epsilon \ln(T/\epsilon)\big)^{\frac{t-s}{T}}
$$
 we have
\begin{align*}
&|u_k(t)- u^{\epsilon}_k (t)|\\
&\leq \big|  \epsilon G_\epsilon(T,t,k)\Big( k^2{e^{T k^2}}g_k
-\int_0^T k^2{e^{s k^2}}f_k(u)(s)ds\Big)\big|\\
&\quad +\epsilon  G_\epsilon(T,t,k)
\big|\int_0^t k^2{e^{s k^2}}f_k(u)(s)ds\big|
+\int_t^T G_\epsilon(s,t,k)|f_k(u)(s)-f_k( u^{\epsilon})(s)|ds \\
&\leq \epsilon  T_1 \big(\epsilon \ln(T/\epsilon)\big)^{\frac{t-T}{T}}
 \Big(|k^2u_k(0)|+ \int_0^t\big| k^2{e^{s k^2}}f_k(u)(s)\big|ds\Big) \\
&\quad +\int_t^T  T_1  \big(\epsilon \ln(T/\epsilon)\big)^{\frac{t-s}{T}}
   |f_k(u)(s)-f_k( u^{\epsilon})(s)|ds  \\
&= \epsilon T_1  \big(\epsilon \ln(T/\epsilon)\big)^{\frac{t-T}{T}}
 \Big(|k^2u_k(0)|+ \int_0^T\big| k^2{e^{s k^2}}f_k(u)(s)\big|ds\Big) \\
&\quad + \epsilon T_1 \big(\epsilon \ln(T/\epsilon)\big)^{\frac{t-T}{T}}
 \int_t^T \epsilon^{-\frac{s}{T}}
 \big( \ln(T/\epsilon)\big)^{1-\frac{s}{T}}
|f_k(u)(s)-f_k( u^{\epsilon})(s)|ds.
\end{align*}
Applying the inequality $(a+b+c)^2 \leq 3(a^2+b^2+c^2)$, we get
\begin{align*}
&\| u(.,t)- u^{\epsilon}(.,t)\|^2 \\
&= \frac{\pi}{2}T_1^2\sum_{k=1}^\infty |u_k(t)- u^{\epsilon}_k (t)|^2\\
&\leq \frac{3\pi}{2}T_1^2\sum_{k=1}^\infty \epsilon^2
  \big(\epsilon \ln(T/\epsilon)\big)^{\frac{2t-2T}{T}}
 |k^2 u_k(0)|^2+\frac{3\pi}{2}T_1^2 \sum_{k=1}^\infty  \epsilon^2
\big(\epsilon \ln(T/\epsilon)\big)^{\frac{2t-2T}{T}}\\
&\quad\times \Big(\int_0^T| k^2{e^{s k^2}}f_k(u)(s)|ds\Big)^2
 + \frac{3\pi}{2}T_1^2\sum_{k=1}^\infty  \epsilon^2
  \big(\epsilon \ln(T/\epsilon)\big)^{\frac{2t-2T}{T}}\\
&\quad \times \Big(\int_t^T \epsilon^{-\frac{s}{T}}\big( \ln(T/\epsilon)\big)
^{1-\frac{s}{T}}|f_k(u)(s)-f_k( u^{\epsilon})(s)|ds\Big)^2\\
&= T_1^2(I_1+I_2+I_3),
\end{align*}
where
\begin{gather*}
I_1= \frac{3\pi}{2}\sum_{k=1}^\infty \epsilon^2
 \big(\epsilon \ln(T/\epsilon)\big)^{\frac{2t-2T}{T}} |k^2 u_k(0)|^2,\\
I_2= \frac{3\pi}{2} \sum_{k=1}^\infty  \epsilon^2
\big(\epsilon \ln(T/\epsilon)\big)^{\frac{2t-2T}{T}}
\Big(\int_0^T| k^2{e^{s k^2}}f_k(u)(s)|ds\Big)^2, \\
I_3= \frac{3\pi}{2}\sum_{k=1}^\infty  \epsilon^2
  \big(\epsilon \ln(T/\epsilon)\big)^{\frac{2t-2T}{T}}
 \Big(\int_t^T \epsilon^{-\frac{s}{T}}\big( \ln(T/\epsilon)
\big)^{1-\frac{s}{T}}|f_k(u)(s)-f_k( u^{\epsilon})(s)|ds\Big)^2.
\end{gather*}
The terms $I_1,I_2,I_3$ can be estimated as follows:
\begin{gather}
\begin{aligned}
I_1 & \leq  3 \epsilon^2 \big(\epsilon \ln(T/\epsilon)
\big)^{\frac{2t-2T}{T}} \|u_{xx}(0)\|^2  \\
&\leq 3\epsilon^{\frac{2t}{T}}\big(\ln(T/\epsilon)\big)^{\frac{2t}{T}-2}
 \|u_{xx}(0)\|^2.
\end{aligned} \label{pt6}
\\
\begin{aligned}
I_2 &\leq  \frac{3\pi}{2}T   \epsilon^2 \big(\epsilon
\ln(T/\epsilon)\big)^{\frac{2t-2T}{T}}
\int_0^T \sum_{k=1}^\infty\left( k^2{e^{s k^2}}f_k(u)(s)\right)^2ds  \\
&\leq  \frac{3\pi}{2} T  \epsilon^2 \big(\epsilon \ln(T/\epsilon)\big)
^{\frac{2t-2T}{T}} \int_0^T \sum_{k=1}^\infty k^4 e^{2s k^2}f^2_k(u)(s)ds.
\\
&\leq  \frac{3\pi}{2}T \epsilon^2 \big(\epsilon \ln(T/\epsilon)
\big)^{\frac{2t-2T}{T}}
\int_0^T \sum_{k=1}^\infty k^4 e^{2s k^2}f^2_{k}(u)(s)ds.
\end{aligned}\label{pt7}
\\
\begin{aligned}
I_3 &\leq  \frac{3\pi}{2}(T-t)  \epsilon^2
\big(\epsilon \ln(T/\epsilon)\big)^{\frac{2t-2T}{T}}
 \int_t^T \epsilon^{-\frac{2s}{T}}\big(\ln(T/\epsilon)
 \big)^{2-\frac{2s}{T}}\\
 &\quad\times \sum_{k=1}^\infty(f_k(u)(s)-f_k( u^{\epsilon})(s))^2ds \\
&\leq 3(T-t) \epsilon^2 \big(\epsilon \ln(T/\epsilon)
\big)^{\frac{2t-2T}{T}}
  \int_t^T  \epsilon^{-\frac{2s}{T}}\big(\ln(T/\epsilon)
 \big)^{2-\frac{2s}{T}} \\
&\quad \times \|f(.,s,u(.,s))-f(.,s,u^{\epsilon}(.,s))\|^2ds \\
&\leq 3L^2T \epsilon^{\frac{2t}{T}}\big(\ln(T/\epsilon)
 \big)^{\frac{2t}{T}-2}\int_t^T \epsilon^{-\frac{2s}{T}}
 \big(\ln(T/\epsilon)\big)^{2-\frac{2s}{T}}
 \| u(.,s)- u^{\epsilon}(.,s)\|^2ds.
\end{aligned} \label{pt8}
\end{gather}
Combining \eqref{pt6}, \eqref{pt7}, \eqref{pt8}, we obtain
\begin{align*}
&\| u(.,t)- u^{\epsilon}(.,t)\|^2\\
&\leq  T_1^2\epsilon^{\frac{2t}{T}}\big(\ln(T/\epsilon)
 \big)^{\frac{2t}{T}-2}
\Big(3\|u_{xx}(0)\|^2+ \frac{3\pi}{2}T \int_0^T \sum_{k=1}^\infty k^4
e^{2s k^2}f^2_{k}(u)(s)ds \Big)\\
&\quad + T_1^23L^2T  \epsilon^{\frac{2t}{T}}
\big(\ln(T/\epsilon)\big)^{\frac{2t}{T}-2}
\int_t^T \epsilon^{-\frac{2s}{T}}\big(\ln(T/\epsilon)
\big)^{2-\frac{2s}{T}}  \|u(.,s)- u^{\epsilon}(.,s)\|^2ds.
\end{align*}
It follows that
\begin{align*}
& \epsilon^{\frac{-2t}{T}}\big(\ln(T/\epsilon)\big)^{2-\frac{2t}{T}}
 \| u(.,t)- u^{\epsilon}(.,t)\|^2\\
&\leq  MT_1^2+   3 L^2TT_1^2\int_t^T
\epsilon^{-\frac{2s}{T}}\big(\ln(T/\epsilon)\big)^{2-\frac{2s}{T}}
 \|u(.,s)- u^{\epsilon}(.,s)\|^2ds.
\end{align*}
Using Gronwall's inequality, we obtain
\[
\epsilon^{\frac{-2t}{T}}\big(\ln(T/\epsilon)\big)^{2-\frac{2t}{T}}
 \| u(.,t)- u^{\epsilon}(.,t)\|^2 \leq MT_1^2e^{3L^2TT_1^2(T-t)}.
\]
So that
\[
 \| u(.,t)- u^{\epsilon}(.,t)\|^2 \leq MT_1^2e^{3L^2TT_1^2(T-t)}
\epsilon^{\frac{2t}{T}}\big(\ln(T/\epsilon)\big)^{\frac{2t}{T}-2}     .
\]
This completes the proof part (a) in Theorem \ref{thm2.3}.

Proof of part (b) in Theorem \ref{thm2.3}.
 From \eqref{eq32}, we have
\begin{align*}
&|u_k(t)- u^{\epsilon}_k (t)| \\
&\leq \big|\big(e^{T k^2}-\frac{1}{\epsilon k^2+e^{-T k^2}}\big)
\Big({e^{-t k^2}}g_k-\int_t^T {e^{(s-t-T) k^2}}f_k(u)(s)ds\Big)\big|
 \\
&\quad+ \big|\int_t^T  G_\epsilon(s,t,k)(f_k(u^\epsilon)(s)
-f_k( u)(s))ds)\big|
 \\
&\leq \big|\frac{\epsilon k^2 e^{-tk^2}}{\epsilon k^2+e^{-T k^2}}
\Big({e^{T k^2}}g_k
-\int_t^T {e^{s k^2}}f_k(u)(s)ds\Big)\big|
\\
&\quad +\int_t^T  G_\epsilon(s,t,k)|f_k(u^\epsilon)(s)-f_k( u)(s)|ds \\
&\leq \big|\frac{\epsilon  e^{-tk^2}}{\epsilon k^2 +e^{-T k^2}}k^2
e^{tk^2}u_k(t)\big|+\int_t^T  G_\epsilon(s,t,k)|f_k(u)(s)-f_k( u^{\epsilon})(s)|ds \\
&\leq \epsilon T_1
\big(\epsilon \ln(T/\epsilon)\big)^{\frac{t-T}{T}} |k^2
 e^{tk^2}u_k(t)|+\int_t^T    T_1
 \big(\epsilon \ln(T/\epsilon)\big)^{\frac{t-s}{T}}
 |f_k(u)(s)-f_k( u^{\epsilon})(s)|ds.
\end{align*}
This implies
\begin{align*}
&\| u(.,t)- u^{\epsilon}(.,t)\|^2\\
&= \frac{\pi}{2}\sum_{k=1}^\infty |u_k(t)- u^{\epsilon}_k (t)|^2 \\
&\leq \pi \sum_{k=1}^\infty \epsilon^2.  T_1^2
\big(\epsilon \ln(T/\epsilon)\big)^{\frac{2t-2T}{T}}
 |k^2  e^{tk^2}u_k(t)|^2\\
&\quad+ \pi \sum_{k=1}^\infty  \epsilon^2.  T_1^2
 \big(\epsilon \ln(T/\epsilon)\big)^{\frac{2t-2T}{T}}
\Big(\int_t^T   \epsilon^{-\frac{s}{T}}
 \big(\ln(T/\epsilon)\big)^{1-\frac{s}{T}}
|f_k(u)(s)-f_k( u^{\epsilon})(s)|ds\Big)^2.
\end{align*}
This implies
\begin{align*}
&\| u(.,t)- u^{\epsilon}(.,t)\|^2\\
&\leq   T_1^2\epsilon^{\frac{2t}{T}}\big(\ln(T/\epsilon)
 \big)^{\frac{2t}{T}-2}\sum_{k=1}^\infty k^4  e^{2tk^2}u^2_k(t)  \\
&\quad + 2L^2T T_1^2\epsilon^{\frac{2t}{T}}\big(\ln(T/\epsilon)
\big)^{\frac{2t}{T}-2}\int_t^T
\epsilon^{\frac{-2s}{T}}\big(\ln(T/\epsilon)\big)^{2-\frac{2s}{T}}
\|u(.,s)- u^{\epsilon}(.,s)\|^2ds.
\end{align*}
Using again  Gronwall's inequality,
\[
  \epsilon^{\frac{-2t}{T}}\big(\ln(T/\epsilon)\big)^{2-\frac{2t}{T}}
\| u(.,t)- u^{\epsilon}(.,t)\|^2
\leq Qe^{2L^2TT_1^2(T-t)}.
\]
This completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2.4}]
Let $u^\epsilon$ be the solution of \eqref{eq6}-\eqref{eq8}
corresponding to $g$. Recall that  $w^{\epsilon}$ be the
solution of  \eqref{eq6}-\eqref{eq8} corresponding to $g_\epsilon$.

Part (a) of Theorem \ref{thm2.4}: Using Theorem \ref{thm2.2}
 and Theorem \ref{thm2.3}(a) , we have
\begin{align*}
\| w^\epsilon(.,t)-u(.,t)\|
&\leq  \| w^{\epsilon}(.,t)-u^\epsilon(.,t)\|
 +\|u^\epsilon(.,t)-u(.,t)\| \\
&\leq  T_1\exp(L^2T_1^2(T-t)^2) \big(\epsilon \ln(T/\epsilon)
\big)^{\frac{t-T}{T}}\|g_\epsilon-g\| \\
&\quad +\sqrt{MT_1^2e^{3L^2TT_1^2(T-t)}} \epsilon^{t/T}
 \big(\ln(T/\epsilon)\big)^{\frac{t}{T}-1}        \\
&\leq T_1(1+\sqrt{M})\exp\big(\frac{3L^2TT_1^2(T-t)}{2}\big)
\epsilon^{t/T}\big(\ln(T/\epsilon)\big)^{\frac{t}{T}-1} ,
\end{align*}
for every $t\in [0,T]$.
The proof of part (b) Theorem \ref{thm2.4} is similar to part (a) and it
is omitted.
\end{proof}

\section{Numerical experiments}

We consider the equation
\[
-u_{xx} +u_t = f(u) +g(x,t)
\]
where
\begin{gather*}
f(u)=u^4,\quad
g(x,t) = 2e^t \sin x - e^{4t} \sin^4 x,\
u(x,1) = \varphi_0 (x) \equiv e\sin x.
\end{gather*}
The exact solution of this equation is
$u(x,t) = e^t \sin x$.
In particular,
\[
u\big(x,\frac{99}{100}\big)\equiv
u(x) = \exp \big(\frac{99}{100}\big)\sin x.
\]
Let $\varphi_\epsilon(x) \equiv \varphi(x)
= (\epsilon + 1)e\sin x$. We have
\[
\|\varphi_\epsilon -\varphi\|_2
= \Big(\int_0^{\pi}\epsilon^2e^2 \sin^2 x dx\Big)^{1/2}
 = \epsilon e \sqrt{\pi/2}.
\]
We find the regularized solution
$u_\epsilon(x,\frac{99}{100}) \equiv u_\epsilon (x)$ having the form
\[
u_\epsilon(x) = v_m(x) = w_{1,m}\sin x
+ w_{2,m} \sin 2x + w_{3,m} \sin 3x,
\]
where
$v_1(x) = (\epsilon + 1)e \sin x$,
$w_{1,1} = (\epsilon + 1)e$, $w_{2,1} = 0$, $w_{3,1} = 0$,
$a = \frac{1}{{10000}}$, $t_m  = 1 - am$, for $m = 1,2,\dots,100$,
and
\begin{align*}
 w_{i,m + 1}  &= \frac{{e^{ - t_{m + 1} i^2 } }}{{\epsilon i^2
+ e^{ - t_m i^2 } }}w_{i,m}
- \frac{2}{\pi }\int_{t_{m + 1} }^{t_m } {\frac{{e^{ - t_{m + 1} i^2 } }}
{{\epsilon i^2 + e^{ - t_m i^2 } }}} e^{(s-t_{m})i^2}\\
&\quad\times \Big( {\int_0^\pi  {\big( {v_m^4 (x) + g(x,s)} \big)\sin ix\,dx}
 } \Big)ds,
\end{align*}
for $i = 1,2,3$.
Table \ref{table1} shows the the error between
the regularization solution $u_\epsilon$  and the exact solution $u$,
 for three values of $\epsilon$:
\begin{table}[ht]
\caption{}\label{table1}
\begin{center}
\begin{tabular}{|c|c|c|} \hline
$\epsilon$& $u_\epsilon$&  $\|u_\epsilon - u\|$ \\ \hline
 $10^{-5}$ &  $2.685490624\sin(x)-0.00009487155350\sin(3x)$
 & 0.005744631447\\ \hline
 $10^{-7}$ &  $2.691122866\sin(x)+0.00001413193606\sin(3x)$
 & 0.0001124971593\\ \hline
 $10^{-11}$ & $2.691180223\sin(x)+0.00002138991088\sin(3x)$
 & 0.00005831365439\\ \hline
\end{tabular}
\end{center}
\end{table}

Table \ref{table2} shows the error table in \cite[p. 214]{t2}.
\begin{table}[ht]
\caption{}\label{table2}
\begin{center}
\begin{tabular}{|c|c|c|} \hline
$\epsilon$& $u_\epsilon$& $\|u_\epsilon - u\|$ \\ \hline
 $10^{-5}$ & $2.430605996\sin x - 0.0001718460902\sin 3x$
 & 0.3266494251\\ \hline
 $10^{-7}$ & $2.646937077\sin x - 0.002178680692\sin 3x$
 & 0.05558566020\\ \hline
 $10^{-11}$ & $2.649052245 \sin x - 0.004495263004\sin 3x$
 & 0.05316693437\\ \hline
\end{tabular}
\end{center}
\end{table}

By applying the stabilized quasi-reversibility method in \cite{t4},
we have the approximate solution
$u_\epsilon\left(x,\frac{99}{100}\right) \equiv u_\epsilon (x)$
having the  form
\[
u_\epsilon(x) = v_m(x) = w_{1,m}\sin x + w_{6,m} \sin 6x\,,
\]
where
$v_1(x) = (\epsilon + 1)e \sin x$,
$w_{1,1} = (\epsilon + 1)e$,
$w_{6,1} = 0$,
and $a = \frac{1}{{10000}}$, $t_m  = 1 - am$ for $m = 1,2,\dots,100$, and
\begin{align*}
w_{i,m + 1}
&=({{\epsilon  + e^{ - t_{m} i^2 } }) ^ {\frac{t_{m+1}-t_{m}}
{t_{m}}}} w_{i,m}  - \frac{2}{\pi }
\int_{t_{m + 1} }^{t_m } {{e^{ (s - t_{m+1 }) i^2 } }} \\
&\quad \times \Big( {\int_0^\pi  {\big( {v_m^4 (x) + g(x,s)} \big)\sin ix\,
dx} } ds\Big),
\end{align*}
for $i = 1, 6$.
Table 3 shows the approximation error in this case.
\begin{table}[ht]
\caption{}\label{table3}
\begin{center}
\begin{tabular}{|c|c|c|} \hline
$\epsilon$& $u_\epsilon$ &  $\|u_\epsilon - u\|$ \\ \hline
$10^{-5}$ & $2.690989330\sin(x)-0.06078794774\sin(6x)$
 & 0.003940316590\\ \hline
$ 10^{-7}$ &  $2.691002638\sin(x) -0.05797060493 \sin(6 x)$
 & 0.003592425036\\ \hline
$10^{-11}$ & $2.691023938\sin(x)-0.05663820292\sin(6x)$
 & 0.003418420030
\\ \hline
\end{tabular}
\end{center}
\end{table}

By applying the method of integral equation in  \cite{t5},
we find the regularized solution
$u_\epsilon(x,\frac{99}{100}) \equiv u_\epsilon (x)$ having
the form
\[
u_\epsilon(x) = v_m(x) = w_{1,m}\sin x + w_{6,m} \sin 6x
\]
where
\[
v_1(x) = (\epsilon + 1)e \sin x,\quad
w_{1,1} = (\epsilon + 1)e, w_{6,1} = 0,
\]
and
$ a = \frac{1}{{5000}}$, $t_m  = 1 - am$ for $m = 1,2,\dots,5$, and
\begin{align*}
 w_{i,m + 1}
&=({{\epsilon i^2 + e^{ - t_{m} i^2 } }) ^ {\frac{t_{m+1}-t_{m}} {t_{m}}}}\\
&\quad\times \Big(w_{i,m}  - \frac{2}{\pi }
\int_{t_{m + 1} }^{t_m } {{e^{ (s - t_{m }) i^2 } }}
\Big( {\int_0^\pi  {\big( {v_m^4 (x) + g(x,s)} \big)\sin ixdx} }
\Big)ds\Big),
\end{align*}
for $i = 1,6$.
Table \ref{table4} shows the approximation errors in this
case.

\begin{table}[ht]
\caption{}\label{table4}
\begin{center}
\begin{tabular}{|c|c|c|} \hline
$\epsilon$& $u_\epsilon$&  $\|u_\epsilon - u\|$ \\ \hline
 $10^{-5}$ & $2.690968476 \sin(x)-0.05677543898\sin(6x)$ & 0.03489446471\\ \hline
 $10^{-7}$ & $2.690947247 \sin(x) - 0.05809747108$ & 0.003662541146\\ \hline
 $10^{-11}$ & $2.6912344727\sin(x)-0.0060809747108\sin(6x)$ & 0.0003371512534\\ \hline
\end{tabular}
\end{center}
\end{table}

Looking at the four tables,
we  see that the error of the second and third  tables  are smaller
than in the first table.
This shows that our approach has a nice regularizing effect
and give a better approximation than the previous
methods in \cite{t2,t4,t5}.

\subsection*{Acknowledgments}
The authors would like to thank the anonymous referees for their
valuable comments leading to the improvement of our manuscript;
Also to Professor Julio G. Dix for his valuable help in the
presentation of this article.

\begin{thebibliography}{00}

\bibitem{a1} Alekseeva, S. M. and Yurchuk, N. I.,
\emph{The quasi-reversibility method for the problem of the control
of an initial condition for the heat equation with an integral
boundary condition}, Differential Equations \textbf{34}, no. 4,
493-500, 1998.

\bibitem{a2} K. A. Ames, L. E. Payne;
\emph{Continuous dependence on modeling for some well-posed
perturbations of the backward heat equation}, J. Inequal. Appl.,
Vol. \textbf{3} (1999), 51-64.

\bibitem{a3} K. A. Ames, R. J. Hughes;
\emph{Structural Stability for Ill-Posed Problems in Banach
Space}, Semigroup Forum,  Vol. \textbf{70} (2005), N0 1,
127-145.

\bibitem{a4} Ames, K. A. and Payne, L. E.,
\emph{Continuous dependence on modeling for some well-posed
perturbations of the backward heat equation},
J. Inequal. Appl. \textbf{3}, $n^0$ 1, 51-64, 1999.

\bibitem{b1} H. Brezis; \emph{Analyse fonctionelle, Th\'eorie et
application}, Masson (1993).

\bibitem{c1} G. W.  Clark, S. F. Oppenheimer;
\emph{Quasireversibility methods for non-well posed problems},
Elect. J. Diff. Equ., \textbf{1994} (1994) no. 8, 1-9.

\bibitem{d1} M. Denche, K. Bessila;
\emph{A modified quasi-boundary value method for ill-posed problems},
J. Math. Anal. Appl,  Vol.\textbf{301}, 2005, pp.419-426.

\bibitem{e1} R. E. Ewing; \emph{The approximation of certain parabolic
equations backward in time by Sobolev equations}, SIAM J. Math.
Anal., Vol. \textbf{6} (1975), No. 2, 283-294.

\bibitem{g1} H. Gajewski, K. Zaccharias;
 \emph{Zur regularisierung einer
klass nichtkorrekter probleme bei evolutiongleichungen}, J. Math.
Anal. Appl., Vol. \textbf{38} (1972),  784-789.

\bibitem{h1} A. Hassanov, J. L. Mueller;
\emph{A numerical method
for backward parabolic problems with non-selfadjoint elliptic
operator}, Applied Numerical Mathematics,  \textbf{37} (2001),
55-78.


\bibitem{h2} Y. Huang, Q. Zhneg; \emph{Regularization for ill-posed
Cauchy problems associated with generators of analytic
semigroups}, J. Differential Equations, Vol. \textbf{203} (2004),
No. 1, 38-54.

\bibitem{h3} Y. Huang, Q. Zhneg;
\emph{Regularization for a class of ill-posed Cauchy problems},
 Proc. Amer. Math. Soc. \textbf{133} (2005), 3005-3012.

\bibitem{i1} V. K. Ivanov, I. V. Mel'nikova, and F. M. Filinkov;
\emph{Differential-Operator Equations and Ill-Posed problems},
Nauka, Moscow, 1995 (Russian).

\bibitem{j1} F. John; \emph{Continuous dependence on data for solutions
of partial differential equations with a prescribed bound}, Comm.
Pure Appl. Math, \textbf{13} (1960), 551-585.

\bibitem{k1} V. A. Kozlov, V. G. Maz'ya; \emph{On the iterative method for
solving ill-posed boundary value problems that preserve
differential equations}, Leningrad Math. J., \textbf{1} (1990),
No. 5, 1207-1228.

\bibitem{l1} R. Latt\`{e}s, J.-L. Lions; \emph{M\'ethode de
Quasi-r\'eversibilit\'e et Applications}, Dunod, Paris, 1967.

\bibitem{l2} M> Lees,  M. H. Protter;
\emph{Unique continuation for parabolic differential equations
and inequalities },Duke Math.J. \textbf{28}, (1961),369-382.

\bibitem{l3} N. T. Long, A. P. Ngoc. Ding;
\emph{Approximation of a parabolic non-linear evolution equation
backwards in time}, Inv. Problems, \textbf{10} (1994), 905-914.

\bibitem{m1} I. V. Mel'nikova, Q. Zheng and J. Zheng;
\emph{Regularization of weakly ill-posed Cauchy problem}, J.
Inv. Ill-posed Problems, Vol. \textbf{10} (2002), No. 5, 385-393.

\bibitem{m2} I. V. Mel'nikova, S. V. Bochkareva;
\emph{C-semigroups and regularization of an ill-posed Cauchy
problem}, Dok. Akad. Nauk., \textbf{329} (1993), 270-273.

\bibitem{m3} I. V. Mel'nikova, A. I. Filinkov; \emph{The Cauchy
problem. Three approaches}, Monograph and Surveys in Pure and
Applied Mathematics, \textbf{120}, London-New York: Chapman \&
Hall, 2001.

\bibitem{m4} K. Miller;
\emph{Stabilized quasi-reversibility and other
nearly-best-possible methods for non-well posed problems},
Symposium on Non-Well  Posed  Problems and Logarithmic Convexity,
Lecture Notes in Mathematics, \textbf{316} (1973),
Springer-Verlag, Berlin , 161-176.

\bibitem{p1} L. E. Payne;
\emph{Some general remarks on improperly posed problems for
partial differential equations}, Symposium on Non-Well  Posed
Problems and Logarithmic Convexity, Lecture Notes in Mathematics,
\textbf{316} (1973), Springer-Verlag, Berlin, 1-30.

\bibitem{p2} L. E. Payne;
\emph{Imprperely Posed Problems in Partial Differential
Equations}, SIAM, Philadelphia, PA, 1975.

\bibitem{p3} A. Pazy;
\emph{Semigroups of linear operators and application to partial
differential equations}, Springer-Verlag, 1983.

\bibitem{p4} S. Piskarev;
\emph{Estimates for the rate of convergence in the solution of
ill-posed problems for evolution equations}, Izv. Akad. Nauk SSSR
Ser. Mat., \textbf{51} (1987), 676-687.

\bibitem{q1} P. H. Quan, D. D. Trong;
\emph{ A nonlinearly backward
heat problem: uniqueness, regularization and error estimate},
Applicable Analysis, Vol. 85, Nos. 6-7, June-July 2006, pp. 641-657.

\bibitem{r1} M. Renardy, W. J. Hursa and J. A. Nohel;
\emph{Mathematical Problems in Viscoelasticity}, Wiley, New
York, 1987.

\bibitem{s1} R. E. Showalter;
\emph{The final value problem for evolution equations}, J. Math.
Anal. Appl, \textbf{47} (1974), 563-572.

\bibitem{s2} R. E. Showalter;
\emph{Cauchy problem for hyper-parabolic partial differential
equations}, in Trends in the Theory  and Practice of Non-Linear
Analysis, Elsevier 1983.

\bibitem{s3} R. E. Showalter;
\emph{Quasi-reversibility of first and second order parabolic
evolution equations}, Improperly posed boundary value problems
(Conf., Univ. New Mexico, Albuquerque, N. M., 1974), pp. 76-84.
Res. Notes in Math., $n^0$ 1, Pitman, London, 1975.

\bibitem {t1} D. D. Trong, N. H.  Tuan;
 \emph{Regularization and error estimates for nonhomogeneous
backward heat problems},
Electron. J. Diff. Equ., Vol. 2006 , No. 04, 2006, pp. 1-10.

\bibitem{t2} D. D. Trong, P. H.  Quan, T. V. Khanh, N. H. Tuan;
\emph{A nonlinear case of the 1-D backward heat problem:
Regularization and error estimate}, Zeitschrift
Analysis und ihre Anwendungen, Volume 26, Issue 2, 2007, pp. 231-245.

\bibitem{t3} D. D. Trong, N. H. Tuan;
\emph{A nonhomogeneous backward heat problem: Regularization and
error estimates}, Electron. J. Diff. Equ., Vol. 2008 , No. 33,
pp. 1-14.

\bibitem{t4} D. D. Trong, N. H. Tuan;
\emph{Stabilized quasi-reversibility method for a class of
nonlinear ill-posed problems }, Electron. J. Diff. Equ., Vol. 2008 ,
No. 84, pp. 1-12.

\bibitem{t5} Dang Duc Trong,  Nguyen Huy Tuan;
 \emph{Regularization and error estimate for the
nonlinear backward heat problem using a
method of integral equation.},
Nonlinear Analysis, Volume 71, Issue 9, 1 November 2009,
Pages 4167-4176.

\end{thebibliography}

\end{document}