Terminal value problem for nonlinear parabolic equation with Gaussian white noise

1.   Introduction

The forward problem for parabolic equations is finding the distribution at a later time when we know the initial distribution. In geophysical exploration, one is often faced with the problem of determining the temperature distribution in the object or any part of the Earth at a time $t_0 > 0$ from temperature measurements at a time $t_1 > t_0.$ This is the backward in time parabolic problem. Backward parabolic problems arises in several practical areas such as image processing, mathematical finance, and physics (see ^[2,3]). Let $T$ be a positive number and $\Omega$ be an open, bounded and connected domain in $\mathbb{R}^d, d \ge 1$ with a smooth boundary $\partial \Omega$. In this paper, we consider the question of finding the function $\bf u(x, t)$, $(x, t)\in \Omega \times [0, T]$, satisfying the nonlinear problem

where the functions $a(x, t), g(x)$ are given and the source function $F$ will be given later. Here the coefficient $a(x, t)$ is a $C^1$ smooth function and $0 < \overline m \le a(x, t) < M$ for all $(x, t) \in \Omega \times (0, T)$ for some finite constants $\overline m, \ M$. The problem is well-known to be ill-posed in the sense of Hadamard. Hence, a solution corresponding to the data does not always exist, and in the case of existence, it does not depend continuously on the given data. In fact, from small noise contaminated physical measurements, the corresponding solutions will have large errors. Hence, one has to resort to a regularization. In the simple case of deterministic noise, Problem (1.1) with $a = 1$ and $F = 0$ was studied by many authors ^[4,5,6]. However, in the case of random noise, the analysis of regularization methods is still limited. The problem is to determine the initial temperature function $f$ given a noisy version of the temperature distribution $g$ at time $T$

where $\delta > 0$ is the amplitude of the noise and $\xi$ is a Gaussian white noise. In practice, we only observe some finite errors as follows

where the natural number ${\bf N}$ is the number of steps of discrete observations and $\phi_j$ is defined in section 2. The main goal is to find an approximate solution $\widehat {\bf u}_{N}(0)$ for ${\bf u}(0)$ and then investigate the rate of convergence ${\bf E} \| \widehat {\bf u}_{N}(0) - {\bf u}(0)\|$, which is called the mean integrated square error (MISE). Here ${\bf E}$ denotes the expectation w.r.t. the distribution of the data in the model (1.2).

There are two main approaches to considering inverse problem for noise modeling. The first approach is based on a formal technique if one is assuming that the noise is definite and small. The second approach is based on a statistical point of view and in this approach one does not need to assume smalll levels of noise. We consider in this paper a statistical point of view for the backward parabolic equation. Our aim is to reconstruct the initial function from the disturbance measurements of the final values in a statistical inverse problem framework. There are many different types of random noise, but we are interested in Gaussian noise here. The model (1.2) and (1.3) were considered in some recent papers; see ^{[7,8,9,10,11]}. In signal processing, Gaussian white noise is a random signal of equal intensity at different frequencies, giving it a constant power spectral density and this term is used in physics, acoustic engineering, telecommunications and statistical forecasting.

The inverse problem with random noise has a long history. The simple case of (1.1) is the homogeneous linear parabolic equation of finding the initial data $u_0: = u(x, 0)$ that satisfies

This equation is a special form of statistical inverse problems and it can be transformed by a linear operator with random noise

where $K$ is a bounded linear operator that does not have a continuous inverse. Fomula (1.5) is interpreted as $Ku_0$ deviated from function $g$ by a random error.

Problem (1.4) was studied by well-known methods including spectral cut-off (or called truncation method) ^[7,9,12,13], the Tiknonov method ^[14], iterative regularization methods ^[15], the Bayes estimation method ^[16,17], and the Lavrentiev regularization method ^[18]. In some parts of these works, the authors show that the error ${\bf E} \| \widehat {\bf u}_{N}(0) - {\bf u}(0)\|$ tend to zero when ${\bf N}$ is suitably chosen according to the value of $\delta$ and $\delta \to 0$. For more details, we refer the reader to ^[19].

To the best of our knowledge, there are no results for the backward problem for nonlinear parabolic equations with Gaussian white noise. There are two types of difficulty in solving our problem. The first difficulty occurs because the problem is nonlinear and nonlinear problems with random noise is more difficult since one cannot apply well known methods. The second is the random noise data, which makes the problem computationally complex. The problem of computation with random data requires some knowledge of the stochastic process, so one has to consider the expectation.

Very recently, in ^[20], the authors studied the discrete random model for backward nonlinear parabolic problems. However, the problem considered in ^[20] is in a rectangular domain which is limited in practice. The present paper uses another random model and also gives approximation of the solution in the case of more general bounded and smooth domains $\Omega$. Our task in this paper is to show that the expectation between the solution and the approximate solution converges to zero when ${\bf N}$ tends to infinity.

This paper is organized as follows. In section 2, we give a couple of preliminary results. In section 3, we give an explanation for ill-posedness of the problem. To help the reader, we divide the problem into three cases under various assumptions on the coefficient $a$, and the source function $F$. Case 1: $a: = a(x, t)$ is a constant and $F$ is a globally Lipschitz function. In section 4, we will study this case and give convergence rates in $L^2$ and $H^p$ norms for $p > 0$. The method here is the well-known spectral method. The main idea is to approximate the final data $g$ by the approximate data and use this function to establish a regularized problem by the truncation method.

Case 2: $a: = a(x, t)$ depends on $x$ and $t$ and $F$ is a locally Lipschitz function. This problem is more difficult. In most practical problems, the function $F$ is often a locally Lipschitz function. The difficulty here is in the fact that the solution cannot be transformed into a Fourier series and therefore, we cannot apply well-known methods to find an approximate solution. In Section 5, we will study a new form of the quasi-reversibility method to construct a regularized solution and obtain the convergence rate. Our method is new and very different than the method of Lattes and Lions ^[21]. We approximate the locally Lipschitz function by a sequence of globally Lipschitz functions and use some new techniques to obtain the convergence rate.

Case 3 Various assumptions on $F$. In practice there are many functions that are not locally Lipschitz. Hence our analysis in section 4 cannot applied in section 6. Our method in section 6 is also the quasi-reversibility method and is very similar to the method in section 4. However in section 6, we do not approximate $F$ as we do in section 4. This leads to a convergence rate that is better than the one in section 4. One difficulty that occurs in this section is showing the existence and uniqueness of the regularized solution. To prove the existence of the regularized solution, we do not follow previously mentioned methods. Instead, we use the Faedo – Galerkin method, and the compactness method introduced by Lions ^[22]. To the best of our knowledge, this is the first result where $F$ is not necessarily a locally Lipschitz function. Finally, in section 7, we give some specific equations which can be applied by our method.

2.   Preliminaries

To give some details on this random model (1.2), we give the following definitions (See ^[12,19]):

Definition 2.1. Let $\mathcal{H}$ be a Hilbert space. Let $g, g_\delta \in \mathcal{H}$ satisfy (1.2). We understand the equal relationship in fomula

as follows:

here $\delta$ is the amplitude of the noise. We also assume that $\xi$ is a zero-mean Gaussian random process indexed by $\mathcal H$ on a probability space. $\left \langle \xi, \chi \right \rangle \sim \mathcal{N} (0, \|\chi\|^2_\mathcal{H})$. Moreover, given $\chi_1, \chi_2 \in H$ then

Definition 2.2. The stochastic error is a Hilbert-space process, i.e., a bounded linear operator $\xi : \mathcal{H} \to L^2({\bf \Omega}, \mathcal{A}, P)$ where $({\bf \Omega}, \mathcal{A}, P)$ is the underlying probability space and $L^2(., .)$ is the space of all square integrable measurable functions.

Let us recall that the eigenvalue problem

admits a family of eigenvalues $0 < \lambda_1 \le \lambda_2 \le \lambda_3 \le...\le \lambda_j\le...\nonumber$ and eigenfunctions $\{\phi_{j}\}$ and $\lambda_j \to \infty$ as $j \to \infty$; see page 335 in ^[23].

Next, we introduce the abstract Gevrey class of functions of index $\sigma > 0$, see, e.g., ^[24], defined by

which is a Hilbert space equipped with the inner product

its corresponding norm is $\left\Vert v\right\Vert _{\mathcal{W}_{\sigma}} = \sqrt{\sum_{j = 1}^{\infty}e^{2\sigma\lambda_{j}}\big|\big\langle v, \phi_{j}\big\rangle_{L^2(\Omega)} \big|^{2}} < \infty.$

3.   The ill-posedness of the nonlinear parabolic equation with random noise

The ill-posedness of the backward heat equation is well known and has appeared in many previous articles. However, in the random case, we need to give an example to illustrate the ill-posedness. From the appearance of the expected component, the evaluation of the nonconformity of the random model is much more complicated than the deterministic model. Therefore, we have to choose a simple case to find a suitable example. In this section, for a special case of Eq (1.1), we show that the nonlinear parabolic equation with random noise is ill-posed in the sense of Hadamard.

Theorem 3.1. Problem (1.1) is ill-posed in the special case when $a = 1, \Omega = (0, \pi)$.

Proof. Let $\Omega = (0, \pi)$ and $a(x, t) = 1$, Then $\lambda_{\bf N} = {\bf N}^2$. Let us consider the following parabolic equation

where $F_0$ is

for any $v \in L^2(\Omega)$, and $\phi_j(x) = \sqrt{\frac{2}{\pi}} \sin (jx)$. Let us choose $G_{\delta, {\bf N(\delta) }} \in \mathcal L^2(\Omega)$ such that

where $g_\delta$ is defined by

By the usual MISE decomposition which involves a variance term and a bias term, we get

The solution of Problem (3.1) is given by the Fourier series (see ^[29])

We show that Problem (3.6) has unique solution ${\bf V}_{\delta, {\bf N(\delta) }} \in C([0, T]; L^2(\Omega))$. Let us consider

For any $v_1, v_2 \in C([0, T]; L^2(\Omega))$, using Hölder inequality, we have for all $t \in [0, T]$

Hence, we obtain that

Thus $\Phi$ is a contraction. Using the contraction principle and we conclude that the equation $\Phi(w) = w$ has a unique solution ${\bf V}_{\delta, {\bf N(\delta) }} \in C([0, T]; L^2(\Omega))$. Using the inequality $a^2+b^2 \ge \frac{1}{2} (a-b)^2, \; \; a, b \in \mathbb{R}$, we have the following estimate

First, using Hölder's inequality, we get

We have the lower bound for $I_1$:

Combining (3.10), (3.11), (3.12), and we obtain

By taking supremum of both sides on $[0, T]$, we get

Choosing ${\bf N}: = {\bf N}(\delta) = \sqrt{ \frac{1}{2T} \ln (\frac{1}{\delta}) }$, we obtain

and

From (3.15) and (3.16), we can conclude that Problem (1.1) is ill-posed.

4.   Regularization result with constant coefficient and globally Lipschitz source function

In this section, we consider the question of finding the function $\bf u(x, t)$, $(x, t)\in \Omega \times [0, T]$, that satisfies the problem

In this section, we assume there exists a constant $K > 0$ with

where $(x, t) \in \Omega \times [0, T]$ and $u, \, v \in \mathbb R$.

Lemma 4.1. Let $\overline G_{\delta, {\bf N}(\delta) } \in L^2(\Omega)$ be such that

Assume that $g \in H^{2\gamma}(\Omega)$. Then we have the following estimate

for any $\gamma \ge 0$. Here ${\bf N}$ depends on $\delta$ and satisfies $\lim_{\delta \to 0} {\bf N}(\delta) = +\infty$ and {$\lim_{\delta \to 0} \delta^2 {\bf N}(\delta) = 0$}.

Remark 4.1. Consider the right hand side of (4.3). In order for the right-hand side of (4.3) to converge to zero we require $\lim_{\delta \to 0} \delta^2 {\bf N}(\delta) = 0$ and the condition

Since the fact that $\lambda_k \sim k^{2/d}$, we see that

and to verify the condition (4.4) we need the condition $\lim_{\delta \to 0} {\bf N}(\delta) = +\infty$.

Proof. For the following proof, we consider the genuine model (1.3). By the usual MISE decomposition which involves a variance term and a bias term, we get

Since $\xi_{j} = \langle \xi, \phi_j \rangle \stackrel {iid}{\sim} N(0, 1)$, it follows that ${\bf E} \xi_j^2 = 1$, so

Using the truncation method, we give a regularized problem for Problem (1.1) as follows

where $\alpha_{\bf N(\delta) }$ is regularization parameter and ${\bf J}_{ \alpha_{\bf N (\delta)}}$ is the following operator

Our main result in this section is as follows

Theorem 4.1. Problem (4.7) has a unique solution ${\bf u}^\delta_{\bf N (\delta) } \in C([0, T];L^2(\Omega))$ which satisfies

Assume that problem (1.1) has unique solution ${\bf u}$ such that

Choose $\alpha_{\bf N(\delta) }$ such that

Then the following estimate holds

Remark 4.2. 1. From the theorem above, it is easy to see that ${\bf E} \left\| {\bf u}^\delta_{\bf N(\delta)} (x, t)-\bf u(x, t) \right\|^2_{L^2(\Omega)}$ is of the order

2. Now, we give one example for the choice of ${\bf N} (\delta)$ which satisfies condition (4.11). Since $\lambda_{\bf N} \sim {\bf N}^{\frac{2}{d}}$, see ^[25], we choose $\alpha_{\bf N}$ such that $e^ {kT \alpha_{\bf N(\delta)} } = |{\bf N(\delta)}|^a$ for any $0 < a < \frac{2\gamma}{d}$. Then we have $\alpha_{\bf N(\delta)} = \frac{a}{kT} \log (\bf N(\delta))$. The number ${\bf N(\delta)}$ is chosen as

for $0 < b < 1$. With ${\bf N} (\delta)$ chosen as above, ${\bf E} \left\| {\bf u}^\delta_{\bf N(\delta)} (x, t)-\bf u(x, t) \right\|^2_{L^2(\Omega)}$ is of the order $\left(\frac{1}{\delta}\right)^{ \frac{-(ba+\frac{b}{2})at} {KT} }$

3. The existence and uniqueness of the solution of Eq (1.1) is an open problem, and we do not investigate this problem here. The case considered in Theorem 3.1 gives the existence of the solution of Problem (1.1) in a special case. The uniqueness of the backward parabolic problem has attracted the attention of many authors (see, for example, ^[26,27,28]) and this is also a challenging open problem.

Proof of Theorem 4.1. We divide the proof into a number of parts.

Part 1. Problem (4.7) has a unique solution ${\bf u}^\delta_{\bf N(\delta)} \in C([0, T];L^2(\Omega))$. The proof is similar to ^[29] (see Theorem 3.1, page 2975 ^[29]). Hence, we omit it here.

Part 2. Estimate the expectation of the error between the exact solution $u$ and the regularized solution ${\bf u}^\delta_{\bf N(\delta)}$.

Let us consider the following integral equation

We have

Taking the expectation of both sides of the last inequality, we get

Multiplying both sides with $e^{2t \alpha_{\bf N} }$, we obtain

Applying Gronwall's inequality, we get

Hence, using Lemma 4.1, we deduce that

Now, we continue to estimate $\| {\bf u}(., t)- {\bf v}^\delta_{\bf N(\delta)}(., t)\|_{L^2(\Omega)}$. Indeed, using Hölder's inequality and globally Lipschitz property of $F$, we get

above, we have used the mild solution of $u$ as follows

Multiplying both sides with $e^{2t \alpha_{\bf N(\delta)} }$, we obtain

Gronwall's inequality implies that

This together with the estimate (4.19) leads to

where $A'$ is given in Eq (4.10). This completes our proof.

The next theorem provides an error estimate in the Sobolev space $H^p (\Omega)$ which is equipped with a norm defined by

To estimate the error in $H^p$ norm, we need a stronger assumption of the solution $u$.

Theorem 4.2. Assume that problem (1.1) has unique solution ${\bf u}$ such that

for any $r > 0$. Choose $\alpha_{\bf N(\delta) }$ such that

Then the following estimate holds

Proof. First, we have

Next, we continue to estimate ${\bf E} \| {\bf u}^\delta_{\bf N(\delta)}(., t)- {\bf u}(., t)\|_{L^2(\Omega)}^2$ with assumption (4.24). Recall ${\bf v}^\delta_{\bf N(\delta)}$ from (4.14). The expectation of the error between ${\bf u}^\delta_{\bf N(\delta)}$ and ${\bf v}^\delta_{\bf N(\delta)}$ is given in the estimate (4.19) as

We only need to estimate $\| {\bf u}(., t)- {\bf v}^\delta_{\bf N(\delta)}(., t)\|_{L^2(\Omega)}$. Indeed, using Hölder's inequality and the globally Lipschitz property of $F$, we get

Multiplying both sides with $e^{2t \alpha_{\bf N(\delta)} }$, we obtain

Gronwall's inequality implies that

This last estimate together with the estimate (4.29) leads to

On the other hand, consider the function

The derivative of $G$ is $G'(\xi) = \xi^{p-1} e^{-D\xi} (p-D\xi)$. Hence we know that $G$ is strictly decreasing when $D\xi \ge p$. Since $\lim_{\delta \to 0} \alpha_{\bf N(\delta)} = +\infty$, we see that if $\delta$ is small enough then $2r \alpha_{\bf N(\delta)} \ge p$. Put $D = 2(t+r), \; \xi = \alpha_{\bf N(\delta)}$ into (4.33), and we obtain for $\lambda_j > \alpha_{\bf N(\delta)}$

The latter equality leads to

where we use assumption (4.24) for the last inequality. Combining (4.28), (4.32) and (4.34), and we deduce that

which completes the proof.

Remark 4.3. In the above Theorem, to obtain the error estimate, we require strong assumptions on ${\bf u}$. This is a limitation of Theorem 4.1, because there are only certain types of functions ${\bf u}$ satisfying these conditions. To remove this limitation, we need to find a new estimator. The convergence rate in the case of weak assumptions of ${ u}$ is a difficult problem. Indeed, in the next Theorem, we give a regularization result in the case of a weaker assumption for ${ u}$, i.e., ${ u} \in C([0, T];L^2(\Omega))$. This is one of the first results in this case.

4.1.   The second regularized solution and the error estimate

To help the reader, we describe our analysis and methods in this subsection. To obtain the approximate solution when the solution ${ u}$ is in $C([0, T];L^2(\Omega))$, we don't use a regularized solution as in Theorem 4.1. Since $\overline G_{\delta, {\bf N}(\delta) }$ is an approximation of $G$, we know that it is an observed data. It can also be called the "input data". Recall that $K$ is the Lipschitz constant of $F$. We divide our results in Theorem 4.3 into two cases:

Case 1: $KT < 1$. By the way the input data $\overline G_{\delta, {\bf N}(\delta) }$ is defined, we construct a new regularized solution. Then we obtain the error between the new regularized solution and the sought solution ${\bf u}$.

Case 2: $KT > 1$. In this case, the construction of the regularized solution is more difficult. To apply the known result in Case 1, we need to divide $[0, T]$ into a collection of sub intervals $[T_{h}, T_{h'}]$ where $K(T_{h'}-T_{h}) < 1$. From the given input data $\theta$ and appropriate parameter regularization $\zeta$, we set the output function ${\bf Y}_{T_{h}, T_{h'}}^{\zeta}(f)({\bf x}, t)$ satisfies the nonlinear integral equation (4.37). The existence of ${\bf Y}_{T_{h}, T_{h'}}^{\zeta}(f)$ in $C([ T_{h}, T_{h'}]; L^2(\Omega))$ holds if $K(T_{h'}-T_{h}) < 1$. From (4.56), we have an important result: If $\zeta$ is suitably chosen and $\theta$ is an approximate function of ${u}(x, T_{h'})$ then the function ${\bf Y}_{T_{h}, T_{h'}}^{\zeta}(f)({ x}, t)$ is an approximate solution of the sought solution ${ u}$ in all intervals $[T_{h}, T_{h'}]$. Let $s$ be a positive integer such that $s > KT$. Define a sequence of points $\{ T_l \}, \; \; l = 0, 1, ...2s$ such that

where $\overline h = \frac{1}{2s}$. In all the intervals $[T_{i}, T_{i+1}], \; i = \overline{0, 2s-1}$, we construct different regularized solutions and combine them into a final regularized solution. More details are as follows:

● In the first step, to construct an approximate solution on $[T_{2s-1}, T]$, we use the input data $\overline G_{\delta, {\bf N}(\delta) }$ and parameter regularization $\zeta_{2s}$ to establish a function ${\bf Y}_{T_{2s-2}, T_{2s}}^{\zeta_{2s}}\left(\overline G_{\delta, {\bf N}(\delta) } \right)({x}, t)$. Then we define a regularized solution ${\mathcal U}_{\delta}(x, t) = {\bf Y}_{T_{2s-2}, T_{2s}}^{\zeta_{2s}}\left(\overline G_{\delta, {\bf N}(\delta) } \right)({ x}, t)$ for all $t \in [T_{2s-1}, T]$.

● In the second step, to construct an approximate solution on $[T_{2s-2}, T_{2s-1}]$, we use the input data ${\mathcal U}_{n}(x, T_{2s-1})$ (which is computed in the first step) and parameter regularization $\zeta_{2s-1}$ to establish a function ${\bf Y}_{T_{2s-2}, T_{2s}}^{\zeta_{2s}}\left(\overline G_{\delta, {\bf N}(\delta) } \right)({ x}, t)$. Then we define a regularized solution

for all $t \in [T_{2s-2}, T_{2s-1}]$.

● We continue similarly for the remaining steps. Finally, we obtain the regularized solution in (4.61) and (4.62).

Now, we consider the following lemma.

Lemma 4.2. Let $0 \le T_{h} < T_{h'} \le T$. For $f \in C([ T_{h}, T_{h'}]; L^2(\Omega))$, we consider the following nonlinear integral equation

for $\zeta > 0$. Assume that $K (T_{h'}-T_{h}) < 1$. Then Problem (4.37) has a unique solution ${\bf Y}_{T_{h}, T_{h'}}^{\zeta}(f) \in C([ T_{h}, T_{h'}]; L^2(\Omega)).$ Moreover, we have the following estimate

for all $t \in [ T_{h}, T_{h'}]$ and $q_0$ satisfies $0 < q_0 < \frac{1}{K^2(T_{h'}-T_{h})^2}-1$.

Proof. Part A. We will begin by showing that Eq (4.37) has a unique solution in $C([ T_{h}, T_{h'}]; L^2(\Omega)).$ Our analysis here is similar to the one in ^[29]. Define on $C([ T_{h}, T_{h'}]; L^2(\Omega))$ the following Bielecki norm

for all $v \in C([ T_{h}, T_{h'}]; L^2(\Omega))$. It is easy to check that $\|.\|_1$ is a norm of $C([ T_{h}, T_{h'}]; L^2(\Omega))$. Now, let $f$ be in $L^2(\Omega)$. We want to show that the map given by

for $w(f) \in C([ T_{h}, T_{h'}]; L^2(\Omega))$, is a contraction on $C([ T_{h}, T_{h'}]; L^2(\Omega))$ with the condition $K (T_{h'}-T_{h}) < 1$. Indeed, we shall prove that, for every $w_1, w_2 \in C([ T_{h}, T_{h'}]; L^2(\Omega))$,

First, by using the Hölder inequality and the global Lipschitz property of $F$, we have the following estimates for all $t\in [T_{h_1}, T_{h_2}]$, namely

Noting that if $\lambda_j > \zeta$ then $e^{(\tau-t) \lambda_j } \le e^{(\tau-t) \zeta }$ for $T_h \le \tau \le t$, it follows that

From the definition of $\mathcal I$ in (4.40), we have

Combining (4.42), (4.42), (4.42) and using the inequality $(a + b)^2 \le (1 + \theta_0)a^2 + \left(1 + \frac{1}{\theta_0} \right)b^2$ for any real numbers $a, b$ and $\theta_0 > 0$, we get the following estimate for all $t \in (T_{h}, T_{h'})$

By choosing $\theta_0 = \frac{T_{h'}-t}{t}$, we obtain {for all} $t \in (T_{h}, T_{h'})$

On the other hand, letting $t = T_{h'}$ in (4.42), we get

By letting $t = T_{h}$ in (4.42), we obtain

Combining (4.42), (4.43) and (4.44), we deduce that for all $T_{h}\le t \le T_{h'}$

which leads to (4.41). Since $K(T_{h'}-T_{h}) < 1$, it follows that $\mathcal I$ is a well-defined contraction on $C([ T_{h}, T_{h'}]; L^2(\Omega))$. By the Banach fixed point theorem, it therefore has a unique fixed point, i.e., the equation $\mathcal {I}(w) = w$ has a unique solution which we denote by ${\bf Y}_{T_{h}, T_{h'}}^{\zeta}(f) \in C([ T_{h}, T_{h'}]; L^2(\Omega))$.

Part B. The error estimate $\mathbb{\bf E} \| {\bf Y}_{T_{h}, T_{h'}}^{\zeta}(f)(., t)- {\bf u}(., t)\|^2_{L^2(\Omega)}$.

By a similar technique as in the proof of Theorem 4.1, we obtain

This leads to

The last equality implies that after some simple transformation

Using the last equality and (4.47), we get

We have

This implies that

Hence, using Parseval's identity and the inequality

for any real numbers $c_1, c_2, c_3$ and $q_0 > 0$ we have

Multiplying both sides of the last inequality by $e^{2(t-T_{h}) \zeta }$, and using the global Lipschitz property of $F$, we obtain

Since ${\bf Y}_{T_{h}, T_{h'}}^{\zeta}(f), \; {\bf u} \in C([ T_{h}, T_{h'}]; L^2(\Omega))$ we obtain that the function

is continuous on $[T_{h}, T_{h'} ]$. Therefore, the following is a finite positive constant

This implies that

Hence

Since by assumption $0 < q_0 < \frac{1}{K^2(T_{h'}-T_{h})^2}-1$, it follows that the term on the left hand-side that is in parenthesis is positive. This implies that for all $t \in [T_{h}, T_{h'}]$

Hence for all $t \in [T_{h}, T_{h'}]$ we conclude that

Our main result in this subsection is as follows.

Theorem 4.3. Let $g$ be as in Theorem 4.1. Assume that ${\bf u}$ is the unique solution of Problem (1.1).

(a) Assume that $KT < 1$, where $K$ is the Lipschitz constant of $F$. A new regularized solution is given as follows

Let us choose $\zeta{(\delta) }$ such that

If ${\bf u} \in C([0, T];L^2(\Omega))$ then as $|{\delta}| \to 0$

(b) Suppose that $KT > 1$ and let us assume that ${\bf u} \in C([0, T];L^2(\Omega))$. Let

We construct a regularized solution $\widehat U_{\delta }$ as follows

and

where ${\bf Y}_{T_{h_1}, T_{h_2}}^{\zeta(\delta)}(\theta)({ x}, t)$ is defined in (4.37). Then we have

● If $t \in [T_k, T_{k+1}]$ and $k = \overline {1, 2s-1}$ then

● If $t \in [0, T_1]$ then

Remark 4.4. In ^[29], we only need the regularization result for $0 < KT < 1$. Our Theorem 4.3 extends this result for any $K > 0$.

Proof of part (a) of Theorem 4.3. By setting $T_{h} = 0$ and $T_{h'} = T$ $f = \overline G_{\delta, {\bf N}(\delta) }$ then ${\bf Y}_{T_{h}, T_{h'}}^{\zeta}(f)$ given by (4.37) in Lemma 4.2 is equal to $\widehat U_{\delta }$ given by (4.57). Then apply the result from (4.38). Since $KT < 1$, applying Lemma 4.2, we obtain

This completes the proof of part (a).

Proof of part (b) of Theorem 4.3

By Theorem 3.1, we have ${\bf E} \| \overline G_{\delta, {\bf N} (\delta) } -g \|_{L^2(\Omega)}^2 \le \widetilde C \mathcal \xi^2({\delta}).{\nonumber}$ where $\widetilde C = 1+ \|g\|_{H^{2\gamma}}^2.$ We will estimate the error for time variable in interval $[T_{l}, T_{l+1}]$ for $l = \overline{0, 2s}$.

Case 1. Let $t \in [T_{2s-1}, T]$. Since $\zeta_{2s}(\delta) = \frac{s}{T} \log\left(\frac{1}{\mathcal \xi (\delta) }\right)$, by Lemma 4.2 we get

which we note that $e^{2(T_{2s-2}-t) \zeta_{2s}({\delta})} \le e^{2(T_{2s-2}-T_{2s-1}) \zeta_{2s}({\delta}) } = \mathcal \xi ({\delta})$ and

Case 2. Let $t \in [T_{2s-2}, T_{2s-1}]$. Since $\zeta_{2s-1}({\delta}) = \frac{s}{2T} \log\left(\frac{1}{\mathcal \xi ({\delta}) }\right),$ by Lemma 4.2 we get

where we used the following result from (4.65):

Therefore, repeating the argument as in the above cases and using the induction method, we can prove the following estimate

for all $t \in [T_k, T_{k+1}]$ and $k = \overline {1, 2s-1}.$

If $t \in [0, T_1]$, then by a similar technique as above, we obtain the error estimate

5.   Regularization result with locally Lipschitz source

Section 4 addressed a problem in which $F$ is a global Lipschitz function. In this section we extend the analysis to a locally Lipschitz function $F$. Results for the locally Lipschitz case are difficult. Here, we have to find another regularization method to study the problem with a locally Lipschitz source.

Assume that $a$ is noisy by the observation data $a_\delta^{{\rm{obs}}} :\Omega \times [0, T] \to \mathbb{R}$ as follows

where $\delta > 0$ and $\psi \in L^\infty(0, T)$ such that

where ${\overline M} > 0$. In the case when $a$ is not disturbed, we can use the method in the previous sections (the case when $a$ is not disturbed is simpler than the case $a$ is noisy). If $a$ is disturbed by random data, it is difficult to use the old method and we need a new approach, as outlined below.

Assume that for each ${\mathcal R} > 0$, there exists $K_{\mathcal R} > 0$ such that

where $(x, t) \in \Omega \times [0, T]$ and

We note that $K_{\mathcal R}$ is increasing and $\lim_{{\mathcal R} \rightarrow +\infty} K_{\mathcal R} = +\infty$. Now, we outline our idea to construct a regularization for problem (1.1). For all $\mathcal R > 0$, we approximate $F$ by $\mathcal{F}_{\mathcal R}$ defined by

For each $\delta > 0$, we consider a parameter $\mathcal R(\delta) \rightarrow +\infty$ as $\delta \rightarrow 0^+$. Let us denote the operator $\mathbb{P} = M \Delta$, where $M$ is a positive number such that $M > {\bf a}_\delta^{{\rm{obs}}}(x, t)$ for all $(x, t) \in \Omega \times (0, T)$. Define the following operator

where

for any function $v \in L^2(\Omega)$. Here ${\bf N}(\delta)$ is defined in Lemma (4.1).

We introduce the main idea to solve problem (1.1) with a generalized case of source term defined by (5.4), and we consider the problem:

Here $\overline G_{\delta, {\bf N(\delta)}}(x)$ is defined in Eq (4.2). Now, we introduce some Lemmas which will be useful for our main results. First, we recall the abstract Gevrey class of functions of index $\sigma > 0$, see, e.g., ^[24], defined by

which is a Hilbert space equipped with the inner product

and the corresponding norm is $\left\Vert v\right\Vert _{\mathcal{W}_{\sigma}} = \sqrt{\sum_{n = 1}^{\infty}e^{2\sigma\lambda_{n}}\big|\big\langle v, \phi_{n}\big\rangle_{L^2(\Omega)} \big|^{2}} < \infty.$

Lemma 5.1. For $\mathcal{F}_{\mathcal R} \in L^\infty(\Omega \times [0, T] \times \mathbb{R})$, we have

Proof. See the proof of Lemma 2.4 in ^[35].

Lemma 5.2. 1. Let $M, T > 0$. For any $v \in \mathcal{W}_{MT}(\Omega)$, we have

2. Let $\beta_{\bf N(\delta)} < 1- e^{-MT \lambda_1}$. For any $v \in L^2(\Omega)$, we have

Proof. Using the inequality $\ln (1+a) \leq a, \; \forall a > 0$, we have

Since $\beta_{\bf N(\delta)} < 1- e^{-MT \lambda_1}$, we know that $\beta_{\bf N(\delta)} + e^{-MT \lambda_j} < 1$. Using Parseval's equality, we easily get

Theorem 5.1. Problem (5.6) has a unique solution ${\bf u}^\delta_{\bf N(\delta)} \in C\left(\left[0, T\right]; L^{2}\left(\Omega\right)\right).$ Assume that the problem (1.1) has a unique solution $\bf u$ satisfying $\bf u(\cdot, t) \in \mathcal{W}_{MT}$. Choose $\beta_{\bf N(\delta)}$ such that

Choose $\mathcal{R}_\delta$ such that

Then we have the following estimate

Here $\widetilde C (\delta)$ is

and assume that $\Omega$ is one dimensional domain.

Remark 5.1. 1. Under asumption (5.11), the right hand side of Eq (5.12) converges to zero when $t > 0$.

2. Choose $\beta_{\bf N(\delta)} = {\bf N(\delta)}^{-c}$ for any $0 < c < \min(\frac{1}{2}, \frac{2\gamma}{d})$, and ${\bf N(\delta)}$ is chosen as

Choose $\mathcal R_\delta$ such that

Then ${\bf E} \left\|{\bf u}^\delta_{\bf N(\delta)}(x, t)-\textbf{u}(x, t) \right\|^2_{L^2(\Omega)}$ is of the order $\delta^{ m c(\frac{1}{2}-c)\frac{t}{T} } \ln (\frac{1}{\delta}).$

Proof of Theorem 5.1. The proof is divided into two Steps.

Step 1. The existence and uniqueness of the solution to the regularized problem (5.6).

Let $b(x, t)$ be defined by $b(x, t) = M-a(x, t)$. It is clear that $0 < b(x, t) < M$. Then from (5.6), we obtain

for $(x, t) \in \Omega\times (0, T).$

Let ${\bf v}^\delta_{\bf N(\delta)}$ be the function defined by ${\bf v}^\delta_{\bf N(\delta)}(x, t) = {\bf u}^\delta_{\bf N(\delta)}(x, T-t).$ Then we have

and

This implies that ${\bf v}^\delta_{\bf N(\delta)}$ satisfies the problem

where $\mathcal{G}$ is defined by

for any $v \in C \left([0, T];L^2(\Omega) \right)$.

Since

and using Parseval's identity, we obtain for any $v_1, v_2 \in L^2(\Omega)$,

Thus $\mathcal{G}$ is a Lipschitz function. Using the results of Theorem 12.2 in ^[32], we complete the proof of Step 1.

Step 2. Error estimate

We consider the error estimate between the regularized solution of problem (5.6) and the exact solution of problem (1.1).

For $(x, t) \in \Omega \times (0, T)$, we begin by establishing that the functions $b(x, t), {\bf b}_\delta^{{\rm{obs}}}(x, t)$ satisfy

and

The functions ${\bf u}^\delta_{\bf N(\delta)} (x, t)$ and $\textbf{u}(x, t)$ solve the following equations

and

For $\rho_\delta > 0$, we put $\bf{V}^\delta_{\bf N(\delta)} (x, t) = e^{\rho_\delta (t-T)} \left[ {\bf u}^\delta_{\bf N(\delta)}(x, t)-\textbf{u}(x, t) \right].$ Then for $(x, t) \in \Omega\times (0, T)$

and

By taking the inner product on both sides of Eq (5.21) with $\bf{V}^\delta_{\bf N(\delta)}$ and noting the equality

we obtain

First, thanks to inequality (5.8), the expectation of $\widetilde{A_4}$ is estimated as follows:

Next, using the inequality (5.7) and the Hölder inequality, we have

For estimating the expectation of $\big|\widetilde{A_6}\big|$, we use the Green's formula to get the equality

then using Hölder's inequality and noting the fact that

we obtain

here in the last inequality, we have used the fact that ${\bf E } |\psi(s)|^2 = s$ since $\psi$ is Brownian motion. Finally, since $\lim_{\delta \rightarrow 0^{+}} {\mathcal R}_\delta = +\infty$, for a sufficiently small $\delta > 0$, there is an ${\mathcal R}_\delta > 0$ such that ${\mathcal R}_\delta \geq \|\textbf{u}\|_{L^\infty([0, T];L^2(\Omega))}.$ For this value of ${\mathcal R}_\delta$ we have

Using the global Lipschitz property of $\mathcal{F}_{{\mathcal R}}$ (see Lemma 3), one obtains similarly the estimate

Combining (5.22), (5.23), (5.24), (5.25) and (5.26), and we obtain

Thus,

Since $\bf{V}^\delta_{\bf N(\delta)}(x, t) = e^{\rho_\delta(t-T)} \Big({\bf u}^\delta_{\bf N(\delta)}(x, t)-\textbf{u}(x, t)\Big)$ and applying Lemma 4.1, we observe that

Gronwall's lemma allows us to obtain

By choosing $\rho_\delta = \frac{1}{T} \ln \left(\frac{1}{\beta_{\bf N_\delta}} \right) > 0$ we have

The proof of Theorem 5.1 is complete.

6.   Regularization result with more general source term

In most previous works on backward nonlinear problems the assumption, that the source is global or locally Lipschitz, is required. To the best of our knowledge, this section is the first result when the source term $F$ is not necessarily a locally Lipschitz source. We will solve the problem (1.1) with a special generalized case of source term defined by (5.4). Our regularized problem is different to the one in section 4 because we do not approximate the source function $F$. Indeed, we have the following regularized problem

We make the following assumptions on $F \in C^0 (\mathbb R)$ in the following: There exists $C_1$ and $C'_1, C_2$ and $p > 1$ and $\overline \gamma$ such that

It is easy to check that the function {$F(x, t, z) = z^{\frac{1}{3}}$ }satisfies conditions (6.2), (6.3) and (6.4). Note here that this function is not locally Lipschitz.

Now we have the following result

Theorem 6.1. Let us assume that $F$ satisfies (6.2), (6.3) and (6.4). Then, there exists a unique weak solution ${\bf u}^\delta_{\bf N(\delta)}$ of problem (6.1) such that

Assume that the problem (1.1) has a unique solution $\bf u$ satisfying $\bf u(\cdot, t) \in \mathcal{W}_{MT}$. Choose $\beta_{\bf N_\delta}$ as in Theorem 5.1. Then we have the following estimate

where $\widetilde C (\delta)$ is defined in (6.51).

Remark 6.1. Our method in this theorem give the convergence rate (6.5) which is better than the error rate in (5.12). Indeed, since $\lim_{\delta \to 0} K({\mathcal R}_\delta) = +\infty$, we have

when $\delta \to 0$.

6.1.   Proof of Theorem 6.1

6.1.1.   Proof of the existence of solution of Problem (6.1)

First, by changing variable ${\bf v}^\delta_{\bf N(\delta)} (x, t) = {\bf u}^\delta_{\bf N(\delta)}(x, T-t)$, we transform Problem (6.1) into the initial value problem

where ${\bf b}_\delta^{{\rm{obs}}}(x, t) = M- {\bf a}_\delta^{{\rm{obs}}}(x, t)$.

The weak formulation of the initial boundary value problem (6.7) can then be given in the following manner: Find ${\bf v}^\delta_{{\bf N(\delta)}}(t)$ defined in the open set $(0, T)$ such that ${\bf v}^\delta_{{\bf N(\delta)}}$ satisfies the following variational problem

for all $\varphi \in H^{1},$ and the initial condition

Proof of the existence of solution of Problem (6.1). The main technique of this proof is learned from the article ^[34]. The proof consists of several steps.

Step 1: The Faedo – Galerkin approximation (introduced by Lions ^[22]).

In the space $H^1(\Omega)$, we take a basis $\{e_{j} \}_{j = 1}^\infty$ and define the finite dimensional subspace

Let $\overline G_{\delta, {\bf N(\delta)}, m}$ be an element of $V_m$ such that

as $m \to +\infty$. We can express the approximate solution of the problem (6.7) in the form

where the coefficients $c^\delta_{mj}$ satisfy the system of linear differential equations

with $i = \overline {1, m}$ and the initial conditions

The existence of a local solution of system (6.12)–(6.13) is guaranteed by Peano's theorem on the existence of solutions. For each $m$ there exists a solution ${\bf v}^\delta_{\bf N(\delta), m}(t)$ in the form (6.11) which satisfies (6.12) and (6.13) almost everywhere on $0\leq t\leq T_{m}$ for some $T_{m},$ $0 < T_{m}\leq T.$ The following estimates allow one to take $T_{m} = T$ for all $m.$

Step 2. A priori estimates.

a) The first estimate. Multiplying the $i^{th}$ equation of (6.12) by $c^\delta_{mi}(t)$ and summing up with respect to $i,$ afterwards, integrating by parts with respect to the time variable from $0$ to $t,$ we get after some rearrangements

From (6.10), we have

where $B_{0}(\delta)$ depends on $\overline G_{\delta, {\bf N(\delta)}}$ and is independent of $m$.

Using the lower bound of ${\bf b}_\delta^{{\rm{obs}}}(x, t)$, we have the following estimate

Using the assumption on $F$, we have

and

Hence, it follows from (6.15)–(6.17) that

Let

Using the fact that $\int\nolimits_{0}^{t} \left\Vert {\bf v}^\delta_{{\bf N(\delta)}, m} (s) \right\Vert_{L^2(\Omega)}^{2}ds \le \int\nolimits_{0}^{t} {\bf S}_m^\delta (s) ds$, we know from (6.18) that

Applying Gronwall's lemma, and we obtain

for all $m\in \mathbb{N},$ for all $t,$ $0\leq t\leq T_{m}\leq T,$ i.e., $T_{m} = T,$ where $C_{T}$ always indicates a bound depending on $T.$

b) The second estimate. Multiplying the $i^{th}$ equation of (6.12) by $t^2 \frac{d}{dt} c^\delta_{mi}(t)$ and summing up with respect to $i,$ we have

It is easy to check that for any $u \in H^1(\Omega)$

The equality (6.22) is equivalent to

By integrating the last equality from $0$ to $t$, we get

Estimate $I_1$. Since the assumption ${\bf b}_\delta^{{\rm{obs}}}(x, t) \ge b_0$, we know that

Estimate $I_2$. To estimate $I_2$, we need the following Lemma

Lemma 6.1. Let $\mu_0 = \left(\frac{C'_1}{C_1} \right)^{1/p}, \; \overline m = \int_{-\mu_0}^{+\mu_0} |F (\xi)|d\xi, \; \; \widetilde F(z) = \int_0^z F(y)dy, \; \; z \in \mathbb{R}.$ Then we get

The proof of Lemma 6.1 is easy and we omit it here. Now we return to estimate $I_2$. By a simple computation and then using Lemma 6.1, we have

Estimate $I_3$. Using (6.19), we have the following estimate

Estimate $I_4$. Let us set

and then $I_4$ is bounded by

Estimate $I_5$. Using Lemma 5.2, we obtain the following estimate for $I_5$:

Combining (6.26), (6.28), (6.29), (6.30), we obtain

Let

and then since

together with (6.32), we deduce that

where

Applying Gronwall's inequlality, we obtain that

where $B(4, \delta)$ depends only on $\delta, T$ and does not depend on $m$.

Step 3. The limiting process.

Combining (6.19), (6.21) and (6.34), we deduce that, there exists a subsequence of $\{ {\bf v}^\delta_{{\bf N(\delta)}, m} \}$ still denoted by $\{ {\bf v}^\delta_{{\bf N(\delta)}, m} \}$ such that (see ^[22]), say,

here $Q_{T} = \Omega \times (0, T)$. Using a compactness lemma (^[22], Lions, p. 57) applied to $(6.36)$, we can extract from the sequence $\{ {\bf v}^\delta_{{\bf N(\delta)}, m} \}$ a subsequence still denoted by $\{ {\bf v}^\delta_{{\bf N(\delta)}, m} \}$ such that

By the Riesz-Fischer theorem, we can extract from $\{ {\bf v}^\delta_{{\bf N(\delta)}, m} \}$ a subsequence still denoted by $\{ {\bf v}^\delta_{{\bf N(\delta)}, m} \}$ such that

Because $F$ is continuous, then

On the other hand, using (6.3), (6.19), (6.21), we obtain

where $B_{5} (\delta, T)$ is a constant independent of $m.$ We shall now require the following lemma, the proof of which can be found in ^[22] (see Lemma 1.3).

Lemma 6.2. Let $Q$ be a bounded open subset of $\mathbb{R} ^{N}$ and $G_{m},$ $G\in L^{q}(Q),$ $1 < q < \infty,$ such that

and

Then

Applying Lemma 6.2 with $q = p^{\prime } = \frac{p}{p-1}, \ G_{m} = F \left({\bf v}^\delta_{{\bf N(\delta)}, m}(x, t) \right),$ $G = F \left({\bf v}^\delta_{{\bf N(\delta)}}(x, t) \right),$ we deduce from (6.38) and (6.39) that

Passing to the limit in (6.12) and (6.10) by (6.35) and (6.41), we have established a solution of Problem (6.1).

6.1.2.   Proof of the uniqueness of solution of Problem (6.1)

Assume that the Problem (6.1) has two solution ${\bf v}^\delta_{\bf N(\delta)}$ and ${\bf w}^\delta_{\bf N(\delta)}$. We have to show that ${\bf v}^\delta_{\bf N(\delta)} = {\bf w}^\delta_{\bf N(\delta)}$. We recall that

For $\overline R_\delta > 0$, we put

Then for $(x, t) \in \Omega\times (0, T)$, we get

and

By taking the inner product of both sides of (6.43) with $\bf{W}^\delta_{\bf N(\delta)}$ then taking the integral from $t$ to $T$ and noting the equality

we deduce

By the assumption we have

Using the inequality (5.8), we get the following estimate

Combine equations (6.44), (6.48), (6.46) and choose

to obtain

This implies that for all $t \in [0, T]$ then $\|\bf{W}^\delta_{\bf N(\delta)} (., t)\|^2_{L^2(\Omega)} = 0$ since $\bf{W}^\delta_{\bf N(\delta)} (x, T) = 0$. The proof is completed.

6.1.3.   Convergence estimate

Our analysis and proof is short and similar to the proof of Theorem 5.1. Indeed, let us also set

By using some of the above steps we obtain

The terms $\widetilde A_4, \widetilde A_5, \widetilde A_6$ are similar to (5.22). Now, we consider $\widetilde A_8$. By assumption (6.4), we have

After using the results of the proof of Theorem 5.1, we get

Since

and applying Lemma 4.1, we observe that

Gronwall's lemma allows us to obtain

By choosing $\rho_\delta = \frac{1}{T} \ln \left(\frac{1}{\beta_{\bf N_\delta}} \right) > 0$ we have

7.   Application to some specific equations

7.1.   Ginzburg-Landau equation

Here we consider a special source function $F (u) = u-u^3$ for Problem (1.1). This is called the Ginzburg-Landau equation. This function satisfies the condition of section 5 and does not satisfy the condition in section 4. For all $\mathcal R > 0$, we approximate $F$ by $\mathcal{F}_{\mathcal R}$ defined by

We consider the problem

It is easy to see that $K (\mathcal R_\delta) = 1+3 \mathcal R_\delta^2$. Choose $\beta_{\bf N(\delta)} = {\bf N(\delta)}^{-c}$ for any $0 < c < \min(\frac{1}{2}, \frac{2\gamma}{d})$, and ${\bf N(\delta)}$ is chosen as

Choose $\mathcal R_\delta$ such that

Then applying Theorem 5.1, the error ${\bf E} \left\|{\bf u}^\delta_{\bf N(\delta)}(x, t)-\textbf{u}(x, t) \right\|^2_{L^2(\Omega)}$ is of the order $\ln^2 \left(\frac{1}{\delta} \right) \Big(\delta \Big)^{2mc(\frac{1}{2}-c)\frac{t}{T} }.$

7.2.   The nonlinear Fisher–KPP equation

In this subsection, we are concerned with the backward problem for a nonlinear parabolic equation of Fisher–Kolmogorov–Petrovsky–Piskunov type

with the following condition

By Skellam ^[33], Eq (7.4) has many applications in population dynamics and periodic environments. In these references, the quantity ${\bf u} (x, t)$ generally stands for a population density, and the coefficients $a(x, t)$, $\gamma(x), \; \mu(x)$ respectively, correspond to the diffusion coefficient, the intrinsic growth rate coefficient and a coefficient measuring the effects of competition on the birth and death rates. Our method that can be applied to this model is similar to example 7.1. However, since the ideas of Example 7.1 and 7.2 are the same, we only state the model without giving the errors.

7.3.   The second equation

Taking the function $F(u) = u^{\frac{1}{3}}$ it is easy to see that $F$ satisfy (6.2), (6.3) and (6.4). Moreover, we can show that $F$ is not locally Lipschitz function. So, we cannot regularize the problem in this case with Problem (5.6). We consider the problem

Choose $\beta_{\bf N_\delta}$ and ${\bf N_\delta}$ as in subsection 6.1. Applying Theorem 5.1, the error between the solution of Problem (7.6) and $\bf u$, ${\bf E} \left\|{\bf u}^\delta_{\bf N(\delta)}(x, t)-\textbf{u}(x, t) \right\|^2_{L^2(\Omega)}$, is of the order $\delta ^{2mc(\frac{1}{2}-c)\frac{t}{T} }.$

Remark 7.1. In the following, we give a comparison of the method and results in this paper with the results in ^[30,31]. All methods are truncation methods, but our problem is complicated due to the data being noised by random data. We need Lemma 4.1 to determine the correct set up according to the measured data. The coefficients $N(\delta)$ should be chosen appropriately so that the error between the sought solution and the correct solution converges. There are two advantages to this article that were not explored in ^[30,31]

● In Theorem 4.3, we give a regularization result in the case of a weaker assumption for ${ u}$, i.e., ${ u} \in C([0, T];L^2(\Omega))$. This is one of the first results obtained in this case and was not considered in ^[30,31]. In those papers, to investigate the error, the exact solution is assumed in a Gevrey space, which limits the number of functions than if one considered the function space $C([0, T];L^2(\Omega))$.

● In ^[30,31], the source functions must satisfy a global lipschitz condition. However, in our article, we deal with a fairly broad function class, consisting of the local Lipschitz function class and some local non-Lipschitz function class (see Section 6).

Acknowledgments

Nguyen Huy Tuan is thankful to the Van Lang University. This research is funded by Thu Dau Mot University, Binh Duong Province, Vietnam under grant number DT.21.1-011.

Conflict of interest

The authors declare there is no conflicts of interest.