On a final value problem for a nonlinear fractional pseudo-parabolic equation

1.   Introduction

We consider the final value problem:

where $m>0$, and $\Omega \subset \mathbb R^d, (d \geq 1)$ is a bounded domain with smooth boundary $\partial \Omega$, the operator $(-\Delta)^\beta$ with is the fractional Laplace operator with $0 < \beta \neq 1$ and the final data $\varphi \in L^2(\Omega)$. Pseudo-parabolic equations have many applications in science and technology, especially in physical phenomena such as seepage of homogeneous fluids through a fissured rock, aggregation of populations, ... see e.g. Ting [24], R. Xu [17,31,32,33] and references therein.

For $\beta = 1$, the direct problem is

with conditions $u(0, x) = u_0(x), \; \; x \in \Omega$ and $u(t, x) = 0, \; \; (t, x) \in (0, T] \times \partial \Omega$. Problem (1.1) has been studied by many authors. Specifically,

$1.$ $f = 0$, see e.g. [11,22,24], the existence and uniqueness of solutions is established. Moreover, the asymptotic behavior and regularity are investigated.

$2.$ ${f(u)} = u^p, \; \; p\geq 1$, in [5], the authors investigate large time behavior of solutions. R. Xu et al. [32] proved the invariance of some sets, global existence, nonexistence and asymptotic behavior of solutions with initial energy $J(u_0) \leq d$ and finite time blow-up with high initial energy $J(u_0) > d$ and some related works [18,34]. For the case of $f(u) = |u|^{p-2}u$, there are other results on the large time behavior of solutions of the pseudo-parabolic see [7,28,29,30,35,36] and their references.

$3.$ When the source term is a logarithmic nonlinearity ${f(u)} = |u|^{p}u\log|u|$, very recently, the work [10] focus on the initial conditions, which ensure the solutions to exist globally, blow up in finite time and blow up at infinite time. The asymptotic behaviour for the solutions has been considered in [4,6,8,12,34] and the references therein.

$4.$ For nonlocal source, ${f(u)} = |u|^p \int_{\Omega} G(x, y)|u|^{p+1}(y)\mathrm d y, \; y \in \Omega$, the authors of [19] considered blow-up time for solutions, obtained a lower bound as well as an upper bound for the blow-up time under different conditions, respectively. Also, they investigated a nonblow-up criterion and compute an exact exponential decay, see also [9,23].

For $0<\beta \neq 1$, [14] considered the Cauchy problem

supplemented with initial condition $u(0, x) = u_0(x), \; x \in \Omega$ and Dirichlet boundary condition $u(t, x) = 0, \; (t, x) \in \mathbb R_+ \times \partial \Omega$. The paper established the global existence and time-decay rates for small-amplitude solutions.

As mention above, initial value problems of nonlinear pseudo-parabolic equations have been considered in many papers see [1,4,5,6,7,8,9,10,11,12,13,14,15,19,21,22,23,24,32,35,36,37]. However, there are not many results devoted to Problem (${\mathbb{P}_T}$). Our approach includes as special cases all previously on the reaction terms. In this work, we consider two cases; first, the source f is globally Lipschitz and in the second case, we consider $f$ is general locally Lipschitz function (a coercive-type condition). At this point, we remark that there exists some locally Lipschitz functions, but we cannot determine its specific Lipschitz coefficient e.g. $f = u(a-bu^2), b>0$ of the Ginzburg-Landau equation. Hence, we have to find another method to study the problem with the locally Lipschitz source which is similar to the Ginzburg-Landau equation, etc. (see Subsection 4.2.2 for more details).

The solution of Problem (${\mathbb{P}_T}$) is of the regularity-loss structure for $0<\beta <1$, $x \in \Omega, t\in (0, T]$, we consider the existence and regularity of Problem (${\mathbb{P}_T}$). In the case $\beta > 1$, the regularity-gain type for $x \in \Omega, t\in (0, T]$ and the Problem (${\mathbb{P}_T}$) is ill-posed. So the regularization methods are required. As we know, there are many regularization methods to suit each problem [2,3,16,20,25,26,27]. For problem (${\mathbb{P}_T}$), we propose two methods to regularize solution: Modified Lavrentiev regularization (MLR) method and Fourier truncated regularization (FTR) method.

The plan of the paper is as follows. Section 2 contains notations and formulation of a solution of Problem (${\mathbb{P}_T}$) and the proof of its instability. In Section 3, the case $0<\beta < 1$, well-posedness of solutions of Problem (${\mathbb{P}_T}$) is established. In Section 4, the case $\beta > 1$, the proof that Problem (${\mathbb{P}_T}$) is ill-posed and the well-posedness of the regularized problem are presented. We also propose two regularization methods: MLR method and FTR method for the globally Lipschitz or locally Lipschitz reaction terms, respectively.

2.   Preliminaries

2.1.   Relevant notations

Let us recall that the spectral problem

admits a family of eigenvalues

The notation $\|\cdot\|_{B}$ stands for the norm in the Banach space $B$. We denote by $L^q (0, T ; B), {1 \leq q \leq \infty},$ the Banach space of real-valued measurable functions $w: (0; T ) \to B$ with norm

The norm of the function space $C^k ([0, T ] ;B), 0 \leq k \leq \infty$ is denoted by

For any $\nu \geq 0$, we define the space

where $\langle\cdot, \cdot \rangle_{L^2(\Omega)}$ is the inner product in $L^2(\Omega)$; $H^{\nu} (\Omega))$ is a Hilbert space with the norm

The Gevrey of order $\beta$ class of functions with index $\eta_1, \eta_2>0$, defined by the spectrum of the Laplacian is denoted by

and its norm given by

Next, we give the formulation of solution of the problem (${\mathbb{P}_T}$).

2.2.   Mild solution of the Problem (${\mathbb{P}_T}$)

Now, assume that Problem (${\mathbb{P}_T}$) has a unique solution, then we find its the form. Let $u(t, x) = \sum_{j = 1}^\infty u_{j}(t) e_j(x)$ be the decomposition of $u$ in $L^2(\Omega)$ with $u_j(t) = { \left<u(t, \cdot), e_j \right>_{L^2(\Omega)} }$. From (${\mathbb{P}_T}$), taking the inner product of both sides of (${\mathbb{P}_T}$) with $e_j(x)$, we obtain the ordinary differential equation

where $u_j'(t) = \dfrac{\mathrm d}{\mathrm d t} { \left<u(t, \cdot), e_j \right>_{L^2(\Omega)} }, f_j(u)(t) = { \left<f(t, \cdot;u), e_j \right>_{L^2(\Omega)} }$, whose solution is

where ${ \varphi_j = \left<\varphi, e_j \right>_{L^2(\Omega)} }$.

Definition 2.1 (Mild solution of Problem (${\mathbb{P}_T}$)). A function $u$ is a mild solution of (${\mathbb{P}_T}$) if $u \in C([0, T];L^2(\Omega))$ and satisfies the following integral equation

for all $(t, x) \in (0, T) \times \Omega$, and $\beta >0$.

Now we introduce the main results in this paper.

3.   The case $0<\beta < 1$: Well-posedness of solutions to the Problem (${\mathbb{P}_T}$)

In this section, we prove that the Problem (${\mathbb{P}_T}$) is well-posed. First prove that for the Problem (${\mathbb{P}_T}$) exists a unique mild solution, then the regularity of the solution is established.

We will make the following assumptions:

$({\bf{A_1}})$ Assume that $f$ satisfy the global Lipschitz condition:

with $K >0$ independent of $t, x, w_1, w_2$, and $(t, x) \in [0, T) \times \Omega, w_i \in C([0, T]; L^2(\Omega)), i = 1, 2$.

$({\bf{A_2}})$ We set $f(0): = f(t, x;0) = 0, \; (t, x) \in [0, T) \times \Omega$ and

Theorem 3.1 ($L^\infty$-Existence). Let $0 < \beta < 1$, assume that $f$ satisfy the assumption $(\bf A_1)$. Then, the integral equation (2.2) has a unique mild solution in $L^\infty(0, T; L^2(\Omega))$.

Proof. We prove the existence of the solution $u \in L^\infty(0, T; L^2(\Omega))$ of the integral equation (2.2). For $w \in L^\infty(0, T; L^2(\Omega))$, we consider the following operator:

and we aim to show that $\mathscr H^k$ is a contraction mapping on the space $L^\infty(0, T; L^2(\Omega))$. In fact, we will prove that for every $w_1, w_2 \in L^\infty(0, T; L^2(\Omega))$, it holds

We will prove (3.4) by induction. For $k = 1$, using Parseval's relation and assumption $(\bf{ A_1})$, one obtains

Assume now that (3.4) is satisfied for $k = k_0$, let us prove that it is satisfied for $k = k_0+1$. It holds

Therefore, by the induction principle we get (3.4). Since the right hand side of (3.5) is independent of $t$, we deduce that

When $k$ is large enough, we have $\frac{1}{k_1!} \left(\frac{KT}{m \lambda_1} \exp \left(\frac{T}{m \lambda_1} \right) \right)^{k_1} \to 0$. Hence there exists $k_1$ such that

We claim that the mapping $\mathscr H^{k_1}$ is a contraction, i.e. $\mathscr H^{k_1}(u) = u$. We have

Due to the uniqueness of the fixed point of $\mathscr H^{k_1}$, it holds $\mathscr H(u) = u$. We conclude that the integral equation (3.3) has a unique solution in $L^\infty(0, T; L^2(\Omega))$.

Theorem 3.2 (Regularity). Let $0 < \beta < 1$, and $f$ only satisfy the assumption $({\bf{A_1}})$, we have the following:

a) If $\varphi \in L^2(\Omega)$ and $f(0) \in L^1(0, T;L^2(\Omega))$, then there exists $C(T, m, \lambda_1)>0$ such that

b) If $\varphi \in H^\beta (\Omega)$, $f(0) \in L^\infty (0, T;L^2(\Omega))$ then there exists $C(T, m, \lambda_1, \beta)>0$ such that

Here, we recall $f(0): = f(t, x;0), \; \forall (t, x) \in [0, T) \times \Omega$.

Proof. First, we set

a) First, using the Parseval's relation, we infer that

where for $\beta \in (0, 1)$, we have $\exp \left(\frac{(T -t) \lambda_j^\beta }{1+m \lambda_j} \right) \leq \exp \left(\frac{T}{m \lambda_1^{1-\beta}} \right), \forall t \in [0, T), j \in \mathbb N^*$. We also obtain

where we have used $\frac{1}{1+m \lambda_j} \leq \frac{1}{m \lambda_1}, \; \forall j \in \mathbb N^*$ and for $\beta \in (0, 1)$ we implies that

Using (3.1), we obtain

then

Combining (3.10) with (3.9), leads to

From (3.8) and (3.11) yields

Thanks to Grönall's inequality

this imlpies (3.6).

b) We observe that for $\beta \in (0, 1)$

We conclude that

Inequality (3.10) associated with (3.13) leads to

Estimates (3.12) and (3.14) lead to

where we have used that $\|w\|_{H^{\beta}(\Omega)} \geq \lambda_1^{\beta}\|w\|_{L^2(\Omega)}$ for $\beta \in (0, 1)$. Grönwall's inequality allows to obtain

we implies (3.7). The proof is complete.

4.   The case $\beta > 1$: Ill-posedness and regularization methods for Problem (${\mathbb{P}_T}$)

4.1.   Ill-posedness of the solution of Problem (${\mathbb{P}_T}$)

Definition 4.1 (Ill-posed). The well-posed problem in the sense of Hadamard is to satisfy the following properties:

ⅰ) a solution exists;

ⅱ) the solution is unique;

ⅲ) the solution's behaviour changes continuously with the initial conditions.

If at least one of the three properties above does not satisfy, the problem is ill-posed.

Next, we give an example which shows that the solution $\widetilde u^{(k)}(t, x)$ (for any $k \in \mathbb{N}^*$) of Problem (${\mathbb{P}_T}$) is not stable (property ⅲ) is unsatisfied). For $\beta >1$, let us set

for $(t, x) \in (0, T) \times \Omega$ and $w \in C([0, T];L^2(\Omega))$. The solution $\widetilde u^{(k)}(t, x)$ satisfies the integral equation

where

and

$\bullet$ Step 1. We show that (4.3) has a unique solution $\widetilde u^{(k)} \in C([0, T]; L^2(\Omega))$.

Indeed, we consider the function

Then for any $w_1, w_2 \in C([0, T]; L^2(\Omega))$, we obtain

where we denote $w_{i, j}(t): = { \left<w_{i}(t, \cdot), e_j \right>_{L^2(\Omega)} }, \; \; i = 1, 2.$

This implies that

Hence $\mathscr{G}$ is a contraction because $m>1$. Using the Banach fixed-point theorem, we conclude that $\mathscr{G} (w) = w$ has a unique solution $\widetilde u^{(k)} \in C([0, T]; L^2(\Omega))$.

$\bullet$ Step 2. The solution of Problem (4.3) is instable. We have

It is easy to see that (here, noting that from (4.2), we have $\widetilde{f}_j(0) = 0$)

Hence

This leads to

We continue to estimate the right hand side of the latter inequality. Indeed, since $\{e_j(x)\}_{j\geq1}$ is a basis of $L^2(\Omega)$, i.e.,

we have

Since the function $\chi(t) : = \exp \left(\dfrac{(T -t) \lambda_k^\beta }{1+m \lambda_k} \right)$ is a decreasing function with respect to variable $t \in [0, T]$ and $\beta >1$, we deduce that

Combining (4.4) and (4.5) yields

As $k \nearrow \infty$, we see that

but

Thus, it is shown that Problem (${\mathbb{P}_T}$)is ill-posed in the Hadamard sense in $L^2$-norm for $\beta >1$.

4.2.   Regularization and error estimate

In order to obtain the stable numerical solutions, we propose two regularization methods to solve the Problem (${\mathbb{P}_T}$) in two cases of $f$:

● $f$ is globally Lipschitz: MLR method.

● $f$ is locally Lipschitz: FTR method.

4.2.1.   MLR method: Globally Lipschitz source term

In this subsection, the functions $f(t, x; u)$ is satisfy the globally Lipschitz $({\bf{A_1}})$. To approximate $u$, we introduce the regularized solutions $\mathcal U_{ \alpha}^{ \varepsilon}$ given by MLR method as follows

where we set

and the coefficient $\alpha: = \alpha( \varepsilon)$ satifies $\lim_{ \varepsilon \to 0^+} \alpha = 0$; it plays the role of a regularization parameter.

The following technical lemma plays the key role in our analysis.

Lemma 4.2. For $0\leq t\leq\tau \leq T$, we have

Proof. $a)$. First, from (4.7), it holds

Using the inequality

for $\nu_i>0, \; i = 1, 2$ and $0 < \zeta < e\nu_2$, if $\alpha < eT$, we obtain

Whereupon

Using (4.12) into (4.10), we obtain (4.8).

With the same argument as in the proof of (4.8), we obtain (4.9). This concludes the proof of the lemma.

A. The well-posedness of the regularized solution (4.6). In this subsection, we will obtain the existence and regularity results for the regularized solution (4.6).

Theorem 4.3 (Existence-uniqueness). Suppose that $f$ satisfying the assumption $({\bf{A_1}})$ then the regularized solution (4.6) has unique weak solution $\mathcal U_ \alpha^ \varepsilon \in C \left([0, T]; \right. \left. L^2(\Omega) \right)$.

Proof. For any $\mathscr F \in C([0, T];L^2(\Omega))$, we define the function

by

We also define $\mathscr F^k$ as follows

We shall prove by induction, for any couple $\vartheta_1, \vartheta_2 \in C([0, T];L^2(\Omega))$, that

For $k = 1$, by using $({\bf{A_1}})$ and Lemma 4.2 and noting that $\frac{ \alpha^{-1}T}{\log \left( \alpha^{-1}T \right)} \geq 1$, it holds

then we have

Hence, (4.14) holds for $k = 1$.

Assume that (4.14) holds for $k = N$. We show now that (4.14) holds for $k = N+1$. In fact, we have

By the induction principle, we deduce that (4.14) holds for all $k\in \mathbb N^*$. Notice that, as $\alpha$ is fixed, then $\dfrac{1}{k!} \left(K \dfrac{ \alpha^{-1}T}{m \log \left( \alpha^{-1}T \right)} (T-t) \right)^{k}$ tends to $0$ when $k \nearrow \infty$, so there exists a positive integer $k_0$ such that

It means that $\mathscr F^{k_0}$ is a contraction. Finally, it follows the desired conclusion that the problem (4.6) has a unique solution $\mathcal U_ \alpha^ \varepsilon \in C([0, T];L^2(\Omega))$. The proof is complete.

Given a constant $\alpha \in (0, 1)$ (which will be assumed from now on) and a function $w \in C([0, T];L^2(\Omega))$, we denote the scaling with $\alpha$ as follows:

Theorem 4.4 (Regularity). Assume that $f$ satisfy the assumptions $({\bf{A_1}})$ and $({\bf{A_2}})$. We have the following results:

a) If $\varphi^ \varepsilon \in L^2(\Omega)$, then $\mathcal U_ \alpha^ \varepsilon$ given by (4.6) satisfies

b) If $\varphi^ \varepsilon \in H^1(\Omega)$, then

Proof. We rewrite (4.6) as

a) Using (3.10) (noting that $f(t, x;0) = 0,$ for all $(t, x) \in (0, T) \times \Omega$) and Lemma 4.2b), we first observe that

where we have used $\left(\frac{T} {\log \left( \alpha^{-1}T \right)} \right)^{\frac{\tau - t}{T}} \leq \frac{T}{\log(T)},$ for $\alpha \in (0, 1), 0\leq t \leq \tau \leq T$.

The next observation, from Lemma 4.2a), is that

noting that $\left(\frac{T} {\log \left( \alpha^{-1}T \right)} \right)^{\frac{\tau - t}{T}} \leq \frac{T}{\log(T)},$ for all $0\leq t \leq T, 0< \alpha<1$.

Combining (4.18)-(4.20) and multiplying the two sides of of the result obtained by $\alpha^{-\frac{t}{T}}$, we deduce that

Using Grönwall's inequality, we get

Since the right side of (4.21) does not depend on $t$, we have

This leads to (4.16).

b) From Lemma 4.2a), we first observe that

The next observation, using Lemma 4.2b), is that

where we have used $\alpha^{\frac{t-\tau}{T}} \left(\frac{T} {\log \left( \alpha^{-1}T \right)} \right)^{\frac{\tau - t}{T}} \leq \frac{ \alpha^{-1}T}{\log(T)}, \; 0 \leq t \leq \tau \leq T$ for $0< \alpha<1$.

From (4.22) and (4.23), we deduce that

where $\|w\|_{L^2(\Omega)} \leq \frac{\|w\|_{H^{1}(\Omega)}}{ \lambda_1}$. Thanks to Grönwall's inequality, we get

which implies (4.17). This concludes the proof.

B. Error estimate In this subsection, by using MLR method, the error between the exact solution and the regularized solution is obtained. Now, we can formulate the main theorem.

Theorem 4.5 (Error estimate). Let $\alpha : = \alpha ( \varepsilon)$ satisfy

Assume that $f$ satisfy the assumptions $({\bf{A_1}}) - ({\bf{A_2}})$ and Problem (${\mathbb{P}_T}$) has a unique solution $u$ satisfying

for some known positive constants $P_0$ and $\eta_1 \geq 2, \eta_2 \geq \frac{2T}{m \lambda_1}$. Then (for all $t \in [0, T]$)

Remark 1. The error estimate in (4.26) is of order $\mathcal O( \varepsilon) = \alpha^{\frac{t}{T} } \left(\frac{T} {\log \left( \alpha^{-1}T \right)} \right)^{1-\frac{t}{T}}, t \in [0, T]$.

$\bullet$ If $t \approx T$, then $\mathcal O( \varepsilon) = \alpha$ tends to zero according to (4.24).

$\bullet$ If $t \approx 0$, then $\mathcal O( \varepsilon) = T\log^{-1} \left( \alpha^{-1}T \right)$ tends to zero as $\varepsilon \to 0^+$.

Remark 2. Choose $\alpha = \varepsilon ^r \stackrel{ \varepsilon \to 0^+}{\longrightarrow} 0$, for some $0<r<1$, then the estimate in (4.26) is of order

Proof. For all $t\in (0, T)$, the triangle inequality gives

where $\mathcal U_ \alpha(t, x)$ is the solution of Problem (4.6) corresponding to exact datum $\varphi(x)$. In order to estimate $\mathcal M_7^{ \alpha, \varepsilon}(t, x)$ and $\mathcal M_8^{ \alpha, \varepsilon}(t, x)$, we need to divide the proof in two steps.

Step 1. Estimate $\mathcal M_7^{ \alpha, \varepsilon}(t, x)$.

Thanks to Parseval's relation and basic inequality $(a+b)^2 \leq 2a^2 + 2b^2, \; \; \forall a, b \geq 0$, we obtain

To estimate $\mathcal M_{7a}^{ \alpha, \varepsilon}(t)$, let us use Lemma 4.2a) yields

Next, we estimate $M_{7b}^{ \alpha, \varepsilon}(t)$. Using Hölder's inequality, $({\bf{A_1}})$ and Lemma 4.2b), one obtains

Combining the results in (4.28)-(4.30), we get

Multiplying by $\alpha^{\frac{-2t}{T} } \left(\frac{T} {\log \left( \alpha^{-1}T \right)} \right)^{\frac{2t}{T}}$ both sides, we deduce that

Grönwall's inequality allows to obtain

Similar calculations yield

Step 2. Estimate $\mathcal M_8^{ \alpha, \varepsilon}(t)$. Let us define an operator

It clearly follows that

The triangle inequality allows to write

We estimate for $\mathcal M_{8a}^{ \alpha}(t)$ as follows (similarly as in (4.30)):

The term $\mathcal M_{8b}^{ \alpha}(t)$ is estimated by using (4.8) as

where we have used $\dfrac{ \lambda_j^\beta }{1+m \lambda_j} \leq \dfrac{ \lambda_j^\beta}{m \lambda_1}$, for all $j \in \mathbb N^*$ and $\eta_1 \geq 2, \eta_2 \geq \frac{2T}{m \lambda_1}$.

Combining all these inequalities, we deduce

Multiplying by $\alpha^{\frac{-2t}{T} } \left(\frac{T} {\log \left( \alpha^{-1}T \right)} \right)^{\frac{2t}{T}}$, we obtain

Using Grönwall's inequality, we thus obtain

Combining (4.27), (4.31) and (4.32), we complete the proof of Theorem 4.5.

4.2.2.   FTR method: Locally Lipschitz source term

In Subsection 4.2.1 has addressed the Problem (${\mathbb{P}_T}$) in which $f$ is a globally Lipschitz function, in the rest of this paper, we extend the analysis to a locally Lipschitz function $f$. Up to the present, the results of Problem (${\mathbb{P}_T}$) for the locally Lipschitz cases are still very scarce. Hence, we have to find another regularization method to study the problem with the locally Lipschitz source. Thus, the FTR method is a very effective method for this case.

We begin by establishing the locally Lipschitz properties of $f$ by the following assumption:

$({\bf{A_3}})$ Assume that for each $\varrho> 0$, there exists $K_\varrho > 0$ such that

and

$K_\varrho$ is a non-decreasing function of $\varrho$. We assume that $\lim_{\varrho \to \infty} K_\varrho = \infty$.

Example 1. Let $f_1(u) = u|u|^2.$ Easy calculations show that

Clearly, $f_1$ is not globally Lipschitz. For $\varrho \geq \max \{|u|, |v|\},$ from (4.33), one can compute explicitely the coefficient $K_\varrho = 3\varrho^2$.

Example 2. Let $f_2(u) = u(a-bu^2)$ (Ginzburg-Landau type), with $a \in \mathbb R, b>0.$ It can be easily seen that $f_2$ is locally Lipschitz source. Moreover, we are possible to verify computationally the coefficient $K_\varrho = 3\varrho^2 \max\{|a|, b\}$. In order to solve the problem with the locally Lipschitz sources as above (and some other types), we outline our ideas to construct an approximation of the function $f$. For all $\varrho>0$, we define

With this definition, we claim that $f_{\varrho}$ is global Lipschitz function. Before stating the main theorem, we first consider the following lemma.

Lemma 4.6. Let $f_{\varrho^ \varepsilon} \in L^\infty([0, T] \times \Omega \times \mathbb{R})$ given as (4.34). Then we have

where $\lim_{ \varepsilon \to 0^+} \varrho^ \varepsilon = \lim_{ \varepsilon \to 0^+} K_{\varrho^ \varepsilon} = \infty.$

Proof. The proof can be found in [2].

Remark 3. For $\varrho^ \varepsilon>0$ satisfying $\lim\limits_{ \varepsilon \to 0^+} \varrho^ \varepsilon = \infty$, we implies that

Based on the above analysis, we propose the regularized solution by using FTR method as follows

where $f_{\varrho^ \varepsilon, j}(\mathcal V^{ \varepsilon}_{\varLambda^ \varepsilon})(\tau) = \left<f_{\varrho^ \varepsilon}(\tau, \cdot;\mathcal V^{ \varepsilon}_{\varLambda^ \varepsilon}), e_j \right>_{L^2(\Omega)}$.

The regularity estimates of the solution $\mathcal V_{\varPi^ \alpha}^{ \varepsilon}$ given by (4.36) is possible that but we will not develop this point here because it is an argument analogous to the previous one. We continue with the error estimate result.

Theorem 4.7 (Error estimate). Suppose that we can choose $\varrho^ \varepsilon, \varLambda^ \varepsilon >0$ such that $\lim_{ \varepsilon \to 0^+}\varLambda^ \varepsilon = \lim_{ \varepsilon \to 0^+}\varrho^ \varepsilon = \infty$ and satisfying

and let us choose

Let $f_{\varrho^ \varepsilon}$ satisfy Lemma 4.6. Then nonlinear integral system (4.36) has a unique solution $\mathcal V^{ \varepsilon}_{\varLambda^ \varepsilon} \in C([0, T];L^2(\Omega))$. Assuming further that equation (4.36) has a unique exact solution $u$ satisfying

and

Then the following stability estimate holds for any $t \in [0, T], \; \delta >0$,

Remark 4. We can choose $\varLambda^ \varepsilon = \frac{m^{1/\beta} \lambda_1^{1/\beta} \sqrt[\beta\;]{\log( \varepsilon^{-r})}}{T^{1/\beta}} \stackrel{ \varepsilon \to 0 ^+}{\longrightarrow}\infty$, for some $r \in (0, 1)$, then the condition in (4.37) is fulfiled. Then the estimate in (4.38) is of order

Remark 5. Obviously, from (4.26), one can raise the question about $I_0$ in (4.25) is large, then the estimate in (4.26) is not exist naturally. By using FTR method, this problem is improved as in (4.38) by the term $\frac{I_0 }{(\varLambda^ \varepsilon)^{\beta+\delta}} \to 0$ as $\varepsilon$ goes to $0^+$.

Proof. We divide the proof into two parts. In Part a, we prove that the integral equation (4.36) has a unique solution $\mathcal V^{ \varepsilon}_{\varLambda^ \varepsilon} \in C([0, T];L^2(\Omega))$. In Part b, the error between the exact solution and regularized solution is obtained.

Part a. The existence and uniqueness of a solution of the integral equation (4.36).

For $\zeta \in C([0, T];L^2(\Omega))$, we consider the following function

Let us define $\mathscr L^k$ as follows

To explore the proof, we only need to prove that there exists $k_0 \in \mathbb N^*$ such that the operator $\mathscr L^{k_0}$ which also maps $C([0, T];L^2(\Omega))$ into itself, is a contraction. In fact, we prove by mathematical induction that for any couples $\zeta_1, \zeta_2 \in C([0, T];L^2(\Omega))$ then

for $\zeta_1, \zeta_2 \in C([0, T];L^2(\Omega))$. The proof of the latter inequality is similar to that of Theorem 4.3 and thus it is omitted.

Part b. The error estimate between the exact solution $u$ and the regularized solution $\mathcal V^{ \varepsilon}_{\varLambda^ \varepsilon}$. Using the triangle inequality, we have

where

To estimate (4.40), we divide the right-hand side of (4.40) into two steps

$\bullet$ Step 1. Estimate $\mathcal M_9^{\varLambda^ \varepsilon, \varepsilon}(t)$. One has

Using the Hölder's inequality with Parseval's relation, we obtain

Multiplying both sides (4.42) by $\exp \left( \frac{2(\varLambda^ \varepsilon)^{\beta}}{m \lambda_1} t \right)$, we get

Applying Grönwall's inequality yields

Consequently,

$\bullet$ Step 2. Estimate of $\mathcal M_{10}^{\varLambda^ \varepsilon}(t)$. One has

To estimate $\mathcal M_{10a}^{\varLambda^ \varepsilon}(t)$, we proceed as follows for $\delta >0$ be arbitrary

Since $\lim_{ \varepsilon \to 0^+}\varrho^ \varepsilon = \infty$, for a sufficiently small $\varepsilon > 0$, there is an $\varrho^ \varepsilon > 0$ such that $\varrho^ \varepsilon \geq \|u\|_{L^\infty(0, T;L^2(\Omega))}$. For this value of $\varrho^ \varepsilon$ (from (3.54)) we have $f_{\varrho^ \varepsilon}(t, x; u) \approx f (t, x; u)$. We estimate of $\mathcal M_{10b}^{\varLambda^ \varepsilon}(t)$ as follows

Combining (4.45) and (4.46), we get

Multiplying by $\exp \left(2t \frac{(\varLambda^ \varepsilon)^\beta}{m \lambda_1} \right)$ both sides and using Grönwall's inequality, we obtain

Combining (4.40), (4.43) and (4.48), we obtain (4.38). This completes the proof of Theorem 4.7.

5.   Conclusion

The paper investigate a final boundary value problem for a class of pseudo-parabolic partial differential equations with nonlinear reaction term. For $0<\beta < 1,$ the well-posedness of solution is established. For $\beta >1$, the problem is ill-posed. Thus, we propose two methods to regularize the problem. The error estimates are given in the cases of globally or locally Lipschitzian source term.