On the solvability of an initial-boundary value problem for a non-linear fractional diffusion equation

Özge Arıbaş; İsmet Gölgeleyen; Mustafa Yıldız; Özge Arıbaş; İsmet Gölgeleyen; Mustafa Yıldız

doi:10.3934/math.2023273

AIMS Mathematics

2023, Volume 8, Issue 3: 5432-5444. doi: 10.3934/math.2023273

Previous Article Next Article

Research article

On the solvability of an initial-boundary value problem for a non-linear fractional diffusion equation

Department of Mathematics, Faculty of Science, Zonguldak Bülent Ecevit University, Zonguldak 67100, Turkey

Received: 27 September 2022 Revised: 16 November 2022 Accepted: 12 December 2022 Published: 16 December 2022
MSC : 26A33, 35C15

In this paper, we consider an initial-boundary value problem for a non-linear fractional diffusion equation on a bounded domain. The fractional derivative is defined in Caputo's sense with respect to the time variable and represents the case of sub-diffusion. Also, the equation involves a second order symmetric uniformly elliptic operator with time-independent coefficients. These initial-boundary value problems arise in applied sciences such as mathematical physics, fluid mechanics, mathematical biology and engineering. By using eigenfunction expansions and Banach fixed point theorem, we establish the existence, uniqueness and regularity properties of the solution of the problem.

Keywords:

solvability,
non-linear fractional diffusion equation,
initial-boundary value problem,
eigenfunction expansions,
Banach fixed point theorem

Citation: Özge Arıbaş, İsmet Gölgeleyen, Mustafa Yıldız. On the solvability of an initial-boundary value problem for a non-linear fractional diffusion equation[J]. AIMS Mathematics, 2023, 8(3): 5432-5444. doi: 10.3934/math.2023273

Related Papers:

[1]	Yige Zhao, Yibing Sun, Zhi Liu, Yilin Wang . Solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type. AIMS Mathematics, 2020, 5(1): 557-567. doi: 10.3934/math.2020037
[2]	Yujun Cui, Chunyu Liang, Yumei Zou . Existence and uniqueness of solutions for a class of fractional differential equation with lower-order derivative dependence. AIMS Mathematics, 2025, 10(2): 3797-3818. doi: 10.3934/math.2025176
[3]	Said Mesloub, Faten Aldosari . Well posedness for a singular two dimensional fractional initial boundary value problem with Bessel operator involving boundary integral conditions. AIMS Mathematics, 2021, 6(9): 9786-9812. doi: 10.3934/math.2021569
[4]	Wei Fan, Kangqun Zhang . Local well-posedness results for the nonlinear fractional diffusion equation involving a Erdélyi-Kober operator. AIMS Mathematics, 2024, 9(9): 25494-25512. doi: 10.3934/math.20241245
[5]	Mohammed Al-Refai, Dumitru Baleanu . Comparison principles of fractional differential equations with non-local derivative and their applications. AIMS Mathematics, 2021, 6(2): 1443-1451. doi: 10.3934/math.2021088
[6]	Hasanen A. Hammad, Hassen Aydi, Maryam G. Alshehri . Solving hybrid functional-fractional equations originating in biological population dynamics with an effect on infectious diseases. AIMS Mathematics, 2024, 9(6): 14574-14593. doi: 10.3934/math.2024709
[7]	Kishor D. Kucche, Sagar T. Sutar, Kottakkaran Sooppy Nisar . Analysis of nonlinear implicit fractional differential equations with the Atangana-Baleanu derivative via measure of non-compactness. AIMS Mathematics, 2024, 9(10): 27058-27079. doi: 10.3934/math.20241316
[8]	Xiaoping Li, Dexin Chen . On solvability of some $ p $-Laplacian boundary value problems with Caputo fractional derivative. AIMS Mathematics, 2021, 6(12): 13622-13633. doi: 10.3934/math.2021792
[9]	Varaporn Wattanakejorn, Sotiris K. Ntouyas, Thanin Sitthiwirattham . On a boundary value problem for fractional Hahn integro-difference equations with four-point fractional integral boundary conditions. AIMS Mathematics, 2022, 7(1): 632-650. doi: 10.3934/math.2022040
[10]	Yasir Nadeem Anjam . The qualitative analysis of solution of the Stokes and Navier-Stokes system in non-smooth domains with weighted Sobolev spaces. AIMS Mathematics, 2021, 6(6): 5647-5674. doi: 10.3934/math.2021334

Abstract

1. Introduction

The classical diffusion equation describes the random motion of particles suspended in a medium. But when the particles do not obey a certain law, it may not give a correct result. Anomalous diffusion which occurs in a very heterogeneous aquifer, is an example of this case. Here, fractional diffusion equations can be used as an accurate model ^[1,2,3]. In recent years, these equations have been getting more attention, since they are able to create new models for a wide range of natural processes in physics, medicine, and so on ^[4].

Some of the significant works for initial-boundary value problems (IBVP) involving fractional-order diffusion equations are as follows: Luchko ^[2] solved this kind of problem for the homogeneous equation and by the same method, the inhomogeneous case was studied in ^[3]. As for the non-linear equations with fractional order $0 < \alpha < 1$ and Laplace operator, Luchko et al. ^[5] and Jin et al. ^[6,7] considered existence, uniqueness and regularity of the solutions. Kian and Yamamoto ^[8] studied a more general operator for $1 < \alpha < 2.$ In ^[4,9], some systems involving non-linear fractional diffusion equations were examined. In this work, we generalize these results by taking a more general elliptic operator with $0 < \alpha < 1$ .

As for the recent algorithms for the non-linear partial differential equations which include fractional derivatives and integrals, we refer to ^[10,11,12].

2. Preliminaries

We use eigenfunction expansions in order to establish the weak solution of the problem, which is a fundamental technique for finding solutions of IBVP for partial differential equations. In the literature, this method was used by ^[1,2] for a homogeneous linear fractional diffusion equation. In ^[2], since the operator in the equation is a symmetric uniformly elliptic operator with time-independent coefficients, the problem for a fractional partial differential equation was solved by transforming into two different problems for fractional ordinary differential equations. Their solutions were obtained by using Laplace transform and the theory of boundary value problems for elliptic equations ^[13]. By using the result of ^[2] in the case of inhomogeneous classical partial differential equations, Sakamoto and Yamamoto ^[3] obtained the solution of the IBVP for an inhomogeneous linear fractional diffusion equation. Then, Luchko et al. ^[5] and Jin et al. ^[7] used this idea for a non-linear fractional diffusion equation.

In order to investigate the existence, uniqueness and regularity properties of the solution of the problem, we use a priori estimates in $L^{2}\left(\Omega\right)$ . Sakamoto and Yamamoto ^[3] proved regularity of solution of an IBVP for an inhomogeneous linear diffusion equation. For non-linear fractional diffusion equations, Jin ^[7] applied a generalized method of Bielecki ^[14] which is widely used in the theory of functional equations.

Let $d\in \left\{ 1, 2, 3\right\}$ and

$\begin{equation} \Omega = \left\{ \begin{array}{lc} \left( 0,r_{1}\right) , & \text{if }d = 1, \\ \left( 0,r_{1}\right) \times \left( 0,r_{2}\right) , & \text{if }d = 2, \\ \left( 0,r_{1}\right) \times \left( 0,r_{2}\right) \times \left( 0,r_{3}\right) , & \text{if }d = 3 \end{array} \right. \end{equation}$

(2.1)

be a bounded domain in $\mathbb{R} ^{d}$ such that $\ r_{1}, r_{2}, r_{3}$ are positive real numbers. See Table 1 for the symbols and notations used in the paper.

Table 1. Nomenclature.

Type	Notation	Description
Symbol	$d$	dimension of the space domain,
	$k$	a fixed positive number,
	$r_{1}, r_{2}, r_{3}$	positive real numbers,
	$t$	the time variable,
	$x$	the space variable,
	$A, L$	the second order symmetric uniformly elliptic operators,
	$C_{j}$	positive constants for $1\leq j\leq 29$ ,
	$E_{\alpha, \beta }$	the Mittag-Leffer function,
	$T$	the upper limit of time domain,
	$\partial _{t}^{\alpha }$	$\alpha$ -th Caputo fractional derivative.
Greek Symbol	$\alpha$	the order of the fractional derivative,
	$\beta$	a positive real number,
	$\delta, \tau$	parameters,
	$\lambda _{j}$	the $j$ -th eigenvalue,
	$\mu$	a parameter for Mittag-Leffler function,
	$\xi$	the vector $\xi =\left(\xi _{1}, ..., \xi _{n}\right) \in \mathbb{R} ^{n},$
	$\varphi _{j}$	the $j$ -th eigenfunction,
	$\Gamma$	the Gamma function,
	$\Sigma$	the summation symbol,
	$\Omega$	domain of the solution according to space coordinates.
Abbreviation	IBVP	Initial-Boundary Value Problem

| Show Table

DownLoad: CSV

We consider a partial differential equation with the Caputo fractional derivative in time $t$ for $0 < \alpha < 1$ , and the fractional derivative is defined by

$\begin{equation} \partial _{t}^{\alpha }u\left( x,t\right) = \frac{1}{\Gamma \left( 1-\alpha \right) }\int\limits_{0}^{t}\frac{\partial _{\tau }u\left( x,\tau \right) }{ \left( t-\tau \right) ^{\alpha }}d\tau , \end{equation}$

(2.2)

see ^[15,16]. Let us assume that

$\begin{equation} F(x,t,u(x,t)) = g(u(x,t))+s(t)p(x), \end{equation}$

(2.3)

and $g$ , $s$ , $p$ are given functions in $\Omega \times \left(0, T\right)$ and $T > 0$ is a fixed value.

We aim to solve the equation

$\begin{equation} \partial _{t}^{\alpha }u\left( x,t\right) = Lu\left( x,t\right) +F\left( x,t,u\left( x,t\right) \right) ,{\ }\left( x,t\right) \in \Omega \times \left( 0,T\right) \end{equation}$

(2.4)

satisfying the following initial and boundary conditions:

$\begin{equation} u(x,0) = b\left( x\right) ,\text{ }x\in \overline{\Omega }, \end{equation}$

(2.5)

$\begin{equation} u(x,t) = 0,\text{ }\left( x,t\right) \in \partial \Omega \times \left[ 0,T \right] . \end{equation}$

(2.6)

We also define the second order symmetric uniformly elliptic operator by

$\begin{equation} Lu = \sum\limits_{i,j = 1}^{d}\partial _{x_{i}}\left( a_{ij}\left( x\right) \partial _{x_{j}}u\right) ,x\in \overline{\Omega }. \end{equation}$

(2.7)

Here, we assume that the coefficients have the following properties:

$\begin{equation} a_{ij}\in C^{\infty }\left( \overline{\Omega }\right) ,\text{ }a_{ij} = a_{ji}, \end{equation}$

(2.8)

for every integer $0\leq i, j\leq d$ , and there exists a constant $v > 0$ satisfying

$\begin{equation} v\sum\limits_{i = 1}^{d}\xi _{i}^{2}\leq \sum\limits_{i,j = 1}^{d}a_{ij}\left( x\right) \xi _{i}\xi _{j}. \end{equation}$

(2.9)

In this paper, $H_{0}^{1}\left(\Omega\right)$ and $H^{2}\left(\Omega \right)$ denote the usual Sobolev spaces ^[17,18]. Additionally, the space $\widetilde{H}^{s}\left(\Omega \right)$ is associated with the elliptic operator

$\begin{equation} A:H^{2}\left( \Omega \right) \cap H_{0}^{1}\left( \Omega \right) \rightarrow L^{2}\left( \Omega \right), \end{equation}$

(2.10)

where it is assumed that $s\geq 0$ is a real number, $A$ is defined as $A = -L$ . The spectrum of $A$ consists entirely of eigenvalues $\left\{ \lambda _{j}\right\} _{j = 1}^{\infty }$ . By Section 6.5 of ^[19], there exists an orthonormal basis $\left\{ \varphi _{j}\right\} _{j = 1}^{\infty }$ of $L^{2}\left(\Omega \right)$ such that

$\begin{equation} A\varphi _{j} = \lambda _{j}\varphi _{j},\left. \varphi _{j}\right\vert _{\partial \Omega } = 0, \end{equation}$

(2.11)

thus, $\varphi _{j}\in \widetilde{H}^{2}\left(\Omega \right)$ is an eigenfunction corresponding to $j$ -th eigenvalue $\lambda _{j}$ .

For any $s\geq 0$ , the space $\widetilde{H}^{s}\left(\Omega \right)$ is defined by

$\begin{equation} \widetilde{H}^{s}\left( \Omega \right) = \left\{ v\in L^{2}\left( \Omega \right) :\sum\limits_{j = 1}^{\infty }\lambda _{j}^{s}\left\vert \left( v,\varphi _{j}\right) \right\vert ^{2} < \infty \right\}, \end{equation}$

(2.12)

and it is a Hilbert space with the norm

$\begin{equation} \left\Vert v\right\Vert _{\widetilde{H}^{s}\left( \Omega \right) }^{2} = \sum\limits_{j = 1}^{\infty }\lambda _{j}^{s}\left\vert \left( v,\varphi _{j}\right) \right\vert ^{2} = \left\Vert A^{s/2}v\right\Vert _{L^{2}\left( \Omega \right) }^{2}. \end{equation}$

(2.13)

Moreover, we have $\widetilde{H}^{s}\left(\Omega \right) \subset H^{s}\left(\Omega \right)$ for $s > 0$ and

$\begin{equation} \widetilde{H}^{2}\left( \Omega \right) = H^{2}\left( \Omega \right) \cap H_{0}^{1}\left( \Omega \right) . \end{equation}$

(2.14)

Since $\widetilde{H}^{s}\left(\Omega \right) \subset L^{2}\left(\Omega \right)$ , by identifying the dual $\left(L^{2}\left(\Omega \right) \right) ^{\prime }$ with $L^{2}\left(\Omega \right)$ , we have

$\begin{equation} \widetilde{H}^{s}\left( \Omega \right) \subset L^{2}\left( \Omega \right) \subset \left( \widetilde{H}^{s}\left( \Omega \right) \right) ^{\prime }, \end{equation}$

(2.15)

see ^[3,7]. Here, we can write

$\begin{equation} \widetilde{H}^{-s}\left( \Omega \right) = \left\{ v\in L^{2}\left( \Omega \right) :\sum\limits_{j = 1}^{\infty }\frac{1}{\lambda _{j}^{s}}\left\vert \left( v,\varphi _{j}\right) \right\vert ^{2} < \infty \right\} = \left( \widetilde{H}^{s}\left( \Omega \right) \right) ^{\prime }. \end{equation}$

(2.16)

Since our solution will be written by using the orthonormal basis, we will work in the space $L^{2}\left(\Omega \right)$ .

Furthermore, for $0 < \alpha < 1$ , the space $C^{0, \alpha }\left(\left[ 0, T \right]; L^{2}\left(\Omega \right) \right)$ is defined as

$\begin{equation} \left\{ u\in C\left( \left[ 0,T\right] ;L^{2}\left( \Omega \right) \right) :\sup\limits_{0\leq t < s\leq T}\frac{\left\Vert u\left( .,t\right) -u\left( .,s\right) \right\Vert _{L^{2}\left( \Omega \right) }}{\left\vert t-s\right\vert ^{\alpha }} < \infty \right\} \end{equation}$

(2.17)

with the norm

$\begin{equation} \left\Vert u\right\Vert _{C^{0,\alpha }\left( \left[ 0,T\right] ;L^{2}\left( \Omega \right) \right) } = \left\Vert u\right\Vert _{C\left( \left[ 0,T\right] ;L^{2}\left( \Omega \right) \right) }+\sup\limits_{0\leq t < s\leq T}\frac{\left\Vert u\left( .,t\right) -u\left( .,s\right) \right\Vert _{L^{2}\left( \Omega \right) }}{\left\vert t-s\right\vert ^{\alpha }}, \end{equation}$

(2.18)

see ^[3].

In this work, $E_{\alpha, \beta }\left(z\right)$ denotes the Mittag-Leffler function, for $\alpha, \beta > 0$ and $z\in \mathbb{C}$ . The function $E_{\alpha, \beta }\left(z\right)$ satisfies the following properties (see Section 1.2 of ^[16]):

(i) Let $0 < \alpha < 2$ , $\beta$ be an arbitrary real number and $\mu$ satisfy $\pi \alpha < \mu < \min \left\{ \pi, \pi \alpha \right\}$ . Then there exists a real constant $C_{1} = C_{1}\left(\alpha, \beta, \mu \right)$ such that

$\begin{equation} \left\vert E_{\alpha ,\beta }\left( z\right) \right\vert \leq \frac{C_{1}}{ 1+\left\vert z\right\vert },\text{ }\mu \leq \left\vert \arg \left( z\right) \right\vert \leq \pi . \end{equation}$

(2.19)

(ii) We have

$\begin{equation} \frac{d^{m}}{dt^{m}}E_{\alpha ,1}\left( -\lambda _{n}t^{\alpha }\right) = -\lambda _{n}t^{\alpha -m}E_{\alpha ,\alpha -m+1}\left( -\lambda _{n}t^{\alpha }\right) \end{equation}$

(2.20)

for $t, \alpha, \lambda _{n} > 0$ and positive $m\in \mathbb{Z}$ .

From now on, we assume that $C_{j}, 1\leq j\leq 29$ are positive constants which are independent of the function $F$ in (2.4) and the initial condition $b$ in (2.5). But it may depend on the fractional order $\alpha$ , the coefficients of the operator $L$ and the domain of the solution.

3. Main result

By the method of eigenfunction expansions, the solution is sought in the form of

$\begin{equation} u\left( x,t\right) = \sum\limits_{n = 1}^{\infty }u_{n}\left( t\right) \varphi _{n}\left( x\right) , \end{equation}$

(3.1)

where the functions $\varphi _{n}\left(x\right)$ are the solution of the following problem:

$\begin{equation} L\varphi _{n} = -\lambda _{n}\varphi _{n},\left. \varphi _{n}\right\vert _{\partial \Omega } = 0. \end{equation}$

(3.2)

From (3.1), we see that

$\begin{equation} u_{n}\left( t\right) = \left( u\left( .,t\right) ,\varphi _{n}\right) _{L^{2}\left( \Omega \right) }, \end{equation}$

(3.3)

where $\left(., .\right) _{L^{2}\left(\Omega \right) }$ denotes the usual inner product of the space $L^{2}\left(\Omega \right)$ . We multiply both sides of Eq (2.4) by $\varphi _{n}$ and integrate it with respect to the space variable and we get

$\begin{equation} \left( \partial _{t}^{\alpha }u\left( .,t\right) ,\varphi _{n}\right) _{L^{2}\left( \Omega \right) } = \left( Lu\left( .,t\right) ,\varphi _{n}\right) _{L^{2}\left( \Omega \right) }+\left( F\left( .,t,u\left( .,t\right) \right) ,\varphi _{n}\right) _{L^{2}\left( \Omega \right) },\text{ }t\in \left( 0,T\right) . \end{equation}$

(3.4)

By analyzing the terms in (3.4), we have

$\begin{equation} \left( \partial _{t}^{\alpha }u\left( .,t\right) ,\varphi _{n}\right) _{L^{2}\left( \Omega \right) } = \partial _{t}^{\alpha }u_{n}\left( t\right) , \end{equation}$

(3.5)

$\begin{equation} \left( Lu\left( .,t\right) ,\varphi _{n}\right) _{L^{2}\left( \Omega \right) } = -\lambda _{n}u_{n}\left( t\right) . \end{equation}$

(3.6)

For simplicity, we can denote

$\begin{equation} F_{n}\left( u\left( t\right) \right) = \left( F\left( .,t,u\left( .,t\right) \right) ,\varphi _{n}\right) _{L^{2}\left( \Omega \right) }. \end{equation}$

(3.7)

Similarly, for the initial condition (2.5), we can write

$\begin{equation} u_{n}\left( 0\right) = \left( b,\varphi _{n}\right) _{L^{2}\left( \Omega \right) }. \end{equation}$

(3.8)

Now, $u_{n}\left(t\right)$ can be found by the means of Laplace transform, and then we have

$\begin{equation} u_{n}\left( t\right) = \left( b,\varphi _{n}\right) E_{\alpha ,1}\left( -\lambda _{n}t^{\alpha }\right) +\int\limits_{0}^{t}\left( t-\tau \right) ^{\alpha -1}E_{\alpha ,\alpha }\left( -\lambda _{n}\left( t-\tau \right) ^{\alpha }\right) F_{n}\left( u\left( \tau \right) \right) d\tau . \end{equation}$

(3.9)

By substituting (3.9) into (3.1), solution of the problem can be written as follows:

$\begin{equation} u\left( x,t\right) = I_{1}\left( x,t\right) +I_{2}\left( x,t\right) , \end{equation}$

(3.10)

where

$\begin{equation} I_{1}\left( x,t\right) = \sum\limits_{n = 1}^{\infty }\left( b,\varphi _{n}\right) E_{\alpha ,1}\left( -\lambda _{n}t^{\alpha }\right) \varphi _{n}\left( x\right) , \end{equation}$

(3.11)

$\begin{equation} I_{2}\left( x,t\right) = \sum\limits_{n = 1}^{\infty }\left( \int\limits_{0}^{t}\left( t-\tau \right) ^{\alpha -1}E_{\alpha ,\alpha }\left( -\lambda _{n}\left( t-\tau \right) ^{\alpha }\right) F_{n}\left( u\left( \tau \right) \right) d\tau \right) \varphi _{n}\left( x\right) . \end{equation}$

(3.12)

Since solution (3.10) is in the form of an integral equation, we can use the Banach fixed point theorem, see ^[20,21].

The main result of this paper is given in the following theorem:

Theorem 3.1. Let $b\in \widetilde{H}^{2}\left(\Omega \right)$ , $s\in C\left[ 0, T\right]$ and $p\in L^{2}\left(\Omega \right).$ We also assume that for any $u, v\in C\left(\left[ 0, T\right]; L^{2}\left(\Omega\right) \right)$ there exists a positive real constant $C_{2}$ such that

$\begin{equation} \left\Vert g\left( u\left( .,t\right) \right) -g\left( v\left( .,t\right) \right) \right\Vert _{L^{2}\left( \Omega \right) }\leq C_{2}\left\Vert u\left( .,t\right) -v\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }. \end{equation}$

(3.13)

Then, for problem (2.4)–(2.6), there exists a unique solution $u$ and a constant $C_{3} > 0$ such that

$\begin{eqnarray} &&\left\Vert u\right\Vert _{C\left( \left( 0,T\right] ;\widetilde{H} ^{2}\left( \Omega \right) \right) }+\left\Vert \partial _{t}u\right\Vert _{C\left( \left( 0,T\right] ;L^{2}\left( \Omega \right) \right) }+\left\Vert \partial _{t}^{\alpha }u\right\Vert _{C\left( \left( 0,T\right] ;L^{2}\left( \Omega \right) \right) }+\left\Vert u\right\Vert _{C^{0,\alpha }\left( \left[ 0 ,T\right] ;L^{2}\left( \Omega \right) \right) } \\ &\leq &C_{3}\left( \left\Vert b\right\Vert _{H^{2}\left( \Omega \right) }+\left\Vert s\right\Vert _{C\left[ 0,T\right] }\left\Vert p\right\Vert _{L^{2}\left( \Omega \right) }\right). \end{eqnarray}$

(3.14)

Moreover, we have

$\begin{equation} u\in C^{0,\alpha }\left( \left[ 0,T\right] ;L^{2}\left( \Omega \right) \right) \cap C\left( \left( 0,T\right] ;\widetilde{H}^{2}\left( \Omega \right) \right) , \end{equation}$

(3.15)

$\begin{equation} \partial _{t}u\in C\left( \left( 0,T\right] ;L^{2}\left( \Omega \right) \right) ,\partial _{t}^{\alpha }u\in C\left( \left( 0,T\right] ;L^{2}\left( \Omega \right) \right) . \end{equation}$

(3.16)

In the proof, we will use the same method as ^[7]. It can be seen as a generalization of Bielecki's method ^[14] which is used to investigate solvability of initial value problems for ordinary differential equations.

Proof. We will show the existence of the solution by defining a map in the following form:

$\begin{equation} M:C\left( \left[ 0,T\right] ;L^{2}\left( \Omega\right) \right) \rightarrow C\left( \left[ 0,T\right] ;L^{2}\left( \Omega\right) \right) , \end{equation}$

(3.17)

$\begin{equation} M\left(u\left( x,t\right)\right) = u\left( x,t\right) . \end{equation}$

(3.18)

Here, instead of the usual norm of $C\left(\left[ 0, T\right]; L^{2}\left(\Omega\right) \right)$ , we consider

$\begin{equation} \left\Vert u\right\Vert _{k} = \max\limits_{t\in\left[ 0,T\right] }\left\{ \left\Vert e^{-kt}u\left( t\right) \right\Vert _{L^{2}\left( \Omega\right) }\right\} \end{equation}$

(3.19)

for any fixed $k > 0$ . In order to use the Banach fixed point theorem, we will determine the value of $k$ later. The function $u$ is a solution of problem (2.4)–(2.6) if and only if $u$ is a fixed point of the map $M$ . We have

$\begin{eqnarray} &&\left\Vert M\left( u\left( .,t\right) \right) -M\left( v\left( .,t\right) \right) \right\Vert _{L^{2}\left( \Omega \right) }^{2} \\ & = &\sum\limits_{n = 1}^{\infty }\left\vert \int\limits_{0}^{t}\left( t-\tau \right) ^{\alpha -1}E_{\alpha ,\alpha }\left( -\lambda _{n}\left( t-\tau \right) ^{\alpha }\right) \left( g\left( u\left( .,\tau \right) \right) -g\left( v\left( .,\tau \right) \right) ,\varphi _{n}\right) _{L^{2}\left( \Omega \right) }d\tau \right\vert ^{2}, \end{eqnarray}$

(3.20)

and using the Cauchy-Schwarz inequality, we get

$\begin{eqnarray} e^{-2kt}\left\Vert M\left( u\left( .,t\right) \right) -M\left( v\left( .,t\right) \right) \right\Vert _{L^{2}\left( \Omega \right) }^{2} &\leq &\sum\limits_{n = 1}^{\infty }\left\vert \int\limits_{0}^{t}\left[ \left( t-\tau \right) ^{\alpha -1}E_{\alpha ,\alpha }\left( -\lambda _{n}\left( t-\tau \right) ^{\alpha }\right) \right] ^{2}d\tau \right\vert \left\Vert \varphi _{n}\right\Vert _{L^{2}\left( \Omega \right) }^{2} \\ &&\times \left\vert e^{-2kt}\int\limits_{0}^{t}\left\Vert g\left( u\left( .,\tau \right) \right) -g\left( v\left( .,\tau \right) \right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}d\tau \right\vert \\ & = &I_{3}\left( t\right) \times I_{4}\left( t\right) , \end{eqnarray}$

(3.21)

for any $u, v\in C\left(\left[ 0, T\right]; L^{2}\left(\Omega \right) \right)$ . Using (2.19), the properties for function $\varphi _{n}$ and the fact that $\lambda _{n}\geq C_{4}n^{2/d}, \; n\in \mathbb{N}$ by ^[3], we evaluate

$\begin{eqnarray} I_{3}\left( t\right) & = &\sum\limits_{n = 1}^{\infty }\left\vert \int\limits_{0}^{t}\left[ w^{\alpha -1}E_{\alpha ,\alpha }\left( -\lambda _{n}w^{\alpha }\right) \right] ^{2}dw\right\vert \left\Vert \varphi _{n}\right\Vert _{L^{2}\left( \Omega \right) }^{2} \\ &\leq &C_{1}^{2}\sum\limits_{n = 1}^{\infty }\frac{1}{\left( \lambda _{n}\right) ^{2\left( \alpha -1\right) /\alpha }}\left\vert \int\limits_{0}^{t}\left[ \frac{\left( \lambda _{n}w^{\alpha }\right) ^{\left( \alpha -1\right) /\alpha }}{1+\lambda _{n}w^{\alpha }}\right] ^{2}dw\right\vert \left\Vert \varphi _{n}\right\Vert _{L^{2}\left( \Omega \right) }^{2} \\ &\leq &tC_{5}^{2}. \end{eqnarray}$

(3.22)

Now we consider $I_{4}$ with (3.13), (3.19), we get

$\begin{eqnarray} I_{4}\left( t\right) &\leq &e^{-2kt}\int\limits_{0}^{t}C_{2}^{2}\left\Vert u\left( .,\tau \right) -v\left( .,\tau \right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}d\tau \\ &\leq &C_{2}^{2}e^{-2kt}\int\limits_{0}^{t}\max\limits_{\tau \in \left[ 0,T\right] }\left\{ e^{-2k\tau }\left\Vert u\left( .,\tau \right) -v\left( .,\tau \right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}\right\} e^{2k\tau }d\tau \\ & = &C_{2}^{2}\frac{1}{2k}\left( 1-\frac{1}{e^{2kt}}\right) \left\Vert u-v\right\Vert _{k}^{2}. \end{eqnarray}$

(3.23)

Multiplying the terms $I_{3}$ and $I_{4}$ , we get

$\begin{eqnarray} \left\Vert M\left( u\right) -M\left( v\right) \right\Vert _{k}^{2} & = &\max\limits_{t\in \left[ 0,T\right] }\left\{ e^{-2kt}\left\Vert M\left( u\left( .,t\right) \right) -M\left( v\left( .,t\right) \right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}\right\} \\ &\leq &\max\limits_{t\in \left[ 0,T\right] }\left\{ tC_{5}^{2}C_{2}^{2}\frac{1}{2k} \left( 1-\frac{1}{e^{2kt}}\right) \left\Vert u-v\right\Vert _{k}^{2}\right\} \\ &\leq &C_{5}^{2}TC_{2}^{2}\frac{1}{2k}\left( 1-\frac{1}{e^{2kT}}\right) \left\Vert u-v\right\Vert _{k}^{2} \\ &\leq &C_{6}^{2}\frac{T}{k}\left\Vert u-v\right\Vert _{k}^{2}. \end{eqnarray}$

(3.24)

With the choice of

$\begin{equation} k > C_{6}^{2}T, \end{equation}$

(3.25)

inequality (3.24) becomes a contraction on the space $C\left(\left[ 0, T\right]; L^{2}\left(\Omega\right)\right)$ with the norm $\left\Vert.\right\Vert _{k}$ . From the Banach fixed point theorem, we conclude that the transform $M$ has a fixed point, which is the solution $u$ of the integral equation (3.10).

Next, we will show the uniqueness of the solution. Let us assume that $u$ and $\tilde{u}$ are two solutions of initial-value problem (2.4)–(2.6). We set

$\begin{equation} C_{7}^{2} = C_{6}^{2}\frac{T}{k}. \end{equation}$

(3.26)

By (3.18), we can write

$\begin{equation} \left\Vert M\left( u\right) -M\left( \tilde{u}\right) \right\Vert _{k} ^{2} = \left\Vert u-\tilde{u}\right\Vert _{k}^{2}\leq C_{7}^{2}\left\Vert u-\tilde{u}\right\Vert _{k}^{2}, \end{equation}$

(3.27)

and

$\begin{equation} \left( 1-C_{7}\right) \left\Vert u-\tilde{u}\right\Vert _{k}\leq0. \end{equation}$

(3.28)

Therefore, we have

$\begin{equation} \left\Vert u-\tilde{u}\right\Vert _{k} = 0, \end{equation}$

(3.29)

which implies $u = \tilde{u}$ . Thus, the solution $u$ is unique.

Finally, we will examine the regularity property of the solution. We can divide this part of the proof into five steps.

Step 1. From (3.10), we write

$\begin{equation} \left\Vert u\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}\leq 2\left\Vert I_{1}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}+2\left\Vert I_{2}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}. \end{equation}$

(3.30)

Here, using (2.19), (2.20) and the Cauchy-Schwarz inequality, we get

$\begin{eqnarray} \left\Vert I_{1}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}& = &\sum\limits_{n = 1}^{\infty }\left\vert \left( b,\varphi _{n}\right) E_{\alpha ,1}\left( -\lambda _{n}t^{\alpha }\right) \right\vert ^{2} \\ &\leq& C_{1}^{2}\left\Vert -Lb\right\Vert _{L^{2}\left( \Omega \right) }^{2}\sum\limits_{n = 1}^{\infty }\frac{1}{\lambda _{n}^{2}} \\ &\leq& C_{8}^{2}\left\Vert b\right\Vert _{H^{2}\left( \Omega \right) }^{2} \end{eqnarray}$

(3.31)

and

$\begin{eqnarray} \left\Vert I_{2}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2} & = &\sum\limits_{n = 1}^{\infty }\left\vert \int\limits_{0}^{t}\left( t-\tau \right) ^{\alpha -1}E_{\alpha ,\alpha }\left( -\lambda _{n}\left( t-\tau \right) ^{\alpha }\right) F_{n}\left( u\left( \tau \right) \right) d\tau \right\vert ^{2} \\ &\leq &\left\Vert F\left( u\right) \right\Vert _{C\left( \left[ 0,T\right] ;L^{2}\left( \Omega \right) \right) }^{2}\sum\limits_{n = 1}^{\infty }\left\vert \int\limits_{0}^{t}\left( t-\tau \right) ^{\alpha -1}E_{\alpha ,\alpha }\left( -\lambda _{n}\left( t-\tau \right) ^{\alpha }\right) d\tau \right\vert ^{2} \\ & = &\left\Vert F\left( u\right) \right\Vert _{C\left( \left[ 0,T\right] ;L^{2}\left( \Omega \right) \right) }^{2}\sum\limits_{n = 1}^{\infty }\left\vert \int\limits_{0}^{t}\frac{d}{dw}\left[ -\frac{1}{\lambda _{n}} E_{\alpha ,1}\left( -\lambda _{n}w^{\alpha }\right) \right] dw\right\vert ^{2} \\ &\leq &2\left\Vert F\left( u\right) \right\Vert _{C\left( \left[ 0,T\right] ;L^{2}\left( \Omega \right) \right) }^{2}\left[ \sum\limits_{n = 1}^{\infty } \frac{1}{\lambda _{n}^{2}}\left[ 1+\left( E_{\alpha ,1}\left( -\lambda _{n}t^{\alpha }\right) \right) ^{2}\right] \right] \\ &\leq &C_{9}^{2}\left\Vert F\left( u\right) \right\Vert _{C\left( \left[ 0,T \right] ;L^{2}\left( \Omega \right) \right) }^{2}. \end{eqnarray}$

(3.32)

Now, we examine the right hand side of (3.32). By the hypotheses of Theorem 3.1, we can write

$\begin{eqnarray} \left\Vert F\left( u\left( .,t\right) \right) \right\Vert _{L^{2}\left( \Omega \right) }^{2} & = &\left\Vert F\left( u\left( .,t\right) \right) -F\left( u\left( 0,0\right) \right) +F\left( u\left( 0,0\right) \right) \right\Vert _{L^{2}\left( \Omega \right) }^{2} \\ &\leq &2\left\Vert s\left( t\right) p\left( x\right) +g\left( 0\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}+2\left\Vert g\left( u\left( .,t\right) \right) -g\left( u\left( 0,0\right) \right) \right\Vert _{L^{2}\left( \Omega \right) }^{2} \\ &\leq &4\left[ s\left( t\right) \right] ^{2}\left\Vert p\right\Vert _{L^{2}\left( \Omega \right) }^{2}+4\left[ g\left( 0\right) \right] ^{2}\left\Vert 1\right\Vert _{L^{2}\left( \Omega \right) }^{2}+2C_{2}^{2}\left\Vert u\left( .,t\right) -u\left( 0,0\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2} \\ &\leq &C_{10}^{2}+4\left[ s\left( t\right) \right] ^{2}\left\Vert p\right\Vert _{L^{2}\left( \Omega \right) }^{2}+2C_{2}^{2}\left\Vert u\left( .,t\right) -u\left( 0,0\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}, \end{eqnarray}$

(3.33)

and by (3.19), (3.24), we have

$\begin{equation} e^{-2kt}\left\Vert u\left( .,t\right) -\tilde{u}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}\leq \max\limits_{t\in \left[ 0,T \right] }\left\{ e^{-2kt}\left\Vert u\left( .,t\right) -\tilde{u}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}\right\} \leq C_{6}^{2}\frac{T}{k}\left\Vert u-\tilde{u}\right\Vert _{k}^{2}. \end{equation}$

(3.34)

Taking a sufficiently large $k$ implies

$\begin{equation} e^{-2kt}\left\Vert u\left( .,t \right) -\tilde{u}\left( .,t \right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}\leq C_{11}^{2}, \end{equation}$

(3.35)

and

$\begin{equation} \left\Vert u\left( .,t \right) -\tilde{u}\left( .,t \right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}\leq C_{11}^{2}e^{2kt}\leq C_{12}^{2}. \end{equation}$

(3.36)

Therefore, it yields

$\begin{equation} \left\Vert F\left( u\left( .,t\right) \right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}\leq C_{10}^{2}+4\left[ s\left( t\right) \right] ^{2}\left\Vert p\right\Vert _{L^{2}\left( \Omega \right) }^{2}+2C_{2}^{2}C_{12}^{2}, \end{equation}$

(3.37)

and taking the maximum with respect to the time variable $t$ on $\left[ 0, T \right]$ , we obtain

$\begin{equation} \left\Vert F\left( u\right) \right\Vert _{C\left( \left[ 0,T\right] ;L^{2}\left( \Omega \right) \right) }\leq C_{13}\left\Vert s\right\Vert _{C \left[ 0,T\right] }\left\Vert p\right\Vert _{L^{2}\left( \Omega \right) }. \end{equation}$

(3.38)

From (3.31) and (3.32), we get

$\begin{equation} \left\Vert u\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }\leq C_{14}\left\{ \left\Vert b\right\Vert _{H^{2}\left( \Omega \right) }+\left\Vert s\right\Vert _{C\left[ 0,T\right] }\left\Vert p\right\Vert _{L^{2}\left( \Omega \right) }\right\} , \end{equation}$

(3.39)

which results in

$\begin{equation} \left\Vert u\right\Vert _{C\left( \left[ 0,T\right] ;L^{2}\left( \Omega \right) \right) }\leq C_{15}\left( \left\Vert b\right\Vert _{H^{2}\left( \Omega \right) }+\left\Vert s\right\Vert _{C\left[ 0,T\right] }\left\Vert p\right\Vert _{L^{2}\left( \Omega \right) }\right) \end{equation}$

(3.40)

and $u\in C\left(\left[ 0, T\right]; L^{2}\left(\Omega \right) \right)$ .

Step 2. In this step, we will show the inequality

$\begin{equation} \left\Vert u\right\Vert _{C\left( \left( 0,T\right] ;\widetilde{H}^{2}\left( \Omega \right) \right) } \leq C_{16}\left( \left\Vert b\right\Vert _{H^{2}\left( \Omega \right) }+\left\Vert s\right\Vert _{C\left[ 0,T\right] }\left\Vert p\right\Vert _{L^{2}\left( \Omega \right) }\right) . \end{equation}$

(3.41)

We apply the second order operator $L$ to both sides of (3.10) by using (3.2), then we have

$\begin{equation} \left\Vert Lu\left( .,t\right) \right\Vert _{L^{2}\left( \Omega\right) }^{2}\leq 2\left\Vert I_{5}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}+2\left\Vert I_{6}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}, \end{equation}$

(3.42)

where

$\begin{equation} \left\Vert I_{5}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2} = \sum\limits_{n = 1}^{\infty}\left\vert -\lambda _{n}\left( b,\varphi_{n}\right) E_{\alpha,1}\left( -\lambda_{n}t^{\alpha }\right) \right\vert ^{2} \end{equation}$

(3.43)

and

$\begin{equation} \left\Vert I_{6}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2} = \sum\limits_{n = 1}^{\infty}\left\vert -\lambda _{n}\int\limits_{0}^{t}\left( t-\tau\right) ^{\alpha-1}E_{\alpha,\alpha }\left( -\lambda_{n}\left( t-\tau\right) ^{\alpha}\right) F_{n}\left( u\left( \tau\right) \right) d\tau\right\vert ^{2}. \end{equation}$

(3.44)

On the other hand, with a similar technique to the one used in the previous steps, we can write

$\begin{equation} \left\Vert I_{5}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2} \leq C_{17}^{2}t^{-2\alpha}\left\Vert b\right\Vert _{H^{2}\left( \Omega\right) }^{2},\text{ }t > 0, \end{equation}$

(3.45)

$\begin{equation} \left\Vert I_{6}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}\leq C_{18}^{2}\left\Vert F\left( u\right) \right\Vert _{C\left( \left[ 0,T\right] ;L^{2}\left( \Omega \right) \right) }^{2}\leq C_{19}^{2}\left\Vert s\right\Vert _{C\left[ 0,T\right] }^{2}\left\Vert p\right\Vert _{L^{2}\left( \Omega \right) }^{2}. \end{equation}$

(3.46)

By adding (3.45) and (3.46) and taking the maximum of both sides with respect to $t$ , we reach (3.41) and therefore we obtain

$\begin{equation} u\in C\left( \left( 0,T\right] ;H^{2}\left( \Omega\right) \cap H_{0} ^{1}\left( \Omega\right) \right) . \end{equation}$

(3.47)

Step 3. Here, we will prove the inequality

$\begin{equation} \left\Vert \partial _{t}u\right\Vert _{C\left( \left( 0,T\right] ;L^{2}\left( \Omega \right) \right) }\leq C_{20}\left( \left\Vert b\right\Vert _{H^{2}\left( \Omega \right) }+\left\Vert s\right\Vert _{C\left[ 0,T\right] }\left\Vert p\right\Vert _{L^{2}\left( \Omega \right) }\right) . \end{equation}$

(3.48)

By calculating the classical derivative of (3.10) with respect to $t$ and making use of the Leibnitz integral rule, we get

$\begin{equation} \left\Vert \partial_{t}u\left( .,t\right) \right\Vert _{L^{2}\left( \Omega\right) }^{2}\leq 2\left\Vert I_{7}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}+2\left\Vert I_{8}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}, \end{equation}$

(3.49)

where

$\begin{equation} \left\Vert I_{7}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2} = \sum\limits_{n = 1}^{\infty}\left\vert \left( b,\varphi_{n}\right) \left( -\lambda_{n}\right) t^{\alpha-1}E_{\alpha ,1}\left( -\lambda_{n}t^{\alpha}\right) \right\vert ^{2} \end{equation}$

(3.50)

and

$\begin{equation} \left\Vert I_{8}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2} = \sum\limits_{n = 1}^{\infty}\left\vert \int\limits_{0}^{t}\partial_{t}\left[ \left( t-\tau\right) ^{\alpha -1}E_{\alpha,\alpha}\left( -\lambda_{n}\left( t-\tau\right) ^{\alpha }\right) \right] F_{n}\left( u\left( \tau\right) \right) d\tau\right\vert ^{2}. \end{equation}$

(3.51)

Then, we see that

$\begin{equation} \left\Vert I_{7}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2} \leq C_{21}^{2}t^{-2}\left\Vert b\right\Vert _{H^{2}\left( \Omega\right) }^{2},\text{ }t > 0, \end{equation}$

(3.52)

and

$\begin{equation} \left\Vert I_{8}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2} \leq C_{22}^{2}t^{-2}\left\Vert F\left( u\right) \right\Vert _{C\left( \left[ 0,T\right] ;L^{2}\left( \Omega\right) \right) }^{2},\text{ }t > 0. \end{equation}$

(3.53)

By (3.52), (3.53), considering (3.38) and taking the maximum of both sides with respect to $t$ , we get (3.48).

Step 4. For this step, we will obtain (3.14), (3.16) and the inequality

$\begin{equation} \left\Vert \partial _{t}^{\alpha }u\right\Vert _{C\left( \left( 0,T\right] ;L^{2}\left( \Omega \right) \right) }\leq C_{23}\left( \left\Vert b\right\Vert _{H^{2}\left( \Omega \right) }+\left\Vert s\right\Vert _{C\left[ 0,T\right] }\left\Vert p\right\Vert _{L^{2}\left( \Omega \right) }\right). \end{equation}$

(3.54)

Since, the terms on the right-hand side of Eq (2.4) are examined in the first and second step, we can easily write inequality (3.54). As for (3.16), the first part is obvious from (2.14) and (3.47). We can write the second part by (3.54).

Step 5. Finally, at this step, we will prove

$\begin{equation} \left\Vert u\right\Vert _{C^{0,\alpha }\left( \left[ 0,T\right] ;L^{2}\left( \Omega \right) \right) }\leq C_{24}\left( \left\Vert b\right\Vert _{H^{2}\left( \Omega \right) }+\left\Vert s\right\Vert _{C\left[ 0,T\right] }\left\Vert p\right\Vert _{L^{2}\left( \Omega \right) }\right) \end{equation}$

(3.55)

and (3.15). By taking $s = t+h$ , we can rewrite the norm of the space as

$\begin{equation} \left\Vert u\right\Vert _{C^{0,\alpha }\left( \left[ 0,T\right] ;L^{2}\left( \Omega \right) \right) } = \left\Vert u\right\Vert _{C\left( \left[ 0,T\right] ;L^{2}\left( \Omega \right) \right) }+\sup\limits_{0\leq t < s\leq T}\frac{\left\Vert u\left( .,t+h\right) -u\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }}{h^{\alpha }}. \end{equation}$

(3.56)

We know from the previous steps that the first term on the right hand side is finite. Now, we consider the second one and we set

$\begin{equation} u\left( .,t+h\right) -u\left( .,t\right) = I_{9}\left( .,t\right) h^{\alpha}+I_{10}\left( .,t\right) h^{\alpha}+I_{11}\left( .,t\right) h^{\alpha}, \end{equation}$

(3.57)

where

$\begin{equation} I_{9}\left( .,t\right) h^{\alpha} = \sum\limits_{n = 1}^{\infty}\left( b,\varphi_{n}\right) \left[ E_{\alpha,1}\left( -\lambda_{n}\left( t+h\right) ^{\alpha}\right) -E_{\alpha,1}\left( -\lambda_{n}t^{\alpha }\right) \right] \varphi_{n}\left( x\right) , \end{equation}$

(3.58)

$\begin{equation} I_{10}\left( .,t\right) h^{\alpha } = \sum\limits_{n = 1}^{\infty }\left( \int\limits_{t}^{t+h}\left( t+h-\tau \right) ^{\alpha -1}E_{\alpha ,\alpha }\left( -\lambda _{n}\left( t+h-\tau \right) ^{\alpha }\right) F_{n}\left( u\left( \tau \right) \right) d\tau \right) \varphi _{n}\left( x\right) , \end{equation}$

(3.59)

$\begin{equation} I_{11}\left( .,t\right) h^{\alpha} = \sum\limits_{n = 1}^{\infty}\left( \int\limits_{0}^{t}W\left( t,\tau\right) \cdot F_{n}\left( u\left( \tau\right) \right) d\tau\right) \varphi_{n}\left( x\right), \end{equation}$

(3.60)

and

$\begin{equation} W\left( t,\tau\right) = \left[ \left( t+h-\tau\right) ^{\alpha-1} E_{\alpha,\alpha}\left( -\lambda_{n}\left( t+h-\tau\right) ^{\alpha }\right) -\left( t-\tau\right) ^{\alpha-1}E_{\alpha,\alpha}\left( -\lambda_{n}\left( t-\tau\right) ^{\alpha}\right) \right] . \end{equation}$

(3.61)

Since

$\begin{equation} \left\vert E_{\alpha ,1}\left( -\lambda _{n}\left( t+h\right) ^{\alpha }\right) -E_{\alpha ,1}\left( -\lambda _{n}t^{\alpha }\right) \right\vert ^{2} = \left\vert \int\limits_{t}^{t+h}\frac{-\lambda _{n}\tau ^{\alpha -1}}{ 1+\lambda _{n}\tau ^{\alpha }}d\tau \right\vert ^{2}\leq C_{25}\left( \frac{h }{t}\right) ^{2\alpha }\leq C_{25}\left( \frac{h}{\delta }\right) ^{2\alpha }, \end{equation}$

(3.62)

where $\delta$ is a number such that $0 < \delta \leq t \leq T$ , and

$\begin{equation} \left\vert W\left( t,\tau \right) \right\vert = \left\vert \int\limits_{t-\tau }^{t+h-\tau }\tau ^{\alpha -2}E_{\alpha ,\alpha -1}\left( -\lambda _{n}\tau ^{\alpha }\right) d\tau \right\vert \leq \frac{ C_{26}h}{\lambda _{n}\left( t-\tau \right) \left( t+h-\tau \right) }, \end{equation}$

(3.63)

we can evaluate (3.58)–(3.60) and obtain the following results.

There are two cases for the term $I_{9}$ , which are

$\begin{equation} \left\Vert I_{9}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega\right) }^{2}h^{2\alpha}\leq\dfrac{C_{27}^{2}\left\Vert b\right\Vert _{H^{2}\left( \Omega\right) }^{2}h^{2\alpha}}{\delta^{2\alpha}} \end{equation}$

(3.64)

for $0 < \delta \leq t \leq T$ and

$\begin{equation} \left\Vert I_{9}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega \right) }^{2}h^{2\alpha }\leq \frac{4C_{1}^{2}}{h^{2\alpha }}\left\Vert b\right\Vert _{H^{2}\left( \Omega \right) }^{2}h^{2\alpha } \end{equation}$

(3.65)

for $t = 0$ . We also have

$\begin{equation} \left\Vert I_{10}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega\right) }^{2}h^{2\alpha}\leq C_{28}^{2}\left\Vert F\left( u\right) \right\Vert _{C\left( \left[ 0,T\right] ;L^{2}\left( \Omega\right) \right) } ^{2}h^{2\alpha} \end{equation}$

(3.66)

and

$\begin{equation} \left\Vert I_{11}\left( .,t\right) \right\Vert _{L^{2}\left( \Omega\right) }^{2}h^{2\alpha}\leq C_{29}^{2}\left\Vert F\left( u\right) \right\Vert _{C\left( \left[ 0,T\right] ;L^{2}\left( \Omega\right) \right) } ^{2}h^{2\alpha}. \end{equation}$

(3.67)

By considering (3.64)–(3.67), we get (3.55). Therefore, we can write

$\begin{equation} u\in C^{0,\alpha}\left( \left[ 0,T\right] ;L^{2}\left( \Omega\right) \right). \end{equation}$

(3.68)

Using (3.41), (3.48), (3.54) and (3.55), we obtain (3.14). Additionally, taking into account of (3.47) and (3.68), we have (3.15).

This completes the proof of the theorem.

4. Conclusions

In this study, we consider an initial-boundary value problem for a non-linear fractional diffusion equation. We prove the existence, uniqueness and regularity properties of the solution under some conditions on the non-linear function $F$ and the initial condition.

Conflict of interest

The authors declare no conflicts of interest.

References

[1]	J. Cheng, J. Nakagawa, M. Yamamoto, T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Probl., 25 (2009), 1–16. https://doi.org/10.1088/0266-5611/25/11/115002 doi: 10.1088/0266-5611/25/11/115002
[2]	Y. Luchko, Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Comput. Math. Appl., 59 (2010), 1766–1772. https://doi.org/10.1016/j.camwa.2009.08.015 doi: 10.1016/j.camwa.2009.08.015
[3]	K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426–447. https://doi.org/10.1016/j.jmaa.2011.04.058 doi: 10.1016/j.jmaa.2011.04.058
[4]	L. Li, L. Y. Jin, S. M. Fang, Existence and uniqueness of the solution to a coupled fractional diffusion system, Adv. Differ. Equ., 370 (2015), 1–14. https://doi.org/10.1186/s13662-015-0707-0 doi: 10.1186/s13662-015-0707-0
[5]	Y. Luchko, W. Rundell, M. Yamamoto, L. H. Zuo, Uniqueness and reconstruction of an unknown semilinear term in a time-fractional reaction-diffusion equation, Inverse Probl., 29 (2013), 1–16. https://doi.org/10.1088/0266-5611/29/6/065019 doi: 10.1088/0266-5611/29/6/065019
[6]	B. T. Jin, B. Y. Li, Z. Zhou, Numerical analysis of nonlinear subdiffusion equations, SIAM J. Numer. Anal., 56 (2018), 1–23. https://doi.org/10.1137/16M1089320 doi: 10.1137/16M1089320
[7]	B. T. Jin, Fractional differential equations, Cham: Springer, 2021. https://doi.org/10.1007/978-3-030-76043-4
[8]	Y. Kian, M. Yamamoto, On existence and uniqueness of solutions for semilinear fractional wave equations, Fract. Calc. App. Anal., 20 (2017), 117–138. https://doi.org/10.1515/fca-2017-0006 doi: 10.1515/fca-2017-0006
[9]	T. B. Ngoc, N. H. Tuan, D. O'Regan, Existence and uniqueness of mild solutions for a final value problem for nonlinear fractional diffusion systems, Commun. Nonlinear Sci. Numer. Simul., 78 (2019), 104882. https://doi.org/10.1016/j.cnsns.2019.104882 doi: 10.1016/j.cnsns.2019.104882
[10]	M. H. Tiwana, K. Maqbool, A. B. Mann, Homotopy perturbation Laplace transform solution of fractional non-linear reaction diffusion system of Lotka-Volterra type differential equation, Eng. Sci. Technol. Int. J., 20 (2017), 672–678. https://doi.org/10.1016/j.jestch.2016.10.014 doi: 10.1016/j.jestch.2016.10.014
[11]	W. L. Qiu, D. Xu, J. Guo, J. Zhou, A time two-grid algorithm based on finite difference method for the two-dimensional nonlinear time-fractional mobile/immobile transport model, Numer. Algorithms, 85 (2020), 39–58. https://doi.org/10.1007/s11075-019-00801-y doi: 10.1007/s11075-019-00801-y
[12]	W. L. Qiu, X. Xiao, K. X. Li, Second-order accurate numerical scheme with graded meshes for the nonlinear partial integrodifferential equation arising from viscoelasticity, Commun. Nonlinear Sci. Numer. Simul., 116 (2023), 106804. https://doi.org/10.1016/j.cnsns.2022.106804 doi: 10.1016/j.cnsns.2022.106804
[13]	V. S. Vladimirov, Equations of mathematical physics, New York: Marcel Dekker, 1971.
[14]	E. U. Tarafdar, M. S. R. Chowhury, Topological methods for set-valued nonlinear analysis, Singapore: World Scientific, 2008. https://doi.org/10.1142/6347
[15]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
[16]	I. Podlubny, Fractional differential equations, Academic Press, 1998.
[17]	R. A. Adams, J. J. F. Fournier, Sobolev spaces, 2 Eds., Elsevier, 2003.
[18]	H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, New York: Springer, 2011. https://doi.org/10.1007/978-0-387-70914-7
[19]	L. C. Evans, Partial differential equations, Providence, RI: American Mathematical Society, 1998.
[20]	A. Henrot, Extremum problems for eigenvalues of elliptic operators, Basel, Switzerland: Birkhauser, 2006.
[21]	J. K. Hunter, B. Nachtergaele, Applied analysis, Singapore: World Scientific, 2001.

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(1850) PDF downloads(140) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Tables(1)

AIMS Mathematics

On the solvability of an initial-boundary value problem for a non-linear fractional diffusion equation

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. Main result

4. Conclusions

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Mathematics

On the solvability of an initial-boundary value problem for a non-linear fractional diffusion equation

Related Papers:

Abstract

1. Introduction

2. Preliminaries

3. Main result

4. Conclusions

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog