Investigation of Well-Posedness for a Direct Problem for a Nonlinear Fractional Diffusion Equation and an Inverse Problem

Abstract

In this paper, we consider a direct problem and an inverse problem involving a nonlinear fractional diffusion equation, which can be applied to many physical situations. The equation contains a Caputo fractional derivative, a symmetric uniformly elliptic operator and a source term consisting of the sum of two terms, one of which is linear and the other is nonlinear. The well-posedness of the direct problem is examined and the results are used to investigate the stability of an inverse problem of determining a function in the linear part of the source. The main tools in our study are the generalized eigenfunction expansions theory for nonlinear fractional diffusion equations, contraction mapping, Young’s convolution and generalized Grönwall’s inequalities. We present a stability estimate for the solution of the inverse source problem by means of observation data at a given point in the domain.

1. Introduction

The diffusion of microscopic particles, which can be seen in a variety of natural and man-made processes, is often characterized by chaotic movements with unpredictable behavior. Anomalous diffusion can occur, which describes slower or faster diffusion than normal [1]. Fractional diffusion equations depict this type of diffusion. They have been used for a number of physical situations, including thin saturated areas within porous media, protein-folding models for non-Markovian dynamical phenomena and anomalous transport in disordered systems [2]. Due to their widespread applications in nature, such as physics, geology, complex viscoelastic materials, hydrology, and ecology, fractional diffusion equations have attracted much interest over the last two decades [3].

Let $\mathsf{\Omega }\subset {\mathbb{R}}^d$, $d\in \left\{1,2,3\right\}$,

$$\mathsf{\Omega }=\left\{\begin{array}{cc}\left(0,b_1\right),\hfill & \mathrm{for}d=1,\\ \left(0,b_1\right)\times \left(0,b_2\right),\hfill & \mathrm{for}d=2,\\ \left(0,b_1\right)\times \left(0,b_2\right)\times \left(0,b_3\right),\hfill & \mathrm{for}d=3,\end{array}\right.$$

for positive $b_1,b_2,b_3\in \mathbb{R}$ and $T>0$ be a fixed value. In this work, we deal with the nonlinear fractional diffusion equation

$${\mathsf{\partial }}_t^{\alpha }u\left(x,t\right)=\left(Lu\right)\left(x,t\right)+p\left(x\right)r\left(t\right)+g\left(u\left(x,t\right)\right),\left(x,t\right)\in \mathsf{\Omega }\times \left(0,T\right),$$

(1)

where $\alpha \in \left(0,1\right)$ and ${\mathsf{\partial }}_t^{\alpha }u\left(x,t\right)$ denote the $\alpha$-th order Caputo fractional derivative of $u\left(x,t\right)$ with respect to t. If $\alpha =1$ and $L,p,r,g$ are chosen in a particular way, as in [4], there exist some biochemical models of enzyme systems for Equation (1), such as artificial membranes coupled with electrodes and a glucose oxidase membrane.

The definition of the fractional derivative is given as follows:

$${\mathsf{\partial }}_t^{\alpha }v\left(t\right)=\left\{\begin{array}{cc}\frac{1}{\Gamma \left(m-\alpha \right)}\underset{0}{\overset{t}{\int }}\frac{{\mathsf{\partial }}_{\tau }^{m}v\left(\tau \right)}{{\left(t-\tau \right)}^{\alpha +1-m}}d\tau ,\hfill & 0\le m-1<\alpha <m,\hfill \\ {\mathsf{\partial }}_t^{m}v\left(t\right),\hfill & \alpha =m\in \mathbb{N}\hfill \end{array}\right.$$

(2)

(see [5]). Here, $\Gamma$ indicates the Gamma function. The operator L is defined as

$$Lu=\sum _{i,j=1}^d\frac{\mathsf{\partial }}{\mathsf{\partial }{x}_i}\left(a_{ij}\left(x\right)\frac{\mathsf{\partial }u}{\mathsf{\partial }{x}_j}\right),x\in \overline{\mathsf{\Omega }},$$

(3)

where we suppose that

$$\begin{array}{cc}{a}_{ij}\in C^{\infty }\left(\overline{\mathsf{\Omega }}\right),& a_{ij}=a_{ji}\end{array}$$

(4)

for every integer $0\le i,j\le d$ and the inequality

$$\delta \sum _{i=1}^d{\xi }_i^2\le \sum _{i,j=1}^d{a}_{ij}\left(x\right){\xi }_i{\xi }_j$$

(5)

holds for a constant $\delta >0$. Moreover, we consider the boundary and initial conditions

$$\begin{array}{ccc}\hfill u(x,t)& =& 0,\left(x,t\right)\in \mathsf{\partial }\mathsf{\Omega }\times \left(0,T\right),\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill u(x,0)& =& 0,x\in \mathsf{\Omega }.\hfill \end{array}$$

(7)

In this paper, we discuss the following problems:

Problem 1.

Find $u\left(x,t\right)$, satisfying Equation (1) with boundary condition (6) and initial condition (7). Here, $\alpha \in (0,1)$; p, r and g are given functions.

Problem 2.

Determine the pair of functions $\left(u,r\right)$ satisfying the initial-boundary value problem (1), (6), (7) by the additional data $u\left(x_0,t\right)=h\left(t\right)$ for $0<t<T$. Here, $\alpha \in (0,1)$; p and g are given functions; $x_0\in \mathsf{\Omega }$ is a given point.

Problem 1 is a direct problem, while Problem 2 is an inverse problem. We note that when $g\left(u\left(x,t\right)\right)=0$ in (1), we obtain the same direct and inverse problems considered in [6].

The method we use to solve the initial-boundary value problem involving a fractional-order diffusion equation relies on the Laplace transform and the theory of boundary-value problems for elliptic equations. It was introduced by [7,8] for a homogeneous equation, further investigated by [9] and was applied to an inhomogeneous case by [6]. Then, the technique for examining the existence, uniqueness and regularity of the solution for a nonlinear model was developed by [10] and generalized by [11,12,13,14,15]. We note that the last five works employ a contraction mapping method by assuming the solution as the fixed point.

With regard to the theory of inverse problems for partial differential equations, there are many studies on the stability investigation for different equations, for instance, [16,17,18,19]. Recently, inverse problems for nonlinear fractional partial differential equations have become a widely studied subject. In [10,20,21,22,23], numerical techniques are employed to obtain the solution of some inverse problems for nonlinear time-fractional diffusion equations. On the other hand, the theoretical aspects have rarely been investigated. In [3], the inverse problems of determining the fractional order and determining the function that defines the nonlinear term in a nonlinear fractional diffusion equation are considered.

Our main objectives are to investigate the existence, uniqueness and regularity of the solution of the direct problem and to prove that the solution of the inverse problem is stable. For this purpose, with the help of tools used in [3,10] for nonlinear fractional differential equations, we use the approach of [6] and generalize to a nonlinear equation in our study. First, we estimate the solution of the direct problem, and then we use it to investigate the stability of the corresponding inverse problem. However, since [6] has no nonlinear terms, the assumptions in [6] are insufficient to solve our problem.

The outline of the paper is as follows: The next Section 2 provides the fundamental theoretical tools which are necessary in our proofs. Section 3 is devoted to our first main result Theorem 1 for the initial-boundary value problem. In Section 4, using the results of the previous section, we show the stability for the solution of the inverse problem, which is our second main result: Theorem 2. Finally, Section 5 concludes the paper with our final remarks on the Problems 1 and 2.

2. Materials and Methods

In this work, $L^2\left(\mathsf{\Omega }\right)$ denotes the Lebesgue space for $p=2$, $H_0^1\left(\mathsf{\Omega }\right)$ and $H^s\left(\mathsf{\Omega }\right),$ $s\in {\mathbb{Z}}^{+}$ denote the usual Sobolev spaces. When $s\in {\mathbb{R}}^{+}$, $H^s\left(\mathsf{\Omega }\right)$ denotes the fractional Sobolev spaces, for which we refer to [14,24,25]. Definition of the spaces $C\left(\left[0,T\right];L^2\left(\mathsf{\Omega }\right)\right)$ can be seen from Section 5.9 of [26].

The spaces ${\tilde{H}}^s\left(\mathsf{\Omega }\right)$, $s\ge 0$ are defined by the association of an elliptic operator. Due to $\mathsf{\Omega }\subset {\mathbb{R}}^d$, $d\in \left\{1,2,3\right\}$ and the Sobolev imbedding theorem, we can write

$$-L:H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)\to L^2\left(\mathsf{\Omega }\right).$$

(8)

The intersection written on (8) is given in Section 8.3 of [27]. Considering (3)–(5), the theory in Section 6.5 of [26] can be used. There exists an orthonormal basis ${\left\{{\phi }_n\right\}}_{n=1}^{\infty }$ of $L^2\left(\mathsf{\Omega }\right)$, ${\phi }_n\in H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)$ denotes an eigenfunction corresponding to ${\lambda }_n$ and

$$\left\{\begin{array}{cc}-L{\phi }_n={\lambda }_n{\phi }_n,\hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ {\phi }_n=0,\hfill & \mathrm{on}\mathsf{\partial }\mathsf{\Omega }\hfill \end{array}\right.$$

(9)

for $n=1,2,\dots$. Here, we have $0<{\lambda }_1\le {\lambda }_2\le {\lambda }_3\le \dots$ and ${\lambda }_n\to \infty$ when $n\to \infty$ for the eigenvalues. The space of ${\tilde{H}}^s\left(\mathsf{\Omega }\right)$ is a Hilbert space and it is defined by

$${\tilde{H}}^s\left(\mathsf{\Omega }\right)=\{w\in L^2\left(\mathsf{\Omega }\right):\sum _{n=1}^{\infty }{\lambda }_n^s|(w,{\phi }_n){|}^2<\infty \},$$

with the norm

$${∥w∥}_{{\tilde{H}}^s\left(\mathsf{\Omega }\right)}^2=\sum _{n=1}^{\infty }{\lambda }_n^s{\left|\left(w,{\phi }_n\right)\right|}^2$$

(10)

for real $s\ge 0$, which corresponds to the space $D\left({\left(-L\right)}^{s/2}\right)$ given by [6]. By Section 5.4 of [27], it is known that

$$\begin{array}{ccc}\hfill w& =& \sum _{n=1}^{\infty }\left(w,{\phi }_n\right){\phi }_n,\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill {∥w∥}_{L^2\left(\mathsf{\Omega }\right)}^2& =& \sum _{n=1}^{\infty }{\left|\left(w,{\phi }_n\right)\right|}^2,\hfill \end{array}$$

(12)

for every $w\in L^2\left(\mathsf{\Omega }\right)$. We can write

$${∥w∥}_{{\tilde{H}}^s\left(\mathsf{\Omega }\right)}^2=\sum _{n=1}^{\infty }{\left|\left({\left(-L\right)}^{s/2}w,{\phi }_n\right)\right|}^2={∥{\left(-L\right)}^{s/2}w∥}_{L^2\left(\mathsf{\Omega }\right)}^2.$$

(13)

We have ${\tilde{H}}^s\left(\mathsf{\Omega }\right)\subset H^s\left(\mathsf{\Omega }\right)$ for any $s>0$,

$${\tilde{H}}^0\left(\mathsf{\Omega }\right)=L^2\left(\mathsf{\Omega }\right),{\tilde{H}}^1\left(\mathsf{\Omega }\right)=H_0^1\left(\mathsf{\Omega }\right),{\tilde{H}}^2\left(\mathsf{\Omega }\right)=H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right),$$

$${\tilde{H}}^s\left(\mathsf{\Omega }\right)\subset L^2\left(\mathsf{\Omega }\right)\subset {\left({\tilde{H}}^s\left(\mathsf{\Omega }\right)\right)}^{\prime }$$

(14)

and

$${\tilde{H}}^{-s}\left(\mathsf{\Omega }\right)=\left\{w\in L^2\left(\mathsf{\Omega }\right):\sum _{n=1}^{\infty }\frac{1}{{\lambda }_n^s}{\left|\left(w,{\phi }_n\right)\right|}^2<\infty \right\}={\left({\tilde{H}}^s\left(\mathsf{\Omega }\right)\right)}^{\prime },$$

(see [14]).

We have the following lemmata which are necessary in the proceeding sections:

Lemma 1

([26]). Suppose that $u_1\in H^s\left({\mathbb{R}}^d\right)$,

$$u_1=\left\{\begin{array}{cc}{u}_2,\hfill & \mathrm{in}\mathsf{\Omega },\hfill \\ 0,\hfill & \mathrm{in}{\mathbb{R}}^d\setminus \mathsf{\Omega },\hfill \end{array}\right.$$

and $s>d/2$. Then, we have $u_1\in L^{\infty }\left({\mathbb{R}}^d\right)$ and

$${∥u_1∥}_{L^{\infty }\left({\mathbb{R}}^d\right)}\le C_1{∥u_1∥}_{H^s\left({\mathbb{R}}^d\right)},$$

(15)

where the constant $C_1$ depends on only d and s.

Lemma 2

([6]). If all eigenvalues of operator $-L$ are represented by the set ${\left\{{\lambda }_n\right\}}_{n\in \mathbb{N}}$, then we have ${\lambda }_n\ge C_2{n}^{2/d}$ for every $n\in \mathbb{N}$.

We consider the two-parameter Mittag–Leffler functions

$$E_{\alpha ,\beta }\left(z\right)=\sum _{n=0}^{\infty }\frac{z^n}{\Gamma (\alpha n+\beta )},\alpha ,\beta >0,z\in \mathbb{C},$$

which are important tools in fractional analysis and the generalized form of the significant function $f\left(z\right)=e^z$ for the theory of classical differential equations, [5,14].

Lemma 3

([5,6,14,28]).

(i)

We have

$$\frac{d^m}{d{t}^m}{E}_{\alpha ,1}\left(-\mu t^{\alpha }\right)=-\mu t^{\alpha -m}{E}_{\alpha ,\alpha -m+1}\left(-\mu t^{\alpha }\right)$$

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

(ii)

$E_{\alpha ,1}\left(-\mu \right)$ is completely monotonic for $\mu \in {\mathbb{R}}^{+}$ and $\alpha \in \left(0,1\right)$.

(iii)

Suppose that $0<\alpha <2$, β is a real number and $\eta \in \left(\pi \alpha ,min\left\{\pi ,\pi \alpha \right\}\right)$. Then, the inequality

$$\left|E_{\alpha ,\beta }\left(\mu \right)\right|\le \frac{C_3}{1+\left|\mu \right|},\eta \le \left|arg\left(\mu \right)\right|\le \pi$$

(16)

is satisfied for a constant $C_3=C_3\left(\alpha ,\beta ,\eta \right)$ and for $\left|\mu \right|\ge 0$.

(iv)

When $\alpha ,\beta ,\mu \in \mathbb{C},$ $Re\left(\alpha \right)>0,$ $n\in \mathbb{N}$, we have the Laplace transform

$$\mathcal{L}\left[t^{\beta -1}{E}_{\alpha ,\beta }\left(\mu t^{\alpha }\right)\right]\left(s\right)=\frac{s^{\alpha -\beta }}{s^{\alpha }-\mu },Re\left(s\right)>0,\left|\mu s^{-\alpha }\right|<1.$$

We also use the well-known Young’s convolution inequality and the generalized Grönwall inequality, which can be found in Appendix A of [29]. For Lebesgue’s theorem, we refer to Section 2.1.7 of [30]. From the first chapter of [29], we know the formula

$$\underset{\theta }{\overset{t}{\int }}{\left(\tau -\theta \right)}^{q-1}{\left(t-\tau \right)}^{z-1}d\tau ={\displaystyle \frac{\Gamma \left(q\right)\Gamma \left(z\right)}{\Gamma \left(q+z\right)}}{\left(t-\theta \right)}^{q+z-1}$$

(17)

for $q,z>0$ and $\theta <t$.

Using the eigenfunction expansions, the weak solution of Problem 1 is sought in the following form

$$u\left(x,t\right)=\sum _{n=1}^{\infty }{u}_n\left(t\right){\phi }_n\left(x\right),$$

(18)

where ${\phi }_n\left(x\right)$ are the solutions of (9) and

$$u_n\left(t\right)={\left(u\left(.,t\right),{\phi }_n\right)}_{L^2\left(\mathsf{\Omega }\right)}.$$

Multiplying (1) with ${\phi }_n\left(x\right)$, integrating the result on $\mathsf{\Omega }$ and analyzing the terms, yields the equation

$${\mathsf{\partial }}_t^{\alpha }{u}_n\left(t\right)=-{\lambda }_n{u}_n\left(t\right)+F_n\left(u\left(t\right)\right),0<t<T,$$

where

$$F_n\left(u\left(t\right)\right)=r\left(t\right){\left(p,{\phi }_n\right)}_{L^2\left(\mathsf{\Omega }\right)}+{\left(g\left(u\left(·,\tau \right)\right),{\phi }_n\right)}_{L^2\left(\mathsf{\Omega }\right)}.$$

Therefore, considering (7), the Problem 1 is converted into solving the system of Problems (9) and

$$\left\{\begin{array}{c}{\mathsf{\partial }}_t^{\alpha }{u}_n\left(t\right)=-{\lambda }_n{u}_n\left(t\right)+F_n\left(u\left(t\right)\right),0<t<T,\hfill \\ u_n\left(0\right)=0.\hfill \end{array}\right.$$

(19)

Then, using the Laplace transform with Lemma 3, the solution of the initial value Problem (19) can be written as

$$u_n\left(t\right)=\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)F_n\left(u\left(t\right)\right)d\tau .$$

(20)

Writing (20) in (18), the solution of Problem 1 is obtained as

$$\begin{array}{c}u\left(x,t\right)=\sum _{n=1}^{\infty }\left(\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)r\left(t\right)\left(p,{\phi }_n\right)d\tau \right){\phi }_n\left(x\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+\sum _{n=1}^{\infty }\left(\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)\left(g\left(u\left(·,\tau \right)\right),{\phi }_n\right)d\tau \right){\phi }_n\left(x\right).\hfill \end{array}$$

(21)

For further details, see [13,15].

3. Solvability of Problem 1

In this section, we present some estimates for the solution u of Problem 1 and the nonlinear term $g\left(u\right)$, which are useful for investigating the stability of the solution of Problem 2.

Since representation (21) is an integral equation, to prove Theorem 1, we define a map which enables us to apply fixed point theory. The existence and uniqueness of the solution are investigated by employing the method of [14], which can be considered as a generalization of Bielecki’s method (see Section 2.4 in [31]). In that investigation, we apply Banach’s fixed-point theorem, which can be seen from Section 9.2 in [26]. Finally, we derive useful inequalities for both the solution u and the nonlinear term $g\left(u\right)$.

Now, we present our main result for Problem 1.

Theorem 1.

Let $p\in L^2\left(\mathsf{\Omega }\right),$ $r\in C\left[0,T\right]$, $g\left(0\right)=0$ and

$$\left|\underset{\varphi \in \mathbb{R}}{sup}\frac{d^{i}g}{d{\varphi }^i}\left(\varphi \right)\right|<\infty ,i=0,1.$$

(22)

Then, there exists a unique solution of Problem 1 satisfying (21) and $u\in C\left(\left[0,T\right];H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)\right)$ with ${\mathsf{\partial }}_t^{\alpha }u\in C\left(\left[0,T\right];L^2\left(\mathsf{\Omega }\right)\right).$ Additionally, we have the inequality

$${∥u∥}_{C\left(\left[0,T\right];L^2\left(\mathsf{\Omega }\right)\right)}\le C_4{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}{∥r∥}_{C\left[0,T\right]},$$

(23)

for a positive constant $C_4$.

It should be noted that for $g\in C^1\left(\mathbb{R}\right)$,

$$g\left(x\right)=\frac{x}{1+x^2}$$

(24)

is an example for satisfying the conditions $g\left(0\right)=0$ and (22).

The proof of Theorem 1 is lengthy and is separated with different steps by Lemmata 4–7. In Lemma 4, we show the existence and uniqueness of the solution. By Lemma 5, we obtain the inequality (23). With Lemma 6, we obtain an estimate for the nonlinear term. As a result of Lemma 7, we have $u\in C\left(\left[0,T\right];H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)\right)$. Finally, we complete the proof by showing that ${\mathsf{\partial }}_t^{\alpha }u\in C\left(\left[0,T\right];L^2\left(\mathsf{\Omega }\right)\right)$.

Lemma 4.

Under the hypotheses of Theorem 1, there exists a unique solution of Problem 1 satisfying (21).

Proof of Lemma 4.

Since the representation of the solution to Problem 1 is in the form of (21) and it is an integral equation, we can employ the technique of [14]. We denote $C{\left(\left[0,T\right];L^2\left(\mathsf{\Omega }\right)\right)}_k$ by the space $C\left(\left[0,T\right];L^2\left(\mathsf{\Omega }\right)\right)$ equipped with the norm

$${∥u∥}_k=\underset{t\in \left[0,T\right]}{max}\left\{{∥e^{-kt}u\left(t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}\right\},$$

(25)

and from [14,31], we know that (25) is equivalent to the standard norm of $C\left(\left[0,T\right];L^2\left(\mathsf{\Omega }\right)\right)$ for any fixed $k>0$. We define a map

$$M:C{\left(\left[0,T\right];L^2\left(\mathsf{\Omega }\right)\right)}_k\to C{\left(\left[0,T\right];L^2\left(\mathsf{\Omega }\right)\right)}_k$$

by

$$\begin{array}{ccc}\hfill M\left(u\left(x,t\right)\right)& :=& \sum _{n=1}^{\infty }\left(\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)r\left(t\right)\left(p,{\phi }_n\right)d\tau \right){\phi }_n\left(x\right)\hfill \\ & & +\sum _{n=1}^{\infty }\left(\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)\left(g\left(u\left(·,\tau \right)\right),{\phi }_n\right)d\tau \right){\phi }_n\left(x\right).\hfill \end{array}$$

(26)

For any $k>0$, $u\in C\left(\left[0,T\right];L^2\left(\mathsf{\Omega }\right)\right)$ is a solution of Problem 1 if, and only if, u is a fixed point of the map M. Therefore, we need to prove that for some $k>0$, the map M has a unique fixed point. For this purpose, we set the notations

$$I_1\left(\tau \right):={\left(g\left(u\left(·,\tau \right)\right)-g\left(v\left(·,\tau \right)\right),{\phi }_n\right)}_{L^2\left(\mathsf{\Omega }\right)},$$

$$I_2\left(t\right):=e^{-2kt}{∥M\left(u\left(·,t\right)\right)-M\left(v\left(·,t\right)\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2$$

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

$$I_3\left(t\right):=\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)I_1\left(\tau \right)d\tau .$$

Now, we start to evaluate the term $I_2$. By implementing Lebesgue’s theorem to the series, using the Cauchy–Schwarz inequality, (12), the properties of ${\left\{{\phi }_n\right\}}_{n=1}^{\infty }$ , we have

$$\begin{array}{ccc}\hfill I_2\left(t\right)& =& e^{-2kt}{∥\sum _{n=1}^{\infty }\left(\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)I_1\left(\tau \right)d\tau \right){\phi }_n\left(x\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & =& \sum _{n=1}^{\infty }{\left|e^{-kt}{I}_3\left(t\right)\right|}^2.\hfill \end{array}$$

(27)

For examining the right-hand side of (27), we use (16) with the notation

$$I_4:=\underset{0\le \tau \le t}{max}\left\{e^{-k\tau }{I}_1\left(\tau \right)\right\}$$

and we write

$$\begin{array}{ccc}\hfill I_3\left(t\right)& =& \underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)I_1\left(\tau \right)e^{k\tau -k\tau }d\tau \hfill \\ & \le & I_4\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)e^{-k\left(t-\tau \right)}d\tau \hfill \\ & \le & C_5{I}_4\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{e}^{-k\left(t-\tau \right)}d\tau =C_5{I}_4\underset{0}{\overset{t}{\int }}{\tau }^{\alpha -1}{e}^{-k\tau }d\tau .\hfill \end{array}$$

(28)

By the change of variable $\epsilon =\tau /t$, we obtain

$$\begin{array}{ccc}\hfill \underset{0}{\overset{t}{\int }}{\tau }^{\alpha -1}{e}^{-k\tau }d\tau & =& t^{\alpha }\underset{0}{\overset{1}{\int }}{\epsilon }^{\alpha -1}{e}^{-kt\epsilon }dw\hfill \\ & =& k^{1-\alpha }t\underset{0}{\overset{1}{\int }}{\left(kt\epsilon \right)}^{\alpha -1}{e}^{-kt\epsilon }dw\hfill \\ & \le & k^{1-\alpha }t\underset{0}{\overset{1}{\int }}\left[\underset{\epsilon \in \left[0,1\right]}{max}\frac{{\left(kt\epsilon \right)}^{\alpha /2}}{e^{kt\epsilon }}\right]{\left(kt\epsilon \right)}^{\left(\alpha /2\right)-1}d\epsilon \hfill \\ & =& \frac{C_6{t}^{\alpha /2}}{k^{\alpha /2}}\underset{0}{\overset{1}{\int }}{\epsilon }^{\left(\alpha /2\right)-1}d\epsilon =\frac{2C_6}{\alpha }{\left(\frac{t}{k}\right)}^{\alpha /2}\hfill \end{array}$$

(29)

and considering (27)–(29) with the Cauchy–Schwarz inequality, we obtain

$$\begin{array}{ccc}\hfill I_2\left(t\right)& \le & {\left[\frac{C_{17}{t}^{\alpha /2}}{k^{\alpha /2}}\right]}^2\underset{0\le \tau \le T}{max}\left|\frac{{\displaystyle \sum _{n=1}^{\infty }}{\left(g\left(u\left(·,\tau \right)\right)-g\left(v\left(·,\tau \right)\right),{\phi }_n\right)}^2}{e^{2k\tau }}\right|\hfill \\ & =& \frac{C_7^2{t}^{\alpha }}{k^{\alpha }}\underset{0\le \tau \le T}{max}\left|\frac{{∥g\left(u\left(·,\tau \right)\right)-g\left(v\left(·,\tau \right)\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2}{e^{2k\tau }}\right|.\hfill \end{array}$$

(30)

By using the mean value theorem, there exists a $\theta \left(x,t\right)\in \left(min\left\{v,w\right\},max\left\{v,w\right\}\right)$ for any $v,w\in C\left(\left[0,T\right];L^2\left(\mathsf{\Omega }\right)\right)$ satisfying

$${∥g\left(v\right)-g\left(w\right)∥}_{L^2\left(\mathsf{\Omega }\right)}={∥g^{\prime }\left(\theta \right)\left[v-w\right]∥}_{L^2\left(\mathsf{\Omega }\right)}.$$

With (22), we obtain

$${∥g\left(v\right)-g\left(w\right)∥}_{L^2\left(\mathsf{\Omega }\right)}\le C_0{∥v-w∥}_{L^2\left(\mathsf{\Omega }\right)}$$

(31)

for any $v,w\in C\left(\left[0,T\right];L^2\left(\mathsf{\Omega }\right)\right)$ and for a constant $C_0>0$ independent from $x,t,v,w$. By (25) and (31), inequality (30) becomes

$$I_2\left(t\right)\le \frac{C_0^2{C}_7^2{t}^{\alpha }}{k^{\alpha }}\underset{0\le \tau \le T}{max}\left|\frac{{∥u\left(·,\tau \right)-v\left(·,\tau \right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2}{e^{2k\tau }}\right|=\frac{C_0^2{C}_7^2{t}^{\alpha }}{k^{\alpha }}{∥u-v∥}_k^2.$$

(32)

By taking the maximum of the inequality (32) with respect to t and writing (25), we have

$${∥M\left(u\right)-M\left(v\right)∥}_k\le \frac{C_0{C}_7{T}^{\alpha /2}}{k^{\alpha /2}}{∥u-v∥}_k.$$

(33)

With the choice of $k^{\alpha /2}>C_0{C}_7{T}^{\alpha /2}$, inequality (33) shows that M is a contraction map. By Banach’s fixed-point theorem, the map M has a unique fixed point in $C{\left(\left[0,T\right];L^2\left(\mathsf{\Omega }\right)\right)}_k$. Therefore, the solution u exists.

As for the uniqueness of the solution u, it is obvious by the selection of k, definition (26), inequality (33) and the fact that every norm is nonnegative. □

Lemma 5.

Under the hypotheses of Theorem 1, we have (23) and $u\in C\left(\left[0,T\right];L^2\left(\mathsf{\Omega }\right)\right)$.

Proof of Lemma 5.

By using the notations

$$\begin{array}{ccc}\hfill I_5\left(x,t\right)& :=& \sum _{n=1}^{\infty }\left(\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)r\left(\tau \right)\left(p,{\phi }_n\right)d\tau \right){\phi }_n\left(x\right)\hfill \\ \hfill I_6\left(x,t\right)& :=& \sum _{n=1}^{\infty }\left(\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)\left(g\left(u\left(·,\tau \right)\right),{\phi }_n\right)d\tau \right){\phi }_n\left(x\right),\hfill \\ \hfill I_7\left(x,t\right)& :=& \underset{0}{\overset{t}{\int }}\left[\sum _{n=1}^{\infty }{\tau }^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\tau }^{\alpha }\right)\left(g\left(u\left(·,t-\tau \right)\right),{\phi }_n\right){\phi }_n\left(x\right)\right]d\tau ,\hfill \end{array}$$

we write

$${∥u\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\le 2{∥I_5\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2+2{∥I_6\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2.$$

(34)

With the help of (12), (16), (22) for $i=0$ and Lebesgue’s theorem, it can be shown that the order of integrations and summations of the terms on the right-hand side of (34) can change. Indeed, for the terms $I_5$ and $I_6$, we obtain

$$\begin{array}{ccc}\hfill \sum _{n=1}^{\infty }{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)r\left(\tau \right)\left(p,{\phi }_n\right){\phi }_n\left(x\right)& \le & C_8{\left(t-\tau \right)}^{\alpha -1}r\left(\tau \right)\sum _{n=1}^{\infty }p\left(x\right){\phi }_n\left(x\right)\hfill \\ & \le & C_8{\left(t-\tau \right)}^{\alpha -1}r\left(\tau \right)p\left(x\right),\hfill \end{array}$$

(35)

$$\begin{array}{ccc}\hfill \sum _{n=1}^{\infty }{\tau }^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\tau }^{\alpha }\right)\left(g\left(u\left(·,t-\tau \right)\right),{\phi }_n\right){\phi }_n\left(x\right)& \le & \left|C_9{\tau }^{\alpha -1}\sum _{n=1}^{\infty }\left(g\left(u\left(·,t-\tau \right)\right),{\phi }_n\right){\phi }_n\left(x\right)\right|\hfill \\ & =& \left|C_9{\tau }^{\alpha -1}g\left(u\left(x,t-\tau \right)\right)\right|\hfill \end{array}$$

(36)

and the right-hand sides of (35)–(36) are integrable on the domain of t by

$$\begin{array}{ccc}\hfill C_{8}p\left(x\right)\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}r\left(\tau \right)d\tau & \le & C_{8}p\left(x\right){∥r∥}_{C\left[0,T\right]}\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}d\tau \hfill \\ & =& C_{8}p\left(x\right){∥r∥}_{C\left[0,T\right]}\frac{T^{\alpha }}{\alpha }<\infty \hfill \end{array}$$

$$\begin{array}{ccc}\hfill C_9\underset{0}{\overset{t}{\int }}\left|{\tau }^{\alpha -1}g\left(u\left(x,t-\tau \right)\right)\right|d\tau & \le & C_9{∥g∥}_{L^{\infty }\left(\mathbb{R}\right)}\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}d\tau \hfill \\ & =& C_9{∥g∥}_{L^{\infty }\left(\mathbb{R}\right)}\frac{T^{\alpha }}{\alpha }<\infty \hfill \end{array}$$

Now, we can estimate the term $I_5$. By Lemma 3 and (12), we have

$$\begin{array}{ccc}\hfill {∥I_5\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}& =& {∥\underset{0}{\overset{t}{\int }}\sum _{n=1}^{\infty }{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)r\left(\tau \right)\left(p,{\phi }_n\right){\phi }_n\left(x\right)d\tau ∥}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ & \le & \underset{0}{\overset{t}{\int }}{∥\sum _{n=1}^{\infty }{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)r\left(\tau \right)\left(p,{\phi }_n\right){\phi }_n\left(x\right)∥}_{L^2\left(\mathsf{\Omega }\right)}d\tau \hfill \\ & =& \underset{0}{\overset{t}{\int }}{\left[{∥\sum _{n=1}^{\infty }{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)r\left(\tau \right)\left(p,{\phi }_n\right){\phi }_n\left(x\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right]}^{1/2}d\tau \hfill \\ & =& \underset{0}{\overset{t}{\int }}{\left[\sum _{n=1}^{\infty }{\left|{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)r\left(\tau \right)\left(p,{\phi }_n\right)\right|}^2\right]}^{1/2}d\tau \hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {∥I_5\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}& \le & C_{10}\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}r\left(\tau \right){\left[\sum _{n=1}^{\infty }{\left|\left(p,{\phi }_n\right)\right|}^2\right]}^{1/2}d\tau \hfill \\ & =& C_{10}{I}_8\left(t\right)\hfill \end{array}$$

(37)

for

$$I_8\left(t\right):=\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}r\left(\tau \right){∥p∥}_{L^2\left(\mathsf{\Omega }\right)}d\tau .$$

In order to estimate the term $I_6$, we obtain

$$\begin{array}{cc}\hfill {∥I_6\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2& ={∥I_7\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ \hfill & ={\left(\underset{0}{\overset{t}{\int }}\sum _{n=1}^{\infty }{\tau }^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\tau }^{\alpha }\right)\left(g\left(u\left(·,t-\tau \right)\right),{\phi }_n\right){\phi }_n\left(x\right)d\tau ,I_7\left(·,t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ \hfill & =\underset{0}{\overset{t}{\int }}{\left(\sum _{n=1}^{\infty }{\tau }^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\tau }^{\alpha }\right)\left(g\left(u\left(·,t-\tau \right)\right),{\phi }_n\right){\phi }_n\left(x\right),I_7\left(·,t\right)\right)}_{L^2\left(\mathsf{\Omega }\right)}d\tau \hfill \\ \hfill & \le {∥I_7\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}\underset{0}{\overset{t}{\int }}{∥\sum _{n=1}^{\infty }{\tau }^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\tau }^{\alpha }\right)\left(g\left(u\left(·,t-\tau \right)\right),{\phi }_n\right){\phi }_n\left(x\right)∥}_{L^2\left(\mathsf{\Omega }\right)}d\tau \hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {∥I_7\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}& \le & \underset{0}{\overset{t}{\int }}{∥\sum _{n=1}^{\infty }{\tau }^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\tau }^{\alpha }\right)\left(g\left(u\left(·,t-\tau \right)\right),{\phi }_n\right){\phi }_n\left(x\right)∥}_{L^2\left(\mathsf{\Omega }\right)}d\tau \hfill \\ & =& I_9\left(t\right),\hfill \end{array}$$

(38)

by using the Fubini theorem, the Cauchy–Schwarz inequality and the notation

$$I_9\left(t\right):=\underset{0}{\overset{t}{\int }}{\left[{∥\sum _{n=1}^{\infty }{\tau }^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\tau }^{\alpha }\right)\left(g\left(u\left(·,t-\tau \right)\right),{\phi }_n\right){\phi }_n\left(x\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right]}^{1/2}d\tau .$$

Considering (12), (16) and the properties of ${\left\{{\phi }_n\right\}}_{n=1}^{\infty }$, the term $I_9\left(t\right)$ can be written as

$$\begin{array}{ccc}\hfill I_9\left(t\right)& =& \underset{0}{\overset{t}{\int }}{\left[\sum _{n=1}^{\infty }{\left|{\left({\tau }^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\tau }^{\alpha }\right){\left(g\left(u\left(·,t-\tau \right)\right),{\phi }_n\right)}_{L^2\left(\mathsf{\Omega }\right)}{\phi }_n,{\phi }_n\right)}_{L^2\left(\mathsf{\Omega }\right)}\right|}^2\right]}^{1/2}d\tau \hfill \\ & =& \underset{0}{\overset{t}{\int }}{\left[\sum _{n=1}^{\infty }{\left|{\tau }^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\tau }^{\alpha }\right)\left(g\left(u\left(·,t-\tau \right)\right),{\phi }_n\right)\right|}^2\right]}^{1/2}d\tau \hfill \\ & \le & C_{11}\underset{0}{\overset{t}{\int }}{\tau }^{\alpha -1}{\left[\sum _{n=1}^{\infty }{\left|\left(g\left(u\left(·,t-\tau \right)\right),{\phi }_n\right)\right|}^2\right]}^{1/2}d\tau \hfill \\ & =& C_{11}\underset{0}{\overset{t}{\int }}{\tau }^{\alpha -1}{∥g\left(u\left(·,t-\tau \right)\right)∥}_{L^2\left(\mathsf{\Omega }\right)}d\tau .\hfill \end{array}$$

(39)

Writing inequalities (37), (38) and (39) in (34), we obtain

$${∥u\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}\le C_{12}{I}_8\left(t\right)+C_{13}\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{∥g\left(u\left(·,\tau \right)\right)∥}_{L^2\left(\mathsf{\Omega }\right)}d\tau .$$

(40)

On the other hand, by hypothesis $g\left(0\right)=0$ and (31), we obtain

$$\begin{array}{ccc}\hfill {∥g\left(u\left(·,t\right)\right)∥}_{L^2\left(\mathsf{\Omega }\right)}& \le & C_0{∥u\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}\hfill \end{array}$$

(41)

for $0<t<T$ and by (40), (41), we write

$${∥u\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}\le C_{12}{I}_8\left(t\right)+C_0{C}_{13}\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{∥u\left(·,\tau \right)∥}_{L^2\left(\mathsf{\Omega }\right)}d\tau .$$

(42)

With Young’s convolution inequality, it can be shown that

$$\begin{array}{ccc}\hfill {∥I_8\left(t\right)∥}_{L^1\left(0,T\right)}& =& {∥p∥}_{L^2\left(\mathsf{\Omega }\right)}{∥I_{10}\ast r∥}_{L^1\left(0,T\right)}\hfill \\ & \le & {∥p∥}_{L^2\left(\mathsf{\Omega }\right)}{∥I_{10}∥}_{L^1\left(0,T\right)}{∥r∥}_{L^1\left(0,T\right)}\hfill \\ & =& {∥p∥}_{L^2\left(\mathsf{\Omega }\right)}{∥I_{10}∥}_{L^1\left(0,T\right)}\underset{0}{\overset{T}{\int }}\left|r\left(\tau \right)\right|d\tau \hfill \\ & \le & \frac{T^{\alpha +1}}{\alpha }{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}{∥r∥}_{C\left[0,T\right]}<\infty \hfill \end{array}$$

(43)

for

$$I_{10}\left(\tau \right):={\tau }^{\alpha -1}.$$

By (43), we have $C_{12}{I}_8\left(t\right)\in L^1\left(0,T\right)$ and using the generalized Grönwall’s inequality, we obtain

$${∥u\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}\le C_{12}{I}_8\left(t\right)+C_{12}{C}_{14}{e}^{C_{15}t}\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{I}_8\left(\tau \right)d\tau$$

(44)

from (42). Here, for the second term in the right-hand side of (44), by using formula (17), we write

$$\begin{array}{ccc}\hfill \underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{I}_8\left(\tau \right)d\tau & =& {∥p∥}_{L^2\left(\mathsf{\Omega }\right)}\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}\underset{0}{\overset{\tau }{\int }}{\left(\tau -\theta \right)}^{\alpha -1}r\left(\theta \right)d\theta d\tau \hfill \\ & =& \underset{0}{\overset{t}{\int }}\left[\underset{\theta }{\overset{t}{\int }}{\left(\tau -\theta \right)}^{\alpha -1}{\left(t-\tau \right)}^{\alpha -1}d\tau \right]r\left(\theta \right)d\theta \hfill \\ & =& \frac{{\left(\Gamma \left(\alpha \right)\right)}^2}{\Gamma \left(2\alpha \right)}\underset{0}{\overset{t}{\int }}{\left(t-\theta \right)}^{2\alpha -1}r\left(\theta \right)d\theta .\hfill \end{array}$$

(45)

With (45), inequality (44) becomes

$$\begin{array}{ccc}\hfill {∥u\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}& \le & C_{12}{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}\left[\underset{0}{\overset{t}{\int }}{\left(t-\theta \right)}^{\alpha -1}r\left(\theta \right)d\theta +C_{14}{e}^{C_{15}t}\frac{{\left(\Gamma \left(\alpha \right)\right)}^2}{\Gamma \left(2\alpha \right)}\underset{0}{\overset{t}{\int }}{\left(t-\theta \right)}^{2\alpha -1}r\left(\theta \right)d\theta \right]\hfill \\ & \le & C_{12}{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}{∥r∥}_{C\left[0,T\right]}\left[\underset{0}{\overset{t}{\int }}{\left(t-\theta \right)}^{\alpha -1}d\theta +C_{14}{e}^{C_{15}t}\frac{{\left(\Gamma \left(\alpha \right)\right)}^2}{\Gamma \left(2\alpha \right)}\underset{0}{\overset{t}{\int }}{\left(t-\theta \right)}^{2\alpha -1}d\theta \right]\hfill \\ & =& \frac{C_{12}}{\alpha }{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}{∥r∥}_{C\left[0,T\right]}\left[t^{\alpha }+C_{14}{e}^{C_{15}t}\frac{{\left(\Gamma \left(\alpha \right)\right)}^2}{\Gamma \left(2\alpha \right)}\frac{t^{2\alpha }}{2}\right].\hfill \end{array}$$

(46)

Finally, taking the maximum of inequality (46) with respect to t on $\left[0,T\right]$, we obtain (23) and $u\in C\left(\left[0,T\right];L^2\left(\mathsf{\Omega }\right)\right).$  □

Lemma 6.

Under the hypotheses of Theorem 1, we have

$${∥g\left(u\left(·,t\right)\right)∥}_{L^2\left(\mathsf{\Omega }\right)}\le \frac{C_0{C}_{16}}{\alpha }\left(t^{\alpha }+\frac{t^{2\alpha }}{2}\right){∥p∥}_{L^2\left(\mathsf{\Omega }\right)}{∥r∥}_{C\left[0,T\right]},0<t<T.$$

(47)

Proof of Lemma 6.

By (41), (46) and the notation

$$I_{11}\left(t\right):={∥g\left(u\left(·,t\right)\right)∥}_{L^2\left(\mathsf{\Omega }\right)},$$

we obtain

$$\begin{array}{ccc}\hfill I_{11}\left(t\right)& \le & C_0{C}_{12}{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}\left[\underset{0}{\overset{t}{\int }}{\left(t-\theta \right)}^{\alpha -1}r\left(\theta \right)d\theta +C_{14}{e}^{C_{15}t}\frac{{\left(\Gamma \left(\alpha \right)\right)}^2}{\Gamma \left(2\alpha \right)}\underset{0}{\overset{t}{\int }}{\left(t-\theta \right)}^{2\alpha -1}r\left(\theta \right)d\theta \right]\hfill \\ & \le & C_0{C}_{12}{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}{∥r∥}_{C\left[0,T\right]}\hfill \\ & & \times \left[\underset{0}{\overset{t}{\int }}{\left(t-\theta \right)}^{\alpha -1}d\theta +C_{14}{e}^{C_{15}t}\frac{{\left(\Gamma \left(\alpha \right)\right)}^2}{\Gamma \left(2\alpha \right)}\underset{0}{\overset{t}{\int }}{\left(t-\theta \right)}^{2\alpha -1}d\theta \right]\hfill \end{array}$$

(48)

for $0<t<T$, which leads to (47). □

Lemma 7.

Under the hypotheses of Theorem 1, we have $u\in C\left(\left[0,T\right];H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)\right)$.

Proof of Lemma 7.

By considering the term $Lu$ with (12), (47), Lemma 3, Lemma 6 and the notations

$$\begin{array}{ccc}\hfill I_{12}\left(x,t\right)& :=& L\left(\sum _{n=1}^{\infty }\left(\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)r\left(\tau \right)\left(p,{\phi }_n\right)d\tau \right){\phi }_n\left(x\right)\right),\hfill \\ \hfill I_{13}\left(x,t\right)& :=& L\left(\sum _{n=1}^{\infty }\left(\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)\left(g\left(u\left(·,\tau \right)\right),{\phi }_n\right)d\tau \right){\phi }_n\left(x\right)\right),\hfill \end{array}$$

we obtain

$$\begin{array}{ccc}\hfill {∥Lu\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2& \le & 2{∥I_{12}\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2+2{∥I_{13}\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & =& 2{∥\sum _{n=1}^{\infty }\left(\underset{0}{\overset{t}{\int }}-{\mathsf{\partial }}_{\tau }\left[E_{\alpha ,1}\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)\right]r\left(\tau \right)\left(p,{\phi }_n\right)d\tau \right){\phi }_n\left(x\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & & +2{∥\sum _{n=1}^{\infty }\left(\underset{0}{\overset{t}{\int }}-{\mathsf{\partial }}_{\tau }\left[E_{\alpha ,1}\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)\right]\left(g\left(u\left(·,\tau \right)\right),{\phi }_n\right)d\tau \right){\phi }_n\left(x\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & =& 2\sum _{n=1}^{\infty }{\left|\underset{0}{\overset{t}{\int }}-{\mathsf{\partial }}_{\tau }\left[E_{\alpha ,1}\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)\right]r\left(\tau \right)\left(p,{\phi }_n\right)d\tau \right|}^2\hfill \\ & & +2\sum _{n=1}^{\infty }{\left|\underset{0}{\overset{t}{\int }}-{\mathsf{\partial }}_{\tau }\left[E_{\alpha ,1}\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)\right]\left(g\left(u\left(·,\tau \right)\right),{\phi }_n\right)d\tau \right|}^2\hfill \\ & \le & 2{∥r∥}_{C\left[0,T\right]}^2\sum _{n=1}^{\infty }{\left|\left(p,{\phi }_n\right)\right|}^2{\left|\underset{0}{\overset{t}{\int }}{\mathsf{\partial }}_{\tau }\left[E_{\alpha ,1}\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)\right]d\tau \right|}^2\hfill \\ & & +2\sum _{n=1}^{\infty }\underset{0\le \tau \le T}{max}{\left|\left(g\left(u\left(·,\tau \right)\right),{\phi }_n\right)\right|}^2{\left|\underset{0}{\overset{t}{\int }}{\mathsf{\partial }}_{\tau }\left[E_{\alpha ,1}\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)\right]d\tau \right|}^2\hfill \\ & \le & 2C_{17}^2\left[{∥r∥}_{C\left[0,T\right]}^2{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}^2+\underset{0\le \tau \le T}{max}{∥g\left(u\left(·,\tau \right)\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\right]\hfill \\ & \le & 2C_{17}^2{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}^2{∥r∥}_{C\left[0,T\right]}^2\hfill \\ & & +2C_{17}^2\underset{0\le \tau \le T}{max}{\left[\frac{C_0{C}_{16}}{\alpha }{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}{∥r∥}_{C\left[0,T\right]}\left({\tau }^{\alpha }+\frac{{\tau }^{2\alpha }}{2}\right)\right]}^2\hfill \\ & \le & C_0^2{C}_{18}^2{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}^2{∥r∥}_{C\left[0,T\right]}^2.\hfill \end{array}$$

(49)

By (12) and (13), inequality (49) becomes

$${∥u\left(·,t\right)∥}_{H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)}\le C_0{C}_{18}{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}{∥r∥}_{C\left[0,T\right]}.$$

(50)

On $\left[0,T\right]$, taking the maximum of (50) with respect to the time variable, we write  $u\in C\left(\left[0,T\right];H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)\right)$. □

Now, we can prove Theorem 1.

Proof of Theorem 1.

The proof of the existence and uniqueness of solution (21) can be seen from Lemma 4. Inequality (23) is shown in Lemma 5. Considering Lemmata 5 and 7, $u\in C\left(\left[0,T\right];H^2\left(\mathsf{\Omega }\right)\cap H_0^1\left(\mathsf{\Omega }\right)\right)$ is obtained. Now, we only need to show that ${\mathsf{\partial }}_t^{\alpha }u\in C\left(\left[0,T\right];L^2\left(\mathsf{\Omega }\right)\right)$. By considering Equation (1) with inequalities (47) and (49), we obtain

$$\begin{array}{ccc}\hfill {∥{\mathsf{\partial }}_t^{\alpha }u\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2& \le & 4{∥Lu\left(·,t\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2+4{\left[r\left(t\right)\right]}^2{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}^2+2{∥g\left(u\left(·,t\right)\right)∥}_{L^2\left(\mathsf{\Omega }\right)}^2\hfill \\ & \le & C_0^2{C}_{19}^2{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}^2{∥r∥}_{C\left[0,T\right]}^2\hfill \end{array}$$

(51)

and by taking the maximum of (51) with respect to the variable t on $\left[0,T\right]$, we have ${\mathsf{\partial }}_t^{\alpha }u\in C\left(\left[0,T\right];L^2\left(\mathsf{\Omega }\right)\right)$, which concludes the proof. □

4. Stability of the Inverse Problem

In this section, we consider Problem 2 and analyze the stability of the inverse problem’s solution. From the previous Section 3, we use the solution representation (21) of Problem 1 and Lemma 6, where the estimate (23) for Problem 1 plays a key role. With additional data and conditions, we obtain a stability estimate for the solution of Problem 2.

Let us remark that the inverse problem by [6] is a special case of Problem 2 and that the problem has a Lipschitz stable solution. For the solution of Problem 2, we also obtain a stability estimate of Lipschitz type.

Now, we present our main result for Problem 2.

Theorem 2.

Suppose that $g\left(0\right)=0$ and we have (22) for the nonlinear term. Let

$$p\in {\tilde{H}}^{2\beta }\left(\mathsf{\Omega }\right),\beta >1+\frac{3d}{4}$$

(52)

and let u satisfy (1), (6), (7) for $r\in C\left[0,T\right]$. We also assume that $p\left(x_0\right)\ne 0$ . Then, there exists a positive constant $C_{20}$ satisfying

$${∥r∥}_{C\left[0,T\right]}\le C_{20}\left({∥{\mathsf{\partial }}_t^{\alpha }h∥}_{C\left[0,T\right]}+{∥g\left(h\right)∥}_{C\left[0,T\right]}\right).$$

(53)

Proof of Theorem 2.

From Theorem 1, the solution u is in the form of (21). By writing the solution (21) in Equation (1), we obtain

$$\begin{array}{ccc}& & {\mathsf{\partial }}_t^{\alpha }u\left(x,t\right)\hfill \\ & =& \sum _{n=1}^{\infty }\left(\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)r\left(\tau \right)\left(p,{\phi }_n\right)d\tau \right)\left(-{\lambda }_n{\phi }_n\left(x\right)\right)\hfill \\ & & +\sum _{n=1}^{\infty }\left(\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)\left(g\left(u\left(·,\tau \right)\right),{\phi }_n\right)d\tau \right)\left(-{\lambda }_n{\phi }_n\left(x\right)\right)\hfill \\ & & +p\left(x\right)r\left(t\right)+g\left(u\left(x,t\right)\right)\hfill \end{array}$$

(54)

and by taking the maximum norm of (54) from both sides with respect to the variable t on $\left[0,T\right]$, we obtain

$${∥{\mathsf{\partial }}_t^{\alpha }u\left(x,·\right)∥}_{C\left[0,T\right]}^2\le 4\left\{{\left[J_1\left(x\right)\right]}^2+{\left[J_2\left(x\right)\right]}^2+{\left[p\left(x\right)\right]}^2{∥r∥}_{C\left[0,T\right]}^2+{∥g\left(u\left(x,·\right)\right)∥}_{C\left[0,T\right]}^2\right\},$$

(55)

where

$$\begin{array}{c}\hfill J_1\left(x\right):={∥\sum _{n=1}^{\infty }\left(\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)r\left(\tau \right)\left(p,{\phi }_n\right)d\tau \right){\lambda }_n{\phi }_n\left(x\right)∥}_{C\left[0,T\right]},\\ \hfill J_2\left(x\right):={∥\sum _{n=1}^{\infty }\left(\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)\left(g\left(u\left(·,\tau \right)\right),{\phi }_n\right)d\tau \right){\lambda }_n{\phi }_n\left(x\right)∥}_{C\left[0,T\right]}.\end{array}$$

In order to write $x=x_0$ in (55), we need to investigate the convergence of the inequality. Since $p\left(x_0\right)$, ${\mathsf{\partial }}_t^{\alpha }u\left(x_0,t\right)={\mathsf{\partial }}_t^{\alpha }h\left(t\right)$ and $g\left(u\left(x_0,t\right)\right)=g\left(h\left(t\right)\right),$ $0<t<T$ are known, we consider the terms $J_1$ and $J_2$. For the notations

$$J_3:=\left|esssu{p}_{x\in \mathsf{\Omega }}{\lambda }_n\left(p,{\phi }_n\right){\phi }_n\left(x\right)\right|,$$

$$J_4\left(t\right):=\left|\underset{0}{\overset{t}{\int }}{\tau }^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\tau }^{\alpha }\right)r(t-\tau )d\tau \right|,$$

we have

$$\begin{array}{ccc}\hfill J_1\left(x\right)& =& \underset{0\le t\le T}{max}\sum _{n=1}^{\infty }\left({\lambda }_n{\phi }_n\left(x\right)\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)r\left(\tau \right)\left(p,{\phi }_n\right)d\tau \right)\hfill \\ & \le & \underset{0\le t\le T}{max}\sum _{n=1}^{\infty }\left|esssu{p}_{x\in \mathsf{\Omega }}{\lambda }_n\left(p,{\phi }_n\right){\phi }_n\left(x\right)\right|\left|\underset{0}{\overset{t}{\int }}{\tau }^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\tau }^{\alpha }\right)r(t-\tau )d\tau \right|\hfill \\ & =& \underset{0\le t\le T}{max}\sum _{n=1}^{\infty }{J}_3·J_4\left(t\right)\hfill \end{array}$$

(56)

and

$${∥{\phi }_n∥}_{L^{\infty }\left(\mathsf{\Omega }\right)}\le C_{21}{∥{\phi }_n∥}_{H^{2\beta -2-d}\left(\mathsf{\Omega }\right)}\le C_{22}{∥{\phi }_n∥}_{{\tilde{H}}^{2\beta -2-d}\left(\mathsf{\Omega }\right)}=C_{22}{∥{(-L)}^{\beta -1-\frac{d}{2}}{\phi }_n∥}_{L^2\left(\mathsf{\Omega }\right)}$$

(57)

by using the hypothesis (52), inequality (15), Lemma 1 with the properties of ${\tilde{H}}^s\left(\mathsf{\Omega }\right)$ spaces. From (57) and the properties of ${\left\{{\phi }_n\right\}}_{n=1}^{\infty }$, we write

$$\begin{array}{ccc}\hfill J_3& =& {\lambda }_n\left(p,{\phi }_n\right){∥{\phi }_n∥}_{L^{\infty }\left(\mathsf{\Omega }\right)}\hfill \\ & \le & C_{22}{\lambda }_n\left(p,{\phi }_n\right){∥{(-L)}^{\beta -1-d/2}{\phi }_n∥}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ & =& C_{22}{\lambda }_n\left(p,{\phi }_n\right){∥{\lambda }_n^{\beta -1-d/2}{\phi }_n∥}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ & =& C_{22}{\lambda }_n^{-d/2}{∥\left({\lambda }_n^{\beta }p,{\phi }_n\right){\phi }_n∥}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ & =& C_{22}{\lambda }_n^{-d/2}{\left({(-L)}^{\beta }p,{\phi }_n\right)}_{L^2\left(\mathsf{\Omega }\right)}.\hfill \end{array}$$

(58)

Using Lemma 3, we obtain

$$\begin{array}{ccc}\hfill J_4\left(t\right)& \le & \underset{0}{\overset{t}{\int }}{\tau }^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\tau }^{\alpha }\right)\left|r(t-\tau )\right|d\tau \hfill \\ & \le & \frac{{∥r∥}_{C\left[0,T\right]}}{{\lambda }_n}\underset{0}{\overset{t}{\int }}-\frac{d}{d\tau }{E}_{\alpha ,1}\left(-{\lambda }_n{\tau }^{\alpha }\right)d\tau \hfill \\ & \le & \frac{{∥r∥}_{C\left[0,T\right]}}{{\lambda }_n}{\left[2{\left[E_{\alpha ,1}\left(-{\lambda }_n{t}^{\alpha }\right)\right]}^2+2{\left[E_{\alpha ,1}\left(0\right)\right]}^2\right]}^{1/2}\hfill \\ & \le & \frac{C_{23}{∥r∥}_{C\left[0,T\right]}}{{\lambda }_n}.\hfill \end{array}$$

(59)

Writing inequalities (58)–(59) in (56) and by using (13), Lemma 2, the Cauchy–Schwarz inequality with the assumption $d\in \left\{1,2,3\right\}$, we obtain

$$\begin{array}{ccc}\hfill J_1\left(x\right)& \le & \sum _{n=1}^{\infty }{C}_{22}\left({(-L)}^{\beta }p,{\phi }_n\right)\frac{C_{23}{∥r∥}_{C\left[0,T\right]}}{{\lambda }_n^{1+d/2}}\hfill \\ & \le & \frac{C_{22}{C}_{23}}{C_{24}}{∥r∥}_{C\left[0,T\right]}{\left[\sum _{n=1}^{\infty }{\left|\left({(-L)}^{\beta }p,{\phi }_n\right)\right|}^2\right]}^{1/2}{\left[\sum _{n=1}^{\infty }\frac{1}{{\left(n^{\left(2/d\right)+1}\right)}^2}\right]}^{1/2}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill J_1\left(x\right)& \le & \frac{C_{22}{C}_{23}}{C_{24}}{∥r∥}_{C\left[0,T\right]}{∥p∥}_{{\tilde{H}}^{2\beta }\left(\mathsf{\Omega }\right)}{\left[\sum _{n=1}^{\infty }\frac{1}{n^{\left(4/d\right)+2}}\right]}^{1/2}\hfill \\ & \le & C_{25}{∥r∥}_{C\left[0,T\right]}.\hfill \end{array}$$

A similar investigation was performed for the term $J_2$. Using the inequalities of Cauchy–Schwarz and (48), we have

$$\begin{array}{ccc}\hfill J_2\left(x\right)& \le & \underset{0\le t\le T}{max}\sum _{n=1}^{\infty }\left({\lambda }_n{\phi }_n\left(x\right)\right)\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right){∥g\left(u\left(·,\tau \right)\right)∥}_{L^2\left(\mathsf{\Omega }\right)}d\tau \hfill \\ & \le & \underset{0\le t\le T}{max}\sum _{n=1}^{\infty }\left\{\frac{C_0{C}_{16}}{\alpha }{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}{∥r∥}_{C\left[0,T\right]}\left({\lambda }_n{\phi }_n\left(x\right)\right)\right.\hfill \\ & & \left.\times \left[\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)\left({\tau }^{\alpha }+\frac{{\tau }^{2\alpha }}{2}\right)d\tau \right]\right\}\hfill \\ & \le & \frac{C_0{C}_{16}{∥r∥}_{C\left[0,T\right]}}{\alpha }\underset{0\le t\le T}{max}\left(t^{\alpha }+\frac{t^{2\alpha }}{2}\right)\sum _{n=1}^{\infty }\left\{J_5·J_6\left(t\right)\right\}\hfill \end{array}$$

(60)

for

$$J_5:=\left|{\lambda }_n{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}{\phi }_n\left(x\right)\right|$$

and

$$J_6\left(t\right):=\left|\underset{0}{\overset{t}{\int }}{\tau }^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\tau }^{\alpha }\right)d\tau \right|.$$

Using (14) and the hypothesis $p\in {\tilde{H}}^{2\beta }\left(\mathsf{\Omega }\right)$, we write $p\in L^2\left(\mathsf{\Omega }\right)$. Thus, we obtain

$$\begin{array}{ccc}\hfill J_5& \le & C_{26}{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}{\lambda }_n{∥{(-L)}^{\beta -1-d/2}{\phi }_n∥}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ & =& C_{26}{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}{∥{\lambda }_n^{\beta -d/2}{\phi }_n∥}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ & =& C_{26}{\lambda }_n^{-d/2}{∥{\left[\sum _{n=1}^{\infty }{\lambda }_n^{2\beta }{\left|\left(p,{\phi }_n\right)\right|}^2\right]}^{1/2}{\phi }_n∥}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ & =& C_{26}{\lambda }_n^{-d/2}{∥p∥}_{{\tilde{H}}^{2\beta }\left(\mathsf{\Omega }\right)},\hfill \end{array}$$

(61)

by (10), (13) and (57). Considering (59), it can be easily seen that

$$J_6\left(t\right)\le \frac{C_{27}}{{\lambda }_n}.$$

(62)

Writing (61), (62) in (60) and taking into account Lemma 2, we obtain the inequality

$$\begin{array}{ccc}\hfill J_2\left(x\right)& \le & C_0{C}_{28}{∥p∥}_{{\tilde{H}}^{2\beta }\left(\mathsf{\Omega }\right)}{∥r∥}_{C\left[0,T\right]}\underset{0\le t\le T}{max}\sum _{n=1}^{\infty }\frac{1}{{\lambda }_n^{(2/d)+1}}\hfill \\ & \le & C_0\frac{C_{28}}{C_{29}}{∥p∥}_{{\tilde{H}}^{2\beta }\left(\mathsf{\Omega }\right)}{∥r∥}_{C\left[0,T\right]}\underset{0\le t\le T}{max}\sum _{n=1}^{\infty }\frac{1}{n^{(4/d)+2}}\hfill \\ & \le & C_0{C}_{30}{∥r∥}_{C\left[0,T\right]}.\hfill \end{array}$$

Therefore, in equality (54), the series are convergent in $C\left(\overline{\mathsf{\Omega }}\times \left[0,T\right]\right)$ and we can write

$$\begin{array}{ccc}\hfill {\mathsf{\partial }}_t^{\alpha }u\left(x_0,t\right)& =& \sum _{n=1}^{\infty }-{\lambda }_n\left(\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)r\left(\tau \right)\left(p,{\phi }_n\right)d\tau \right){\phi }_n\left(x_0\right)\hfill \\ & & +\sum _{n=1}^{\infty }-{\lambda }_n\left(\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)\left(g\left(u\left(·,\tau \right)\right),{\phi }_n\right)d\tau \right){\phi }_n\left(x_0\right)\hfill \\ & & +p\left(x_0\right)r\left(t\right)+g\left(u\left(x_0,t\right)\right)\hfill \end{array}$$

(63)

for every t in the domain. In order to estimate (63), we set

$$\begin{array}{ccc}\hfill J_7(x,\tau )& :=& \sum _{n=1}^{\infty }-{\lambda }_n{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\tau }^{\alpha }\right)\left(p,{\phi }_n\right){\phi }_n\left(x\right),\hfill \\ \hfill J_8(x;t,\tau )& :=& \sum _{n=1}^{\infty }-{\lambda }_n{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)\left(g\left(u\left(·,\tau \right)\right),{\phi }_n\right){\phi }_n\left(x\right),\hfill \\ \hfill J_9(x,\tau )& :=& \sum _{n=1}^{\infty }\left|{\lambda }_n\left(g\left(u\left(·,\tau \right)\right),{\phi }_n\right){\phi }_n\left(x\right)\right|,\hfill \\ \hfill J_{10}\left(\tau \right)& :=& \left|\underset{0}{\overset{\tau }{\int }}\left[{\left(\tau -\theta \right)}^{\alpha -1}+{\left(\tau -\theta \right)}^{2\alpha -1}\right]r\left(\theta \right)d\theta \right|.\hfill \end{array}$$

By equalities (10), (12), (13) and the inequalities of Cauchy–Schwarz, (48), (58), (61) with Lemmata 1, 2, 3, we obtain

$$\begin{array}{ccc}\hfill {∥J_7\left(x,·\right)∥}_{C\left[0,T\right]}& \le & \underset{0\le \tau \le T}{max}\left|\sum _{n=1}^{\infty }-{\lambda }_n\left(p,{\phi }_n\right)E_{\alpha ,\alpha }\left(-{\lambda }_n{\tau }^{\alpha }\right){\phi }_n\left(x\right)\right|\hfill \\ & \le & \underset{0\le \tau \le T}{max}\left|\sum _{n=1}^{\infty }{∥{\lambda }_n\left(p,{\phi }_n\right){\phi }_n∥}_{L^{\infty }\left(\mathsf{\Omega }\right)}\left|E_{\alpha ,\alpha }\left(-{\lambda }_n{\tau }^{\alpha }\right)\right|\right|\hfill \\ & \le & \left|C_{31}\sum _{n=1}^{\infty }\frac{1}{{\lambda }_n^{d/2}}{\left({(-L)}^{\beta }p,{\phi }_n\right)}_{L^2\left(\mathsf{\Omega }\right)}\right|\hfill \\ & \le & \left|\frac{C_{31}}{C_{32}}{\left[\sum _{n=1}^{\infty }\frac{1}{n^2}\right]}^{1/2}{\left[\sum _{n=1}^{\infty }{\left({(-L)}^{\beta }p,{\phi }_n\right)}^2\right]}^{1/2}\right|\hfill \\ & \le & C_{33}{∥p∥}_{{\tilde{H}}^{2\beta }\left(\mathsf{\Omega }\right)}\hfill \\ & \le & C_{34},\hfill \end{array}$$

(64)

$$\begin{array}{ccc}\hfill \left|J_8(x;t,\tau )\right|& \le & \sum _{n=1}^{\infty }\left|-{\lambda }_n{E}_{\alpha ,\alpha }\left(-{\lambda }_n{\left(t-\tau \right)}^{\alpha }\right)\left(g\left(u\left(·,\tau \right)\right),{\phi }_n\right){\phi }_n\left(x\right)\right|\hfill \\ & \le & C_{35}{J}_9(x,\tau )\hfill \end{array}$$

(65)

and

$$\begin{array}{ccc}\hfill J_9(x,\tau )& \le & \left|{\left[\sum _{n=1}^{\infty }{\left[{\lambda }_n{\phi }_n\left(x\right)\right]}^2\right]}^{1/2}{\left[\sum _{n=1}^{\infty }{\left|\left(g\left(u\left(·,\tau \right)\right),{\phi }_n\right)\right|}^2\right]}^{1/2}\right|\hfill \\ & =& \left|{\left[\sum _{n=1}^{\infty }{\left[{\lambda }_n{\phi }_n\left(x\right)\right]}^2\right]}^{1/2}{∥g\left(u\left(·,\tau \right)\right)∥}_{L^2\left(\mathsf{\Omega }\right)}\right|\hfill \\ & \le & \left|C_0{C}_{12}{\left[\sum _{n=1}^{\infty }{∥{\lambda }_n{∥p∥}_{L^2\left(\mathsf{\Omega }\right)}{\phi }_n\left(x\right)∥}_{L^{\infty }\left(\mathsf{\Omega }\right)}^2\right]}^{1/2}\right.\hfill \\ & & \times \left.\left[\underset{0}{\overset{\tau }{\int }}{\left(\tau -\theta \right)}^{\alpha -1}r\left(\theta \right)d\theta +C_{14}{e}^{C_{15}\tau }\frac{{\left(\Gamma \left(\alpha \right)\right)}^2}{\Gamma \left(2\alpha \right)}\underset{0}{\overset{\tau }{\int }}{\left(\tau -\theta \right)}^{2\alpha -1}r\left(\theta \right)d\theta \right]\right|\hfill \\ & \le & C_0{C}_{36}{∥p∥}_{{\tilde{H}}^{2\beta }\left(\mathsf{\Omega }\right)}{\left[\sum _{n=1}^{\infty }{\left(\frac{1}{{\lambda }_n^{d/2}}\right)}^2\right]}^{1/2}\hfill \\ & & \times \left|\underset{0}{\overset{\tau }{\int }}\left[{\left(\tau -\theta \right)}^{\alpha -1}+{\left(\tau -\theta \right)}^{2\alpha -1}\right]r\left(\theta \right)d\theta \right|\hfill \\ & \le & C_{37}{J}_{10}\left(\tau \right)\hfill \end{array}$$

(66)

for any $0<\tau <T$. Using the notation

$$J_{11}\left(\tau \right):={\tau }^{\alpha -1}+{\tau }^{2\alpha -1},$$

and Young’s convolution inequality, we also obtain

$$\begin{array}{ccc}\hfill {∥J_{10}∥}_{L^1\left(0,t\right)}& =& {∥J_{11}\ast r∥}_{L^1\left(0,t\right)}\hfill \\ & \le & {∥J_{11}∥}_{L^1\left(0,t\right)}{∥r∥}_{L^1\left(0,t\right)}\hfill \\ & \le & \left(\frac{T^{\alpha }}{\alpha }+\frac{T^{2\alpha }}{2\alpha }\right)\underset{0}{\overset{T}{\int }}\left|r\left(\tau \right)\right|d\tau \hfill \\ & \le & \left(\frac{T^{\alpha +1}}{\alpha }+\frac{T^{2\alpha +1}}{2\alpha }\right){∥r∥}_{C\left[0,T\right]}.\hfill \end{array}$$

(67)

By (64)–(67), we have $J_7,J_9\in C\left[0,T\right]$ for any $x\in \mathsf{\Omega }$ and in particular $J_7\left(x_0,t\right),J_9(x_0,t)\in C\left[0,T\right]$. We can use Lebesgue’s theorem and the order of integrations and summations in (63) can be changed. Now, by (54) and the additional data $h\left(t\right)=u\left(x_0,t\right)$, we can write

$$\begin{array}{ccc}\hfill {\mathsf{\partial }}_t^{\alpha }h\left(t\right)& =& \underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{J}_7\left(x_0,t-\tau \right)r\left(\tau \right)d\tau +\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{J}_8\left(x_0;t,\tau \right)d\tau \hfill \\ & & +p\left(x_0\right)r\left(t\right)+g\left(h\left(t\right)\right)\hfill \end{array}$$

(68)

for $0<t<T$. Using the hypothesis $p\left(x_0\right)\ne 0$ in (68), we have

$$\begin{array}{ccc}\hfill r\left(t\right)& =& \frac{1}{p\left(x_0\right)}\left[{\mathsf{\partial }}_t^{\alpha }h\left(t\right)-\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{J}_7\left(x_0,t-\tau \right)r\left(\tau \right)d\tau \right.\hfill \\ & & \left.-\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{J}_8(x_0;t,\tau )d\tau -g\left(h\left(t\right)\right)\right]\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\left|r\left(t\right)\right|}^2& \le & \frac{2}{{\left[p\left(x_0\right)\right]}^2}\left[2{\left|{\mathsf{\partial }}_t^{\alpha }h\left(t\right)\right|}^2+2{\left|g\left(h\left(t\right)\right)\right|}^2\right.\hfill \\ & & \left.+{\left|\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{J}_7(x_0,t-\tau )r\left(\tau \right)d\tau +\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{J}_8\left(x_0;t,\tau \right)d\tau \right|}^2\right].\hfill \end{array}$$

By the change of variable $\epsilon =t-\tau$, we obtain

$$\begin{array}{ccc}\hfill {\left|r\left(t\right)\right|}^2& \le & \frac{2}{{\left[p\left(x_0\right)\right]}^2}\left[2\underset{0\le t\le T}{max}{\left|{\mathsf{\partial }}_t^{\alpha }h\left(t\right)\right|}^2+2\underset{0\le t\le T}{max}{\left|g\left(h\left(t\right)\right)\right|}^2\right.\hfill \\ & & \left.+{\left|\underset{0}{\overset{t}{\int }}{\epsilon }^{\alpha -1}\left[\underset{0\le \epsilon \le T}{max}{J}_7(x_0,\epsilon )\right]r(t-\epsilon )d\epsilon +C_{35}\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}{J}_9(x_0,\tau )d\tau \right|}^2\right]\hfill \\ & \le & \frac{2}{{\left[p\left(x_0\right)\right]}^2}\left[2\underset{0\le t\le T}{max}{\left|{\mathsf{\partial }}_t^{\alpha }h\left(t\right)\right|}^2+2\underset{0\le t\le T}{max}{\left|g\left(h\left(t\right)\right)\right|}^2\right.\hfill \\ & & +\left|{∥J_7\left(x_0,·\right)∥}_{C\left[0,T\right]}\underset{0}{\overset{t}{\int }}{(t-\tau )}^{\alpha -1}\left|r\left(\tau \right)\right|d\tau \right.\hfill \\ & & \left.{\left.+C_{35}{C}_{36}\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}\underset{0}{\overset{\tau }{\int }}\left[{\left(\tau -\theta \right)}^{\alpha -1}+{\left(\tau -\theta \right)}^{2\alpha -1}\right]\left|r\left(\theta \right)\right|d\theta d\tau \right|}^2\right].\hfill \end{array}$$

(69)

Using the notation

$$\begin{array}{c}\hfill J_{12}\left(t\right):=\underset{0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}\underset{0}{\overset{\tau }{\int }}\left[{\left(\tau -\theta \right)}^{\alpha -1}+{\left(\tau -\theta \right)}^{2\alpha -1}\right]r\left(\theta \right)d\theta d\tau ,\end{array}$$

the last term on the right-hand side of (69) can be written as

$$\begin{array}{ccc}\hfill J_{12}\left(t\right)& =& \underset{0}{\overset{t}{\int }}r\left(\theta \right)\underset{\theta }{\overset{t}{\int }}{\left(\tau -\theta \right)}^{\alpha -1}{\left(t-\tau \right)}^{\alpha -1}d\tau d\theta \hfill \\ & & +\underset{0}{\overset{t}{\int }}r\left(\theta \right)\underset{\theta }{\overset{t}{\int }}{\left(\tau -\theta \right)}^{\alpha -1}{\left(t-\tau \right)}^{2\alpha -1}d\tau d\theta ,\hfill \end{array}$$

(70)

and the terms on the right-hand side of equality (70) yield two special cases of formula (17). By writing $q=z=\alpha$ and $q=\alpha ,$$z=2\alpha$ in (17), equality (70) becomes

$$\begin{array}{ccc}\hfill J_{12}\left(t\right)& =& \frac{{\left(\Gamma \left(\alpha \right)\right)}^2}{\Gamma \left(2\alpha \right)}\underset{0}{\overset{t}{\int }}{\left(t-\theta \right)}^{2\alpha -1}r\left(\theta \right)d\theta +\frac{\Gamma \left(\alpha \right)\Gamma \left(2\alpha \right)}{\Gamma \left(3\alpha \right)}\underset{0}{\overset{t}{\int }}{\left(t-\theta \right)}^{3\alpha -1}r\left(\theta \right)d\theta \hfill \\ & \le & \frac{{\left(\Gamma \left(\alpha \right)\right)}^2}{\Gamma \left(2\alpha \right)}\underset{0}{\overset{t}{\int }}\left[\underset{0\le \theta \le t}{max}{\left(t-\theta \right)}^{\alpha }\right]{\left(t-\theta \right)}^{\alpha -1}r\left(\theta \right)d\theta \hfill \\ & & +\frac{\Gamma \left(\alpha \right)\Gamma \left(2\alpha \right)}{\Gamma \left(3\alpha \right)}\underset{0}{\overset{t}{\int }}\left[\underset{0\le \theta \le t}{max}{\left(t-\theta \right)}^{2\alpha }\right]{\left(t-\theta \right)}^{\alpha -1}r\left(\theta \right)d\theta \hfill \end{array}$$

and

$$J_{12}\left(t\right)\le \left[\frac{{\left(\Gamma \left(\alpha \right)\right)}^2{t}^{\alpha }}{\Gamma \left(2\alpha \right)}+\frac{\Gamma \left(\alpha \right)\Gamma \left(2\alpha \right)t^{2\alpha }}{\Gamma \left(3\alpha \right)}\right]\underset{0}{\overset{t}{\int }}{\left(t-\theta \right)}^{\alpha -1}r\left(\theta \right)d\theta .$$

(71)

Considering (69) with (71), we obtain

$$\begin{array}{ccc}\hfill \left|r\left(t\right)\right|& \le & C_{38}\left({∥{\mathsf{\partial }}_t^{\alpha }h∥}_{C\left[0,T\right]}+{∥g\left(h\right)∥}_{C\left[0,T\right]}\right)\hfill \\ & & +C_{38}\left({∥J_7\left(x_0,·\right)∥}_{C\left[0,T\right]}+\frac{{\left(\Gamma \left(\alpha \right)\right)}^2{t}^{\alpha }}{\Gamma \left(2\alpha \right)}+\frac{\Gamma \left(\alpha \right)\Gamma \left(2\alpha \right)t^{2\alpha }}{\Gamma \left(3\alpha \right)}\right)J_{13}\left(t\right)\hfill \end{array}$$

(72)

for $t\in (0,T)$, where

$$J_{13}\left(t\right):=\underset{0}{\overset{t}{\int }}{\left(t-\theta \right)}^{\alpha -1}\left|r\left(\theta \right)\right|d\theta .$$

Since, we have

$$C_{38}\left({∥{\mathsf{\partial }}_t^{\alpha }h∥}_{C\left[0,T\right]}+{∥g\left(h\right)∥}_{C\left[0,T\right]}\right)\in L^1\left(0,T\right),$$

the generalized Grönwall inequality can be used for (72). Therefore, we write

$$\begin{array}{ccc}\hfill \left|r\left(t\right)\right|& \le & C_{38}\left({∥{\mathsf{\partial }}_t^{\alpha }h∥}_{C\left[0,T\right]}+{∥g\left(h\right)∥}_{C\left[0,T\right]}\right)\hfill \\ & & +C_{38}{C}_{39}{e}^{C_{40}t}\underset{0}{\overset{t}{\int }}{(t-\tau )}^{\alpha -1}\left({∥{\mathsf{\partial }}_t^{\alpha }h∥}_{C\left[0,T\right]}+{∥g\left(h\right)∥}_{C\left[0,T\right]}\right)d\tau \hfill \\ & =& C_{38}\left({∥{\mathsf{\partial }}_t^{\alpha }h∥}_{C\left[0,T\right]}+{∥g\left(h\right)∥}_{C\left[0,T\right]}\right)\left(1+C_{39}{e}^{C_{40}t}\underset{0}{\overset{t}{\int }}{\left(t-\theta \right)}^{\alpha -1}d\theta \right)\hfill \end{array}$$

(73)

and

$$\underset{0}{\overset{t}{\int }}{\left(t-\theta \right)}^{\alpha -1}d\theta =\frac{t^{\alpha }}{\alpha }\le \frac{T^{\alpha }}{\alpha }$$

(74)

for any t in the domain. With (74), inequality (73) becomes

$$\left|r\left(t\right)\right|\le C_{38}\left(1+C_{39}{e}^{C_{40}T}\frac{T^{\alpha }}{\alpha }\right)\left({∥{\mathsf{\partial }}_t^{\alpha }h∥}_{C\left[0,T\right]}+{∥g\left(h\right)∥}_{C\left[0,T\right]}\right).$$

(75)

Finally, taking the maximum with respect to t on both sides of (75), we obtain the stability inequality (53) for the solution $(u,r)$ of Problem 2.

This completes the proof. □

5. Concluding Remarks

In this study, we first consider a direct problem for a nonlinear time-fractional partial differential equation with initial and boundary conditions. We study the well-posedness of the problem by the methodology of [15]. Then, we apply the results to an inverse problem with additional data and we obtain the stability of the solution $(u,r)$. These two problems are generalizations of the problems that were previously discussed by [6].

In this paper, we consider the case of $p\left(x_0\right)\ne 0$. On the other hand, as far as we know, the case of $p\left(x_0\right)=0$ is still an open problem, which is important in applications. In the linear case, it was studied in [32]. For further research, this paper can be used to explore other direct and inverse problems for nonlinear fractional partial differential equations.