On the Solvability of a Singular Time Fractional Parabolic Equation with Non Classical Boundary Conditions

Abstract

This paper deals with a singular two dimensional initial boundary value problem for a Caputo time fractional parabolic equation supplemented by Neumann and non-local boundary conditions. The well posedness of the posed problem is demonstrated in a fractional weighted Sobolev space. The used method based on some functional analysis tools has been successfully showed its efficiency in proving the existence, uniqueness and continuous dependence of the solution upon the given data of the considered problem. More precisely, for proving the uniqueness of the solution of the posed problem, we established an energy inequality for the solution from which we deduce the uniqueness. For the existence, we proved that the range of the operator generated by the considered problem is dense.

1. Introduction

In recent decades, the study of IBVPs characterized by non-local conditions within the realm of both integer and fractional order evolution differential equations has garnered substantial academic interest. Notably, problems featuring boundary conditions of integral type, commonly referred to as energy specifications, have been crucial due to their practical relevance in simulating and analyzing a myriad of physical phenomena. These phenomena span across various domains, including but not limited to chemical engineering, thermoelasticity, aerodynamics, heat diffusion, plasma physics, subterranean water flow, and fluid mechanics, along with numerous other physical and biological systems (refer to sources [1,2,3,4,5,6,7,8,9]. It is important to recognize that while integer order differential equations are adept at modeling local phenomena, fractional order differential equations are more suited for capturing non-local phenomena. This distinction underscores the limitations of classical mathematics in fully representing real-world phenomena, which often rely on both current and historical states. References [10,11,12,13,14,15,16,17] provide insights into integer order cases, whereas [18,19,20,21,22,23,24] delve into fractional order scenarios. Despite this, the exploration and in-depth study of IBVPs in nonlinear singular fractional differential equations remains an area ripe for further research.

Recent findings in the field of two-dimensional singular time fractional partial differential equations, particularly those associated with initial and non-local conditions, have shown significant advancements. These equations, characterized by their fractional derivatives in time, are critical for modeling various phenomena in science and engineering. For IBVPs for singular fractional parabolic equations with nonlocal conditions have been explored in various research studies. We cite some notable references in this direction: The paper [25] constructs a theory of the well posedness for parabolic problems with singular data, also considering initial data and a forcing term. A study in [26] examines the global existence and blow-up of solutions for an IBVP for a nonlinear parabolic equation with non-local boundary conditions. The results obtained in this paper are dependent on the behavior of variable coefficients over time. Another paper [27] focuses on a nonlocal 1-D IBVP for a singular fractional equation in the Caputo sense, subject to Neumann and weighted integral conditions. The study in this article demonstrates the application of the energy inequality methods for obtaining a priori bounds for the solution of fractional boundary problems. In the paper [27], the authors studied a parabolic problem associated with nonlocal conditions, they proved the local and global solution by using the Banach’s fixed point theorem. Last but not least, the paper [28] studies similarity solutions of fractional parabolic problems with uncertainty, deriving existence results using Schauder’s fixed point theorem. Finally, for the farctinoal case were $\sigma \in (1,2]$ in (1), the reader can refer to [29].

These references offer a diverse range of approaches and findings in the field of IBVPs for singular fractional parabolic equations with non-local conditions. For the one dimensional case, the reader could refer to [30].

For advanced Numerical Methods, there has been progress in developing more efficient and accurate numerical methods for solving these equations. Techniques like spectral methods [31,32], finite difference methods [33,34,35], and mesh-free methods [36] have been refined to handle the singular behavior and non-local properties of these partial differential equations. These methods are designed to cope with the challenges posed by the fractional time derivative and the non-local boundary conditions, ensuring better stability and convergence. On the analytical side of solutions, researchers have made strides in finding analytical solutions to certain classes of two-dimensional singular time fractional partial differential equations. These solutions are crucial for validating numerical methods and understanding the behavior of solutions under various initial and boundary conditions. Methods like the separation of variables, integral transform techniques, and Green’s function methods have been employed to derive these solutions. Researchers used regularization techniques, to deal with the singular nature of these equations, regularization techniques have been developed. These techniques involve transforming the singular PDE into a regularized version, which is easier to handle analytically and numerically. This approach has enabled researchers to better understand the impact of singularities on the solution’s behavior.

The methodology employed in this paper for demonstrating the existence and uniqueness of solutions for the problems addressed relies on the energy inequality method. This approach is fundamentally based on a priori estimates and the range density of the operators formulated by the problem. The literature features a limited number of studies that utilize energy inequalities to establish the existence and uniqueness of solutions in fractional initial-boundary value problems (see references [37,38,39,40,41,42]). The novelty of this paper, resides in developing the used method for a singular fractional order partial differential equation with a non local condition of integral type. In our model, the presence of singularities ($x=0$, $y=0$) in the coefficients necessitates the development of specialized analytical methods such as the used energy inequality method. The novelty arises from the creation of techniques that accurately deal with these singularities such as adapted weighted fractional Sobolev spaces. Also, in our model, the boundary integral condition represents a significant generalization compared to the classical boundary conditions. This is particularly relevant in material science and biological systems, where the state at a point can be influenced by non local interactions.

The outline of this article is set in the following way. In section two, we pose the problem and we introduce needed functional spaces. In section three, we give a preliminary lemmas. Section four concerns the proof of uniqueness of the solution of (1)–(4), that is we establish an a priori bound for the solution from which the uniqueness and continuous dependence of the solution upon the given data follow. In section five, we provide the main result for the existence of the solution of the given problem (1)–(4).

2. Problem Position and Needed Function Spaces

Let $T>0$, and $\mathrm{\Lambda }=\mathrm{\Omega }\times [0,T],$ where $\mathrm{\Omega }=(0,a)\times (0,b)\subset {\mathbb{R}}^2$. We search for a function $\mathcal{M}:\mathrm{\Lambda }\to \mathbb{R}$ satisfying the two dimensional singular Caputo fractional PDE with Bessel operator [43]

$$\mathcal{L}\left(\mathcal{M}\right)={\partial }_t^{\sigma }\mathcal{M}-\frac{1}{xy}div(xy\nabla \mathcal{M})+\mathcal{M}=H_1(x,y,t),$$

(1)

where $\sigma \in (0,1],$ supplemented by the initial condition

$$\mathrm{\Gamma }\mathcal{M}=\mathcal{M}(x,y,0)=f(x,y),$$

(2)

the Neumann boundary conditions

$${\mathcal{M}}_x(a,y,t)=0.{\mathcal{M}}_y(x,b,t)=0,$$

(3)

and the non local weighted boundary integral conditions

$$\underset{0}{\overset{a}{\int }}x\mathcal{M}(x,y,t)dx=0.\underset{0}{\overset{b}{\int }}y\mathcal{M}(x,y,t)dy=0,$$

(4)

with the compatibility conditions

$$f_x(a,y,t)=0.f_y(x,b,t)=0,\underset{0}{\overset{a}{\int }}xfdx=0,\underset{0}{\overset{b}{\int }}yfdx=0.$$

(5)

Here $H_1\in L^2(0,T,L_{\rho }^2\left(\mathrm{\Omega }\right)),$ $f\in W_{2,\rho }^1\left(\mathrm{\Omega }\right)$. The symbol ${\partial }_t^{\sigma }$ denotes the time fractional derivative operator in the Caputo sense of order $0<\sigma \le 1$. It is defined by

$${\partial }_t^{\sigma }V(x,t)=\frac{1}{\mathrm{\Gamma }(1-\sigma )}\underset{0}{\overset{t}{\int }}\frac{V_{\tau }(x,\tau )}{{(t-\tau )}^{\sigma }}d\tau ,$$

(6)

for a certain function V [20], where $\mathrm{\Gamma }(.)$ is the Gamma function.

We first introduce the functional frame, where we seek the solution of problem (1)–(4). Let $L_{\rho }^2\left(\mathrm{\Omega }\right)$ to be the well known Hilbert space with weight $\rho =xy$ and with scalar product, and associated norm respectively

$${(\vartheta ,V)}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}=\underset{\mathrm{\Omega }}{\int }xy\vartheta Vdxdy,{\parallel \vartheta \parallel }_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2=\underset{\mathrm{\Omega }}{\int }xy{\vartheta }^{2}dxdy.$$

(7)

Let $L^2(0,T;Y)$ be the set of all functions $\mathcal{M}:(0,T)\to Y$, where Y is the Banach space with finite norm

$${\parallel \vartheta \parallel }_{L^2(0,T;Y)}^2=\underset{0}{\overset{T}{\int }}{\parallel \vartheta (.,.,t)\parallel }_Y^{2}dt<\infty .$$

(8)

The function space $L^2(0,T;Y)$ is a Hilbert space, since Y is a Hilbert space. We design by $C(0,T;Y)$ the continuous functions $\vartheta :[0,T]\to Y$ such that

$${\parallel \vartheta \parallel }_{C(0,T;Y)}=\underset{t\in [0,T]}{max}{\parallel \vartheta (.,.,t)\parallel }_Y,$$

(9)

is finite. We denote by $W_{2,\rho }^1\left(\mathrm{\Omega }\right)$ the weighted Sobolev space having the norm

$${\parallel \vartheta \parallel }_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}^2={\parallel \vartheta \parallel }_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{\parallel {\vartheta }_x\parallel }_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{\parallel {\vartheta }_y\parallel }_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2.$$

(10)

We introduce the Hilbert space $L^2(0,T;W_{2,\rho }^{1,\gamma }\left(\mathrm{\Omega }\right))$ of abstract strongly measurable functions $\vartheta$ on the interval $[0,T]$ into $W_{2,\rho }^{1,\gamma }\left(\mathrm{\Omega }\right)$ such that

$$\begin{array}{ccc}\hfill {\parallel \vartheta \parallel }_{L^2(0,T;W_{2,\rho }^{1,\gamma }\left(\mathrm{\Omega }\right))}^2& =& \underset{0}{\overset{T}{\int }}{\parallel \vartheta (.,.,t)\parallel }_{W_{2,\rho }^{1,\gamma }\left(\mathrm{\Omega }\right)}^{2}dt\hfill \\ & =& \underset{0}{\overset{T}{\int }}\left({\parallel \vartheta \parallel }_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}^2+{\parallel {\partial }_t^{\sigma }\vartheta \parallel }_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2\right)dt<\infty .\hfill \end{array}$$

(11)

Let $W_{2,\rho }^{1,\gamma }\left(\mathrm{\Omega }\right)$ be the weighted Hilbert space having the norm

$${\parallel \vartheta \parallel }_{W_{2,\rho }^{1,\gamma }\left(\mathrm{\Omega }\right)}^2={\parallel \vartheta \parallel }_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}^2+{\parallel {\partial }_t^{\sigma }\vartheta \parallel }_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2.$$

(12)

We can view the problem (1)–(4) as solving the equation $\mathcal{PM}=(H_1,f),$ where $\mathcal{PM}=W=(\mathcal{LM},\mathrm{\Gamma }\mathcal{M})$ for all $\mathcal{M}\in D\left(\mathcal{P}\right)$ where $\mathcal{P}$ is the operator given by $\mathcal{P}=(\mathcal{L},\mathrm{\Gamma })$ and $D\left(\mathcal{P}\right)$ is constituting of functions $\mathcal{M}\in L_{\rho }^2\left(\mathrm{\Lambda }\right):{\partial }_t^{\sigma }\mathcal{M}$, ${\mathcal{M}}_x$, ${\mathcal{M}}_{xx}$, ${\mathcal{M}}_y$, ${\mathcal{M}}_{yy}$, ${\mathcal{M}}_t\in L_{\rho }^2\left(\mathrm{\Lambda }\right)$ and such that $\mathcal{M}$ verifies (2)–(4). The operator $\mathcal{P}:$ B $\to Y$, where B is the Banach space having the finite norm

$${\parallel \mathcal{M}\parallel }_B^2={\parallel \mathcal{M}(.,.,\tau )\parallel }_{L^2(0,T;W_{2,\rho }^{1,\gamma }\left(\mathrm{\Omega }\right))}^2.$$

(13)

Observe that the functions $\mathcal{M}$ $\in B$ are $C[0,T]$ with values in $W_{2,\rho }^{1,\gamma }\left(\mathrm{\Omega }\right),$ consequently the applications

$$\mathrm{\Gamma }:B\ni \mathcal{M}\to \mathrm{\Gamma }\mathcal{M}=\mathcal{M}(x,y,0)\in W_{2,\rho }^1\left(\mathrm{\Omega }\right)$$

are continuous on the space B. And Y is the Hilbert space $L^2(0,T,L_{\rho }^2\left(\mathrm{\Omega }\right))\times W_{2,\rho }^1\left(\mathrm{\Omega }\right)$ of functions $W=(H_1,f)$ having the finite norm

$${\parallel W\parallel }_Y^2={\parallel H_1\parallel }_{L^2(0,T;L_{\rho }^2\left(\mathrm{\Omega }\right))}^2+{\parallel f\parallel }_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}^2.$$

In the following section, we shall state some lemmas dealing with fractional and integer differential inequalities, that are crucial in proving different results concerning the posed problem.

3. Preliminary Lemmas

Lemma 1 

([44]). Let $Z\left(s\right)$ be nonnegative and absolutely continuous on $[0,T]$, and for all $s\in [0,T]$ verifies

$$\frac{dZ}{ds}\le A_1\left(s\right)Z\left(t\right)+B_1\left(s\right),$$

(14)

where the functions $A_1\left(s\right)$ and $B_1\left(s\right)$ are nonnegative on $[0,T].$ Then

$$Z\left(s\right)\le exp\left(\underset{0}{\overset{s}{\int }}{A}_1\left(t\right)dt\right)\left(Z\left(0\right)+\underset{0}{\overset{s}{\int }}{B}_1\left(t\right)dt\right).$$

(15)

Lemma 2 

([39]). Let a nonnegative absolutely continuous function $S\left(t\right)$ satisfy the inequality

$${\partial }_t^{\sigma }S\left(t\right)\le b_{1}S\left(t\right)+b_2\left(t\right),0<\sigma <1,$$

(16)

for almost all $t\in [0,T],$ where $b_1$ is a positive constant and $b_2\left(t\right)$ is an integrable nonnegative function on $[0,T].$ Then

$$S\left(t\right)\le S\left(0\right)E_{\sigma }\left(b_1{t}^{\sigma }\right)+\mathrm{\Gamma }\left(\sigma \right)E_{\sigma ,\sigma }\left(b_1{t}^{\sigma }\right)D_t^{-\sigma }{b}_2\left(t\right),$$

(17)

where

$$E_{\sigma }\left(x\right)=\sum _{n=0}^{\infty }\frac{x^n}{\mathrm{\Gamma }(\sigma n+1)}and{E}_{\sigma ,\mu }\left(x\right)=\sum _{n=0}^{\infty }\frac{x^n}{\mathrm{\Gamma }(\sigma n+\mu )},$$

are the Mittag-Leffler functions.

Lemma 3 

([39]). For any absolutely continuous function $J\left(t\right)$ on the interval $[0,T],$ the following inequality holds

$$J\left(t\right){\partial }_t^{\sigma }J\left(t\right)\ge \frac{1}{2}{\partial }_t^{\sigma }{J}^2\left(t\right),0<\sigma <1$$

(18)

We are now in a position to prove the first main result concerning the well posedness of our posed problem.

4. Main Result of Uniqueness

Theorem 1. 

For any function $\mathcal{M}\in D\left(\mathcal{P}\right)$ we have

$${∥\mathcal{M}∥}_{L^2(0,T;W_{2,\rho }^{1,\gamma }\left(\mathrm{\Omega }\right))}^2\le d_2(C+1)\left({\parallel H_1\parallel }_{L^2(0,T;L_{\rho }^2\left(\mathrm{\Omega }\right))}^2+{∥f∥}_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}^2\right),$$

(19)

where $d_2$, and C are positive constants independent of the function $\mathcal{M}$ given respectively by (34) and (39).

Proof. 

We first observe that

$$\begin{array}{ccc}& & {\partial }_t^{\sigma }\mathcal{M}-\frac{1}{xy}div(xy\nabla \mathcal{M})+\mathcal{M}\hfill \\ & =& {\partial }_t^{\sigma }\mathcal{M}-\frac{1}{x}{\left(x{\mathcal{M}}_x\right)}_x-\frac{1}{y}{\left(y{\mathcal{M}}_y\right)}_y+\mathcal{M}=H_1(x,y,t).\hfill \end{array}$$

(20)

Consider the inner product in $L_{\rho }^2\left(\mathrm{\Omega }\right)$ of the PDE (20) and the differential operator $\mathcal{NM}={\partial }_t^{\sigma }\mathcal{M}+\mathcal{M}+{\Im }_{xy}^2\left(\xi \eta \mathcal{M}\right),$ where

$${\Im }_{xy}^2\left(\xi \eta \mathcal{M}\right)=\underset{0}{\overset{x}{\int }}\underset{0}{\overset{y}{\int }}\underset{0}{\overset{\xi }{\int }}\underset{0}{\overset{\eta }{\int }}{\xi }_1{\eta }_1\mathcal{M}({\xi }_1,{\eta }_1)d{\eta }_{1}d{\xi }_{1}d\eta d\xi ,$$

then we have

$$\begin{array}{ccc}& & {({\partial }_t^{\sigma }\mathcal{M},\mathcal{NM})}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}-{(\frac{1}{x}{\left(x{\mathcal{M}}_x\right)}_x,\mathcal{NM})}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}-{(\frac{1}{y}{\left(y{\mathcal{M}}_y\right)}_y,\mathcal{NM})}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}\hfill \\ \hfill & +& {(\mathcal{M},\mathcal{NM})}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}\hfill \\ \hfill & =& {(H_1,\mathcal{NM})}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}.\hfill \end{array}$$

(21)

Computation of each term in (21) yields

$$\begin{array}{ccc}\hfill {({\partial }_t^{\sigma }\mathcal{M},\mathcal{NM})}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}& =& {∥{\partial }_t^{\sigma }\mathcal{M}∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{({\partial }_t^{\sigma }\mathcal{M},\mathcal{M})}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}\hfill \\ & & +({\partial }_t^{\sigma }{({\Im }_{xy}\left(\xi \eta \mathcal{M}\right),{\Im }_{xy}\left(\xi \eta \mathcal{M}\right))}_{L^2\left(\mathrm{\Omega }\right)},\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill -{(\frac{1}{x}{\left(x{\mathcal{M}}_x\right)}_x,\mathcal{NM})}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}& =& -{({\left(x{\mathcal{M}}_x\right)}_x,{\partial }_t^{\sigma }\mathcal{M})}_{L^2\left(\mathrm{\Omega }\right)}-{({\left(x{\mathcal{M}}_x\right)}_x,\mathcal{M})}_{L^2\left(\mathrm{\Omega }\right)}\hfill \\ & & -{({\left(x{\mathcal{M}}_x\right)}_x,{\Im }_{xy}^2\left(\xi \eta \mathcal{M}\right))}_{L^2\left(\mathrm{\Omega }\right)}\hfill \\ & =& {({\partial }_t^{\sigma }{\mathcal{M}}_x,{\mathcal{M}}_x)}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}+{∥{\mathcal{M}}_x∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & & +{({\mathcal{M}}_x,{\Im }_{xyy}\left(\xi \eta \mathcal{M}\right))}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}.\hfill \end{array}$$

(23)

By symmetry, we have

$$\begin{array}{ccc}\hfill {-(\frac{1}{y}{\left(y{\mathcal{M}}_y\right)}_y,\mathcal{NM})}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}& =& -{({\left(y{\mathcal{M}}_y\right)}_y,{\partial }_t^{\sigma }\mathcal{M})}_{L^2\left(\mathrm{\Omega }\right)}-{({\left(y{\mathcal{M}}_y\right)}_y,\mathcal{M})}_{L^2\left(\mathrm{\Omega }\right)}\hfill \\ & & -{({\left(y{\mathcal{M}}_y\right)}_y,{\Im }_{xy}^2\left(\xi \eta \mathcal{M}\right))}_{L^2\left(\mathrm{\Omega }\right)}\hfill \\ & =& {({\partial }_t^{\sigma }{\mathcal{M}}_y,{\mathcal{M}}_y)}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}+{∥{\mathcal{M}}_y∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & & +{({\mathcal{M}}_y,{\Im }_{xxy}\left(\xi \eta \mathcal{M}\right))}_{L_{\rho }^2\left(\mathrm{\Omega }\right)},\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill {(\mathcal{M},\mathcal{NM})}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}& =& {∥\mathcal{M}∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{({\partial }_t^{\sigma }\mathcal{M},\mathcal{M})}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}\hfill \\ & & +{∥{\Im }_{xy}(\xi \eta \mathcal{M}∥}_{L^2\left(\mathrm{\Omega }\right)}^2.\hfill \end{array}$$

(25)

If we combine Equations (21)–(25), we obtain

$$\begin{array}{ccc}& & {∥{\partial }_t^{\sigma }\mathcal{M}∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+2{({\partial }_t^{\sigma }\mathcal{M},\mathcal{M})}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}+{({\partial }_t^{\sigma }{\mathcal{M}}_x,{\mathcal{M}}_x)}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}+{∥{\mathcal{M}}_x∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & & +{({\partial }_t^{\sigma }{\mathcal{M}}_y,{\mathcal{M}}_y)}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}+{∥{\mathcal{M}}_y∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{∥\mathcal{M}∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & & +({\partial }_t^{\sigma }{({\Im }_{xy}\left(\xi \eta \mathcal{M}\right),{\Im }_{xy}\left(\xi \eta \mathcal{M}\right))}_{L^2\left(\mathrm{\Omega }\right)}+{∥{\Im }_{xy}(\xi \eta \mathcal{M}∥}_{L^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & =& -{({\mathcal{M}}_x,{\Im }_{xyy}\left(\xi \eta \mathcal{M}\right))}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}-{({\mathcal{M}}_y,{\Im }_{xxy}\left(\xi \eta \mathcal{M}\right))}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}\hfill \\ & & {(H_1,{\partial }_t^{\sigma }\mathcal{M})}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}+{(H_1,\mathcal{M})}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}+{(H_1,{\Im }_{xy}^2\left(\xi \eta \mathcal{M}\right))}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}.\hfill \end{array}$$

(26)

In the light of Lemma 3, the Cauchy Epsilon inequality, the Poincare’ type inequalities

$$\left\{\begin{array}{c}{∥{\Im }_{xxy}(\xi \eta U∥}_{L^2\left(\mathrm{\Omega }\right)}^2\le \frac{a^2}{2}{∥{\Im }_{xy}(\xi \eta U∥}_{L^2\left(\mathrm{\Omega }\right)}^2,\\ {∥{\Im }_{xyy}(\xi \eta U∥}_{L^2\left(\mathrm{\Omega }\right)}^2\le \frac{b^2}{2}{∥{\Im }_{xy}(\xi \eta U∥}_{L^2\left(\mathrm{\Omega }\right)}^2,\end{array}\right.$$

(27)

we have

$$\begin{array}{ccc}& & {∥{\partial }_t^{\sigma }\mathcal{M}∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+\frac{1}{2}{\partial }_t^{\sigma }{∥\mathcal{M}∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+\frac{1}{2}{\partial }_t^{\sigma }{∥{\mathcal{M}}_x∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+\frac{1}{2}{\partial }_t^{\sigma }{∥{\mathcal{M}}_y∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & & +{∥\mathcal{M}∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{∥{\mathcal{M}}_x∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{∥{\mathcal{M}}_y∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & & +\frac{1}{2}{\partial }_t^{\sigma }{∥{\Im }_{xy}(\xi \eta \mathcal{M}∥}_{L^2\left(\mathrm{\Omega }\right)}^2+{∥{\Im }_{xy}(\xi \eta \mathcal{M}∥}_{L^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & \le & {\epsilon }_1{∥{\mathcal{M}}_x∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+\frac{b^2}{2{\epsilon }_1}{∥{\Im }_{xy}(\xi \eta \mathcal{M}∥}_{L^2\left(\mathrm{\Omega }\right)}^2+{\epsilon }_2{∥{\mathcal{M}}_y∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & & +\frac{a^2}{2{\epsilon }_2}{∥{\Im }_{xy}(\xi \eta \mathcal{M}∥}_{L^2\left(\mathrm{\Omega }\right)}^2+\frac{3}{2}{∥H_1∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+\frac{1}{2}{∥{\partial }_t^{\sigma }\mathcal{M}∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & & +\frac{1}{2}{∥\mathcal{M}∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+\frac{a^2{b}^2}{4}{∥{\Im }_{xy}(\xi \eta \mathcal{M}∥}_{L^2\left(\mathrm{\Omega }\right)}^2.\hfill \end{array}$$

(28)

Let ${ϵ}_1={ϵ}_2=\frac{1}{2}$ in (28), and discard the last term on the LHS of (28), then it follows that

$$\begin{array}{ccc}& & {∥\mathcal{M}∥}_{W_{2,\rho }^{1,\gamma }\left(\mathrm{\Omega }\right)}^2+{\partial }_t^{\sigma }{∥\mathcal{M}∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{\partial }_t^{\sigma }{∥{\mathcal{M}}_x∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & & +{\partial }_t^{\sigma }{∥{\mathcal{M}}_y∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{\partial }_t^{\sigma }{∥{\Im }_{xy}(\xi \eta \mathcal{M}∥}_{L^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & \le & d_1\left({∥H_1∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{∥{\Im }_{xy}(\xi \eta \mathcal{M}∥}_{L^2\left(\mathrm{\Omega }\right)}^2\right),\hfill \end{array}$$

(29)

where

$$d_1=max\left\{\frac{3}{2},\frac{4a^2+4b^2+a^2{b}^2}{4}\right\}.$$

(30)

Now replace t by $\tau$ and integrate both sides of (30) from 0 to t with respect to $\tau$, we obtain

$$\begin{array}{ccc}& & {∥\mathcal{M}∥}_{L^2(0,t;W_{2,\rho }^{1,\gamma }\left(\mathrm{\Omega }\right))}^2+D_t^{\sigma -1}{\parallel \mathcal{M}\parallel }_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}^2+D_t^{\sigma -1}\parallel {\Im }_{xy}{(\xi \eta \mathcal{M}\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & \le & d_1\left({\int }_0^t{∥H_1∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^{2}d\tau +{\int }_0^t{∥{\Im }_{xy}(\xi \eta \mathcal{M}∥}_{L^2\left(\mathrm{\Omega }\right)}^{2}d\tau \right)\hfill \\ & & +\frac{T^{1-\sigma }}{(1-\sigma )\mathrm{\Gamma }(1-\sigma )}\left({∥f∥}_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}^2+{∥{\Im }_{xy}(\xi \eta f∥}_{L^2\left(\mathrm{\Omega }\right)}^2\right).\hfill \end{array}$$

(31)

Since

$${∥{\Im }_{xy}(\xi \eta f∥}_{L^2\left(\mathrm{\Omega }\right)}^2\le \frac{a^2{b}^2}{4}{∥f∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2\le \frac{a^2{b}^2}{4}{∥f∥}_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}^2,$$

(32)

then (31) reduces to

$$\begin{array}{ccc}& & {∥\mathcal{M}∥}_{L^2(0,t;W_{2,\rho }^{1,\gamma }\left(\mathrm{\Omega }\right))}^2+D_t^{\sigma -1}{\parallel \mathcal{M}\parallel }_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}^2+D_t^{\sigma -1}\parallel {\Im }_{xy}{(\xi \eta \mathcal{M}\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2\hfill \\ \hfill & \le & d_2\left({\int }_0^t{∥{\Im }_{xy}(\xi \eta \mathcal{M}∥}_{L^2\left(\mathrm{\Omega }\right)}^{2}d\tau +{∥H_1∥}_{L^2(0,T;L_{\rho }^2\left(\mathrm{\Omega }\right))}^2+{∥f∥}_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}^2\right),\hfill \end{array}$$

(33)

where

$$d_2=max\left\{d_1,\frac{T^{1-\sigma }}{(1-\sigma )\mathrm{\Gamma }(1-\sigma )}max\left(1,1+\frac{a^3{b}^3}{16}\right)\right\}.$$

(34)

Now, if we omit the first two terms on the LHS of (33), and we apply Lemma 2 by taking

$$S\left(t\right)=\underset{0}{\overset{t}{\int }}{\parallel {\Im }_{xy}\left(\xi \eta \mathcal{M}\right)\parallel }_{L^2\left(\mathrm{\Omega }\right)}^{2}d\tau ,$$

$${\partial }_t^{\sigma }S\left(t\right)=D_t^{\sigma -1}{\parallel {\Im }_{xy}\left(\xi \eta \mathcal{M}\right)\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2,\mathrm{and}S\left(0\right)=0,$$

we obtain the inequality

$$S\left(t\right)\le d_2\mathrm{\Gamma }\left(\sigma \right)E_{\sigma ,\sigma }\left(d_2{T}^{\sigma }\right)D_t^{-\sigma }\left({∥H_1∥}_{L^2(0,T;L_{\rho }^2\left(\mathrm{\Omega }\right))}^2+{∥f∥}_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}^2\right).$$

(35)

Since

$$D_t^{-\sigma }\underset{0}{\overset{t}{\int }}{∥H_1∥}_{L^2(0,T;L_{\rho }^2\left(\mathrm{\Omega }\right))}^{2}d\tau \le \frac{T^{\sigma }}{\sigma \mathrm{\Gamma }\left(\sigma \right)}\underset{0}{\overset{t}{\int }}{\parallel H_1\parallel }_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^{2}d\tau ,$$

(36)

and

$$D_t^{-\sigma }\underset{0}{\overset{t}{\int }}{∥f∥}_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}^{2}d\tau \le \frac{T^{\sigma +1}{∥f∥}_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}^2}{\sigma \mathrm{\Gamma }\left(\sigma \right)},$$

(37)

then

$$S\left(t\right)\le C\left(\underset{0}{\overset{t}{\int }}{\parallel H_1\parallel }_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^{2}d\tau +{∥f∥}_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}^2\right),$$

(38)

where

$$C=\frac{d_2{T}^{\sigma }\mathrm{\Gamma }\left(\sigma \right)E_{\sigma ,\sigma }\left(d_2{T}^{\sigma }\right)}{\sigma }max\left\{1,T\right\}.$$

(39)

Hence (33) reduces to

$$\begin{array}{ccc}& & {∥\mathcal{M}∥}_{L^2(0,t;W_{2,\rho }^{1,\gamma }\left(\mathrm{\Omega }\right))}^2+D_t^{\sigma -1}{\parallel \mathcal{M}\parallel }_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}^2+D_t^{\sigma -1}\parallel {\Im }_{xy}{(\xi \eta \mathcal{M}\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & \le & d_2(C+1)\left(\underset{0}{\overset{t}{\int }}{\parallel H_1\parallel }_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^{2}d\tau +{∥f∥}_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}^2\right).\hfill \end{array}$$

(40)

By neglacting the last two terms on the LHS of (40) and taking $t=T$ in (40), it follows that

$${∥\mathcal{M}∥}_{L^2(0,T;W_{2,\rho }^{1,\gamma }\left(\mathrm{\Omega }\right))}^2\le d_2(C+1)\left(\parallel H_1{\parallel }_{L^2(0,T;L_{\rho }^2\left(\mathrm{\Omega }\right))}^2+{∥f∥}_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}^2\right)$$

(41)

□

The image of the operator $\mathcal{P}:$ B $\to Y$ is a subset of Y, so we can extend $\mathcal{P}$ so that the a priori bound (19) is true for the extension and $\overline{\mathcal{P}}\mathcal{M}=Y.$

Proposition 1. 

The operator $\mathcal{P}:$ B $\to Y$ is closable.

Proof. 

The proof can be done as in [14]

We then have

$${\parallel \mathcal{M}\parallel }_B\le d_2(C+1){\parallel \overline{\mathcal{P}}\mathcal{M}\parallel }_Y.\forall \mathcal{M}\in D\left(\overline{\mathcal{P}}\right),$$

(42)

where $\overline{\mathcal{P}}$ denotes the closure of $\mathcal{P}$, and $D\left(\overline{\mathcal{P}}\right)$ is its domain of definition. We conclude from the estimate (42) that the range $R\left(\overline{\mathcal{P}}\right)\subset Y$ and that $R\left(\overline{\mathcal{P}}\right)=\overline{R\left(\mathcal{P}\right)}$. □

5. Solvability of the Problem

Theorem 2. 

Problem (1)–(4) has a unique solution $\mathcal{M}={\mathcal{P}}^{-1}(H_1,f)=\overline{{\mathcal{P}}^{-1}}(H_1,f)$, that depends continuously on $H_1\in L^2(0,T;L_{\rho }^2\left(\mathrm{\Omega }\right))$, and $f\in W_{2,\rho }^1\left(\mathrm{\Omega }\right)$.

To show that (1)–(4) has a unique solution for all elements $W=(H_1,f)\in Y$, we must prove that $\overline{R\left(\mathcal{P}\right)}=Y$. We first need to prove the following result:

Theorem 3. 

If for some function $\omega \in L_{\rho }^2\left(\mathrm{\Lambda }\right)$ and for all $U(x,y,t)\in D\left(\mathcal{P}\right)$ satisfiying the homogeneous initial condition $U(x,y,0)=0$, we have

$${(\mathcal{L}U,\omega )}_{L_{\rho }^2\left(\mathrm{\Lambda }\right)}=0,$$

(43)

then $\omega =0$ a.e in Λ.

Proof. 

Equation (43) implies

$${\left({\partial }_t^{\sigma }U-\frac{1}{xy}div(xy\nabla U)+U,\omega \right)}_{L_{\rho }^2\left(\mathrm{\Lambda }\right)}=0,$$

(44)

that is

$${\left({\partial }_t^{\sigma }U-\frac{1}{x}{\left(x{U}_x\right)}_x-\frac{1}{y}{\left(y{U}_y\right)}_y+U,\omega \right)}_{L_{\rho }^2\left(\mathrm{\Lambda }\right)}=0.$$

(45)

We set

$$U(x,y,t)={\Im }_t^{2}Z=\underset{0}{\overset{t}{\int }}\underset{0}{\overset{s}{\int }}Z(x,y,\tau )d\tau ds,$$

where the function $Z(x,y,t)$ satisfies conditions (3) and (4), such that Z, $Z_x$, $Z_y$, ${\Im }_t^{2}Z$, ${\Im }_t^2{Z}_x$, ${\Im }_t^2{Z}_y$, ${\Im }_{xy}^2\left({\Im }_t^{2}Z\right)$, ${\partial }_t^{\sigma }Z$ belong to $L_{\rho }^2\left(\mathrm{\Lambda }\right)$. We now put

$$\omega (x,y,t)={\Im }_t^{2}Z+{\Im }_{xy}^2\left(\xi \eta {\Im }_t^{2}Z\right).$$

(46)

Integrating first over $\mathrm{\Omega }$, then Equation (45) takes the form

$$\begin{array}{ccc}& & {({\partial }_t^{\sigma }\left({\Im }_t^{2}Z\right),{\Im }_t^{2}Z)}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}+{({\partial }_t^{\sigma }\left({\Im }_t^{2}Z\right),{\Im }_{xy}^2\left(\xi \eta {\Im }_t^{2}Z\right))}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}\hfill \\ \hfill & -& {(\frac{1}{x}{\left(x{\Im }_t^2{Z}_x\right)}_x,{\Im }_t^{2}Z)}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}-{(\frac{1}{x}{\left(x{\Im }_t^2{Z}_x\right)}_x,{\Im }_{xy}^2\left(\xi \eta {\Im }_t^{2}Z\right))}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}\hfill \\ \hfill & & -{(\frac{1}{y}{\left(y{\Im }_t^2{Z}_y\right)}_y,{\Im }_t^{2}Z)}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}-{(\frac{1}{y}{\left(y{\Im }_t^2{Z}_y\right)}_y,{\Im }_{xy}^2\left(\xi \eta {\Im }_t^{2}Z\right))}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}\hfill \\ & & \parallel {\Im }_t^{2}Z{{\parallel }_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+\left({\Im }_t^{2}Z\right),{\Im }_{xy}^2\left(\xi \eta {\Im }_t^{2}Z\right))}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}\hfill \\ \hfill & =& 0.\hfill \end{array}$$

(47)

Using the fact that Z verifies the boundary and initial conditions (2)–(4), we evaluate each term in (47) as follows

$${({\partial }_t^{\sigma }\left({\Im }_t^{2}Z\right),{\Im }_{xy}^2\left(\xi \eta {\Im }_t^{2}Z\right))}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}={({\partial }_t^{\sigma }\left({\Im }_{xy}\left(\xi \eta {\Im }_t^{2}Z\right)\right),{\Im }_{xy}\left(\xi \eta {\Im }_t^{2}Z\right))}_{L^2\left(\mathrm{\Omega }\right)},$$

(48)

$$\begin{array}{ccc}\hfill -{(\frac{1}{x}{\left(x{\Im }_t^2{Z}_x\right)}_x,{\Im }_t^{2}Z)}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}& =& -{(y{\left(x{\Im }_t^2{Z}_x\right)}_x,{\Im }_t^{2}Z)}_{L^2\left(\mathrm{\Omega }\right)}\hfill \\ & =& {∥{\Im }_t^2{Z}_x∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2,\hfill \end{array}$$

(49)

$$\begin{array}{ccc}& & -{(\frac{1}{x}{\left(x{\Im }_t^2{Z}_x\right)}_x,{\Im }_{xy}^2\left(\xi \eta {\Im }_t^{2}Z\right))}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}\hfill \\ & =& -{(y{\left(x{\Im }_t^2{Z}_x\right)}_x,{\Im }_{xy}^2\left(\xi \eta {\Im }_t^{2}Z\right))}_{L^2\left(\mathrm{\Omega }\right)}\hfill \\ & =& {({\Im }_t^2{Z}_x,{\Im }_{xyy}\left(\xi \eta {\Im }_t^{2}Z\right))}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}.\hfill \end{array}$$

(50)

By symmetry we have

$$\begin{array}{ccc}\hfill -{(\frac{1}{y}{\left(y{\Im }_t^2{Z}_y\right)}_y,{\Im }_t^{2}Z)}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}& =& -{(x{\left(y{\Im }_t^2{Z}_y\right)}_y,{\Im }_t^{2}Z)}_{L^2\left(\mathrm{\Omega }\right)}\hfill \\ & =& {∥{\Im }_t^2{Z}_y∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2,\hfill \end{array}$$

(51)

$$\begin{array}{ccc}& & -{(\frac{1}{y}{\left(y{\Im }_t^2{Z}_y\right)}_y,{\Im }_{xy}^2\left(\xi \eta {\Im }_t^{2}Z\right))}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}\hfill \\ & =& -{(x{\left(y{\Im }_t^2{Z}_y\right)}_y,{\Im }_{xy}^2\left(\xi \eta {\Im }_t^{2}Z\right))}_{L^2\left(\mathrm{\Omega }\right)}\hfill \\ & =& {({\Im }_t^2{Z}_y,{\Im }_{xxy}\left(\xi \eta {\Im }_t^{2}Z\right))}_{L_{\rho }^2\left(\mathrm{\Omega }\right)},\hfill \end{array}$$

(52)

$$\left({\Im }_t^{2}Z\right),{\Im }_{xy}^2\left(\xi \eta {\Im }_t^{2}Z\right){)}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}={∥{\Im }_{xy}(\xi \eta {\Im }_t^{2}Z∥}_{L^2\left(\mathrm{\Omega }\right)}^2.$$

(53)

Insertion of Equations (48)–(53) into (47), yields

$$\begin{array}{ccc}& & {({\partial }_t^{\sigma }\left({\Im }_t^{2}Z\right),{\Im }_t^{2}Z)}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}+{({\partial }_t^{\sigma }\left({\Im }_{xy}\left(\xi \eta {\Im }_t^{2}Z\right)\right),{\Im }_{xy}\left(\xi \eta {\Im }_t^{2}Z\right))}_{L^2\left(\mathrm{\Omega }\right)}\hfill \\ & & +\parallel {\Im }_t^2{Z\parallel }_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{∥{\Im }_t^2{Z}_x∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{∥{\Im }_t^2{Z}_y∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & & +{∥{\Im }_{xy}(\xi \eta {\Im }_t^{2}Z∥}_{L^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & =& -{({\Im }_t^2{Z}_x,{\Im }_{xyy}\left(\xi \eta {\Im }_t^{2}Z\right))}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}-{({\Im }_t^2{Z}_y,{\Im }_{xxy}\left(\xi \eta {\Im }_t^{2}Z\right))}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}.\hfill \end{array}$$

(54)

If we discard the last term on the LHS of (54), and Apply Lemma 3, Poincare’ type inequality, and Young’s inequality, we have

$$\begin{array}{ccc}& & \frac{1}{2}{\partial }_t^{\sigma }\parallel {\Im }_t^2{Z\parallel }_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+\frac{1}{2}{\partial }_t^{\sigma }{\parallel {\Im }_{xy}\left(\xi \eta {\Im }_t^{2}Z\right)\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & & +\parallel {\Im }_t^2{Z\parallel }_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{∥{\Im }_t^2{Z}_x∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{∥{\Im }_t^2{Z}_y∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & \le & \frac{{\epsilon }_1}{2}{∥{\Im }_t^2{Z}_x∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}+\frac{{\epsilon }_2}{2}{∥{\Im }_t^2{Z}_y∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}\hfill \\ & & +\left(\frac{a{b}^3}{4{\epsilon }_1}+\frac{a^{3}b}{4{\epsilon }_2}\right){\parallel {\Im }_{xy}\left(\xi \eta {\Im }_t^{2}Z\right)\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2.\hfill \end{array}$$

(55)

Let ${\epsilon }_1={\epsilon }_2=1$ in (55) to obtain

$$\begin{array}{ccc}& & {\partial }_t^{\sigma }\parallel {\Im }_t^2{Z\parallel }_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{\partial }_t^{\sigma }{\parallel {\Im }_{xy}\left(\xi \eta {\Im }_t^{2}Z\right)\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & & +\parallel {\Im }_t^2{Z\parallel }_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{∥{\Im }_t^2{Z}_x∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{∥{\Im }_t^2{Z}_y∥}_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2\hfill \\ & \le & C_1{\parallel {\Im }_{xy}\left(\xi \eta {\Im }_t^{2}Z\right)\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2,\hfill \end{array}$$

(56)

where

$$C_1=max\left\{1,\frac{a{b}^3+a^{3}b}{4}\right\}.$$

(57)

Replace t by $\tau$ in (56) and integrate with respect to $\tau$ over ($0,t)$, we obtain

$$\begin{array}{ccc}& & D_t^{-\sigma }\parallel {\Im }_{xy}\left(\xi \eta {\Im }_t^{2}Z\right){\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2+D_t^{-\sigma }\parallel {\Im }_t^2{Z\parallel }_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{\parallel {\Im }_t^{2}Z\parallel }_{L^2(0,t.,L_{\rho }^2\left(\mathrm{\Omega }\right))}^2\hfill \\ & & +{∥{\Im }_t^2{Z}_x∥}_{L^2(0,t.,L_{\rho }^2\left(\mathrm{\Omega }\right))}^2+{∥{\Im }_t^2{Z}_y∥}_{L^2(0,t.,L_{\rho }^2\left(\mathrm{\Omega }\right))}^2\hfill \\ & \le & {\left.\frac{T^{1-\sigma }\left(\parallel {\Im }_t^2{Z\parallel }_{L_{\rho }^2\left(\mathrm{\Omega }\right)}^2+{\parallel {\Im }_{xy}\left(\xi \eta {\Im }_t^{2}Z\right)\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2\right)}{\mathrm{\Gamma }(1-\sigma )(1-\sigma )}\right|}_{ t=0}+C_1\underset{0}{\overset{t}{\int }}{\parallel {\Im }_{xy}\left(\xi \eta {\Im }_t^{2}Z\right)\parallel }_{L^2\left(\mathrm{\Omega }\right)}^{2}d\tau \hfill \\ & =& C_1\underset{0}{\overset{t}{\int }}{\parallel {\Im }_{xy}\left(\xi \eta {\Im }_t^{2}Z\right)\parallel }_{L^2\left(\mathrm{\Omega }\right)}^{2}d\tau .\hfill \end{array}$$

(58)

If we neglect the last four terms on the LHS of (58), and use Lemma 2, by taking

$$S\left(t\right)=\underset{0}{\overset{t}{\int }}\parallel {\Im }_{xy}\left(\xi \eta {\Im }_t^{2}Z\right){\parallel }_{L^2\left(\mathrm{\Omega }\right)}^{2}d\tau ,{\partial }_t^{\sigma }S\left(t\right)=D_t^{-\sigma }{\parallel {\Im }_{xy}\left(\xi \eta {\Im }_t^{2}Z\right)\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2,$$

(59)

$$\frac{dS\left(t\right)}{dt}={\parallel {\Im }_{xy}\left(\xi \eta {\Im }_t^{2}Z\right)\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2,S\left(0\right)=0,$$

(60)

then

$$\begin{array}{ccc}\hfill S\left(t\right)& =& \underset{0}{\overset{t}{\int }}{\parallel {\Im }_{xy}\left(\xi \eta {\Im }_t^{2}Z\right)\parallel }_{L^2\left(\mathrm{\Omega }\right)}^{2}d\tau \hfill \\ & \le & S\left(0\right)E_{\sigma }\left(C_1{t}^{\sigma }\right)+\mathrm{\Gamma }\left(\sigma \right)E_{\sigma ,\sigma }\left(C_1{t}^{\sigma }\right)D^{-\sigma }\left(0\right)\hfill \\ & =& 0.\hfill \end{array}$$

(61)

Inequality (61) implies that $Z=0$, and hence $\omega =0$ a.e in $\mathrm{\Lambda }.$

To finish with the proof of Theorem 2, we assume that for some $W=({\eta }_1,{\eta }_2)\in R{\left(\mathcal{P}\right)}^{\perp }$ (orthogonal complement of the range of $\mathcal{P}$), we have

$${(\mathcal{LM},{\eta }_1)}_{L^2(0,T;L_{\rho }^2\left(\mathrm{\Omega }\right))}+{(\mathrm{\Gamma }\mathcal{M},{\eta }_2)}_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}=0.$$

(62)

We should show that $W=0$. If we put $\mathcal{M}\in D\left(\mathcal{P}\right)$ satisfying the initial homogeneous condition into (62), we have

$${(\mathcal{LM},{\eta }_1)}_{L^2(0,T;L_{\rho }^2\left(\mathrm{\Omega }\right))}=0.$$

(63)

Then Theorem 3 asserts that ${\eta }_1=0$, and consequently relation (62) becomes

$${(\mathrm{\Gamma }\mathcal{M},{\eta }_2)}_{W_{2,\rho }^1\left(\mathrm{\Omega }\right)}=0.$$

(64)

Since the image of the operator $\mathrm{\Gamma }$ is dense in the Sobolev space $W_{2,\rho }^1\left(\mathrm{\Omega }\right)$, then relation (64) implies that ${\eta }_2=0$. Thus $W=0$ and Theorem 2 follows. □

Remark 1. 

The problem (1)–(4) can be solved in the case where the free term $H_1(x,y,t)$ is nonlinear, that is $H_1=H_1(x,y,t,\mathcal{M},{\mathcal{M}}_x,{\mathcal{M}}_y)$, such that $H_1$ satifies the Lipschitz condition

$$\begin{array}{ccc}& & \left|H_1(x,y,t,U_1,V_1,W_1)-H_1(x,y,t,U_2,V_2,W_2)\right|\hfill \\ & \le & C\left(\left|U_1-U_2\right|+\left|V_1-V_2\right|+\left|W_1-W_2\right|\right).\hfill \end{array}$$

where C is a positive constant.

6. Conclusions

In this study, we examined a singular two-dimensional fractional diffusion equation for an IBVP in the Caputo framework, complemented by a non-local boundary condition and a standard condition. The approach to solving this problem involves seeking solutions within a fractional weighted Sobolev space. We established a priori bounds for the solution, leading to proofs of its uniqueness and its continuous dependence on the initial data. Subsequently, we demonstrated the primary finding regarding the solution’s existence for the problem at hand. Our methodology predominantly utilizes operator theory. This research contributes to the advancement of functional analysis methods for establishing the well-defined nature of fractional order problems. The findings underscore the effectiveness of this approach in resolving the existence and uniqueness of solutions to time fractional order differential equations with non-local conditions.