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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 321903, 11 pages
http://dx.doi.org/10.1155/2011/321903
Research Article

Existence and Uniqueness of the Solution for a Time-Fractional Diffusion Equation with Robin Boundary Condition

Mathematics Division, Department of Electrical and Information Engineering, Faculty of Technology, University of Oulu, PL 4500, 90014 Oulu, Finland

Received 10 January 2011; Revised 1 March 2011; Accepted 8 March 2011

Academic Editor: W. A. Kirk

Copyright © 2011 Jukka Kemppainen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Existence and uniqueness of the solution for a time-fractional diffusion equation with Robin boundary condition on a bounded domain with Lyapunov boundary is proved in the space of continuous functions up to boundary. Since a Green matrix of the problem is known, we may seek the solution as the linear combination of the single-layer potential, the volume potential, and the Poisson integral. Then the original problem may be reduced to a Volterra integral equation of the second kind associated with a compact operator. Classical analysis may be employed to show that the corresponding integral equation has a unique solution if the boundary data is continuous, the initial data is continuously differentiable, and the source term is Hölder continuous in the spatial variable. This in turn proves that the original problem has a unique solution.

1. Introduction

In this paper, we study solvability of the time-fractional diffusion equation (TFDE)𝜕𝛼𝑡Φ(𝑥,𝑡)Δ𝑥Φ(𝑥,𝑡)=𝑓(𝑥,𝑡),in𝑄𝑇],=Ω×(0,𝑇𝜕Φ(𝑥,𝑡)𝜕𝑛(𝑥)+𝛽(𝑥,𝑡)Φ(𝑥,𝑡)=𝑔(𝑥,𝑡),onΣ𝑇],=Γ×(0,𝑇Φ(𝑥,0)=𝜓(𝑥),𝑥Ω,(1.1) where 𝑓,𝑔,𝜓 are any given functions, Ω𝑛, 𝑛2, is a bounded domain with Lyapunov boundary Γ𝒞1+𝜆,  0<𝜆<1, and𝜕𝛼𝑡1𝑢(𝑡)=dΓ(1𝛼)d𝑡𝑡0(𝑡𝜏)𝛼𝑢(𝜏)d𝜏𝑡𝛼𝑢(0)(1.2) is the fractional Caputo time derivative of order 0<𝛼<1. Physically fractional diffusion equations describe anomalous diffusion on complex systems like some amorphous semiconductors or strongly porous materials (see [1] and references therein).

As to the mathematical theory of fractional diffusion equations, only the first steps have been taken. In the literature, mainly the Cauchy problems for these equations have been considered until now (see [25] and references therein). Existence and uniqueness of a generalized solution for an initial-boundary-value problem for the generalized time-fractional diffusion equation is proved in [6]. However, uniqueness and existence of the classical solution is given only in a special 1-dimensional case.

Our model problem is much simpler than those treated for example, in [3, 5]. However, the boundary integral approach used in this paper can be used in more general situations as well. We decided to concentrate on a simple model instead of the more general ones to clarify the basic idea. Boundary integral approach also allows us to study (TFDE) or its generalizations in weaker spaces such as 𝐿𝑝-spaces or in the scale of anisotropic Sobolev spaces.

The paper is organized as follows. In Preliminaries, we recall the definitions of the potentials and the Poisson integral. We introduce their well-known properties from theory of PDEs of parabolic type, which are needed for proving the existence and uniqueness of the solution. That is, we recall the boundary behavior of the single-layer potential. We show that the volume potential solves the nonhomogeneus TFDE with the zero initial condition. Moreover, we prove that the Poisson integral solves the homogeneous TFDE with a given initial datum. The final section is dedicated to the proof of existence and uniqueness of the solution.

2. Preliminaries

Here we recall the potentials and the Poisson integral and their basic properties. In the sequel we shall assume that the functions appearing in the definitions are smooth enough such that the corresponding integrals exist.

The single-layer potential can be defined as(𝑆𝜑)(𝑥,𝑡)=𝑡0Γ𝜕𝑛(𝑦)𝐺(𝑥𝑦,𝑡𝜏)𝜑(𝑦,𝜏)d𝜎(𝑦)d𝜏,𝑥Ω,(2.1) where 𝑛(𝑦) denotes the outward unit normal at 𝑦Γ and𝜋𝐺(𝑥,𝑡)=𝑛/2𝑡𝛼1|𝑥|𝑛𝐻201214|𝑥|2𝑡𝛼(𝛼,𝛼)(𝑛/2,1),(1,1),𝑥𝑛,𝑡>0,0,𝑥𝑛,𝑡<0,(2.2) is the fundamental solution of (TFDE) [3, 79]. Here 𝐻2012 is the Fox 𝐻-function, which is defined via Mellin-Barnes integral representation𝐻2012(𝑧)=𝐻2012=1𝑧(𝛼,𝛼)(𝑛/2,1),(1,1)2𝜋i𝒞Γ(𝑛/2+𝑠)Γ(1+𝑠)𝑧Γ(𝛼+𝛼𝑠)𝑠d𝑠,(2.3) where 𝒞 is an infinite contour on the complex plane circulating the negative real axis counterclockwise.

The volume potential is defined by(𝑉𝜑)(𝑥,𝑡)=𝑡0Ω𝐺(𝑥𝑦,𝑡𝜏)𝜑(𝑦,𝜏)d𝑦d𝜏,𝑥Ω(2.4) for 𝜑 such that supp𝜑(,𝑡)Ω for any 𝑡(0,𝑇].

The Poisson integral is defined as(𝑃𝜑)(𝑥,𝑡)=𝑡0Ω0𝐸(𝑥𝑦,𝑡)𝜑(𝑦)d𝑦,𝑥Ω,(2.5) where Ω0 is some neighborhood of Ω and𝐸(𝑥,𝑡)=𝜋𝑛/2|𝑥|𝑛𝐻201214|𝑥|2𝑡𝛼(2.6) with𝐻2012(𝑧)=𝐻2012=1𝑧(1,𝛼)(𝑛/2,1),(1,1)2𝜋i𝒞Γ(𝑛/2+𝑠)Γ(1+𝑠)𝑧Γ(1+𝛼𝑠)𝑠d𝑠(2.7) and 𝒞 as in the definition of 𝐻2012.

Note that in contrast to classical parabolic partial differential equations, we have a Green matrix {𝐸(𝑥,𝑡),𝐺(𝑥,𝑡)} instead of one fundamental solution. We also emphasize that the Green's functions have singularities both in time and spatial variable unlike in the case of classical parabolic PDEs, where singularity occurs only in time.

Let us now state the basic properties of the aforementioned quantities. Since the proofs are strongly based on the detailed analysis of the Fox 𝐻-functions, we shall recall their basic properties. For further details of these functions, we refer to [3, 10, 11].

In order to simplify the notations, we introduce the following function defined for 𝑧>0:𝐻(𝑝)(1𝑧)=2𝜋i𝒞Γ(𝑛/2+𝑠)Γ(1+𝑠)𝑝Γ(𝛼+𝛼𝑠)Γ(𝑠)𝑝1𝑧𝑠d𝑠,𝑝=1,2.(2.8)

Note that, in particular, 𝐻(1)(𝑧)=𝐻2012(𝑧). The following properties of 𝐻(𝑝) are needed.

Lemma 2.1. For the functions 𝐻(𝑝), there holds (i)differentiation formula (d/d𝑧)𝐻(1)(𝑧)=𝑧1𝐻(2)(z), (ii)the asymptotic behaviour at infinity, ||𝐻(𝑝)||(𝑧)𝐶𝑧𝑛/2exp𝜎𝑧1/(2𝛼),𝜎=𝛼𝛼/2𝛼(2𝛼),(2.9) for 𝑝=1,2 and 𝑧1,(iii)the asymptotic behaviour near zero ||𝐻(𝑝)||𝑧(𝑧)𝐶𝑛/2𝑧if𝑛=2or𝑛=3,2||||𝑧logzif𝑛=4,2if𝑛>4,(2.10) for 𝑝=1,2 and 𝑧1. The constants in (𝑖𝑖) and (𝑖𝑖𝑖) can depend on 𝑛, 𝑝, and 𝛼.

Proof. The proofs follow from the Mellin-Barnes integral representation and the analyticity of the functions 𝐻(𝑝) [3, 10, 12].

Remark 2.2. Above and in the sequel, 𝐶 denotes a generic constant, which may depend on various quantities. The only thing that matters is that in our calculations 𝐶 will be independent of 𝑥 and 𝑡.

Let us now concentrate on the properties of the potentials. We start with the single-layer potential 𝑆𝜑. First of all, standard calculations show that 𝑆𝜑 solves the equation (𝜕𝛼𝑡Δ𝑥)𝑢=0. Moreover, we need to know the boundary behavior of the single-layer potential, which is given in the following result.

Theorem 2.3. Let 𝜑𝒞(Σ𝑇). The single-layer potential defined by (2.1) is continuous in Ω𝑇 with the zero initial value. Moreover, for 𝑥Ω and 𝑥0Γ, 𝑥(𝑆𝜑)(𝑥,𝑡)𝑛(𝑥0) has the following limiting value: 𝜕𝑛(𝑥0)𝑥𝑆𝜑0,𝑡=lim𝑥𝑥0𝑥𝑥(𝑆𝜑)(𝑥,𝑡)𝑛0=12𝜑𝑥0+,𝑡𝑡0Γ𝜕𝑛(𝑥0)𝐺𝑥0𝜑𝑦,𝑡𝜏(𝑦,𝜏)d𝜎(𝑦)d𝜏(2.11) as 𝑥 tends to 𝑥0 nontangentially.

Proof. The proof follows the same lines as in the case of the single-layer potential for the heat equation [13, Chapter 5.2] and is based on a detailed analysis of the kernel 𝐺. Since the proof is rather lengthy, we give only the reference [12, Theorems 1 and 2].

For the volume potential, we have the following result.

Theorem 2.4. Let 𝑓𝒞(Σ𝑇) such that 𝑓(,𝑡) is Hölder continuous uniformly in 𝑡[0,𝑇] and supp𝑓(,𝑡)Ω, 𝑡[0,𝑇]. Then the volume potential 𝑉𝑓 with 𝑉 defined by (2.4) solves 𝜕𝛼𝑡𝑢Δ𝑥𝑢=𝑓 with the zero initial condition.

Proof. The zero initial condition follows since 𝐺 is locally integrable. Indeed, we split the integral 𝑉𝑓 into two parts 𝐼1+𝐼2 depending on whether 𝑧=(1/4)(𝑡𝜏)𝛼|𝑥𝑦|21 or 𝑧1.
If 𝑧1, we use the fact that 𝑧𝛾exp(𝜎𝑧1/(2𝛼)) is uniformly bounded for any 𝛾,𝜎>0. Then Lemma 2.1 together with the definition of 𝐺 yields ||||𝐺(𝑥𝑦,𝑡𝜏)𝐶(𝑡𝜏)𝛼+𝛼𝛾1𝛼𝑛/2||||𝑥𝑦2𝛾.(2.12) If we choose 𝑛/21<𝛾<𝑛/2, we see that lim𝑡0+𝐼1(𝑥,𝑡)=0.
On the other hand, if 𝑧1, then Lemma 2.1 yields ||||𝐺(𝑥𝑦,𝑡𝜏)𝐶(𝑡𝜏)(2𝑛)𝛼/2,𝑛=2,3,(𝑡𝜏)𝛼1|||||||log𝑥𝑦2(𝑡𝜏)𝛼|||+1,𝑛=4,(𝑡𝜏)𝛼1||||𝑥𝑦4𝑛,𝑛>4.(2.13)
Then lim𝑡0+𝐼2(𝑥,𝑡)=0 follows immediately for 𝑛=2,3. If 𝑛=4, we may use the fact that 𝑧𝛾|log𝑧| is bounded in (0,1] for any 𝛾>0. If 𝑛>4, we use 𝑧𝛾1 for any 𝛾>0. In the preceeding two cases, we obtain ||||𝐺(𝑥𝑦,𝑡𝜏)𝐶(𝑡𝜏)𝛼𝛾𝛼1||||𝑥𝑦4𝑛2𝛾.(2.14) If we choose 1<𝛾<2, we see that lim𝑡0+𝐼2(𝑥,𝑡)=0.
For the proof of the first claim, we refer to [3, Sections 5.2 and 5.3], where the proof is given in a much more general case of a time-fractional diffusion equation.

Finally, for the Poisson integral there holds the following theorem.

Theorem 2.5. Let 𝜓 be a continuous function in Ω0. Then the Poisson integral 𝑃𝜓 with 𝑃 defined by (2.5) solves 𝜕𝛼𝑡𝑢Δ𝑥𝑢=0 with 𝑢(𝑥,0)=𝜓(𝑥), 𝑥Ω.

Proof. The fact that 𝑃𝜓 solves 𝜕𝛼𝑡𝑢Δ𝑥𝑢=0 follows from the calculations given in [8]. Note that differentiation inside the integral is allowed because there is no singularity in 𝑡. Therefore, it remains to prove the initial condition.
We proceed as in [13, Proof of Theorem 1.2.1] and consider first the case of constant 𝜓. The integral is divided into two parts (𝑃𝜓)(𝑥,𝑡)=𝐼𝑅+𝐼𝑐𝑅, where 𝐼𝑅 is the integral over the ball 𝐵(𝑥,𝑅) with 𝑅 being so small that 𝐵(𝑥,𝑅) is contained in Ω0 and 𝐼𝑐𝑅 denotes its complementary part. Since 𝑅 is fixed and there is no singularity in the spatial variable, the asymptotic behavior of 𝐺 shows that lim𝑡0+𝐼𝑐𝑅=0. We need to prove that lim𝑡0+𝐼𝑅=𝜓.
Introducing spherical coordinates, we get 𝐼𝑅=𝜋𝑛/2𝜔𝑛𝜓𝑅0𝑟1𝐻201214𝑡𝛼𝑟2d𝑟=𝜋𝑛/2𝜔𝑛𝜓𝑅/2𝑡𝛼/20𝑟1𝐻2012𝑟2d𝑟𝜋𝑛/2𝜔𝑛𝜓0𝑟1𝐻2012𝑟2d𝑟,𝑡0+,(2.15) where 𝜔𝑛 denotes the surface area of the unit sphere in 𝑛.
To evaluate the last integral denoted by 𝐼, we note that the asymptotic behavior of the integrand guarantees the absolute integrability. Through the change of variables 𝑟2=𝑡, we see that the integral is nothing but half of the Mellin transform of 𝐻2012, 1𝐼=2𝐻2012(𝑠)=Γ(𝑛/2+𝑠)Γ(1+𝑠),2Γ(1+𝛼𝑠)(2.16) evaluated at the point 𝑠=0.
Therefore, we may conclude that the claim in the case 𝜓 is constant. In the case of general 𝜓 we may proceed as in [13, Proof of Theorem 1.2.1.]

3. Existence and Uniqueness of the Solution

As it was mentioned in Introduction, we seek the solution in a form of𝑢(𝑥,𝑡)=(𝑆𝜑)(𝑥,𝑡)+(𝑃𝜓)(𝑥,𝑡)+(𝑉𝑓)(𝑥,𝑡),(3.1) where 𝜑 is to be determined. The density 𝜑 is determined by reducing the original problem to a corresponding integral equation.

We assume that 𝛽 is a continuous function on Σ𝑇. We need to calculate the normal derivative of 𝑆𝜑, 𝑉𝑓, and 𝑃𝜓. For 𝑃𝜓, we observe that differentiation inside the integral defining 𝑃 is allowed since there is no singularity in 𝑡. For 𝑉𝑓, the differentiation inside the integral is justified by the calculations given in [8]. Finally, Theorem 2.3 gives the boundary value for the normal derivative of 𝑆𝜑. Then the Robin boundary condition is equivalent with12𝐼+𝑊+𝛽𝑆𝜑=𝐹,(3.2) where(𝑊𝜑)(𝑥,𝑡)=𝑡0Γ𝜕𝑛(𝑥)𝐺(𝑥𝑦,𝑡𝜏)𝜑(𝑦,𝜏)d𝜎(𝑦)d𝜏(3.3) is the integral in Theorem 2.3 and𝐹(𝑥,𝑡)=𝑔(𝑥,𝑡)Ω0𝜕𝑛(𝑥)𝐸(𝑥𝑦,𝑡)𝜓(𝑦)d𝑦𝑡0Ω𝜕𝑛(𝑥)𝐺(𝑥𝑦,𝑡𝜏)𝑓(𝑦,𝜏)d𝑦d𝜏𝛽(𝑥,𝑡)(𝑃𝜓)(𝑥,𝑡)𝛽(𝑥,𝑡)(𝑉𝑓)(𝑥,𝑡).(3.4)

We will prove that (3.2) admits a unique solution for any bounded function 𝐹. Therefore, it is needed to determine the conditions, which guarantee boundedness.

For the second integral on the right-hand side of (3.4), we use the following result.

Lemma 3.1. Let 𝑧=(1/4)|𝑥𝑦|2𝑡𝛼 with 𝑥Γ and 𝑦Ω. The following estimates for the normal derivative of 𝐺 hold: (1)if 𝑧1, then ||𝜕𝑛(𝑥)||𝐺(𝑥𝑦,𝑡)𝐶𝑡𝛼𝑛/21||||𝑥𝑦exp𝜎𝑡𝛼/(2𝛼)||||𝑥𝑦2/(2𝛼);(3.5)(2)if 𝑧1, then ||𝜕𝑛(𝑥)||𝑡𝐺(𝑥𝑦,𝑡)𝐶𝛼1|||||||||||𝑥𝑦log𝑥𝑦2𝑡𝛼|||𝑡ifn=2,𝛼1||||𝑥𝑦3𝑛ifn3.(3.6)

Proof. Applying the differentiation formula in Lemma 2.1, we get 𝜕𝑛(𝑥)𝐺(𝑥𝑦,𝑡)=𝜋𝑛/2𝑥𝑦,𝑛(𝑥)||||𝑥𝑦𝑛+2𝑡𝛼1𝑛𝐻(1)(𝑧)+2𝐻(2),(𝑧)(3.7) where 𝑧=(1/4)|𝑥𝑦|2𝑡𝛼 and , denotes the inner product in 𝑛.
Using the definition of 𝐻(𝑝) and the property Γ(𝑧+1)=𝑧Γ(𝑧) of the Gamma function, it follows that the Mellin transform of 𝑛𝐻(1)+2𝐻(2) is 𝑛𝐻(1)+2𝐻(2)1(𝑠)=2Γ((𝑛+2)/2+𝑠)Γ(1+𝑠),Γ(𝛼+𝛼𝑠)(3.8) which is nothing but half of the Mellin transform of 𝐻(1) with 𝑛 replaced by 𝑛+2.
Using the estimate (2.9) of Lemma 2.1 with 𝑛+2 instead of 𝑛, we obtain the first estimate for 𝑧1.
If 𝑧1, we use the estimate (2.10) of Lemma 2.1 with 𝑛+2=4 and 𝑛+2>4 to obtain the second estimate.

Let us return to the estimation of the second integral on r.h.s. of (3.4). Once again we split the integral into two parts 𝐼1 and 𝐼2 depending on whether 𝑧1 or 𝑧1 with 𝑧=(1/4)|𝑥𝑦|2(𝑡𝜏)𝛼. If 𝑓 is a bounded function and 𝑧1, there holds||𝐼1||𝐶𝑓𝐿(Ω𝑇)𝑡0(𝑡𝜏)𝛼/21d𝜏4𝑟𝑛exp𝜎𝑟2/(2𝛼)d𝑟𝐶𝑓𝐿(Ω𝑇),(3.9) where we have used the spherical coordinates with 𝑟=(𝑡𝜏)𝛼/2|𝑥𝑦|.

If 𝑧1, we have to consider different cases of 𝑛's separately. As an example, let us consider case 𝑛=3. We have||𝐼2||𝐶𝑓𝐿(Ω𝑇)Ω×(0,𝑡){z1}(𝑡𝜏)𝛼𝛾𝛼1||||𝑥𝑦2𝛾d𝑦d𝜏,(3.10) where the fact 𝑧𝛾1 for any 𝛾>0 is used. Choosing 1<𝛾<3/2, we see that 𝐼2 is bounded.

Using the estimates (2.12) and (2.13) in the proof of Theorem 2.4, we see that 𝑉𝑓 is bounded as well.

For the first integral on the right-hand side, we use the following result [3, Proposition 1].

Lemma 3.2. Let 𝑧=(1/4)|𝑥|2𝑡𝛼. For 𝐸 there holds the following: (1)if 𝑧1, then ||𝑥||𝐸(𝑥,𝑡)𝐶𝑡𝛼(𝑛+1)/2exp𝜎𝑡𝛼/(2𝛼)|𝑥|2/(2𝛼);(3.11)(2)if 𝑧1, then ||𝑥||𝐸(𝑥,𝑡)𝐶𝑡𝛼|𝑥|𝑛+1.(3.12)

We split the first integral on right-hand side of (3.4) into two parts 𝐼1 and 𝐼2 depending whether 𝑧=(1/4)|𝑥𝑦|2𝑡𝛼1 or 𝑧1. If 𝜓 is a bounded function, then using Lemma 3.2 we have||𝐼1||𝐶𝑡𝛼+𝛼𝛾𝜓𝐿(Ω0)(3.13) for any 𝛾>0, since 𝑧𝑧𝛾exp(𝜎𝑧𝛽) is uniformly bounded on [1,) for any 𝛽,𝛾,𝜎>0. Similarly, for 𝐼2 there holds||𝐼2||𝐶𝜓𝐿(Ω0)Ω0{𝑧1}𝑡𝛼+𝛼𝛾||||𝑥𝑦𝑛+12𝛾d𝑦(3.14) for any 𝛾>0, since 𝑧𝛾1 for any 𝛾>0. Choosing 𝛾<1/2, we have||𝐼2||𝐶𝑡𝛼+𝛼𝛾𝜓𝐿(Ω0).(3.15)

We see that 𝐼2 blows up as 𝑡0+. Therefore, we have to assume more smoothness on 𝜓 to guarantee boundedness. Assume that 𝜓 is a continuously differentiable function in Ω0. Then integration by parts yields a better kernel 𝐸. Asymptotic behavior of 𝐸 guarantees that the resulting integral is uniformly bounded on 𝑥,𝑡 (see [3, Proposition 1]). We have||||Ω0𝜕𝑛(𝑥)||||𝐸(𝑥𝑦,𝑡)𝜓(𝑦)d𝑦𝐶𝜓𝐿(Ω0)+𝜓𝐿(Ω0).(3.16)

The same reason as above implies that 𝑃𝜓 is bounded.

Now we are ready to prove that (3.2) has a unique solution.

Theorem 3.3. Let 𝑓𝐿(Ω𝑇), 𝑔𝒞(Σ𝑇), and 𝜓𝒞1(Ω0). Then the boundary integral equation (3.2) admits a unique bounded, continuous solution 𝜑.

Proof. Using similar estimates as in Lemma 3.1 and in the proof of Theorem 2.4, we see that 𝑊+𝛽𝑆 is an integral operator with a weakly singular kernel. Note that the estimates for the normal derivative given in Lemma 3.1 can be multiplied by |𝑥𝑦|𝜆 in the estimates for 𝑊 due to the Lyapunov smoothness of the boundary Γ. For details we refer to [12].
We conclude that 𝑊+𝛽𝑆 is a compact operator in 𝒞(Σ𝑇) [14, Theorem 2.22]. Moreover, similarly as in [15] we can prove that there exists an integer 𝑘0 such that (2𝑊+2𝛽𝑆)𝑘0𝑙𝜑𝐿(Σ𝑇)(𝑀𝑇)𝑙𝑙!𝜑𝐿(Σ𝑇)(3.17) for some constant 𝑀 and for all 𝑙. This, in particular, implies that the homogeneous equation ((1/2)𝐼+𝑊+𝛽𝑆)𝜑=0 has a unique solution. Moreover, (2𝑊+2𝛽𝑆)𝑘0𝑙 is a contraction for some 𝑙. Therefore, (1/2)𝐼+𝑊+𝛽𝑆 is invertible and the inverse is given by the Neumann series ((1/2)𝐼+𝑊+𝛽𝑆)1=2𝑘=0(1)𝑘(2𝑊+2𝛽𝑆)𝑘.(3.18)
Since the series is uniformly convergent, we have 𝜑𝐿(Σ𝑇)𝐶𝐹𝐿(Σ𝑇),(3.19) and continuity of 𝜑 follows from that of 𝐹.

In conclusion, 𝑢 defined by (3.1) solves (TFDE) provided 𝜑 solves (3.2). Combining our results with the results in [3, 8], we have proved our main result, which is stated as follows.

Theorem 3.4. Let 𝑔𝒞(Σ𝑇), 𝜓𝒞1(Ω0), and 𝑓𝒞(Σ𝑇) such that 𝑓(,𝑡) is Hölder continuous uniformly in 𝑡[0,𝑇] and supp𝑓(,𝑡)Ω, 𝑡[0,𝑇]. Then (TFDE) admits a unique classical solution and the solution depends continuously on the data in the following sense: 𝑢(𝑥,𝑡)𝒞(Ω𝑇)𝐶𝑓𝒞(Ω𝑇)+𝑔𝒞(Σ𝑇)+𝜓𝒞1(Ω0).(3.20)

If 𝜓 has compact support in Ω, we may relax the smoothness assumption on 𝜓 and proof of Theorem 3.4 implies the following.

Corollary 3.5. Let 𝑔 and 𝑓 satisfy the assumptions in Theorem 3.4, and let 𝜓𝒞(Ω) with compact support. Then (TFDE) admits a unique classical solution and the solution depends continuously on the data in the following sense: 𝑢(𝑥,𝑡)𝒞(Ω𝑇)𝐶𝑓𝒞(Ω𝑇)+𝑔𝒞(Σ𝑇)+𝜓𝒞(Ω).(3.21)

Proof. All the arguments are the same as in Theorem 3.4 except now we can choose 𝛾=1 in the estimates for 𝐼1 and 𝐼2 of Ω𝜕𝑛(𝑥)𝐸(𝑥𝑦,𝑡)𝜓(𝑦)d𝑦=𝐼1+𝐼2(3.22) in Theorem 3.4. Therefore, we obtain ||||Ω𝜕𝑛(𝑥)||||𝐸(𝑥𝑦,𝑡)𝜓(𝑦)d𝑦𝐶1+dist(supp𝜓,Γ)𝑛1𝜓𝐿(Ω)(3.23) and the claim follows.

Remark 3.6. The estimates given for 𝐹 reveal that if 𝜓 is merely continuous, we have |𝐹(𝑥,𝑡)|𝐶𝑡𝛽 for some 1<𝛽<𝛼/2. Then the same technique as in [13, Section 5.3] may be employed to prove that (TFDE) with the initial condition replaced by 𝑢(𝑥,0)=𝜓(𝑥), 𝑥Ω, has a unique solution, which may not be even continuous and can be unbounded near Γ×{0} and therefore is not a classical solution.

Remark 3.7. The same technique as above may be used for more general time-fractional diffusion equations, where Δ is replaced by a uniformly elliptic second-order differential operator in nondivergence form with bounded continuous real-valued coefficients depending on 𝑥.

Remark 3.8. In [15], we have proved existence and uniqueness of the solution of TFDE with the zero initial condition and the zero source term with Dirichlet boundary condition. Using the same technique as above, we may also consider the case of nonzero initial condition and nontrivial source term. Indeed, use of the double-layer ansatz leads to a Volterra integral equation of the second kind as in this paper. Then, using the same arguments as above, we can prove uniqueness and existence of a classical solution without any restrictions on 𝑛 or on boundary conditions such as in [6].

Acknowledgment

The author would like to thank the referee's suggestions for the improvement of this paper.

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