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.

In this paper, we study solvability of the time-fractional diffusion equation (TFDE)

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 [

Our model problem is much simpler than those treated for example, in [

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.

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

The volume potential is defined by

The Poisson integral is defined as

Note that in contrast to classical parabolic partial differential equations, we have a Green matrix

Let us now state the basic properties of the aforementioned quantities. Since the proofs are strongly based on the detailed analysis of the Fox

In order to simplify the notations, we introduce the following function defined for

Note that, in particular,

For the functions

differentiation formula

the asymptotic behaviour at infinity,

the asymptotic behaviour near zero

The proofs follow from the Mellin-Barnes integral representation and the analyticity of the functions

Above and in the sequel,

Let us now concentrate on the properties of the potentials. We start with the single-layer potential

Let

The proof follows the same lines as in the case of the single-layer potential for the heat equation [

For the volume potential, we have the following result.

Let

The zero initial condition follows since

If

On the other hand, if

Then

For the proof of the first claim, we refer to [

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

Let

The fact that

We proceed as in [

Introducing spherical coordinates, we get

To evaluate the last integral denoted by

Therefore, we may conclude that the claim in the case

As it was mentioned in Introduction, we seek the solution in a form of

We assume that

We will prove that (

For the second integral on the right-hand side of (

Let

if

if

Applying the differentiation formula in Lemma

Using the definition of

Using the estimate (

If

Let us return to the estimation of the second integral on r.h.s. of (

If

Using the estimates (

For the first integral on the right-hand side, we use the following result [

Let

if

if

We split the first integral on right-hand side of (

We see that

The same reason as above implies that

Now we are ready to prove that (

Let

Using similar estimates as in Lemma

We conclude that

Since the series is uniformly convergent, we have

In conclusion,

Let

If

Let

All the arguments are the same as in Theorem

The estimates given for

The same technique as above may be used for more general time-fractional diffusion equations, where

In [

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