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.