General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems

Author:

Luchko Yuri1,Yamamoto Masahiro2

Affiliation:

1. Dept. of Mathematics, Physics, and Chemistry, Beuth Technical University of Applied Sciences, Luxemburger Str. 10, Berlin - 13353, GERMANY

2. Dept. of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo - 153, JAPAN

Abstract

Abstract In this paper, we deal with the initial-boundary-value problems for a general time-fractional diffusion equation which generalizes the single- and the multi-term time-fractional diffusion equations as well as the time-fractional diffusion equation of the distributed order. First, important estimates for the general time-fractional derivatives of the Riemann-Liouville and the Caputo type of a function at its maximum point are derived. These estimates are applied to prove a weak maximum principle for the general time-fractional diffusion equation. As an application of the maximum principle, the uniqueness of both the strong and the weak solutions to the initial-boundary-value problem for this equation with the Dirichlet boundary conditions is established. Finally, the existence of a suitably defined generalized solution to the the initial-boundary-value problem with the homogeneous boundary conditions is proved.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,Analysis

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