Existence and uniqueness of the weak solution of initial-boundary-value problems for the time-space fractional diffusion equation

Abstract

This paper aims to study the existence and uniqueness of the weak solution of initial-boundary-value problems for the time-space fractional diffusion equation over a bounded domain Ω × [0, T], Ω ⊂ Rn. We first establish the existence of the weak solution of the initial-boundary-value problem for the time-space fractional diffusion equation and the proof is based on the eigenfunction expansion. Then to prove the uniqueness of the weak solution, a maximum principle for the time-space fractional diffusion equation is presented using the properties of the time fractional derivative and the fractional Laplace operator.