On existence and regularity of a terminal value problem for the time fractional diffusion equation

Abstract

In this paper we consider a final value problem for a diffusion equation with time-space fractional differentiation on a bounded domain D of Rk, k ≥ 1, which includes the fractional power Lβ, 0 < β ≤ 1, of a symmetric uniformly elliptic operator L defined on L 2(D). A representation of solutions is given by using the Laplace transform and the spectrum of Lβ. We establish some existence and regularity results for our problem in both the linear and nonlinear case.