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 ${\mathbb{R}}^k$, $k\ge 1$, which includes the fractional power ${\mathcal{L}}^{\beta }$, $0<\beta \le 1$, of a symmetric uniformly elliptic operator $\mathcal{L}$ defined on $L^2(D)$. A representation of solutions is given by using the Laplace transform and the spectrum of ${\mathcal{L}}^{\beta }$. We establish some existence and regularity results for our problem in both the linear and nonlinear case.