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

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 $\mathcal L^\beta$,$0<\beta\le 1$,ofasymmetricuniformlyellipticoperator$\mathcal L$definedon$L^2(D)$. 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.