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 R k , k ≥ 1, which includes the fractional power L β , 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 L β . We establish some existence and regularity results for our problem in both the linear and nonlinear case.