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In this paper we consider a final value problem for a diffusion equation withtime-space fractional differentiation on a bounded domain $D$ of $\mathbb{R}^{k}$, $k\ge 1$, which includes the fractional power $\mathcalL^\beta$, $0<\beta\le 1$, of a symmetric uniformly elliptic operator $\mathcalL$ defined on $L^2(D)$. A representation of solutions is given by using theLaplace transform and the spectrum of $\mathcal L^\beta$. We establish someexistence and regularity results for our problem in both the linear andnonlinear case.

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