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 $\mathcal L^β$, $0<β\le 1$, 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 $\mathcal L^β$. We establish some existence and regularity results for our problem in both the linear and nonlinear case.

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