Inverse problems for a generalized fractional diffusion equation with unknown history

Abstract Solutions of equations governing nonlocal in time processes depend on history of the processes that may be unknown in various situations. In this paper, a method to exclude the unknown history in identification problems making use of non-analyticity of an input is proposed. The method is applied to inverse problems for a diffusion equation containing a generalized fractional derivative. It is assumed that a source $f$ is unknown for time values $t$ in $(0,t_0)$, vanishes for $t\in (t_0,t_1)$ and has nonzero (generated) values for $t\in (t_1,T)$. Provided that $f|_{(t_1,T)}$ satisfies certain restrictions, it is proved that product of a kernel of the derivative with an elliptic operator as well as the history of $f$ for $t\in (0,t_0)$ are uniquely recovered by a measurement of a state $u$ in $(t_0,T)$. In case of less restrictions on $f$ the uniqueness of the kernel and the history of $f$ is shown. Moreover, in a case when a functional of $u$ in $(t_0,T)$ is given the uniqueness of the kernel is proved under unknown history.