On a terminal value problem for stochastic space‐time fractional wave equations

Abstract

This work is to investigate terminal value problem for a stochastic time fractional wave equation, driven by a cylindrical Wiener process on a Hilbert space. A representation of the solution is obtained by basing on the terminal value data and the spectrum of the fractional Laplacian operator (in a bounded domain , ). First, we show the existence and uniqueness of a mild solution in , for a suitable sub‐space of . A limitation of this result is the lack of time continuity at . Second, we study the inverse problem (IP) of recovering when the terminal value data and the source are given. We give an explanation why the time continuity of the solution at could not derived. The main reason comes from unboundedness of a solution operator, so the problem (IP) is then ill‐posed, that is, recovery cannot be obtained in general. Hence, we propose a truncation regularization method with a suitable choice of the regularization parameter. Finally, we present a numerical example to demonstrate our proposed method.