On a terminal value problem for stochastic space‐time fractional wave equations. (22nd July 2022) - Tran Bao, Ngoc; Nane, Erkan; Huy Tuan, Nguyen
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 u ( T, x ) = φ ( x ) u(T, x)=φ (x) and the spectrum of the fractional Laplacian operator ( − Δ ) s / 2 (-Δ )^s/2 (in a bounded domain 𝕏 ⊂ ℝ d, 0 < s < 2 0<s<2 ). First, we show the existence and uniqueness of a mild solution in L p ( 0, T ; L 2 ( Ω, V ) ) ∩ C ( ( 0, T ] ; L 2 ( Ω, L 2 ( 𝕏 ) ) ), for a suitable sub‐space V V of L 2 ( 𝕏 ) . A limitation of this result is the lack of time continuity at t = 0 t=0 . Second, we study the inverse problem (IP) of recovering u ( 0, x ) u(0, x) when the terminal value data φ φ and the source f f are given. We give an explanation why the time continuity of the solution at t = 0 t=0 could not derived. The main reason comes from unboundedness of a solution operator, so the problem (IP) is then ill‐posed, that is, recovery u ( 0, x ) u(0, x) 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.