New continuity results for a class of time fractional stochastic heat equations in bounded and unbounded domains

Abstract

In this paper, we consider a class of time fractional stochastic heat type equation ${\partial }_t^{\alpha }{u}_{\alpha }={\partial }_{xx}{u}_{\alpha }+I_t^{1-\alpha }\left[\lambda \sigma \left(u_{\alpha }\right)\dot{W}(t,x)\right]$where ${\partial }_t^{\alpha },0<\alpha <1$ is the Caputo fractional derivative, $\sigma :R\to R$ is a Lipschitz continuous function, and $\dot{W}$ is space–time white noise. These equations have significant applications in modeling temperature in thermal materials. Our main purpose in this paper is to study the continuity of solutions of fractional order Equation (1) with respect to $\alpha$. Two interesting questions for our problem are stated as follows. Let $u_{\alpha }$ and $u$ be the solution of Equation (1) for $0<\alpha <1$ and $\alpha =1$, respectively. The first question is that : Does $u_{{\alpha }^{\prime }}\to u_{\alpha }$ in an appropriate sense as $\alpha \to {\alpha }^{\prime }$? The second question is that: Does $u_{\alpha }\to u$ in an appropriate sense as $\alpha \to 1^{-}$? We will give affirmative answers to both of these questions. Furthermore, under some suitable assumptions on the initial datum, we provide the convergence rate estimates between $u_{\alpha }$ and $u_{{\alpha }^{\prime }}$, as well as $u_{\alpha }$ and $u$.