Level of noises and long time behavior of the solution for space-time fractional SPDE in bounded domains

Abstract

In this paper we study the long time behavior of the solution to a certain class of space-time fractional stochastic equations with respect to the level $\lambda$ of a noise and show how the choice of the order $\beta \in (0, \,1)$ of the fractional time derivative affects the growth and decay behavior of their solution. We consider both the cases of white noise and colored noise. Our results extend the main results in "M. Foondun, \textit{Remarks on a fractional-time stochastic equation}, Proc. Amer. Math. Soc. 149 (2021), 2235-2247" to fractional Laplacian as well as higher dimensional cases.