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

TL;DR

This paper investigates how the noise level and fractional time derivative order influence the long-term behavior of solutions to space-time fractional stochastic PDEs in bounded domains, extending previous results to higher dimensions and fractional Laplacians.

Contribution

It extends existing work on fractional-time stochastic equations by analyzing the effects of noise level and fractional derivatives in higher dimensions with fractional Laplacians.

Findings

Solution growth and decay depend on noise level and fractional order.

Results apply to both white and colored noise cases.

Extends prior results to higher-dimensional and fractional Laplacian settings.

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.