Asymptotic properties of some space-time fractional stochastic equations

Abstract

Consider non-linear time-fractional stochastic heat type equations of the following type, $${\partial }_t^{\beta }{u}_t(x)=-\nu (-\Delta {)}^{\alpha /2}{u}_t(x)+I_t^{1-\beta }[\lambda \sigma (u)\stackrel{\cdot }{F}(t,x)]$$ in $(d+1)$ dimensions, where $\nu >0,\beta \in (0,1)$, $\alpha \in (0,2]$. The operator ${\partial }_t^{\beta }$ is the Caputo fractional derivative while $-(-\Delta {)}^{\alpha /2}$ is the generator of an isotropic stable process and $I_t^{1-\beta }$ is the fractional integral operator. The forcing noise denoted by $\stackrel{\cdot }{F}(t,x)$ is a Gaussian noise. And the multiplicative non-linearity $\sigma$ is assumed to be globally Lipschitz continuous. Under suitable conditions on the initial function, we study the asymptotic behaviour of the solution with respect to time and the parameter $\lambda$. In particular, our results are significant extensions of existing results. Along the way, we prove a number of interesting properties about the deterministic counterpart of the equation.

References in corpus (3)

• Fractional Cauchy problems on bounded domains
• On fractional Duhamel's principle and its applications
• Fractional SPDEs driven by spatially correlated noise: existence of the solution and smoothness of its density

Cited by in corpus (1)

• Heat kernel estimates for time fractional equations