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

Asymptotic properties of some space-time fractional stochastic equations