Intermittence and time fractional stochastic partial differential equations

Abstract

We consider time fractional stochastic heat type equation $${\partial }_t^{\beta }u(t,x)=-\nu (-\Delta {)}^{\alpha /2}{u}_t(x)+I_t^{1-\beta }[\sigma (u)\stackrel{\cdot }{W}(t,x)]$$ in $(d+1)$ dimensions, where $\nu >0$, $\beta \in (0,1)$, $\alpha \in (0,2]$, $d<\min\{2,β^{-1}\}\a$, ${\partial }_t^{\beta }$ is the Caputo fractional derivative, $-(-\Delta {)}^{\alpha /2}$ is the generator of an isotropic stable process, $\stackrel{\cdot }{W}(t,x)$ is space-time white noise, and $σ:\RR{R}\to\RR{R}$ is Lipschitz continuous. The time fractional stochastic heat type equations might be used to model phenomenon with random effects with thermal memory. We prove: (i) absolute moments of the solutions of this equation grows exponentially; and (ii) the distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. These results extend the results of Foondun and Khoshnevisan \cite{foondun-khoshnevisan-09} %(Mohammud Foondun and Davar Khoshnevisan, Intermittence and nonlinear parabolic %stochastic partial differential equations, Electron. J. Probab. 14 (2009), no. 21, 548--568) and Conus and Khoshnevisan \cite{conus-khoshnevisan} % (On the existence and position of the farthest peaks of a family of stochastic %heat and wave equations, Probab. Theory Related Fields 152 (2012), no. 3-4, 681--701) on the parabolic stochastic heat equations.

20 pages, Submitted for publication. arXiv admin note: text overlap with arXiv:1409.7366

References in corpus (1)

• Fractional Cauchy problems on bounded domains

Cited by in corpus (1)

• Space-time fractional diffusions in Gaussian noisy environment