Non-linear noise excitation for some space-time fractional stochastic equations in bounded domains

摘要

In this paper we study non-linear noise excitation for the following class ofspace-time fractional stochastic equations in bounded domains:$$\partial^\beta_tu_t(x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\lambda\sigma(u)\stackrel{\cdot}{F}(t,x)]$$ in $(d+1)$ dimensions, where $\nu>0,\beta\in (0,1)$, $\alpha\in (0,2]$. The operator $\partial^\beta_t$ is theCaputo fractional derivative, $-(-\Delta)^{\alpha/2} $ is the generator of anisotropic stable process and $I^{1-\beta}_t$ is the fractional integraloperator. The forcing noise denoted by $\stackrel{\cdot}{F}(t,x)$ is a Gaussiannoise. The multiplicative non-linearity $\sigma:\RR{R}\to\RR{R}$ is assumed tobe globally Lipschitz continuous. These equations were recently introduced byMijena and Nane(J. Mijena and E. Nane. Space time fractional stochastic partialdifferential equations. Stochastic Process Appl. 125 (2015), no. 9,3301--3326). We first study the existence and uniqueness of the solution ofthese equations {and} under suitable conditions on the initial function, we{also} study the asymptotic behavior of the solution with respect to theparameter $\lambda$. In particular, our results are significant extensions ofthose in Foondun et al (M. Foondun, K. Tian and W. Liu. On some properties of aclass of fractional stochastic equations. Preprint available at arxiv.org1404.6791v1.), Foondun and Khoshnevisan (M. Foondun and D. Khoshnevisan.Intermittence and nonlinear parabolic stochastic partial differentialequations, Electron. J. Probab. 14 (2009), no. 21, 548--568.), Nane and Mijena(J. Mijena and E. Nane. Space time fractional stochastic partial differentialequations. Stochastic Process Appl. 125 (2015), no. 9, 3301--3326; J. B.Mijena, and E.Nane. Intermittence and time fractional partial differentialequations. Submitted. 2014).