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

Abstract

Abstract In this paper we study non-linear noise excitation for the following class of space-time fractional stochastic equations in bounded domains: ∂ t β u t ( x ) = − ν ( − Δ ) α / 2 u t ( x ) + I t 1 − β [ λ σ ( u ) F ⋅ ( t , x ) ] $$\partial^\beta_tu_t(x) = -\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\lambda \sigma(u) \stackrel{\cdot}{F}(t,x)] $$ in (d+1) dimensions, where ν > 0, β ∈ (0, 1), α ∈ (0, 2]. The operator ∂ t β $\partial^\beta_t$ is the Caputo fractional derivative, −(−Δ) α/2 is the generator of an isotropic stable process and I 1−β t is the fractional integral operator. The forcing noise denoted by F ⋅ ( t , x ) $\stackrel{\cdot}{F}(t,x)$ is a Gaussian noise. The multiplicative non-linearity σ : ℝ → ℝ is assumed to be globally Lipschitz continuous. These equations were recently introduced by Mijena and Nane [32]. We first study the existence and uniqueness of the solution of these equations and under suitable conditions on the initial function, we also study the asymptotic behavior of the solution with respect to the parameter λ. In particular, our results are significant extensions of those in [14], [16], [32], and [33].