Intermittency and stochastic pseudo-differential equation with spatially inhomogeneous white noise

Abstract

In this paper, we study the intermittent property for the following nonlinear stochastic partial differential equation (SPDE in the sequel) in (1+1)-dimension $$\begin{aligned} \left( \frac{\partial }{\partial t}+q(x,D_x)\right) u(t,x)= g(u(t,x))\frac{\partial ^2 w_\rho }{\partial t\partial x}(t,x),\quad t>0 \quad \mathrm{and} \quad x\in {\mathbb {R}}, \end{aligned}$$ with $$q(x,D_x)$$ is a pseudo-differential operator which generates a stable-like process. The forcing noise denoted by $$\frac{\partial ^2 w_\rho }{\partial t\partial x}(t,x)$$ is a spatially inhomogeneous white noise. Under some mild assumptions on the catalytic measure of the inhomogeneous Brownian sheet $$w_\rho (t,x)$$ , we prove that the solution is weakly full intermittent based on the moment estimates of the solution.