Nonexplosion for a large class of superlinear stochastic parabolic equations, in arbitrary spatial dimension

Abstract

This paper explores the finite time explosion of the stochastic parabolic equation $\frac{\partial u}{\partial t}(t,x)=Au(t,x)+\sigma(u(t,x))\dot{W}(t,x)$ in arbitrary bounded spatial domain with a large class of space-time colored noise under Neumann, periodic or Dirichlet boundary conditions where $A$ is second-order self-adjoint elliptic operator and $\sigma$ grows like $\sigma(u)\approx C(1+|u|^{\chi})$ where $\chi=1+\frac{1-\eta}{2\beta}$ with $\eta$ and $\beta$ are the parameters related to the singularities of heat kernel and noise covariance kernel. We improve upon previous results by proving the theory in arbitrary spatial dimension, general elliptic operator, general space-time colored noise, a larger class of boundary conditions and proves that $\chi$ can reach the level $1+\frac{1-\eta}{2\beta}$.