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

TL;DR

This paper investigates finite time explosion phenomena in a broad class of superlinear stochastic parabolic equations across arbitrary spatial dimensions, considering various boundary conditions and noise types.

Contribution

It extends existing theories by establishing explosion results for general elliptic operators, boundary conditions, and noise in any spatial dimension, with a broader growth rate parameter.

Findings

Proves finite time explosion under superlinear growth conditions.

Extends results to arbitrary spatial dimensions and boundary conditions.

Allows the growth parameter  to reach a specific critical level.

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 $1+\frac{1-\eta}{2\beta}$.