Positive probability of explosion for stochastic heat equation with superlinear accretive reaction term and polynomially growing multiplicative noise

TL;DR

This paper investigates conditions under which solutions to a stochastic heat equation with superlinear reaction and polynomially growing noise can explode in finite time, refining understanding in a complex parameter regime.

Contribution

It identifies new parameter regimes where the stochastic heat equation's solutions can explode with positive probability, advancing the theoretical understanding of such phenomena.

Findings

Solutions explode with positive probability for certain parameter ranges.

Refined the understanding of explosion behavior in previously less understood regimes.

Characterized the interplay between reaction term and noise in explosion phenomena.

Abstract

This paper studies the finite time explosion of the stochastic heat equation $\frac{\partial u}{\partial t}(t,x)=\frac{\partial^2}{\partial x^2} u(t,x)+(u(t,x))^{\beta}+\sigma(u(t,x))\dot{W}(t,x)$. We consider an interval $D=[-\pi,\pi]$ under periodic boundary condition where $\dot{W}(t,x)$ is a space-time white noise and $\sigma(u)\approx u^{\gamma}$ near $\infty$. Our results refine existing results by identifying behavior in a previously less understood regime, where we show that if $\beta\in(1,3),\gamma\in(\frac{\beta}{2},\frac{\beta+3}{4})$ or $\beta>1,\gamma\in(0,\frac{\beta}{2}]$ then mild solutions can explode with positive probability. This paper provides a partial characterization of the explosion behavior in an intermediate parameter regime, and contribute to the understanding of the interplay between the drift and diffusion terms.