On the support of solutions to nonlinear stochastic heat equations

TL;DR

This paper studies when solutions to a nonlinear stochastic heat equation become strictly positive or retain compact support, depending on the behavior of the noise coefficient near zero, extending previous results to more general nonlinearities.

Contribution

It provides new criteria for strict positivity and compact support of solutions based on the local behavior of the noise coefficient function near zero.

Findings

Solutions are strictly positive if v/σ(v) is large near zero.

Solutions have compact support if v/σ(v) is small near zero.

Established uniqueness and comparison principles under certain conditions.

Abstract

We investigate the strict positivity and the compact support property of solutions to the one-dimensional nonlinear stochastic heat equation: $$\partial_t u(t,x) = \frac{1}{2}\partial^2_x u(t,x) + \sigma(u(t,x))\dot{W}(t,x), \quad (t,x)\in \mathbf{R}_+\times\mathbf{R},$$ with nonnegative and compactly supported initial data $u_0$, where $\dot{W}$ is the space-time white noise and $\sigma:\mathbf{R} \to \mathbf{R} $ is a continuous function with $\sigma(0)=0$. We prove that (i) if $v/ \sigma(v)$ is sufficiently large near $v=0$, then the solution $u(t,\cdot)$ is strictly positive for all $t>0$, and (ii) if $v/\sigma(v)$ is sufficiently small near $v= 0$, then the solution $u(t,\cdot)$ has compact support for all $t>0$. These findings extend previous results concerning the strict positivity and the compact support property, which were analyzed only for the case $\sigma(u)\approx u^\gamma$ for $\gamma>0$. Additionally, we establish the uniqueness of a solution and the weak comparison principle in case (i).