On the support of solutions to nonlinear stochastic heat equations

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).