On the support of solutions to nonlinear stochastic heat equations

We investigate the strict positivity and the compact support property of solutions to the one-dimensional nonlinear stochastic heat equation: $${\mathrm{\partial }}_{t}u(t,x)=\frac{1}{2}{\mathrm{\partial }}_x^{2}u(t,x)+\sigma (u(t,x))\dot{W}(t,x),(t,x)\in {\mathbf{R}}_{+}\times \mathbf{R},$$ 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 $v\mathrm{/}\sigma (v)$ is sufficiently large near $v=0$, then the solution $u(t,\cdot )$ is strictly positive for all $t>0$, and (ii) if $v\mathrm{/}\sigma (v)$ 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).