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: ∂ₜ u(t,x) = 1/2∂²ₓ u(t,x) + σ(u(t,x))Ẇ(t,x), (t,x)∈ R₊, with nonnegative and compactly supported initial data u₀, where Ẇ is the space-time white noise and σ:R → R is a continuous function with σ(0)=0. We prove that (i) if v/ σ(v) is sufficiently large near $v=0$, then the solution u(t,·) is strictly positive for all $t>0$, and (ii) if v/σ(v) is sufficiently small near $v= 0$, then the solution u(t,·) 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 σ(u)≈ u^γ for γ>0. Additionally, we establish the uniqueness of a solution and the weak comparison principle in case (i).