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: (Formula presented), 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) ≈ uγ for γ > 0. Additionally, we establish the uniqueness of a solution and the weak comparison principle in case (i).