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) = 12²ₓ u (t, x) + (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: R R is a continuous function with (0) =0. We prove that (i) if v/ (v) is sufficiently large near v=0, then the solution u (t, ) is strictly positive for all t>0, and (ii) if v/ (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 (u) u^ for >0. Additionally, we establish the uniqueness of a solution and the weak comparison principle in case (i).