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: partial derivative(t)u(t,x)=(1)/(2)partial derivative(2)(x)u(t,x)+sigma(u(t,x))W-center dot(t,x),(t,x)is an element of R(+)xR, with nonnegative and compactly supported initial data u(0), where W-center dot is the space-time white noise and sigma:R -> 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,& sdot;) is strictly positive for all t>0, and (ii) if v/sigma(v) is sufficiently small near v=0, then the solution u(t,& sdot;) 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)approximate to u(gamma) for gamma>0. Additionally, we establish the uniqueness of a solution and the weak comparison principle in case (i).