Instantaneous shrinking of supports for stochastic PDEs

We study instantaneous shrinking of supports for nonnegative solutions of the stochastic partial differential equation \[ \partial_t u=a(t,x)\,\partial_x^2 u + b(t,x)\,\partial_x u + c(t,x)\,u +σ(u)\,ξ(t,x), \qquad (t,x)\in(0,\infty)\times\mathbb R, \] where $ξ$ is space-time white noise, the coeffi