Narrow Results

This paper is concerned with the blow-up phenomenon of stochastic nonlocal heat equations. We first establish the sufficient condition to ensure that the stochastic nonlocal heat equations have a unique non-negative solution. Then the problem of blow-up solutions in finite time is considered.

We study two types of one-dimensional stochastic Burgers equations: Burgers equations with stochastic fluxes and Burgers equations perturbed by transport noise. Using the methods of stochastic characteristics and the characteristics perturbed by noise, respectively, we obtain the probabilities for the occurrence of shocks before and after a given time. Compared with the deterministic equations, in case of small initial data the occurrence time of shocks for Burgers equations with stochastic fluxes will be delayed with large probability, but this conclusion is only true for Burgers equations perturbed by the transport noise with large initial data.