Blowup for the multiplicative stochastic heat equation with superlinear drift

Abstract

We consider the stochastic heat equation with multiplicative white noise: ${\partial }_{t}u={\partial }_x^{2}u+b(u)+\ddot{I}ƒ(u)\dot{W}$, both on $[0,1]$ and $\mathbf{R}$. In the case of $[0,1]$ we show that the finite Osgood criterion on $b$ is a necessary and sufficient condition for finite-time blowup, under fairly general conditions on $\ddot{I}ƒ$. In the case of $\mathbf{R}$ we show instantaneous explosion when we start with initial profile $u_0\equiv 1$, extending the work of [10] which dealt with bounded $\ddot{I}ƒ$. The second result follows from the first by a comparison result which shows that the solution on $\mathbf{R}$ stays above the corresponding solution on $[0,1]$ with Dirichlet boundary conditions.

Corrected some errors. Improved the exposition