Blowup for the multiplicative stochastic heat equation with superlinear drift

We consider the stochastic heat equation with multiplicative white noise: $\partial_t u =\partial_x^2u + b(u) +\sigma(u) \dot W$, 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 $\sigma$. 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 $\sigma$. 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.