Blowup for the multiplicative stochastic heat equation with superlinear drift
Mathew Joseph, Shubham Ovhal
TL;DR
The paper analyzes finite-time blowup versus global existence for a stochastic heat equation with multiplicative noise and superlinear drift on a bounded interval and on the real line. It proves that a finite Osgood condition on the drift, expressed as $\int_1^{\infty} \frac{dx}{b(x)} < \infty$, is necessary and sufficient for blowup on [0,1], under broad growth conditions on the diffusion, and shows instantaneous explosion on $\mathbb{R}$ from $u_0\equiv 1$ when the drift satisfies the same criterion and the diffusion is globally Lipschitz. The key techniques combine level-crossing time analysis, a sharp noise-term bound, and a lattice-approximation/monotonicity approach to establish a comparison principle between the Dirichlet problem and the Cauchy problem. A detailed approximation framework on lattices demonstrates convergence of lattice SPDEs to the continuum solutions, enabling the transfer of blowup behavior from the bounded domain to the whole line. Overall, the work extends prior blowup results to unbounded diffusion under controlled growth and strengthens the connection between bounded-domain blowup and global-in-space explosion via a robust comparison principle.
Abstract
We consider the stochastic heat equation with multiplicative white noise: $\partial_t u =\partial_x^2u + b(u) +σ(u) \dot W$, 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 $σ$. 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 $σ$. 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.