Can solutions of the one-dimensional wave equation with nonlinear multiplicative noise blow up?

Carl Mueller, Geordie Richards

Abstract


Original proposers of the open problem: Carl Mueller
The year when the open problem was proposed: 1994
Sponsor of the submission: Sandra Cerrai - University of Maryland
AMS Subject classification: 60
Status of the problem: Open


Consider the stochastic partial differential equation $$u_{tt}=u_{xx}+|u|^\gamma\dot{W},$$ where $x\in \mathbf{S}:=\mathbf{R}/J\mathbf{Z}$, $\dot{W}=\dot{W}(t,x)$ is 2-parameter white noise, and we assume the initial functions $u(0,x)$ and $u_{t}(0,x)$ are continuous. If $\gamma>1$, does there exist a random time $\sigma< \infty$ such that $$P\left(\lim_{t\uparrow \sigma}\sup_{x\in\mathbf{S}} |u(t,x)|=\infty\right)>0?$$

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