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

#### Abstract

Original proposers of the open problem:

The year when the open problem was proposed:

Sponsor of the submission:

AMS Subject classification:

Status of the problem:

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 $$P\left(\lim_{t\uparrow \sigma}\sup_{x\in\mathbf{S}} |u(t,x)|=\infty\right)>0?$$

**Carl Mueller**The year when the open problem was proposed:

**1994**Sponsor of the submission:

**Sandra Cerrai**- University of MarylandAMS 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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