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**The critical parameter for the heat equation with a noise term to blow up in finite time.**
*(English)*
Zbl 1044.60051

Summary: Consider the stochastic partial differential equation \(u_t =u_{xx}+ u^\gamma\dot W\), where \(x\in{\mathbf I}\equiv[0,J]\), \(\dot W=\dot W(t, x)\) is 2-parameter white noise, and we assume that the initial function \(u(0,x)\) is nonnegative and not identically 0. We impose Dirichlet boundary conditions on \(u\) in the interval \({\mathbf I}\). We say that \(u\) blows up in finite time, with positive probability, if there is a random time \(T<\infty\) such that \(P( \lim_{t\uparrow T}\sup_x u(t,x)=\infty)>0.\) It was known that if \(\gamma< 3/2\), then with probability 1, \(u\) does not blow up in finite time. It was also known that there is a positive probability of finite time blowup for \(\gamma\) sufficiently large. We show that if \(\gamma>3/2\), then there is a positive probability that \(u\) blows up in finite time.

### MSC:

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

35R60 | PDEs with randomness, stochastic partial differential equations |

35K05 | Heat equation |

35B40 | Asymptotic behavior of solutions to PDEs |

60H40 | White noise theory |

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