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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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References:

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