The Critical Parameter for the Heat Equation with a Noise Term to Blow Up in Finite Time

Details The Critical Parameter for the Heat Equation with a Noise Term to Blow Up in Finite Time The Critical Parameter for the Heat Equation with a Noise Term to Blow Up in Finite Time

1999-02-22

arxiv.org/abs/math/9902126v1

Ann. Probab., 28, 4, 1735-1746, (2000)

Mathematics

Probability

Scientific paper

Consider the stochastic partial differential equation u_t=u_{xx}+u^gamma dot{W}, where x in [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. We say that u blows up in finite time, with positive probability, if there is a finite random time T such that P(\lim_{t->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 blow-up for gamma sufficiently large. In this paper, we show that if gamma>3/2, then there is a positive probability that u blows up in finite time.

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