The critical parameter for the heat equation with a noise term to blow up in finite time

• 成果类型: Article
• 署名作者: Mueller, C
• 署名单位: University of Rochester
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1019160505
• 发表日期: 2000
• 页码: 1735-1746
• 关键词: Existence

摘要

Consider the stochastic partial differential equation u(t) = u(xx) + u(gamma)(W)over dot, where x epsilon I equivalent to [0, J], (W)over dot = (W)over dot(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 I. We say that u blows up in finite time, with positive probability, if there is a random time T < infinity such that P(lim(t up arrowT) sup(x) u( t, x) = infinity) > 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.