Singular initial conditions for the heat equation with a noise term

• 成果类型: Article
• 署名作者: Mueller, C
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1996
• 页码: 377-398
• 关键词: curvature

摘要

We consider the equation u(t) = u(xx) + u(gamma)W, t > 0, 0 less than or equal to x less than or equal to J, u(0,x) = u(0)(x), u(t,0) = u(t,J) = 0, where W = W(t,x) is two-parameter white noise. We show local existence and uniqueness for unbounded initial conditions satisfying certain conditions. Our results are motivated by earlier work, which showed that, for large gamma, solutions of this equation can blow up. One would wish to show that solutions can be extended beyond blowup, and our results can be viewed as a step in that direction.