SOLUTIONS TO THE STOCHASTIC HEAT EQUATION WITH POLYNOMIALLY GROWING MULTIPLICATIVE NOISE DO NOT EXPLODE IN THE CRITICAL REGIME

• 成果类型: Article
• 署名作者: Salins, Michael
• 署名单位: Boston University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/24-AOP1704
• 发表日期: 2025
• 页码: 223-238
• 关键词: long-time existence; reaction-diffusion equations; global-solutions; blow-up; nonexistence; term

摘要

We investigate the finite time explosion of the stochastic heat equation (partial derivative u)/(partial derivative t)=Delta u(t,x)+sigma(u(t,x))W-center dot(t,x) in the critical setting where sigma grows like sigma(u)approximate to C(1+|u|(gamma)) and gamma=(3)/(2). Mueller previously identified gamma=(3)/(2) as the critical growth rate for explosion and proved that solutions cannot explode in finite time if gamma<(3)/(2) and solutions will explode with positive probability if gamma>(3)/(2). This paper proves that explosion does not occur in the critical gamma=(3)/(2) setting.