GLOBAL SOLUTIONS TO STOCHASTIC REACTION-DIFFUSION EQUATIONS WITH SUPER-LINEAR DRIFT AND MULTIPLICATIVE NOISE

• 成果类型: Article
• 署名作者: Dalang, Robert C.; Khoshnevisan, Davar; Zhang, Tusheng
• 署名单位: Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Utah System of Higher Education; University of Utah; University of Manchester
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1270
• 发表日期: 2019
• 页码: 519-559
• 关键词: systems

摘要

Let xi(t, x) denote space-time white noise and consider a reaction-diffusion equation of the form <(u) over dot>(t, x) = 1/2u ''(t, x) + b(u(t, x)) + sigma(u(t,x))xi(t,x) on R+ x [0, 1], with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists epsilon > 0 such that vertical bar b(z)vertical bar >= vertical bar z vertical bar (log vertical bar z vertical bar)(1+epsilon) for all sufficiently-large values of vertical bar z vertical bar. When sigma equivalent to 0, it is well known that such PDEs frequently have nontrivial stationary solutions. By contrast, Bonder and Groisman [Phys. D 238 (2009) 209-215] have recently shown that there is finite-time blowup when sigma is a nonzero constant. In this paper, we prove that the Bonder-Groisman condition is unimprovable by showing that the reaction-diffusion equation with noise is typically well posed when vertical bar b(z)vertical bar =O(vertical bar vertical bar z vertical bar log(+) vertical bar z vertical bar) as vertical bar z vertical bar -> infinity. We interpret the word typically in two essentially-different ways without altering the conclusions of our assertions.