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

• Published in: The Annals of probability 2019-01, Vol.47 (1), p.519-559
• Main Authors: Dalang, Robert C., Khoshnevisan, Davar, Zhang, Tusheng
• Format: Article
• Language: English
• ISSN: 0091-1798; 2168-894X
• DOI: 10.1214/18-AOP1270

Let ξ(t, x) denote space–time white noise and consider a reaction–diffusion equation of the form u̇(t, x) = ½u″(t, x) + b(u(t, x)) + σ(u(t, x))ξ(t, x), on ℝ₊ × [0, 1], with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists ε > 0 such that |b(z)| ≥ |z|(log |z|)1+ε for all sufficiently-large values of |z|. When σ ≡ 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 σ 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 |b(z)| = O(|z| log₊ |z|) as |z| → ∞. We interpret the word “typically” in two essentially-different ways without altering the conclusions of our assertions.