2017/01/17 by Dalang, Robert C., Khoshnevisan, Davar, Zhang, Tusheng
#35B33 (Secondary) #35B45 #35K57 (Primary) #35R60 #60H15 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1701.04660
Let ξ(t ,x) denote space-time white noise and consider a reaction-diffusion equation of the form u(t ,x)=\tfrac12 u"(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 \vert b(z)\vert ≥|z|(log|z|)1+ε for all sufficiently-large values of |z|. When σ≡ 0, it is well known that such PDEs frequently have non-trivial stationary solutions. By contrast, Bonder and Groisman (2009) have recently shown that there is finite-time blowup when σ is a non-zero constant. In this paper, we prove that the Bonder--Groisman condition is unimproveable by showing that the reaction-diffusion equation with noise is "typically" well posed when \vert b(z) \vert =O(|z|log+|z|) as |z|→∞. We interpret the word "typically" in two essentially-different ways without altering the conclusions of our assertions.