The Annals of Probability

Global solutions to stochastic reaction–diffusion equations with super-linear drift and multiplicative noise

Robert C. Dalang, Davar Khoshnevisan, and Tusheng Zhang

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Abstract

Let $\xi (t,x)$ denote space–time white noise and consider a reaction–diffusion equation of the form \[\dot{u}(t,x)=\frac{1}{2}u"(t,x)+b\big(u(t,x)\big)+\sigma \big(u(t,x)\big)\xi (t,x),\] on $\mathbb{R}_{+}\times [0,1]$, with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists $\varepsilon >0$ such that $\vert b(z)\vert \ge \vert z\vert (\log \vert z\vert )^{1+\varepsilon }$ for all sufficiently-large values of $\vert z\vert $. When $\sigma \equiv 0$, 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 $\vert b(z)\vert =O(\vert z\vert \log_{+}\vert z\vert )$ as $\vert z\vert \to \infty $. We interpret the word “typically” in two essentially-different ways without altering the conclusions of our assertions.

Article information

Source
Ann. Probab., Volume 47, Number 1 (2019), 519-559.

Dates
Received: June 2017
Revised: March 2018
First available in Project Euclid: 13 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1544691627

Digital Object Identifier
doi:10.1214/18-AOP1270

Mathematical Reviews number (MathSciNet)
MR3909975

Zentralblatt MATH identifier
07036343

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 35K57: Reaction-diffusion equations
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 35B45: A priori estimates 35B33: Critical exponents

Keywords
Stochastic partial differential equations reaction–diffusion equations blow-up logarithmic Sobolev inequality

Citation

Dalang, Robert C.; Khoshnevisan, Davar; Zhang, Tusheng. Global solutions to stochastic reaction–diffusion equations with super-linear drift and multiplicative noise. Ann. Probab. 47 (2019), no. 1, 519--559. doi:10.1214/18-AOP1270. https://projecteuclid.org/euclid.aop/1544691627


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