Stochastic wave equations with polynomial nonlinearity
type: Item
instance of: scholarly article
title: Stochastic wave equations with polynomial nonlinearity
zbMATH Open document ID: 1017.60071
published in: The Annals of Applied Probability
publication date: 2003-05-06T00:00:00Z
review text: The author studies the stochastic wave equation with polynomial nonlinearity, \[ \partial^2_t u= \nabla^2 u+ f(u)+ \sigma(u)\partial_\tau W(t,x),x\in\mathbb{R}^d,\;t> 0,\quad u(0,x)= g(x),\quad \partial_t u(0,x)= h(x), \] \(d\leq 3\), where \(W(t,.)\) is a Wiener random field and the nonlinear terms \(f(u)\) and \(\sigma(u)\) are assumed to be polynomials in \(u\). The author proves an existence and uniqueness result of local and global solutions in Sobolev space \(H_1\) for a class of these stochastic equations. The proofs are based on an energy inequality for a linear stochastic wave equation. Nevertheless, in order to show that in general the global solution does not exist, the author shows a cubically nonlinear stochastic wave equation with an explosive solution.
reviewed by: Carles Rovira
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35R60
zbMATH DE Number: 1906187
zbMATH Keywords: stochastic wave equation
zbMATH Keywords: polynomial nonlinearity
author: Pao-Liu Chow
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1015961168
OpenAlex ID: W2006430388
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DOI: 10.1214/AOAP/1015961168
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description: scientific article; zbMATH DE number 1906187