Existence of global mild and strong solutions to stochastic hyperbolic evolution equations driven by a spatially homogeneous Wiener process
type: Item
instance of: scholarly article
title: Existence of global mild and strong solutions to stochastic hyperbolic evolution equations driven by a spatially homogeneous Wiener process
zbMATH Open document ID: 1054.60068
author: Martin Ondreját
published in: Journal of Evolution Equations
publication date: 2004-09-27T00:00:00Z
review text: The author considers a semilinear stochastic hyperbolic equation \[ u_{t,t} = \mathcal{A} u + f(u) + g(u) \dot W, \quad u(0)=u_0, \quad u_t(0)=v_0, \] in a \(d\)-dimensional domain \(D\), driven by an \(n\)-dimensional Wiener process \(W\) and where \(\mathcal{A}\) is a uniformly elliptic second order differential operator. When \(f\) and \(g\) are globally Lipschitz, it is known that this type of equation has a unique global mild solution. The author presents a more general set of sufficient assumptions upon \(f\) and \(g\) that imply that the mild solution to this equation exists globally. Moreover, the author also gives conditions on \(f,g\) and the finite spectral measure of \(W\), implying the existence of a real global strong solution. These results apply to equations with polynomial drift. The sufficient conditions are given in terms of Lyapunov functions for the equation.
reviewed by: Carles Rovira
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 2103743
zbMATH Keywords: hyperbolic evolution equations
zbMATH Keywords: Lyapunov functions
MaRDI profile type: Publication
full work available at URL: s00028-003-0130-y
OpenAlex ID: W265834655
DOI: 10.1007/S00028-003-0130-Y
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description: scientific article; zbMATH DE number 2103743