Existence, uniqueness, and stability of stochastic wave equation with cubic nonlinearities in two dimensions
type: Item
instance of: scholarly article
title: Existence, uniqueness, and stability of stochastic wave equation with cubic nonlinearities in two dimensions
zbMATH Open document ID: 1318.60069
author: Henri Schurz
author: Haziem M. Hazaimeh
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-01-26T00:00:00Z
review text: Assume that \(l_x\) and \(l_y\) are two positive real numbers and \(D=[0,l_x]\times [0,l_y]\). The aim of this paper is to study the nonlinear wave equation \[ u_{tt}=\sigma^2( u_{xx}+ u_{yy}) + A(u,v) + B(u,v) \frac{dW}{dt} \] in which \(A(u,v)= (a_1 - a_2\|u\|^2_{L^2(D)})u -\kappa v \) and \(B(u,v)= b_0 +b_1\|u\|_{L^2(D)} + b_2\|v\|_{L^2(D)}\). The authors assume a two-dimensional Fourier series expansion for the solution and prove that the coefficients satisfy an ordinary stochastic differential equation. Then, they define a finite-dimensional truncated Fourier series and Lyapunov functional for this equation to obtain an existence and uniqueness result for a continuous strong solution. Since the estimates for the Lyapunov functionals of the approximating systems are uniform with respect to the dimension \(N\), the authors can obtain estimates for the total expected energy of the above equation. Moreover, they prove the moment and sample path stability of the solutions by the method of Lyapunov functionals and give a bound for the Lyapunov exponent.
reviewed by: Ruhollah Jahanipur
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 6390525
zbMATH Keywords: nonlinear stochastic wave equation
zbMATH Keywords: additive space-time noise
zbMATH Keywords: energy Lyapunov functional
zbMATH Keywords: strong Fourier series solution
zbMATH Keywords: moment stability
zbMATH Keywords: sample path stability
MaRDI profile type: Publication
full work available at URL: j.jmaa.2014.04.011
OpenAlex ID: W2057442961
cites work: Q4404205
cites work: Q3425601
cites work: Nonlinear stochastic wave equations: blow-up of second moments in \(L^{2}\)-norm
cites work: Stochastic Equations in Infinite Dimensions
cites work: The stochastic wave equation in two spatial dimensions
cites work: Q4086524
cites work: Stochastic wave equations with nonlinear damping and source terms
cites work: Q3764976
cites work: Q4848614
cites work: Q4002114
cites work: A stochastic wave equation in two space dimensions: smoothness of the law
cites work: Stochastic differential equations. An introduction with applications.
cites work: Existence and uniqueness of solutions of semilinear stochastic infinite-dimensional differential systems with \(H\)-regular noise
cites work: Nonlinear stochastic wave equations in \(\mathbb{R}^1\) with power-law nonlinearity and additive space-time noise
cites work: Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise
cites work: Q4929961
cites work: Stochastic wave equations with cubic nonlinearity and Q-regular additive noise in \(\mathbb R^2\)
cites work: Q4223074
cites work: Q3747436
DOI: 10.1016/J.JMAA.2014.04.011
description: scientific article; zbMATH DE number 6390525