Wave equation with stochastic boundary values
type: Item
instance of: scholarly article
title: Wave equation with stochastic boundary values
zbMATH Open document ID: 0784.60061
author: Xuerong Mao
author: L. Markus
published in: Journal of Mathematical Analysis and Applications
publication date: 1994-01-13T00:00:00Z
review text: Consider the transverse displacement \(U(x,t)\) satisfying the linear partial differential equation \[ U_{tt}(x,t)=U_{xx}(x,t) \quad \text{on } 0 \leq x \leq L,\;0 \leq t<\infty, \tag{*} \] with stochastic boundary values of the form \[ \begin{cases} U(0,t)=0 \quad & \text{ on } 0 \leq t<\infty, \\ U_ x(L,t)=\dot B(t) & \text{ on } 0 \leq t<\infty, \\ U_ t(L,t)=\dot B(t) & \text{ on } 0 \leq t<\infty, \end{cases} \tag{**} \] where \(\dot B(t)\) is white noise. The authors prove that the solution of equation (*) with initial data of the form (**) satisfies the asymptotic estimates \[ \pi /2 \sqrt 2 \leq \liminf_{n \to \infty} \bigl( \| U(\cdot,t_ n)\|/ \sqrt{t_ n/ \log t_ n} \bigr) \quad\text{ (a.s.)} \] and \[ \limsup_{n \to \infty} \bigl( \| U(\cdot,t_ n)\|/ \sqrt {t_ n \log t_ n} \bigr) \leq \sqrt 2 \quad\text{ (a.s.)}, \] where \(\| U(\cdot,t)\|=\max_{0 \leq x \leq L} | U(x,t)|\) and \(\{t_ n\}\) is any suitable sequence of real numbers.
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35R60
zbMATH DE Number: 480445
zbMATH Keywords: linear partial differential equation
zbMATH Keywords: white noise
reviewed by: Stanisław Wedrychowicz
MaRDI profile type: Publication
full work available at URL: jmaa.1993.1261
OpenAlex ID: W2013272668
DOI: 10.1006/JMAA.1993.1261
description: scientific article; zbMATH DE number 480445