Stochastic PDE for nonlinear vibration of elastic panels
type: Item
instance of: scholarly article
title: Stochastic PDE for nonlinear vibration of elastic panels
zbMATH Open document ID: 1006.60058
published in: Differential and Integral Equations
publication date: 2003-01-26T00:00:00Z
review text: The paper aims at proving existence and uniqueness of solutions to a nonlinear stochastic hyperbolic equation with a nonlocal term \[ \partial ^2_{t} u - \Biggl [\alpha + \beta \int ^{l}_{0} |\partial _{y}u(t,y)|^2 dy\Biggr ]\partial ^2_{x} u + \gamma \partial ^4_{x} u = F(t,x,\partial _{t}u,\partial _{x}u), \tag{1} \] \(x\in [0,l]\), \(t\geq 0\), with boundary conditions \(u(t,0)=u(t,l) =0\), \(\partial _{x}u(t,0) = \partial _{x}u(t,l) = 0\) and initial conditions \(u(0,\cdot)=\varphi _0\), \(\partial _{t}u(0,\cdot) = \varphi _1\). The constants \(\alpha \), \(\beta \), \(\gamma \) are positive, \(F\) is a random force, \[ F(t,x,r,v)= f(t,x) + \sum ^\infty _{k=1} \sigma _{k}(t,x)\dot w_{k}(t) + g(t,x,r,v) + \sum ^{\infty } _{k=1} \rho _{k}(t,x,r,v)\dot B_{k}(t), \] where \(w_{k}\), \(B_{k}\) are independent standard one-dimensional Wiener processes. Equations of this type arise in nonlinear aeroelasticity; existence and uniqueness of local solutions (up to an explosion time \(\tau \)) follow from standard theory; however, it is not clear at all whether \(\tau =\infty \). In the paper, global solutions are constructed under suitable Lipschitz continuity and linear growth hypotheses upon the functions \(g\), \(\sigma _{k}\), \(\rho _{k}\).
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35R60
zbMATH DE Number: 1858754
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: nonlinear aeroelasticity
zbMATH Keywords: Wiener process
author: Pao-Liu Chow
author: José-Luis Menaldi
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1858754