Velocity dependent boundary conditions for the displacement in a one dimensional viscoelastic material
type: Item
instance of: scholarly article
title: Velocity dependent boundary conditions for the displacement in a one dimensional viscoelastic material
zbMATH Open document ID: 0806.35182
published in: Rocky Mountain Journal of Mathematics
publication date: 1994-09-28T00:00:00Z
review text: In this paper a boundary value problem of one dimensional viscoelastic material is analysed. The governing equation is \[ {\partial^ 2 u \over \partial t^ 2} = - \bigl( P(V) + \alpha (V) u_{tx} \bigr)_ x \tag{1} \] where \(x\) is the Lagrange mass variable, \(t\) is the time, \(u = u(x,t)\) is the displacement, \(V = V(x,t)\) is the specific volume, \(P\) is the pressure and the term \(-\alpha (V) u_{tx}\) gives the viscous stress. We have \[ V(x,t) = {\partial u \over \partial x} + V_ 0 (x), \tag{2} \] here \(V_ 0(x)\) is the initial value of the specific volume \(V\). The boundary conditions are as follows \[ - P \bigl( V(t,0) \bigr) + \alpha \bigl( V(t,0) \bigr) u_{tx} (t,0) = k_ 0 \tag{3} \] \[ - P \bigl( V(t,1) \bigr) + \alpha \bigl( V(t,1) \bigr) u_{tx} (t,1) = k_ 1. \tag{4} \] In this paper \(k_ 0\), \(k_ 1\) are depending on the time and velocity. It is assumed that (a) \(P\) satisfies a global Lipschitz condition, (b) the coefficient of viscosity \(\alpha (V)\) is bounded. Existence, regularity and uniqueness theorems for the global solutions to the boundary value problem (1), (3), (4) are given.
Mathematics Subject Classification ID: 35Q72
Mathematics Subject Classification ID: 74Hxx
Mathematics Subject Classification ID: 35D99
zbMATH DE Number: 645222
zbMATH Keywords: existence
zbMATH Keywords: regularity
zbMATH Keywords: uniqueness
zbMATH Keywords: global solutions
author: K. L. Kuttler
reviewed by: István Ecsedi
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072420
OpenAlex ID: W2021341602
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DOI: 10.1216/RMJM/1181072420
description: scientific article; zbMATH DE number 645222