Global existence and exponential stability of small solutions to nonlinear viscoelasticity

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Abstract

The global existence of smooth solutions to the equations of nonlinear hyperbolic system of 2nd order with third order viscosity is shown for small and smooth initial data in a bounded domain of n-dimensional Euclidean space with smooth boundary. Dirichlet boundary condition is studied and the asymptotic behaviour of exponential decay type of solutions as t tending to ∞ is described. Time periodic solutions are also studied. As an application of our main theorem, nonlinear viscoelasticity, strongly damped nonlinear wave equation and acoustic wave equation in viscous conducting fluid are treated.

Original languageEnglish
Pages (from-to)189-208
Number of pages20
JournalCommunications in Mathematical Physics
Volume148
Issue number1
DOIs
Publication statusPublished - 1992 Aug
Externally publishedYes

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Nonlinear Viscoelasticity
Small Solutions
viscoelasticity
Exponential Stability
Global Existence
wave equations
Nonlinear Hyperbolic Systems
hyperbolic systems
Time-periodic Solutions
Damped Wave Equation
conducting fluids
Euclidean geometry
Nonlinear Wave Equation
Smooth Solution
Acoustic Waves
Exponential Decay
Dirichlet Boundary Conditions
Euclidean space
n-dimensional
Wave equation

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Physics and Astronomy(all)
  • Mathematical Physics

Cite this

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abstract = "The global existence of smooth solutions to the equations of nonlinear hyperbolic system of 2nd order with third order viscosity is shown for small and smooth initial data in a bounded domain of n-dimensional Euclidean space with smooth boundary. Dirichlet boundary condition is studied and the asymptotic behaviour of exponential decay type of solutions as t tending to ∞ is described. Time periodic solutions are also studied. As an application of our main theorem, nonlinear viscoelasticity, strongly damped nonlinear wave equation and acoustic wave equation in viscous conducting fluid are treated.",
author = "Shuichi Kawashima and Yoshihiro Shibata",
year = "1992",
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AB - The global existence of smooth solutions to the equations of nonlinear hyperbolic system of 2nd order with third order viscosity is shown for small and smooth initial data in a bounded domain of n-dimensional Euclidean space with smooth boundary. Dirichlet boundary condition is studied and the asymptotic behaviour of exponential decay type of solutions as t tending to ∞ is described. Time periodic solutions are also studied. As an application of our main theorem, nonlinear viscoelasticity, strongly damped nonlinear wave equation and acoustic wave equation in viscous conducting fluid are treated.

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