### 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 language | English |
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Pages (from-to) | 189-208 |

Number of pages | 20 |

Journal | Communications in Mathematical Physics |

Volume | 148 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1992 Aug |

Externally published | Yes |

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### ASJC Scopus subject areas

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

### Cite this

**Global existence and exponential stability of small solutions to nonlinear viscoelasticity.** / Kawashima, Shuichi; Shibata, Yoshihiro.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Global existence and exponential stability of small solutions to nonlinear viscoelasticity

AU - Kawashima, Shuichi

AU - Shibata, Yoshihiro

PY - 1992/8

Y1 - 1992/8

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=0000100845&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000100845&partnerID=8YFLogxK

U2 - 10.1007/BF02102372

DO - 10.1007/BF02102372

M3 - Article

VL - 148

SP - 189

EP - 208

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -