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Zbl 0779.35066
Kawashima, S.; Shibata, Y.
Global existence and exponential stability of small solutions to nonlinear viscoelasticity. (English)
[J] Commun. Math. Phys. 148, No.1, 189-208 (1992). ISSN 0010-3616; ISSN 1432-0916/e

Summary: The global existence of smooth solutions of the equations of nonlinear hyperbolic system of second 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 $\infty$ 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.

MSC 2000:: *35L55 Higher order hyperbolic systems
35L60 First-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions of PDE
74D99 Materials of strain-rate type and history type, etc.
35A05 General existence and uniqueness theorems (PDE)
35B65 Smoothness of solutions of PDE

Keywords: asymptotic exponential decay; time periodic solutions; exponential stability; global existence; smooth solutions; nonlinear hyperbolic system of second order; Dirichlet boundary condition; nonlinear wave equation

Cited in: Zbl 0836.35156 Zbl 0835.35094

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