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Global existence and exponential stability of small solutions to nonlinear viscoelasticity. (English) Zbl 0779.35066
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:
35L55 Higher-order hyperbolic systems
35L60 First-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
74D99 Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
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