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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 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.

35L55Higher order hyperbolic systems
35L60Nonlinear first-order hyperbolic equations
35B40Asymptotic behavior of solutions of PDE
74D99Materials of strain-rate type and history type, other materials with memory
35A05General existence and uniqueness theorems (PDE) (MSC2000)
35B65Smoothness and regularity of solutions of PDE
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