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Nonlinear transmission problem with a dissipative boundary condition of memory type. (English) Zbl 1112.35021

Summary: We consider a differential equation that models a material consisting of two elastic components. One component is clamped while the other is in a viscoelastic fluid producing a dissipative mechanism on the boundary. So, we have a transmission problem with boundary damping condition of memory type. We prove the existence of a global solution and its uniformly decay to zero as time approaches infinity. More specifically, the solution decays exponentially provided the relaxation function decays exponentially.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L70 Second-order nonlinear hyperbolic equations
74D99 Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials)
35L20 Initial-boundary value problems for second-order hyperbolic equations
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