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Optimal decay rate estimates of a nonlinear viscoelastic Kirchhoff plate. (English) Zbl 1441.35050
Summary: This paper is concerned with a nonlinear viscoelastic Kirchhoff plate \(u_{t t}\left( t\right)-\sigma\Delta u_{t t}\left( t\right)+ \Delta^2u\left( t\right)-{\displaystyle \int_0^t g \left( t - s\right) \Delta^2 u \left( s\right) \text{d} s}=\text{div}F\left( \nabla u \left( t\right)\right).\) By assuming the minimal conditions on the relaxation function \(g\): \( g^\prime\left( t\right)\leq\xi\left( t\right)G\left( g \left( t\right)\right)\), where \(G\) is a convex function, we establish optimal explicit and general energy decay results to the system. Our result holds for \(G\left( t\right)=tp\) with the range \(p\in\left[ 1, 2\right)\), which improves earlier decay results with the range \(p\in\left[ 1, 3 / 2\right)\). At last, we give some numerical illustrations and related comparisons.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35L35 Initial-boundary value problems for higher-order hyperbolic equations
35L76 Higher-order semilinear hyperbolic equations
74K20 Plates
35R09 Integral partial differential equations
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