Communications in Contemporary Mathematics · 2016 · 14 citations · 25 references
Natural SciencesMathematicsPhysicsDifferential EquationsDynamical SystemsMemoryDissipative SystemsNonlinear DynamicsPartial Differential EquationsStability TheoryViscoelastic BehaviorViscoelastic PropertiesDecay Rate EstimationDecay RatesMemory EffectsNonlinear Plate AnalysisNonlinear Plate TheorySecond-order Differential EquationsUniform StabilityViscoelastic Plate Theory
This paper is concerned with uniform stability of the energy corresponding to a class of nonlinear plate equations with memory. It is assumed that the memory kernel [Formula: see text] satisfies the condition [Formula: see text] of Alabau-Boussouira and Cannarsa [A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Acad. Sci. Paris Ser. I 347 (2009) 867–872], where [Formula: see text] is positive, convex, increasing, and satisfies [Formula: see text]. Then, we obtain sharp energy decay rate in the sense that it recovers the decay rate assumed to the memory kernel. To this end we use a recent approach proposed by Lasiecka and Wang [Intrinsic decay rate estimates for semilinear abstract second order equations with memory, in New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer Series INDAM, Vol. 10 (Springer, 2014), pp. 271–303].
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