Sharp time decay rates on a hyperbolic plate model under effects of an intermediate damping with a time-dependent coefficient

Marcello  D'Abbicco; Ruy Coimbra  Charão; Cleverson Roberto da  Luz

doi:10.3934/dcds.2016.36.2419

Article Contents

2016, Volume 36, Issue 5: 2419-2447. Doi: 10.3934/dcds.2016.36.2419

This issue Previous Article Flows of vector fields with point singularities and the vortex-wave system Next Article On the Fibonacci complex dynamical systems

Sharp time decay rates on a hyperbolic plate model under effects of an intermediate damping with a time-dependent coefficient

1.
Departamento de Computação e Matemática, Universidade de São Paulo (USP), FFCLRP, Av. dos Bandeirantes 3900, Ribeirão Preto, SP 14040-901
2.
Department of Mathematics, Federal University of Santa Catarina, Campus Trindade, Florianópolis, SC 88040-900
3.
Department of Mathematics, Federal University of Santa Catarina, Campus Trindade, Florianopolis, SC 88040-900

Received: March 31, 2015

Revised: August 31, 2015

Published: May 2017

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Abstract

In this work we study decay rates for a hyperbolic plate equation under effects of an intermediate damping term represented by the action of a fractional Laplacian operator and a time-dependent coefficient. We obtain decay rates with very general conditions on the time-dependent coefficient (Theorem 2.1, Section 2), for the power fractional exponent of the Laplace operator $(-\Delta)^\theta$, in the damping term, $\theta \in [0,1]$. For the special time-dependent coefficient $b(t)=\mu (1+t)^{\alpha}$, $\alpha \in (0,1]$, we get optimal decay rates (Theorem 3.1, Section 3).

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Mathematics Subject Classification: Primary: 35B40; Secondary: 35L30, 35B35, 74K20.

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