Optimal energy decay rates for some wave equations with double damping terms

Ryo Ikehata; Shingo Kitazaki

doi:10.3934/eect.2019040

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2019, Volume 8, Issue 4: 825-846. Doi: 10.3934/eect.2019040

This issue Previous Article A new energy-gap cost functional approach for the exterior Bernoulli free boundary problem Next Article Exponential stability for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping

Optimal energy decay rates for some wave equations with double damping terms

Ryo Ikehata^, and
Shingo Kitazaki

Department of Mathematics, Graduate School of Education, Hiroshima University, Higashi-Hiroshima 739-8524, Japan

^* Corresponding author: Ryo Ikehata
^* Corresponding author: Ryo Ikehata

Received: October 2018

Revised: February 2019

Early access: June 2019

Published: June 2019

The first author is supported by grant-in-Aid for scientific Research (C)15K04958 of JSPS

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Abstract

We consider the Cauchy problem in $ {\bf R}^{n} $ for some wave equations with double damping terms, that is, one is the frictional damping $ u_{t}(t, x) $ and the other is very strong structural damping expressed as $ (-\Delta)^{\theta}u_{t}(t, x) $ with $ \theta > 1 $. We will derive optimal decay rates of the total energy and the $ L^{2} $-norm of solutions as $ t \to \infty $. These results can be obtained in the case when the initial data have a sufficient high regularity in order to guarantee that the corresponding high frequency parts of such energy and $ L^{2} $-norm of solutions are remainder terms. A strategy to get such results comes from a method recently developed by the first author [11].

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Mathematics Subject Classification: Primary: 35L15, 35L05; Secondary: 35B40.

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