Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions

Mohammad Akil; Ali Wehbe

doi:10.3934/mcrf.2019005

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2019, Volume 9, Issue 1: 97-116. Doi: 10.3934/mcrf.2019005

This issue Previous Article On Algebraic condition for null controllability of some coupled degenerate systems Next Article Insensitizing controls for a semilinear parabolic equation: A numerical approach

Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions

Mohammad Akil and
Ali Wehbe

Lebanese University-Faculty of Sciences, Khawarezmi laboratory for mathematics and applications-(KALMA), Beirut, Lebanon

Received: October 31, 2017

Revised: March 31, 2018

Published: January 2019

This research is supported by the Lebanese University.

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In this paper, we consider a multidimensional wave equation with boundary fractional damping acting on a part of the boundary of the domain. First, combining a general criteria of Arendt and Batty with Holmgren's theorem we show the strong stability of our system in the absence of the compactness of the resolvent and without any additional geometric conditions. Next, we show that our system is not uniformly stable in general, since it is the case of the interval. Hence, we look for a polynomial decay rate for smooth initial data for our system by applying a frequency domain approach combining with a multiplier method. Indeed, by assuming that the boundary control region satisfy some geometric conditions and by using the exponential decay of the wave equation with a standard damping, we establish a polynomial energy decay rate for smooth solutions, which depends on the order of the fractional derivative.

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Mathematics Subject Classification: 35B35, 93B52, 93C20.

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