Asymptotic behavior for coupled abstract evolution equations with one infinite
memory
By:Guesmia, A (Guesmia, Aissa)[ 1 ]
APPLICABLE ANALYSIS
Volume: 94
Issue: 1
Pages: 184-217
Special Issue: SI
DOI: 10.1080/00036811.2014.890708
Published: JAN 2 2015
View Journal Information
Abstract
In this paper, we consider two coupled abstract linear evolution equations with one infinite
memory acting on the first equation. Our work is motivated by the recent results of [42],
where the authors considered the case of two wave equations with one convolution kernel
converging exponentially to zero at infinity, and proved the lack of exponential decay. On
the other hand, the authors of [42] proved that the solutions decay polynomially at infinity
with a decay rate depending on the regularity of the initial data. Under a boundedness
condition on the past history data, we prove that the stability of our abstract system holds
for convolution kernels having much weaker decay rates than the exponential one. The
general and precise decay estimate of solution we obtain depends on the growth of the
convolution kernel at infinity, the regularity of the initial data, and the connection between
the operators describing the considered equations. We also present various applications to
some distributed coupled systems such as wave-wave, Petrovsky-Petrovsky, wavePetrovsky, and elasticity-elasticity.
Keywords
Author Keywords:semigroups theory; well-posedness; memory; energy method; coupled
evolution equations; asymptotic behavior; 35L15; 93D15; 35L05; 35L70
KeyWords Plus:INDIRECT DAMPING MECHANISMS; LINEAR VISCOELASTICITY;
EXPONENTIAL STABILITY; GENERAL DECAY; TIMOSHENKO SYSTEMS; NONLINEAR
SOURCE; WAVE-EQUATIONS; STABILIZATION; ENERGY; EXISTENCE
Author Information
Reprint Address: Guesmia, A (reprint author)
King Fahd Univ Petr & Minerals, Dept Math & Stat, Coll Sci, POB 5005, Dhahran 31261,
Saudi Arabia.
Organization-Enhanced Name(s)
King Fahd University of Petroleum & Minerals
Addresses:
[ 1 ] Univ Lorraine, Elie Cartan Inst Lorraine, UMR 7502, F-57045 Metz 01, France
Organization-Enhanced Name(s)
Centre National de la Recherche Scientifique (CNRS)
University of Lorraine
E-mail Addresses:guesmia@univ-metz.fr
Document Information
Document Type:Article
Language:English
Accession Number: WOS:000347289800014
ISSN: 0003-6811
eISSN: 1563-504X