Some well-posedness and stability results for abstracthyperbolic equations with infinite memory and distributed time delay

Aissa Guesmia; Nasser-eddine Tatar

doi:10.3934/cpaa.2015.14.457

Article Contents

2015, Volume 14, Issue 2: 457-491. Doi: 10.3934/cpaa.2015.14.457

This issue Previous Article Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents Next Article Sharp existence criteria for positive solutions of Hardy--Sobolev type systems

Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay

Aissa Guesmia^1, and
Nasser-eddine Tatar^2,

1.
Department of Mathematics and Statistics, College of Sciences, King Fahd University of Petroleum and Minerals, P.O. Box 5005, Dhahran 31261
2.
Department of Mathematics and Statistics, College of Sciences, King Fahd University of Petroleum and Minerals, P.O.Box. 5005, Dhahran 31261

Received: July 31, 2014

Revised: September 30, 2014

Published: February 2015

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Abstract

In this paper, we consider a class of second order abstract linear hyperbolic equations with infinite memory and distributed time delay. Under appropriate assumptions on the infinite memory and distributed time delay convolution kernels, we prove well-posedness and stability of the system. Our estimation shows that the dissipation resulting from the infinite memory alone guarantees the asymptotic stability of the system in spite of the presence of distributed time delay. The decay rate of solutions is found explicitly in terms of the growth at infinity of the infinite memory and the distributed time delay convolution kernels. An application of our approach to the discrete time delay case is also given.

Keywords:

Mathematics Subject Classification: 35L05, 35L15, 35L70, 93D15.

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