# American Institute of Mathematical Sciences

March  2015, 14(2): 457-491. doi: 10.3934/cpaa.2015.14.457

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

 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, Saudi Arabia

Received  August 2014 Revised  October 2014 Published  December 2014

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.
Citation: Aissa Guesmia, Nasser-eddine Tatar. Some well-posedness and stability results for abstract hyperbolic equations with infinite memory and distributed time delay. Communications on Pure & Applied Analysis, 2015, 14 (2) : 457-491. doi: 10.3934/cpaa.2015.14.457
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