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.
Highlights
Aissa GuesmiaDepartment of Mathematics and Statistics, College of Sciences King Fahd University of Petroleum and Minerals, P.O.Box. 5005, Dhahran 31261, Saudi Arabia and Elie Cartan Institute of Lorraine, UMR 7502 University of Lorraine, Bat. A, Ile du Saulcy, 57045 Metz Cedex 01, France Department of Mathematics and Statistics, College of Sciences King Fahd University of Petroleum and Minerals, P.O.Box. 5005, Dhahran 31261, Saudi Arabia (Communicated by Alain Miranville)
Let H be a real Hilbert space with inner product and related norm denoted by ·, · and ·, respectively
We refer the reader to [13, 12] and [38] for the one-dimensional wave equation with internal and/or boundary feedbak and constant discrete time delay, [3, 33, 34, 35] and [36] for the N -dimensional case, and [15] and [37] for an abstract system with constant or variable discrete time delay. These results show that the damping But(t) is strong enough to stabilize the system (5) in presence of a constant discrete time delay provided that |μ| is small enough
Summary
Department of Mathematics and Statistics, College of Sciences King Fahd University of Petroleum and Minerals, P.O.Box. 5005, Dhahran 31261, Saudi Arabia and Elie Cartan Institute of Lorraine, UMR 7502 University of Lorraine, Bat. A, Ile du Saulcy, 57045 Metz Cedex 01, France Department of Mathematics and Statistics, College of Sciences King Fahd University of Petroleum and Minerals, P.O.Box. 5005, Dhahran 31261, Saudi Arabia (Communicated by Alain Miranville)
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