The model under consideration in this work describes a vibrating structure of an interfacial slip and consists of three coupled hyperbolic equations in one-dimensional bounded interval under mixed homogeneous Dirichlet-Neumann boundary conditions. The first two equations are related to Timo-shenko type systems, and the third one is subject to the dynamics of the slip. The main problem we discuss here is stabilizing the system by a viscoelastic damping generated by an infinite memory and acting only on one equation. First, we prove the existence, uniqueness and regularity of solutions using the semigroup theory. After that, we combine the energy method and the frequency domain approach to show that the infinite memory is capable alone to guarantee the strong and polynomial stability of the model; that is bringing it back to its equilibrium state with a decay rate of type where d is a positive constant depending on the regularity of initial data. Moreover, we prove that, when the infinite memory is effective on the first equation, the model is not exponentially stable independently of the values of the parameters. However, when the infinite memory is effective on the second or the third equation, we prove that the exponential stability is equivalent to the equality between the three speeds of wave propagations. An extension of our results to the frictional damping case is also given. Our results improve and extend some existing results in the literature subject to other types of controls. These results were published in IMA J. Math. Control and Information (2019).
Dr. habil. Guesmia obtained his Master degree in Mathematics at Louis Pasteur University, Strasbourg, France in 07/02/2007, Doctor of Philosophy in Mathematics at Louis Pasteur University, Strasbourg, France in 13/02/2012 and Higher Degree Research in Mathematics at University of Lorraine, Metz, France in 08/02/2017. He started to work as a Teaching Assistant at Louis Pasteur University, Strasbourg, France in 1997, and after two years (in 1999) he became a Lecturer. He joined University of Lorraine, Metz, France in 2001 and since then is an Associate Professor of Mathematics. Guesmia was and Associate Professor at KFUPM, Dhahran, Saudi Arabia from 01/09/2012-31/08/2014 and since 01/09/2019 he is also an Adjunct Professor at KFUPM, Dhahran, Saudi Arabia.