Study of the stability of some elastic systems by a boundary feedback and a delay term

Abstract

The present thesis investigates the global well-posedness and the asymptotic behaviour of the global solution for some problems arising in Mathematical elasticity. The first model considered is an abstract viscoelastic equation that includes several PDEs of hyperbolic type such as the wave or plate equation. The second problem is a non-dissipative wave equation with memory-type boundary condition localized on a part of the boundary. The third system is the one-dimensional Timoshenko beam with a linear strong damping and a strong constant delay acting on the transverse displacement of the system. The fourth one is the Porous system subjected to a nonlinear delayed damping acting on the volume fraction equation. The last one is the Bresse system with three control boundary conditions and interior delay in all the three equations. Some well-posedness results are based on the semigroup theory, whereas the others are obtained by combining the Faedo-Galerkin’s procedure with some energies estimates. Furthermore, to study the solution’s asymptotic behavior we employ the multipiliers method which relies on the construction of a Lyapunov functional satisfying a proper differential inequality that leads to the desired stability estimate. For the first and the fourth problems, we use also some properties of convex functions and some techniques developped in these studies [22, 23, 102].