The asymptotic behavior of some hyperbolic PDE systems

Abstract

The main goal of this dissertation is to discuss the asymptotic behavior of some hyperbolic PDE systems,precisely,we handled the well-posedness and stability of the solutions for the Moore-Gibson-Thompson equation(MGT)employing various types of dissipation.In fact, under an appropriate assumption on the coefficients of the systems together with the energy method in Fourier space we have proved the well-posedness of the systems and built some Lyapunov functionals which allowed us to get control estimates on the Fourier image of the solution and led to the decay rate of the L2-norm of the solution.On the other hand,by comparing the behavior of the resolvent of the Moore-Gibson-Thompson system with the one of the resolvent of the wave equation with a frictional interior damping,we furnish weaker conditions that guarantee exponential,polynomial or even logarithmic decay of the solution of the Moore-Gibson-Thompson system in a bounded domain.