Indirect linear locally distributed damping of coupled systems

Annick Beyrath

doi:10.5269/bspm.v22i2.7478

Abstract

The aim of this paper is to prove indirect internal stabilization results for different coupled systems with linear locally distributed damping (coupled wave equations, wave equations with different speeds of propagation). In our case, a linear local damping term appears only in the first equation whereas no damping term is applied to the second one (this is indirect stabilization, see [11]). Using the piecewise multiplier method we prove that the full system is stabilized and that the total energy of the solution of this system decays polynomially.

Highlights

The aim of this paper is to prove indirect internal stabilization results for different coupled systems with linear locally distributed damping
The problem of stabilization of the wave equation in a bounded domain using a locally distributed damping has been studied by several authors
Martinez [12], using the piecewise multiplier method introduced by Liu [11], weakened the usual geometrical conditions on the localization of the damping

Summary

Indirect linear locally distributed damping of coupled systems

Annick BEYRATH abstract: The aim of this paper is to prove indirect internal stabilization results for different coupled systems with linear locally distributed damping (coupled wave equations, wave equations with different speeds of propagation). Using the piecewise multiplier method we prove that the full system is stabilized and that the total energy of the solution of this system decays polynomially. [3] studied the indirect internal stabilization of weakly coupled systems where the damping is effective in the whole domain. They prove that the behavior of the first equation is sufficient to stabilize the total system and to have polynomial decay for sufficiently smooth solutions. Our purpose in this paper is to study the indirect internal stabilization of coupled systems with a local damping term applied only to the first equation and to prove that the full system is polynomially stabilized. The existence and the regularity of the solution of (1.1) is given by the following theorem

Theorem solution U

Remark that

Ψj Ψj

Note that

Thanks to

We estimate the secSondωterm as dt