Well-posedness and stabilization theory for dispersive systems

Abstract

This work deals with the study of the well-posedness and stabilization of nonlinear disper- sive equations in bounded domains. We start by proving Massera-type theorems for the nonlinear Kawahara equation. More precisely, thanks to the properties of the semigroup of the linear operator associated with the equation studied and the exponential decay of the solutions of the linear system, it was possible to show that the solutions of the Kawahara equation are periodic and quasi-periodic. In a second moment, we study the stabilization problems of this same equation. Precisely, by introducing only one term of infinite memory in the Kawahara equation, which played a role as a damping mechanism, we guarantee the exponential stability of the system solutions. Furthermore, by designing a boundary feedback law for the Kawahara system, which combines a damping term and a finite memory term, we show that the energy associated with this system, with the presence of this feedback law, decays exponentially. Finally, we study another equation, namely, the fourth-order linear Schrödinger equation or biharmonic Schrödinger equa- tion. Here, adding an infinite memory term, we prove that the energy associated with this equation decays at polynomial-type rates.