Stabilization of the Kawahara equation with localized damping

Abstract

Abstract.We study the stabilization of global solutions of the Kawahara (K) equation in a bounded interval, under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model. The proof requires of a unique continuation theorem and the smoothing effect of the (K) equation on the real line, which are proved in this work.Mathematics Subject Classification. 35Q35, 35B40, 35Q53.

xact Controllability and Stabilization for Kuramoto-Sivashinsky System

“…As in [10], [18] and [19], we shall use multipliers techniques. The origin of this method can be found in Zuazua [20], see also Komornik [4].…”

The Linear Kawahara Equation on the Half-Line

“…Dispersive problems have been object of intensive research, notably in exact controllability and the stabilization of the system. About the Kawahara system in bounded domain, we can consider, for instance, the following papers: [13,15,16] and references therein. In the case with periodic boundary conditions, see [14].…”

“…The natural idea to answer the above questions is to follow closely previous articles on Kawahara system as [15,16]. However, the present system is defined on the half-line and therefore others difficulties arise as the regularity of solution and the lack of compactness.…”

Critical set of the Kawahara equation

“…Recently global stabilization of the generalized KdV system have been obtained by Rosier and Zhang [10] and Linares and Pazoto [7] studied the stabilization of the generalized KdV system with critical exponents. For the stabilization of global solutions of the Kawahara under the effect of a localized damping mechanism, see Vasconcellos and Silva [11][12][13].…”

“…Sometimes, while discussing the existence of solutions of certain partial differential equations, it is necessary to establish when a certain quotient of entire functions still turns out to be an entire function (see, for instance, Rosier [9], Vasconcellos and Silva [11]). We have a polynomial p : C → C and a family of functions…”