We study the stabilization of solutions of the Korteweg-de Vries (KdV) equation in a bounded interval under the effect of a localized damping mechanism. Using multiplier techniques we deduce the exponential decay in time of the solutions of the underlying linear equation. A locally uniform stabilization result of the solutions of the nonlinear KdV model is also proved. The proof combines compactness arguments, the smoothing effect of the KdV equation on the line and unique continuation results.

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.

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We analyze the stabilization and the exact controllability of a third order linear equation in a bounded interval. That is, we consider the following equation:where u ¼ uðx; tÞ is a complex valued function defined in ð0; LÞ Â ð0; þlÞ and a, b and g are real constants. Using multiplier techniques, HUM method and a special uniform continuation theorem, we prove the exponential decay of the total energy and the boundary exact controllability associated with the above equation. Moreover, we characterize a set of lengths L, named X, in which it is possible to find non null solutions for the above equation with constant (in time) energy and we show it depends strongly on the parameters a, b and g.