Stabilization of the linear Kawahara equation with localized damping

Article type: Research Article

Authors: Vasconcellos, Carlos F. | da Silva, Patricia N.

Affiliations: Instituto de Matemática – UFRJ, P.O. Box 68530, CEP 21945-970, Rio de Janeiro, RJ, Brasil. E-mail: cfredvasc@gmail.com | Instituto de Matemática e Estatística – UERJ, R. São Francisco Xavier, 524, Sala 6006, Bloco D, CEP 20550-013, Rio de Janeiro, RJ, Brasil. E-mail: nunes@ime.uerj.br

Abstract: We study the stabilization of global solutions of the linear 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. We also prove that the decay of solutions, in absence of damping, fails for some critical values of the length L and we define precisely this countable set. Finally, we include some remarks about nonlinear problem and we analyze the exact boundary control for linear Kawahara system.

DOI: 10.3233/ASY-2008-0895

Journal: Asymptotic Analysis, vol. 58, no. 4, pp. 229-252, 2008