PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 135, Number 5, May 2007, Pages 1515–1522 S 0002-9939(07)08810-7 Article electronically published on January 9, 2007 ON THE EXPONENTIAL DECAY OF THE CRITICAL GENERALIZED KORTEWEG-DE VRIES EQUATION WITH LOCALIZED DAMPING F. LINARES AND A. F. PAZOTO (Communicated by David S. Tartakoff) Abstract. This paper is concerned with the asymptotic behavior of solutions of the critical generalized Korteweg-de Vries equation in a bounded interval with a localized damping term. Combining multiplier techniques and com- pactness arguments it is shown that the problem of exponential decay of the energy is reduced to prove the unique continuation property of weak solutions. A locally uniform stabilization result is derived. 1. Introduction We study the exponential decay of solutions of the critical generalized Korteweg- de Vries equation in the domain (0,L) under the presence of a localized damping, that is, (1.1) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u t + u x + u xxx + u 4 u x + a(x)u =0 in (0,L) × (0, ∞), u(0,t)= u(L, t)=0, t ∈ (0, ∞), u x (L, t)=0, t ∈ (0, ∞), u(x, 0) = u 0 (x), x ∈ (0,L). We assume that the real-valued function a = a(x) satisfies the condition (1.2)  a ∈ L ∞ (0,L) and a(x) ≥ a 0 > 0 a.e. in Ω, where Ω is a nonempty open subset of (0,L). Multiplying (1.1) by u and integrating in (0,L) we get dE dt = −  L 0 a(x)|u(x, t)| 2 dx − 1 2 |u x (0,t)| 2 (1.3) where E(t)= 1 2  L 0 |u(x, t)| 2 dx. (1.4) Received by the editors October 24, 2005 and, in revised form, February 24, 2006. 2000 Mathematics Subject Classification. Primary 93D15, 93B05, 35Q53. Key words and phrases. Exponential decay, stabilization, Korteweg-de Vries equation. The first author was partially supported by CNPq, Brazil. c 2007 American Mathematical Society Reverts to public domain 28 years from publication 1515 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use