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Zbl 1148.35348
Pazoto, Ademir Fernando
Unique continuation and decay for the Korteweg-de Vries equation with localized damping. (English)
[J] ESAIM, Control Optim. Calc. Var. 11, 473-486 (2005). ISSN 1292-8119; ISSN 1262-3377/e

Summary: This work is devoted to prove the exponential decay for the energy of solutions of the Korteweg-de Vries equation in a bounded interval with a localized damping term. Following the method in {\it G. Perla Menzala, C. F. Vasconcellos} and {\it E. Zuazua} [Q. Appl. Math. 60, No. 1, 111--129 (2002; Zbl 1039.35107)] which combines energy estimates, multipliers and compactness arguments the problem is reduced to prove the unique continuation of weak solutions. In Perla Menzala (loc. cit.) the case where solutions vanish on a neighborhood of both extremes of the bounded interval where equation holds was solved combining the smoothing results by {\it T. Kato} [Adv. Math., Suppl. Stud. 8, 93--128 (1983; Zbl 0549.34001)] and earlier results on unique continuation of smooth solutions by {\it J. C. Saut} and {\it B. Scheurer} [J. Differ. Equ. 66, No. 1, 118--139 (1987; Zbl 0631.35044 )]. In this article we address the general case and prove the unique continuation property in two steps. We first prove, using multiplier techniques, that solutions vanishing on any subinterval are necessarily smooth. We then apply the existing results on unique continuation of smooth solutions.

MSC 2000:: *35Q53 KdV-like equations
35B40 Asymptotic behavior of solutions of PDE
35B60 Continuation of solutions of PDE

Keywords: Unique continuation; decay; stabilization; KdV equation; localized damping

Citations: Zbl 1039.35107; Zbl 0549.34001; Zbl 0631.35044

Cited in: Zbl 1114.93080

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