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Unique continuation and decay for the Korteweg-de Vries equation withlocalized damping

Published online by Cambridge University Press:  15 July 2005

Ademir Fernando Pazoto
Ademir Fernando Pazoto*
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, PO Box 68530, CEP 21945-970, Rio de Janeiro, RJ, Brasil; ademir@acd.ufrj.br
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Abstract

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 Menzala (2002) which combines energy estimates, multipliers and compactness arguments the problem is reduced to prove the unique continuation of weak solutions. In Menzala (2002) 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 T. Kato (1983) and earlier results on unique continuation of smooth solutions by J.C. Saut and B. Scheurer (1987). 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.

Keywords

Unique continuationdecaystabilizationKdV equationlocalized damping.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 3 , July 2005 , pp. 473 - 486
Copyright
© EDP Sciences, SMAI, 2005

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References

Bisognin, E., Bisognin, V. and Menzala, G.P., Exponential stabilization of a coupled system of Korteweg-de Vries Equations with localized damping. Adv. Diff. Eq. 8 (2003) 443–469.
Coron, J. and Crepéau, E., Exact boundary controllability of a nonlinear KdV equation with critical lengths. J. Eur. Math. Soc. 6 (2004) 367–398. CrossRef
Dehman, B., Lebeau, G. and Zuazua, E., Stabilization and control for the subcritical semilinear wave equation. Ann. Sci. École Norm. Sup. 36 (2003) 525–551. CrossRef
Gear, J.A. and Grimshaw, R., Weak and strong interaction between internal solitary waves. Stud. Appl. Math. 70 (1984) 235–258. CrossRef
L. Hörmander, Linear partial differential operators. Springer Verlag, Berlin/New York (1976)
L. Hörmander, The analysis of linear partial differential operators (III-IV). Springer-Verlag, Berlin (1985).
O. Yu Imanuvilov and M. Yamamoto, Carleman estimate for a parabolic equation in Sobolev spaces of negative order and its applications, in Control of Nonlinear Distributed Parameter Systems, G. Chen et al. Eds. Marcel-Dekker (2001) 113–137.
T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Stud. Appl. Math. Adv., in Math. Suppl. Stud. 8 (1983) 93–128.
Korteweg, D.J. and de Vries, G., On the change of form of long waves advancing in a retangular canal, and on a new type of long stacionary waves. Philos. Mag. 39 (1895) 422–423. CrossRef
Kruzhkov, S.N. and Faminskii, A.V., Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation. Math. URSS Sbornik 38 (1984) 391–421. CrossRef
J. Lions, Contrôlabilité exacte, perturbations et stabilization de systèmes distribué, Tome 1, Contrôlabilité exacte, Colletion de Recherches en Mathématiques Appliquées, Masson, Paris 8 (1988).
G.P. Menzala, C.F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping. Quarterly Appl. Math. LX (2002) 111–129.
Menzala, G.P. and Zuazua, E., Decay rates for the von Kàrmàn system of thermoelastic plates. Diff. Int. Eq. 11 (1998) 755–770.
Rauch, J. and Taylor, M., Exponential decay of solutions to symmetric hyperbolic equations in bounded domains. Indiana J. Math. 24 (1974) 79–86. CrossRef
Rosier, L., Exact boundary controllability for the Korteweg-de Vries equation on a bonded domain. ESAIM: COCV 2 (1997) 33–55. CrossRef
Ruiz, A., Unique continuation for weak solutions of the wave equation plus a potential. J. Math. Pures Appl. 71 (1992) 455–467.
Saut, J.C. and Scheurer, B., Unique Continuation for some evolution equations. J. Diff. Equations 66 (1987) 118–139. CrossRef
J. Simon, Compact sets in the space Lp(0,T;B). Annali di Matematica Pura ed Appicata CXLVI (IV) (1987) 65–96.
F. Trêves, Linear Partial Differential Equations. Gordon and Breach, New York/London/Paris (1970).
Zhang, B.Y., Unique continuation for the Korteweg-de Vries equation. SIAM J. Math. Anal. 23 (1992) 55–71. CrossRef
Zhang, B.Y., Exact boundary controllability of the Kortewed-de Vries equation. SIAM J. Control Opt. 37 (1999) 543–565. CrossRef
E. Zuazua, Contrôlabilité exacte de quelques modèles de plaques en un temps arbitrairement petit, Appendix I in [11] 465–491.
Zuazua, E., Exponential decay for the semilinear wave equation with locally distributed damping. Comm. Partial Diff. Eq. 15 (1990) 205–235.
C. Zuily, Uniqueness and nonuniqueness in the Cauchy problem. Birkhäuser, Progr. Math. 33 (1983).