# Unique continuation and decay for the Korteweg-de Vries equation with localized damping

ESAIM: Control, Optimisation and Calculus of Variations (2005)

- Volume: 11, Issue: 3, page 473-486
- ISSN: 1292-8119

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topPazoto, Ademir Fernando. "Unique continuation and decay for the Korteweg-de Vries equation with localized damping." ESAIM: Control, Optimisation and Calculus of Variations 11.3 (2005): 473-486. <http://eudml.org/doc/244711>.

@article{Pazoto2005,

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.},

author = {Pazoto, Ademir Fernando},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {unique continuation; decay; stabilization; KdV equation; localized damping; Unique continuation},

language = {eng},

number = {3},

pages = {473-486},

publisher = {EDP-Sciences},

title = {Unique continuation and decay for the Korteweg-de Vries equation with localized damping},

url = {http://eudml.org/doc/244711},

volume = {11},

year = {2005},

}

TY - JOUR

AU - Pazoto, Ademir Fernando

TI - Unique continuation and decay for the Korteweg-de Vries equation with localized damping

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2005

PB - EDP-Sciences

VL - 11

IS - 3

SP - 473

EP - 486

AB - 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.

LA - eng

KW - unique continuation; decay; stabilization; KdV equation; localized damping; Unique continuation

UR - http://eudml.org/doc/244711

ER -

## References

top- [1] E. Bisognin, V. Bisognin and G.P. Menzala, Exponential stabilization of a coupled system of Korteweg-de Vries Equations with localized damping. Adv. Diff. Eq. 8 (2003) 443–469. Zbl1057.35048
- [2] J. Coron and E. Crepéau, Exact boundary controllability of a nonlinear KdV equation with critical lengths. J. Eur. Math. Soc. 6 (2004) 367–398. Zbl1061.93054
- [3] B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation. Ann. Sci. École Norm. Sup. 36 (2003) 525–551. Zbl1036.35033
- [4] J.A. Gear and R. Grimshaw, Weak and strong interaction between internal solitary waves. Stud. Appl. Math. 70 (1984) 235–258. Zbl0548.76020
- [5] L. Hörmander, Linear partial differential operators. Springer Verlag, Berlin/New York (1976) Zbl0321.35001MR404822
- [6] L. Hörmander, The analysis of linear partial differential operators (III-IV). Springer-Verlag, Berlin (1985). Zbl0601.35001MR781536
- [7] 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. Zbl0977.93041
- [8] 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. Zbl0549.34001
- [9] D.J. Korteweg and G. de Vries, 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. Zbl26.0881.02JFM26.0881.02
- [10] S.N. Kruzhkov and A.V. Faminskii, Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation. Math. URSS Sbornik 38 (1984) 391–421. Zbl0549.35104
- [11] J. Lions, Contrôlabilité exacte, perturbations et stabilization de systèmes distribué, Tome 1, Contrôlabilité exacte, Colletion de Recherches en Mathématiques Appliquées, Masson, Paris 8 (1988). Zbl0653.93002MR953547
- [12] 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. Zbl1039.35107
- [13] G.P. Menzala and E. Zuazua, Decay rates for the von Kàrmàn system of thermoelastic plates. Diff. Int. Eq. 11 (1998) 755–770. Zbl1008.35077
- [14] J. Rauch and M. Taylor, Exponential decay of solutions to symmetric hyperbolic equations in bounded domains. Indiana J. Math. 24 (1974) 79–86. Zbl0281.35012
- [15] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bonded domain. ESAIM: COCV 2 (1997) 33–55. Zbl0873.93008
- [16] A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential. J. Math. Pures Appl. 71 (1992) 455–467. Zbl0832.35084
- [17] J.C. Saut and B. Scheurer, Unique Continuation for some evolution equations. J. Diff. Equations 66 (1987) 118–139. Zbl0631.35044
- [18] J. Simon, Compact sets in the space ${L}^{p}(0,T;B)$. Annali di Matematica Pura ed Appicata CXLVI (IV) (1987) 65–96. Zbl0629.46031
- [19] F. Trêves, Linear Partial Differential Equations. Gordon and Breach, New York/London/Paris (1970). Zbl0209.12001
- [20] B.Y. Zhang, Unique continuation for the Korteweg-de Vries equation. SIAM J. Math. Anal. 23 (1992) 55–71. Zbl0746.35045
- [21] B.Y. Zhang, Exact boundary controllability of the Kortewed-de Vries equation. SIAM J. Control Opt. 37 (1999) 543–565. Zbl0930.35160
- [22] E. Zuazua, Contrôlabilité exacte de quelques modèles de plaques en un temps arbitrairement petit, Appendix I in [11] 465–491.
- [23] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping. Comm. Partial Diff. Eq. 15 (1990) 205–235. Zbl0716.35010
- [24] C. Zuily, Uniqueness and nonuniqueness in the Cauchy problem. Birkhäuser, Progr. Math. 33 (1983). Zbl0521.35003MR701544

## Citations in EuDML Documents

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- Eugene Kramer, Ivonne Rivas, Bing-Yu Zhang, Well-posedness of a class of non-homogeneous boundary value problems of the Korteweg-de Vries equation on a finite domain

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