On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping

Linares, F.; Pazoto, A.

doi:10.1090/S0002-9939-07-08810-7

On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping
HTML articles powered by AMS MathViewer

by F. Linares and A. F. Pazoto PDF

Proc. Amer. Math. Soc. 135 (2007), 1515-1522 Request permission

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 compactness 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.

References

Jerry L. Bona, Shu Ming Sun, and Bing-Yu Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. Partial Differential Equations 28 (2003), no. 7-8, 1391–1436. MR 1998942, DOI 10.1081/PDE-120024373
Thierry Colin and Jean-Michel Ghidaglia, An initial-boundary value problem for the Korteweg-de Vries equation posed on a finite interval, Adv. Differential Equations 6 (2001), no. 12, 1463–1492. MR 1858429
Jean-Michel Coron and Emmanuelle Crépeau, Exact boundary controllability of a nonlinear KdV equation with critical lengths, J. Eur. Math. Soc. (JEMS) 6 (2004), no. 3, 367–398. MR 2060480, DOI 10.4171/JEMS/13
A. V. Faminskii, On an initial boundary value problem in a bounded domain for the generalized Korteweg-de Vries equation, Funct. Differ. Equ. 8 (2001), no. 1-2, 183–194. International Conference on Differential and Functional Differential Equations (Moscow, 1999). MR 1949998
J. Holmer, The initial-boundary value problem for the Korteweg-de Vries equation, preprint.
Tosio Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, Adv. Math. Suppl. Stud., vol. 8, Academic Press, New York, 1983, pp. 93–128. MR 759907
Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), no. 4, 527–620. MR 1211741, DOI 10.1002/cpa.3160460405
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969 (French). MR 0259693
G. Perla Menzala, C. F. Vasconcellos, and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quart. Appl. Math. 60 (2002), no. 1, 111–129. MR 1878262, DOI 10.1090/qam/1878262
Frank Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc. 14 (2001), no. 3, 555–578. MR 1824989, DOI 10.1090/S0894-0347-01-00369-1
Ademir Fernando Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping, ESAIM Control Optim. Calc. Var. 11 (2005), no. 3, 473–486. MR 2148854, DOI 10.1051/cocv:2005015
Lionel Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var. 2 (1997), 33–55. MR 1440078, DOI 10.1051/cocv:1997102
Lionel Rosier, Control of the surface of a fluid by a wavemaker, ESAIM Control Optim. Calc. Var. 10 (2004), no. 3, 346–380. MR 2084328, DOI 10.1051/cocv:2004012
L. Rosier, B.-Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, preprint 2005.
Jean-Claude Saut and Bruno Scheurer, Unique continuation for some evolution equations, J. Differential Equations 66 (1987), no. 1, 118–139. MR 871574, DOI 10.1016/0022-0396(87)90043-X
Jacques Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, DOI 10.1007/BF01762360
Bing Yu Zhang, Unique continuation for the Korteweg-de Vries equation, SIAM J. Math. Anal. 23 (1992), no. 1, 55–71. MR 1145162, DOI 10.1137/0523004
Enrike Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations 15 (1990), no. 2, 205–235. MR 1032629, DOI 10.1080/03605309908820684