Hostname: page-component-7479d7b7d-rvbq7 Total loading time: 0 Render date: 2024-07-09T11:33:32.414Z Has data issue: false hasContentIssue false
  • English
  • Français

Stabilization of Regular Solutions for the Zakharov-Kuznetsov Equation Posed on Bounded Rectangles and on a Strip

Published online by Cambridge University Press:  10 April 2015

G. G. Doronin  and
N. A. Larkin
G. G. Doronin
Affiliation:
Departamento de Matemática, Universidade Estadual de Maringá, 87020-900 Maringá, Paraná, Brazil (ggdoronin@uem.br; nlarkine@uem.br)
N. A. Larkin
Affiliation:
Departamento de Matemática, Universidade Estadual de Maringá, 87020-900 Maringá, Paraná, Brazil (ggdoronin@uem.br; nlarkine@uem.br)
Get access Rights & Permissions [Opens in a new window]

Abstract

Initial–boundary-value problems for the two-dimensional Zakharov–Kuznetsov equation posed on bounded rectangles and on a strip are considered. Spectral properties of a linearized operator and critical sizes of domains are studied. An exponential decay rate of regular solutions for the original nonlinear problems is proved.

Keywords

ZK equationstabilizationexponential decaycritical domainsPrimary 35Q5335B35
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 3 , October 2015 , pp. 661 - 682
Copyright
Copyright © Edinburgh Mathematical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bona, J. L. and Smith, R. W., The initial-value problem for the Korteweg-de Vries equation, Phil. Trans. R. Soc. Lond. A 278 (1975), 555–601.Google Scholar
2.Bona, J. L., Sun, S. M. and Zhang, B.-Y., A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Commun. PDEs 28 (2003), 1391–1436.Google Scholar
3.Bona, J. L., Sun, S. M. and Zhang, B.-Y., Nonhomogeneous problems for the Korteweg-de Vries and the Korteweg-de Vries-Burgers equations in a quarter plane, Annales Inst. H. Poincaré Analyse Non Linéaire 25 (2008), 1145–1185.Google Scholar
4.Bourgain, J., On the compactness of the support of solutions of dispersive equations, Int. Math. Res. Not. 9 (1997), 437–447.CrossRefGoogle Scholar
5.Bubnov, B. A., Solvability in the large of nonlinear boundary-value problems for the Korteweg-de Vries equation in a bounded domain, Diff. Eqns 16 (1980), 24–31.Google Scholar
6.Colin, T. and Ghidaglia, J.-M., An initial-boundary-value problem for the Korteweg-de Vries equation posed on a finite interval, Adv. Diff. Eqns 6 (2001), 1463–1492.Google Scholar
7.Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T., Sharp global well-posedness results for periodic and non-periodic KdV and modified KdV on R and T, J. Am. Math. Soc. 16 (2003), 705–749.CrossRefGoogle Scholar
8.Coron, J.-M., Control and nonlinearity, Mathematical Surveys and Monographs, Volume 136 (American Mathematical Society, Providence, RI, 2007).Google Scholar
9.Doronin, G. G. and Larkin, N. A., KdV equation in domains with moving boundaries, J. Math. Analysis Applic. 328 (2007), 503–515.CrossRefGoogle Scholar
10.Faminskii, A. V., The Cauchy problem for the Zakharov-Kuznetsov equation, Diff. Eqns 31 (1995), 1002–1012.Google Scholar
11.Faminskii, A. V., Well-posed initial-boundary value problems for the Zakharov-Kuznetsov equation, Electron. J. Diff. Eqns 127 (2008), 1–23.Google Scholar
12.Faminskii, A. V. and Larkin, N. A., Initial-boundary value problems for quasilinear dispersive equations posed on a bounded interval, Electron. J. Diff. Eqns 2010 (2010), 1–20.Google Scholar
13.Farah, L. G., Linares, F. and Pastor, A., A note on the 2D generalized Zakharov-Kuznetsov equation: local, global, and scattering results, J. Diff. Eqns 253 (2012), 2558–2571.Google Scholar
14.Glass, O. and Guerrero, S., Controllability of the Korteweg-de Vries equation from the right Dirichlet boundary condition, Syst. Contr. Lett. 59(7 (2010), 390–395.CrossRefGoogle Scholar
15.Kato, T., On the Cauchy problem for the (generalized) Korteweg-de Vries equations, in Studies in applied mathematics, Advanced Mathematics: Supplementary Studies, Volume 8, pp. 93–128 (Academic, 1983).Google Scholar
16.Kenig, C. E., Ponce, G. and Vega, L., Well-posedness and scattering results for the generalized Korteweg-de Vries equation and the contraction principle, Commun. Pure Appl. Math. 46 (1993), 527–620.CrossRefGoogle Scholar
17.Kruzhkov, S. N. and Faminskii, A. V., Generalized solutions of the Cauchy problem for the Korteweg–de Vries equation, Mat. Sb. 48 (1984), 391–421.CrossRefGoogle Scholar
18.Ladyzhenskaya, O. A., The boundary value problems of mathematical physics, Applied Mathematical Sciences, Volume 49 (Springer, 1985).Google Scholar
19.Ladyzhenskaya, O. A., Solonnikov, V. A. and Uraltseva, N. N., Linear and quasilinear equations of parabolic type (American Mathematical Society, Providence, RI, 1968).Google Scholar
20.Larkin, N. A., Vries, Korteweg–de and Kuramoto–Sivashinsky equations in bounded domains, J. Math. Analysis Applic. 297 (2004), 169–185.Google Scholar
21.Larkin, N. A. and Tronco, E., Nonlinear quarter-plane problem for the Korteweg–de Vries equation, Electron. J. Diff. Eqns 2011 (2011), 1–22.Google Scholar
22.Larkin, N. A. and Tronco, E., Regular solutions of the 2D Zakharov–Kuznetsov equation on a half-strip, J. Diff. Eqns 254 (2013), 81–101.CrossRefGoogle Scholar
23.Linares, F. and Pastor, A., Local and global well-posedness for the 2D generalized Zakharov–Kuznetsov equation, J. Funct. Analysis 260 (2011), 1060–1085.Google Scholar
24.Linares, F. and Pazoto, A. F., Asymptotic behavior of the Korteweg–de Vries equation posed in a quarter plane, J. Diff. Eqns 246 (2009), 1342–1353.Google Scholar
25.Linares, F. and Saut, J.-C., The Cauchy problem for the 3D Zakharov–Kuznetsov equation, Discrete Contin. Dynam. Syst. A 24 (2009), 547–565.CrossRefGoogle Scholar
26.Linares, F., Pastor, A. and Saut, J.-C., Well-posedness for the ZK equation in a cylinder and on the background of a KdV Soliton, Commun. PDEs 35 (2010), 1674–1689.Google Scholar
27.Perla Menzala, G., Vasconcellos, C. F. and Zuazua, E., Stabilization of the Korteweg–de Vries equation with localized damping, Q. Appl. Math. 60 (2002), 111–129.Google Scholar
28.Rivas, I., Usman, M. and Zhang, B.-Y., Global well-posedness and asymptotic behavior of a class of initial-boundary value problem for the Korteweg–de Vries equation on a finite domain, Math. Control Related Fields 1 (2011), 61–81.Google Scholar
29.Rosier, L., Exact boundary controllability for the Korteweg–de Vries equation on a bounded domain, ESAIM: Control Optim. Calc. Variations 2 (1997), 33–55.Google Scholar
30.Rosier, L., A survey of controllability and stabilization results for partial differential equations, J. Eur. Syst. Autom. 41 (2007), 365–411.Google Scholar
31.Rosier, L. and Zhang, B.-Y., Control and stabilization of the KdV equation: recent progress, J. Syst. Sci. Complexity 22 (2009), 647–682.Google Scholar
32.Saut, J. C., Sur quelques généralisations de l’équation de Korteweg–de Vries, J. Math. Pures Appl. 58 (1979), 21–61 (in French).Google Scholar
33.Saut, J.-C. and Temam, R., An initial boundary-value problem for the Zakharov–Kuznetsov equation, Adv. Diff. Eqns 15 (2010), 1001–1031.Google Scholar
34.Saut, J.-C., Temam, R. and Wang, C., An initial and boundary-value problem for the Zakharov–Kuznetsov equation in a bounded domain, J. Math. Phys. 53 (2012), 115612.CrossRefGoogle Scholar
35.Temam, R., Sur un problème non Linéaire, J. Math. Pures Appl. 48 (1969), 159–172 (in French).Google Scholar
36.Zakharov, V. E. and Kuznetsov, E. A., On three-dimensional solitons, Sov. Phys. JETP 39 (1974), 285–286.Google Scholar
37.Zhang, B.-Y., Exact boundary controllability of the Korteweg–de Vries equation, SIAM J. Control Optim. 37 (1999), 543–565.CrossRefGoogle Scholar