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## Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line (2010)

### Citations

609 | Problemes aux limites non homogenes et applications. - Lions, Magenes - 1968 |

215 |
Vries, On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves
- Korteweg, de
(Show Context)
Citation Context ...ay, Korteweg-de Vries equation, Stabilization. 1 Introduction The Korteweg-de Vries (KdV) equation was first derived as a model for the propagation of small amplitude long water waves along a channel =-=[9, 16, 17]-=-. It has been intensively studied from various aspects for both mathematics and physics since the 1960s when solitons were discovered through solving the KdV equation, and the inverse scattering metho... |

152 |
On the Cauchy problem for the (generalized) Korteweg-de Vries equations,”
- Kato
- 1983
(Show Context)
Citation Context ...-valued function a = a(x) satisfies the condition (4) for some given positive numbers a0, x0. In this paper we investigate the stability properties of (1) in the weighted spaces introduced by Kato in =-=[15]-=-. More precisely, for b > 0 and m ∈ N, we prove that the solution u exponentially decays to 0 in L2b and L 2 (x+1)mdx (if u(0) belongs to one of these spaces), where L2b = {u : R+ → R; ∫ ∞ 0 |u(x)|2e2... |

87 |
The Korteweg–de Vries equation: a survey of results.
- Miura
- 1976
(Show Context)
Citation Context ...s and physics since the 1960s when solitons were discovered through solving the KdV equation, and the inverse scattering method, a so-called nonlinear Fourier transform, was invented to seek solitons =-=[14, 22]-=-. It is now well known that the KdV equation is not only a good model for water waves but also a very useful approximation model in nonlinear studies whenever one wishes to include and balance weak no... |

76 |
An evaluation of a model equation for water waves,
- BONA, PRITCHARD, et al.
- 1981
(Show Context)
Citation Context ...e and balance weak nonlinear and dispersive effects. The initial boundary value problems (IBVP) arise naturally in modeling small-amplitude long waves in a channel with a wavemaker mounted at one end =-=[1, 2, 3, 29]-=-. Such mathematical formulations have received considerable attention in the past, and a satisfactory theory of global wellposedness is available for initial and boundary conditions satisfying physica... |

69 |
Essai sur la théorie des eaux courantes. Mémories présentés par divers savants à l’Académie des Sciences (Paris)
- Boussinesq
- 1877
(Show Context)
Citation Context ...ay, Korteweg-de Vries equation, Stabilization. 1 Introduction The Korteweg-de Vries (KdV) equation was first derived as a model for the propagation of small amplitude long water waves along a channel =-=[9, 16, 17]-=-. It has been intensively studied from various aspects for both mathematics and physics since the 1960s when solitons were discovered through solving the KdV equation, and the inverse scattering metho... |

66 |
Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain
- Rosier
- 1997
(Show Context)
Citation Context ...in (0, δ) for some δ > 0, but (6) may be dropped by replacing the unique continuation property [20, Lemma 2.4] by [30, Theorem 1.6]. The exponential decay of E(t) is obtained following the methods in =-=[23, 25, 26]-=- which combine multiplier techniques and compactness arguments to reduce the problem to some unique continuation property for weak solutions of KdV. Along this work we assume that the real-valued func... |

57 |
A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain,
- Bona, Sun, et al.
- 2003
(Show Context)
Citation Context ...tention in the past, and a satisfactory theory of global wellposedness is available for initial and boundary conditions satisfying physically relevant smoothness and consistency assumptions (see e.g. =-=[1, 4, 6, 7, 11, 12, 13]-=- and the references therein). The analysis of the long-time behavior of IBVP on the quarter-plane for KdV has also received considerable attention over recent years, and a review of some of the result... |

30 |
Generalized solutions of the Cauchy problem for the Kortewegde Vries equation
- Kruzhkov, Faminskii
- 1984
(Show Context)
Citation Context ... KdV equation of the extra terms ux and a(x)u does not cause any serious trouble. On the other hand, choosing a cut-off function in x of the form η(x) = ψ0(x/ε) (instead of η(x) = ψ0(x− x0 + 2) as in =-=[18]-=-) where ψ0 ∈ C∞(R, [0, 1]) satisfies ψ0(x) = 0 for x ≤ 1/2 and ψ0(x) = 1 for x ≥ 1, allows to overcome the fact that u is a solution of (1) on the half-line only. 3.2 Decay in L2b This section is devo... |

30 |
Stabilization of the Korteweg-de Vries equation with localized damping,
- Menzala, Vasconcellos, et al.
- 2002
(Show Context)
Citation Context ...in (0, δ) for some δ > 0, but (6) may be dropped by replacing the unique continuation property [20, Lemma 2.4] by [30, Theorem 1.6]. The exponential decay of E(t) is obtained following the methods in =-=[23, 25, 26]-=- which combine multiplier techniques and compactness arguments to reduce the problem to some unique continuation property for weak solutions of KdV. Along this work we assume that the real-valued func... |

27 |
Control of the surface of a fluid by a wavemaker
- Rosier
(Show Context)
Citation Context ...e and balance weak nonlinear and dispersive effects. The initial boundary value problems (IBVP) arise naturally in modeling small-amplitude long waves in a channel with a wavemaker mounted at one end =-=[1, 2, 3, 29]-=-. Such mathematical formulations have received considerable attention in the past, and a satisfactory theory of global wellposedness is available for initial and boundary conditions satisfying physica... |

24 |
A mathematical model for long waves generated by wavemakers in non-linear dispersive systems,
- Bona, Bryant
- 1973
(Show Context)
Citation Context ...e and balance weak nonlinear and dispersive effects. The initial boundary value problems (IBVP) arise naturally in modeling small-amplitude long waves in a channel with a wavemaker mounted at one end =-=[1, 2, 3, 29]-=-. Such mathematical formulations have received considerable attention in the past, and a satisfactory theory of global wellposedness is available for initial and boundary conditions satisfying physica... |

24 |
The generalized Korteweg-de Vries equation on the half line.
- Colliander, Kenig
- 2002
(Show Context)
Citation Context ...tention in the past, and a satisfactory theory of global wellposedness is available for initial and boundary conditions satisfying physically relevant smoothness and consistency assumptions (see e.g. =-=[1, 4, 6, 7, 11, 12, 13]-=- and the references therein). The analysis of the long-time behavior of IBVP on the quarter-plane for KdV has also received considerable attention over recent years, and a review of some of the result... |

22 | Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line
- Rosier
(Show Context)
Citation Context ...s, and a review of some of the results related to the issues we address here can be found in [5, 7, 19]. For stabilization and controllability issues on the half line, we refer the reader to [20] and =-=[27, 28]-=-, respectively. In this work, we are concerned with the asymptotic behavior of the solutions of the IBVP for the KdV equation posed on the positive half line under the presence of a localized damping ... |

21 |
Unique continuation and decay for the Korteweg-de Vries equation with localized damping
- Pazoto
(Show Context)
Citation Context ...in (0, δ) for some δ > 0, but (6) may be dropped by replacing the unique continuation property [20, Lemma 2.4] by [30, Theorem 1.6]. The exponential decay of E(t) is obtained following the methods in =-=[23, 25, 26]-=- which combine multiplier techniques and compactness arguments to reduce the problem to some unique continuation property for weak solutions of KdV. Along this work we assume that the real-valued func... |

21 | Global stabilization of the generalized Korteweg-de Vries equation - Rosier, Zhang |

16 |
Miura: Method for solving the Korteweg–de Vries equation
- Gardner, Greene, et al.
- 1967
(Show Context)
Citation Context ...s and physics since the 1960s when solitons were discovered through solving the KdV equation, and the inverse scattering method, a so-called nonlinear Fourier transform, was invented to seek solitons =-=[14, 22]-=-. It is now well known that the KdV equation is not only a good model for water waves but also a very useful approximation model in nonlinear studies whenever one wishes to include and balance weak no... |

11 | Forced oscillations of a damped Kortewegde Vries equation in a quarter plane,”
- Bona, Sun, et al.
- 2003
(Show Context)
Citation Context ...ime behavior of IBVP on the quarter-plane for KdV has also received considerable attention over recent years, and a review of some of the results related to the issues we address here can be found in =-=[5, 7, 19]-=-. For stabilization and controllability issues on the half line, we refer the reader to [20] and [27, 28], respectively. In this work, we are concerned with the asymptotic behavior of the solutions of... |

11 | Stabilization of a Boussinesq system of KdV-KdV type - Pazoto, Rosier |

9 |
Boundary smoothing properties of the Korteweg-de Vries equation in a quarter plane and applications,
- Bona, Sun, et al.
- 2006
(Show Context)
Citation Context ...tention in the past, and a satisfactory theory of global wellposedness is available for initial and boundary conditions satisfying physically relevant smoothness and consistency assumptions (see e.g. =-=[1, 4, 6, 7, 11, 12, 13]-=- and the references therein). The analysis of the long-time behavior of IBVP on the quarter-plane for KdV has also received considerable attention over recent years, and a review of some of the result... |

9 |
Nonhomogeneous problems for the Korteweg-de Vries and the Korteweg-de Vries-Burgers equations in a quarter plane
- Bona, Sun, et al.
(Show Context)
Citation Context |

9 |
An initial boundary-value problem in a half-strip for the Korteweg-de Vries equation in fractional-order Sobolev spaces.
- Faminskii
- 2004
(Show Context)
Citation Context |

9 |
The large-time development of the solution to an initial-value problem for the Korteweg-de Vries equation. I. Initial data has a discontinuous expansive step, Nonlinearity 21
- Leach, Needham
- 2008
(Show Context)
Citation Context ...ime behavior of IBVP on the quarter-plane for KdV has also received considerable attention over recent years, and a review of some of the results related to the issues we address here can be found in =-=[5, 7, 19]-=-. For stabilization and controllability issues on the half line, we refer the reader to [20] and [27, 28], respectively. In this work, we are concerned with the asymptotic behavior of the solutions of... |

8 | Rapid exponential stabilization for a linear Korteweg-de Vries equation, Discrete Contin - Cerpa, Crépeau |

6 |
A mixed problem in a semistrip for the Korteweg-de Vries equation and its generalizations
- Faminskii
- 1988
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Citation Context |

6 |
Asymptotic behavior of the Korteweg-de Vries equation posed in a quarter plane
- Linares, Pazoto
(Show Context)
Citation Context ...† February 3, 2010 Abstract Studied here is the large-time behavior of solutions of the Korteweg-de Vries equation posed on the right half-line under the effect of a localized damping. Assuming as in =-=[20]-=- that the damping is active on a set (a0,+∞) with a0 > 0, we establish the exponential decay of the solutions in the weighted spaces L2((x + 1)mdx) for m ∈ N∗ and L2(e2bxdx) for b > 0 by a Lyapunov ap... |

4 |
On the origin of the Korteweg-de Vries equation
- Jager
(Show Context)
Citation Context ...ay, Korteweg-de Vries equation, Stabilization. 1 Introduction The Korteweg-de Vries (KdV) equation was first derived as a model for the propagation of small amplitude long water waves along a channel =-=[9, 16, 17]-=-. It has been intensively studied from various aspects for both mathematics and physics since the 1960s when solitons were discovered through solving the KdV equation, and the inverse scattering metho... |

4 |
A fundamental solution supported in a strip for a dispersive equation
- Rosier
(Show Context)
Citation Context ...s, and a review of some of the results related to the issues we address here can be found in [5, 7, 19]. For stabilization and controllability issues on the half line, we refer the reader to [20] and =-=[27, 28]-=-, respectively. In this work, we are concerned with the asymptotic behavior of the solutions of the IBVP for the KdV equation posed on the positive half line under the presence of a localized damping ... |

2 |
The Korteweg-de Vries equation,posed in a quarter-plane
- Bona, Winther
- 1983
(Show Context)
Citation Context |

2 | and Jiahong Wu, Temporal growth and eventual periodicity for dispersive wave equations in a quarter plane, Discrete Contin - Bona |