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Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain
1. | Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Oh 45221, United States, United States |
2. | Department of Mathematics, University of Dayton, Dayton, OH 45431, United States |
References:
[1] |
J. L. Bona, S. Sun and B.-Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane,, Trans. American Math. Soc., 354 (2001), 427.
doi: 10.1090/S0002-9947-01-02885-9. |
[2] |
J. L. Bona, S. M. Sun and B.-Y. Zhang, Forced oscillations of a damped Korteweg-de Vries equation in a quarter plane,, Comm. Contemp. Math, 5 (2003), 369.
doi: 10.1142/S021919970300104X. |
[3] |
J. L. Bona, S. Sun and B.-Y. Zhang, A nonhomogeneous boundary value problem for the KdV equation posed on a finite domain,, Commun. Partial Differential Eq., 28 (2003), 1391.
doi: 10.1081/PDE-120024373. |
[4] |
J. L. Bona, S. Sun and B.-Y. Zhang, Non-homogeneous boundary value problems for the Korteweg-de Vries and the Korteweg-de Vries-Burgers equations in a quarter plane,, Annales de l'Institut Henri Poincaré - Analyse non linéaire, 25 (2008), 1145.
|
[5] |
J. L. Bona, S. Sun and B.-Y. Zhang, The Korteweg-de Vries equation on a finite domain II,, J. Diff. Eqns, 247 (2009), 2558.
doi: 10.1016/j.jde.2009.07.010. |
[6] |
B. A. Bubnov, Generalized boundary value problems for the Korteweg-de Vries equation in bounded domain,, Differential Equations, 15 (1979), 17.
|
[7] |
B. A. Bubnov, Solvability in the large of nonlinear boundary-value problem for the Korteweg-de Vries equations,, Differential Equations, 16 (1980), 24.
|
[8] |
J. Colliander and C. Kenig, The generalized Korteweg-de Vries equation on the half line,, Comm. Partial Differential Equations, 27 (2002), 2187.
doi: 10.1081/PDE-120016157. |
[9] |
T. Colin and J.-M. Ghidaglia, An initial-boundary-value problem for the Korteweg-de Vries Equation posed on a finite interval,, Adv. Differential Equations, 6 (2001), 1463.
|
[10] |
P. Constantin, C. Foias, B. Nicolaenko and R. Temam, "Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations,", Applied Mathematical Sciences, 70 (1989).
|
[11] |
W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equations,, Comm. Pure Appl. Math., 46 (1993), 1409.
doi: 10.1002/cpa.3160461102. |
[12] |
A. V. Faminskii, The Cauchy problem and the mixed problem in the half strip for equation of Korteweg-de Vries type,, (Russian) Dinamika Sploshn. Sredy, 162 (1983), 152.
|
[13] |
A. V. Faminskii, A mixed problem in a semistrip for the Korteweg-de Vries equation and its generalizations,, (Russian), 258 (1988), 54.
|
[14] |
A. V. Faminskii, Mixed problms for the Korteweg-de Vries equation,, Sbornik: Mathematics, 190 (1999), 903.
doi: 10.1070/SM1999v190n06ABEH000408. |
[15] |
A. V. Faminskii, On an initial boundary value problem in a bounded domain for the generalized Korteweg-de Vries equation,, International Conference on Differential and Functional Differential Equations, 8 (2001), 183.
|
[16] |
A. V. Faminskii, An initial boundary-value problem in a half-strip for the Korteweg-de Vries equation in fractional order Sobelev Spaces,, Comm. Partial Differential Eq., 29 (2004), 1653.
doi: 10.1081/PDE-200040191. |
[17] |
A. V. Faminskii, Global well-posedness of two initial-boundary-value problems for the Korteweg-de Vries equation,, Differential Integral Equations, 20 (2007), 601.
|
[18] |
J.-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite-dimensional dynamical system in the long time,, J. Differential Eqns., 74 (1988), 369.
doi: 10.1016/0022-0396(88)90010-1. |
[19] |
J.-M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations,, J. Differential Eqns., 110 (1994), 356.
doi: 10.1006/jdeq.1994.1071. |
[20] |
J. Holmer, The initial-boundary value problem for the Korteweg-de Vries equation,, Comm. Partial Differential Equations, 31 (2006), 1151.
doi: 10.1080/03605300600718503. |
[21] |
T. Kato, "Perturbation Theory for Linear Operators,", Dir Grundlehren der mathematischen Wissenschaften, 132 (1966). Google Scholar |
[22] |
J. B. Keller and L. Ting, Periodic vibrations of systems governed by nonlinear partial differential equations,, Comm. Pure and Appl. Math., 19 (1966), 371.
doi: 10.1002/cpa.3160190404. |
[23] |
J. U. Kim, Forced vibration of an aero-elastic plate,, J. Math. Anal. Appl., 113 (1986), 454.
doi: 10.1016/0022-247X(86)90317-3. |
[24] |
G. Kramer and B.-Y. Zhang, Nonhomogeneous boundary value problems of the KdV equation on a bounded domain,, Journal Syst. Sci. & Complexity, 23 (2010), 499.
doi: 10.1007/s11424-010-0143-x. |
[25] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).
|
[26] |
I. Rivas, G. Kramer and B.-Y. Zhang, Well-posedness of a class of initial-boundary-value problem for the Kortweg-de Vries equation on a bounded domain,, preprint, (). Google Scholar |
[27] |
P. H. Rabinowitz, Periodic solutions of nonlinear hyperbolic differential equations,, Comm. Pure Appl. Math., 20 (1967), 145.
doi: 10.1002/cpa.3160200105. |
[28] |
P. H. Rabinowitz, Free vibrations for a semi-linear wave equation,, Comm. Pure Appl. Math., 31 (1978), 31.
doi: 10.1002/cpa.3160310103. |
[29] |
G. R. Sell and Y. C. You, Inertial manifolds: the nonselfadjoint case,, J. Diff. Eqns., 96 (1992), 203.
doi: 10.1016/0022-0396(92)90152-D. |
[30] |
L. Tartar, Interpolation non linéaire et régularité,, J. Funct. Anal, 9 (1972), 469.
doi: 10.1016/0022-1236(72)90022-5. |
[31] |
O. Vejvoda, "Partial Differential Equations: Time-Periodic Solutions,", Mrtinus Nijhoff Publishers, (1981).
|
[32] |
C. E. Wayne, Periodic solutions of nonlinear partial differential equations,, Notices Amer. Math. Soc., 44 (1997), 895.
|
[33] |
M. Usman and B.-Y. Zhang, Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability,, J. Systems Sciences and Complexity, 20 (2007), 15.
doi: 10.1007/s11424-007-9025-2. |
[34] |
M. Usman and B.-Y. Zhang, Forced oscillations of a class of nonlinear dispersive wave equations and their stability,, Discrete and Continuous Dynamical Systems, 26 (2010), 1509.
|
[35] |
Y. Yang and B.-Y. Zhang, Forced oscillations of a damped Benjamin-Bona-Mahony equation in a quarter plane,, in, 242 (2005).
|
[36] |
B.-Y. Zhang, Forced oscillations of a regularized long-wave equation and their global stability,, in, (1999), 456. Google Scholar |
[37] |
B.-Y. Zhang, Forced oscillation of the Korteweg-de Vries-Burgers equation and its stability,, in, 218 (1999), 337.
|
show all references
References:
[1] |
J. L. Bona, S. Sun and B.-Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane,, Trans. American Math. Soc., 354 (2001), 427.
doi: 10.1090/S0002-9947-01-02885-9. |
[2] |
J. L. Bona, S. M. Sun and B.-Y. Zhang, Forced oscillations of a damped Korteweg-de Vries equation in a quarter plane,, Comm. Contemp. Math, 5 (2003), 369.
doi: 10.1142/S021919970300104X. |
[3] |
J. L. Bona, S. Sun and B.-Y. Zhang, A nonhomogeneous boundary value problem for the KdV equation posed on a finite domain,, Commun. Partial Differential Eq., 28 (2003), 1391.
doi: 10.1081/PDE-120024373. |
[4] |
J. L. Bona, S. Sun and B.-Y. Zhang, Non-homogeneous boundary value problems for the Korteweg-de Vries and the Korteweg-de Vries-Burgers equations in a quarter plane,, Annales de l'Institut Henri Poincaré - Analyse non linéaire, 25 (2008), 1145.
|
[5] |
J. L. Bona, S. Sun and B.-Y. Zhang, The Korteweg-de Vries equation on a finite domain II,, J. Diff. Eqns, 247 (2009), 2558.
doi: 10.1016/j.jde.2009.07.010. |
[6] |
B. A. Bubnov, Generalized boundary value problems for the Korteweg-de Vries equation in bounded domain,, Differential Equations, 15 (1979), 17.
|
[7] |
B. A. Bubnov, Solvability in the large of nonlinear boundary-value problem for the Korteweg-de Vries equations,, Differential Equations, 16 (1980), 24.
|
[8] |
J. Colliander and C. Kenig, The generalized Korteweg-de Vries equation on the half line,, Comm. Partial Differential Equations, 27 (2002), 2187.
doi: 10.1081/PDE-120016157. |
[9] |
T. Colin and J.-M. Ghidaglia, An initial-boundary-value problem for the Korteweg-de Vries Equation posed on a finite interval,, Adv. Differential Equations, 6 (2001), 1463.
|
[10] |
P. Constantin, C. Foias, B. Nicolaenko and R. Temam, "Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations,", Applied Mathematical Sciences, 70 (1989).
|
[11] |
W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equations,, Comm. Pure Appl. Math., 46 (1993), 1409.
doi: 10.1002/cpa.3160461102. |
[12] |
A. V. Faminskii, The Cauchy problem and the mixed problem in the half strip for equation of Korteweg-de Vries type,, (Russian) Dinamika Sploshn. Sredy, 162 (1983), 152.
|
[13] |
A. V. Faminskii, A mixed problem in a semistrip for the Korteweg-de Vries equation and its generalizations,, (Russian), 258 (1988), 54.
|
[14] |
A. V. Faminskii, Mixed problms for the Korteweg-de Vries equation,, Sbornik: Mathematics, 190 (1999), 903.
doi: 10.1070/SM1999v190n06ABEH000408. |
[15] |
A. V. Faminskii, On an initial boundary value problem in a bounded domain for the generalized Korteweg-de Vries equation,, International Conference on Differential and Functional Differential Equations, 8 (2001), 183.
|
[16] |
A. V. Faminskii, An initial boundary-value problem in a half-strip for the Korteweg-de Vries equation in fractional order Sobelev Spaces,, Comm. Partial Differential Eq., 29 (2004), 1653.
doi: 10.1081/PDE-200040191. |
[17] |
A. V. Faminskii, Global well-posedness of two initial-boundary-value problems for the Korteweg-de Vries equation,, Differential Integral Equations, 20 (2007), 601.
|
[18] |
J.-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite-dimensional dynamical system in the long time,, J. Differential Eqns., 74 (1988), 369.
doi: 10.1016/0022-0396(88)90010-1. |
[19] |
J.-M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations,, J. Differential Eqns., 110 (1994), 356.
doi: 10.1006/jdeq.1994.1071. |
[20] |
J. Holmer, The initial-boundary value problem for the Korteweg-de Vries equation,, Comm. Partial Differential Equations, 31 (2006), 1151.
doi: 10.1080/03605300600718503. |
[21] |
T. Kato, "Perturbation Theory for Linear Operators,", Dir Grundlehren der mathematischen Wissenschaften, 132 (1966). Google Scholar |
[22] |
J. B. Keller and L. Ting, Periodic vibrations of systems governed by nonlinear partial differential equations,, Comm. Pure and Appl. Math., 19 (1966), 371.
doi: 10.1002/cpa.3160190404. |
[23] |
J. U. Kim, Forced vibration of an aero-elastic plate,, J. Math. Anal. Appl., 113 (1986), 454.
doi: 10.1016/0022-247X(86)90317-3. |
[24] |
G. Kramer and B.-Y. Zhang, Nonhomogeneous boundary value problems of the KdV equation on a bounded domain,, Journal Syst. Sci. & Complexity, 23 (2010), 499.
doi: 10.1007/s11424-010-0143-x. |
[25] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).
|
[26] |
I. Rivas, G. Kramer and B.-Y. Zhang, Well-posedness of a class of initial-boundary-value problem for the Kortweg-de Vries equation on a bounded domain,, preprint, (). Google Scholar |
[27] |
P. H. Rabinowitz, Periodic solutions of nonlinear hyperbolic differential equations,, Comm. Pure Appl. Math., 20 (1967), 145.
doi: 10.1002/cpa.3160200105. |
[28] |
P. H. Rabinowitz, Free vibrations for a semi-linear wave equation,, Comm. Pure Appl. Math., 31 (1978), 31.
doi: 10.1002/cpa.3160310103. |
[29] |
G. R. Sell and Y. C. You, Inertial manifolds: the nonselfadjoint case,, J. Diff. Eqns., 96 (1992), 203.
doi: 10.1016/0022-0396(92)90152-D. |
[30] |
L. Tartar, Interpolation non linéaire et régularité,, J. Funct. Anal, 9 (1972), 469.
doi: 10.1016/0022-1236(72)90022-5. |
[31] |
O. Vejvoda, "Partial Differential Equations: Time-Periodic Solutions,", Mrtinus Nijhoff Publishers, (1981).
|
[32] |
C. E. Wayne, Periodic solutions of nonlinear partial differential equations,, Notices Amer. Math. Soc., 44 (1997), 895.
|
[33] |
M. Usman and B.-Y. Zhang, Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability,, J. Systems Sciences and Complexity, 20 (2007), 15.
doi: 10.1007/s11424-007-9025-2. |
[34] |
M. Usman and B.-Y. Zhang, Forced oscillations of a class of nonlinear dispersive wave equations and their stability,, Discrete and Continuous Dynamical Systems, 26 (2010), 1509.
|
[35] |
Y. Yang and B.-Y. Zhang, Forced oscillations of a damped Benjamin-Bona-Mahony equation in a quarter plane,, in, 242 (2005).
|
[36] |
B.-Y. Zhang, Forced oscillations of a regularized long-wave equation and their global stability,, in, (1999), 456. Google Scholar |
[37] |
B.-Y. Zhang, Forced oscillation of the Korteweg-de Vries-Burgers equation and its stability,, in, 218 (1999), 337.
|
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