Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain

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March 2011, 1(1): 61-81. doi: 10.3934/mcrf.2011.1.61

Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain

Ivonne Rivas ^1, , Muhammad Usman ^2, and Bing-Yu Zhang ^1,

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

Received October 2010 Revised January 2011 Published March 2011

Abstract
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In this paper, we study a class of initial boundary value problem (IBVP) of the Korteweg-de Vries equation posed on a finite interval with nonhomogeneous boundary conditions. The IBVP is known to be locally well-posed, but its global $L^2$- a priori estimate is not available and therefore it is not clear whether its solutions exist globally or blow up in finite time. It is shown in this paper that the solutions exist globally as long as their initial value and the associated boundary data are small, and moreover, those solutions decay exponentially if their boundary data decay exponentially.

Keywords: Korteweg-de Vries equation, Global well-posedness, asymptotic behavior..

Mathematics Subject Classification: Primary: 35Q53, 35Q35; Secondary: 35S1.

Citation: Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control and Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61

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), 427-490. 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), 369-400. 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), 1391-1436. 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), 1145-1185.
[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), 2558-2596. 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), 17-21.
[7]	B. A. Bubnov, Solvability in the large of nonlinear boundary-value problem for the Korteweg-de Vries equations, Differential Equations, 16 (1980), 24-30.
[8]	J. Colliander and C. Kenig, The generalized Korteweg-de Vries equation on the half line, Comm. Partial Differential Equations, 27 (2002), 2187-2266. 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), 1463-1492.
[10]	P. Constantin, C. Foias, B. Nicolaenko and R. Temam, "Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations," Applied Mathematical Sciences, 70, Springer-Verlag, New York-Berlin, 1989.
[11]	W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), 1409-1498. 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), 152-158.
[13]	A. V. Faminskii, A mixed problem in a semistrip for the Korteweg-de Vries equation and its generalizations, (Russian), Dinamika Sploshn. Sredy, 258 (1988), 54-94; (English transl. in Trans. Moscow Math. Soc., 51 (1989), 53-91).
[14]	A. V. Faminskii, Mixed problms for the Korteweg-de Vries equation, Sbornik: Mathematics, 190 (1999), 903-935. 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, (Moscow, 1999), Funct. Differ. Equ., 8 (2001), 183-194.
[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), 1653-1695. 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), 601-642.
[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), 369-390. 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), 356-359. 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), 1151-1190. doi: 10.1080/03605300600718503.
[21]	T. Kato, "Perturbation Theory for Linear Operators," Dir Grundlehren der mathematischen Wissenschaften, 132, Springer, New York, 1966.
[22]	J. B. Keller and L. Ting, Periodic vibrations of systems governed by nonlinear partial differential equations, Comm. Pure and Appl. Math., 19 (1966), 371-420. doi: 10.1002/cpa.3160190404.
[23]	J. U. Kim, Forced vibration of an aero-elastic plate, J. Math. Anal. Appl., 113 (1986), 454-467. 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), 499-526. doi: 10.1007/s11424-010-0143-x.
[25]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 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, arXiv:math/1012.1057.
[27]	P. H. Rabinowitz, Periodic solutions of nonlinear hyperbolic differential equations, Comm. Pure Appl. Math., 20 (1967), 145-205. doi: 10.1002/cpa.3160200105.
[28]	P. H. Rabinowitz, Free vibrations for a semi-linear wave equation, Comm. Pure Appl. Math., 31 (1978), 31-68. doi: 10.1002/cpa.3160310103.
[29]	G. R. Sell and Y. C. You, Inertial manifolds: the nonselfadjoint case, J. Diff. Eqns., 96 (1992), 203-255. doi: 10.1016/0022-0396(92)90152-D.
[30]	L. Tartar, Interpolation non linéaire et régularité, J. Funct. Anal, 9 (1972), 469-489. 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), 895-902.
[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), 15-24. 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), 1509-1523.
[35]	Y. Yang and B.-Y. Zhang, Forced oscillations of a damped Benjamin-Bona-Mahony equation in a quarter plane, "Control Theory of Partial Differential Equations," Lect. Notes Pure Appl. Math., 242, Chapman & Hall/CRC, Boca Raton, FL, 2005, 375-386.
[36]	B.-Y. Zhang, Forced oscillations of a regularized long-wave equation and their global stability, in "Differential Equations and Computational Simulations" (Chengdu, 1999), World Sci. Publishing, River Edge, NJ, 2000, 456-463.
[37]	B.-Y. Zhang, Forced oscillation of the Korteweg-de Vries-Burgers equation and its stability, in "Control of Nonlinear Distributed Parameter Systems" (College Station, TX, 1999), Lecture Notes in Pure and Appl. Math., 218, Dekker, New York, 2001, 337-357.

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), 427-490. 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), 369-400. 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), 1391-1436. 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), 1145-1185.
[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), 2558-2596. 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), 17-21.
[7]	B. A. Bubnov, Solvability in the large of nonlinear boundary-value problem for the Korteweg-de Vries equations, Differential Equations, 16 (1980), 24-30.
[8]	J. Colliander and C. Kenig, The generalized Korteweg-de Vries equation on the half line, Comm. Partial Differential Equations, 27 (2002), 2187-2266. 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), 1463-1492.
[10]	P. Constantin, C. Foias, B. Nicolaenko and R. Temam, "Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations," Applied Mathematical Sciences, 70, Springer-Verlag, New York-Berlin, 1989.
[11]	W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math., 46 (1993), 1409-1498. 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), 152-158.
[13]	A. V. Faminskii, A mixed problem in a semistrip for the Korteweg-de Vries equation and its generalizations, (Russian), Dinamika Sploshn. Sredy, 258 (1988), 54-94; (English transl. in Trans. Moscow Math. Soc., 51 (1989), 53-91).
[14]	A. V. Faminskii, Mixed problms for the Korteweg-de Vries equation, Sbornik: Mathematics, 190 (1999), 903-935. 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, (Moscow, 1999), Funct. Differ. Equ., 8 (2001), 183-194.
[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), 1653-1695. 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), 601-642.
[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), 369-390. 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), 356-359. 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), 1151-1190. doi: 10.1080/03605300600718503.
[21]	T. Kato, "Perturbation Theory for Linear Operators," Dir Grundlehren der mathematischen Wissenschaften, 132, Springer, New York, 1966.
[22]	J. B. Keller and L. Ting, Periodic vibrations of systems governed by nonlinear partial differential equations, Comm. Pure and Appl. Math., 19 (1966), 371-420. doi: 10.1002/cpa.3160190404.
[23]	J. U. Kim, Forced vibration of an aero-elastic plate, J. Math. Anal. Appl., 113 (1986), 454-467. 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), 499-526. doi: 10.1007/s11424-010-0143-x.
[25]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 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, arXiv:math/1012.1057.
[27]	P. H. Rabinowitz, Periodic solutions of nonlinear hyperbolic differential equations, Comm. Pure Appl. Math., 20 (1967), 145-205. doi: 10.1002/cpa.3160200105.
[28]	P. H. Rabinowitz, Free vibrations for a semi-linear wave equation, Comm. Pure Appl. Math., 31 (1978), 31-68. doi: 10.1002/cpa.3160310103.
[29]	G. R. Sell and Y. C. You, Inertial manifolds: the nonselfadjoint case, J. Diff. Eqns., 96 (1992), 203-255. doi: 10.1016/0022-0396(92)90152-D.
[30]	L. Tartar, Interpolation non linéaire et régularité, J. Funct. Anal, 9 (1972), 469-489. 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), 895-902.
[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), 15-24. 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), 1509-1523.
[35]	Y. Yang and B.-Y. Zhang, Forced oscillations of a damped Benjamin-Bona-Mahony equation in a quarter plane, "Control Theory of Partial Differential Equations," Lect. Notes Pure Appl. Math., 242, Chapman & Hall/CRC, Boca Raton, FL, 2005, 375-386.
[36]	B.-Y. Zhang, Forced oscillations of a regularized long-wave equation and their global stability, in "Differential Equations and Computational Simulations" (Chengdu, 1999), World Sci. Publishing, River Edge, NJ, 2000, 456-463.
[37]	B.-Y. Zhang, Forced oscillation of the Korteweg-de Vries-Burgers equation and its stability, in "Control of Nonlinear Distributed Parameter Systems" (College Station, TX, 1999), Lecture Notes in Pure and Appl. Math., 218, Dekker, New York, 2001, 337-357.

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