Mixed initial-boundary value problem for Ott-Sudan-Ostrovskiy equation

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February 2012, 32(2): 381-409. doi: 10.3934/dcds.2012.32.381

Mixed initial-boundary value problem for Ott-Sudan-Ostrovskiy equation

Martn P. Árciga Alejandre ^1, and Elena I. Kaikina ^2,

1.	Instituto de Física y Matemáticas, UMSNH, Ediﬁcio C-3, Ciudad Universitaria), Morelia CP 58040, Michoacán, Mexico
2.	Instituto de Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, Mexico

Received August 2010 Revised May 2011 Published September 2011

Abstract
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We consider the mixed initial-boundary value problem for Ott-Sudan-Ostrovskiy equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.

Keywords: Large Time Asymptotics, Dissipative Nonlinear Evolution Equation, Fractional derivative..

Mathematics Subject Classification: Primary: 35Q35; Secondary: 35Q5.

Citation: Martn P. Árciga Alejandre, Elena I. Kaikina. Mixed initial-boundary value problem for Ott-Sudan-Ostrovskiy equation. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 381-409. doi: 10.3934/dcds.2012.32.381

References:

[1]	Hans-Dieter Alber and Peicheng Zhu, Global solutions to an initial boundary value problem for the Mullins equation, J. Partial Differential Equations, 20 (2007), 30-44.
[2]	Ravi P. Agarwal, Donal O'Regan and Svatoslav Staněk, Positive and maximal positive solutions of singular mixed boundary value problem, Cent. Eur. J. Math., 7 (2009), 694-716. doi: 10.2478/s11533-009-0049-9.
[3]	T. Buchukuri, O. Chkadua and D. Natroshvili, Mixed boundary value problems of thermopiezoelectricity for solids with interior cracks, Integral Equations Operator Theory, 64 (2009), 495-537. doi: 10.1007/s00020-009-1694-x.
[4]	Mouffak Benchohra and Samira Hamani, Nonlinear boundary value problems for differential inclusions with Caputo fractional derivative, Topol. Methods Nonlinear Anal., 32 (2008), 115-130.
[5]	R. Brown, I. Mitrea, M. Mitrea and M. Wright, Mixed boundary value problems for the Stokes system, Trans. Amer. Math. Soc., 362 (2010), 1211-1230. doi: 10.1090/S0002-9947-09-04774-6.
[6]	Jean-François Coulombel, Stability of finite difference schemes for hyperbolic initial boundary value problems, SIAM J. Numer. Anal., 47 (2009), 2844-2871. doi: 10.1137/080728342.
[7]	Gilles Carbou and Bernard Hanouzet, Relaxation approximation of the Kerr model for the three-dimensional initial-boundary value problem, J. Hyperbolic Differ. Equ., 6 (2009), 577-614. doi: 10.1142/S0219891609001939.
[8]	I. Chudinovich and C. Constanda, The traction initial-boundary value problem for bending of thermoelastic plates with cracks, Appl. Anal., 88 (2009), 961-975. doi: 10.1080/00036810903042224.
[9]	F. D. Gakhov, Boundary value problems, Pergamon Press, Oxford-New York-Paris, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1966.
[10]	A. S. Fokas, The Davey-Stewartson equation on the half-plane, Comm. Math. Phys., 289 (2009), 957-993. doi: 10.1007/s00220-009-0809-1.
[11]	Yiping Fu and Yongsheng Li, Initial boundary value problem for generalized 2D complex Ginzburg-Landau equation, J. Partial Differential Equations, 20 (2007), 65-70.
[12]	Helmut Friedrich, Initial boundary value problems for Einstein's field equations and geometric uniqueness, Gen. Relativity Gravitation, 41 (2009), 1947-1966. doi: 10.1007/s10714-009-0800-3.
[13]	N. Hayashi, E. I. Kaikina, P. I. Naumkin and I. A. Shishmarev, "Asymptotics for Dissipative Nonlinear Equations," Lecture Notes in Mathematics, 1884, Springer-Verlag, Berlin, 2006.
[14]	Nakao Hayashi and Elena Kaikina, "Nonlinear Theory of Pseudodifferential Equations on a Half-line," North-Holland Mathematics Studies, 194, Elsevier Science B.V., Amsterdam, 2004.
[15]	Elena I. Kaikina, Subcritical pseudodifferential equation on a half-line with nonanalytic symbol, Differential Integral Equations, 18 (2005), 1341-1370.
[16]	Elena I. Kaikina, Pseudodifferential operator with a nonanalytic symbol on a half-line, J. of Mathematical Physics, 48 (2007), 113509, 20 pp. doi: 10.1063/1.2804860.
[17]	Elena I. Kaikina, Critical Ostrovskiy-type equation on a half-line, Differential Integral Equations, 22 (2009), 69-98.
[18]	Elena I. Kaikina, Ott-Sudan-Ostrovskiy type equations on a segment with large initial data, Z. Angew. Math. Phys., 59 (2008), 647-675. doi: 10.1007/s00033-007-6075-1.
[19]	Elena I. Kaikina, Nonlinear pseudoparabolic type equations on a half-line with large initial data, Nonlinear Anal., 67 (2007), 2839-2858. doi: 10.1016/j.na.2006.09.044.
[20]	L. A. Ostrovsky, Short-wave asymptotics for weak shock waves and solitons in mechanics, Int. J. Non-Linear Mechanics, 11 (1976), 401-416. doi: 10.1016/0020-7462(76)90026-3.
[21]	E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic waves with Landau damping, Phys. Fluids, 12 (1969), 2388-2394. doi: 10.1063/1.1692358.
[22]	S. G. Samko, A. A. Kilbas and O. I. Marichev, "Fractional Integrals and Derivatives. Theory and Applications," Gordon and Breach Science Publishers, Yverdon, 1993.
[23]	Zhi-Qiang Shao, Global existence of classical solutions to the mixed initial-boundary value problem for quasilinear hyperbolic systems of diagonal form with large BV data, J. Math. Anal. Appl., 360 (2009), 398-411. doi: 10.1016/j.jmaa.2009.06.066.
[24]	Sheng Zhang, A domain embedding method for mixed boundary value problems, C. R. Math. Acad. Sci. Paris, 343 (2006), 287-290.

show all references

References:

[1]	Hans-Dieter Alber and Peicheng Zhu, Global solutions to an initial boundary value problem for the Mullins equation, J. Partial Differential Equations, 20 (2007), 30-44.
[2]	Ravi P. Agarwal, Donal O'Regan and Svatoslav Staněk, Positive and maximal positive solutions of singular mixed boundary value problem, Cent. Eur. J. Math., 7 (2009), 694-716. doi: 10.2478/s11533-009-0049-9.
[3]	T. Buchukuri, O. Chkadua and D. Natroshvili, Mixed boundary value problems of thermopiezoelectricity for solids with interior cracks, Integral Equations Operator Theory, 64 (2009), 495-537. doi: 10.1007/s00020-009-1694-x.
[4]	Mouffak Benchohra and Samira Hamani, Nonlinear boundary value problems for differential inclusions with Caputo fractional derivative, Topol. Methods Nonlinear Anal., 32 (2008), 115-130.
[5]	R. Brown, I. Mitrea, M. Mitrea and M. Wright, Mixed boundary value problems for the Stokes system, Trans. Amer. Math. Soc., 362 (2010), 1211-1230. doi: 10.1090/S0002-9947-09-04774-6.
[6]	Jean-François Coulombel, Stability of finite difference schemes for hyperbolic initial boundary value problems, SIAM J. Numer. Anal., 47 (2009), 2844-2871. doi: 10.1137/080728342.
[7]	Gilles Carbou and Bernard Hanouzet, Relaxation approximation of the Kerr model for the three-dimensional initial-boundary value problem, J. Hyperbolic Differ. Equ., 6 (2009), 577-614. doi: 10.1142/S0219891609001939.
[8]	I. Chudinovich and C. Constanda, The traction initial-boundary value problem for bending of thermoelastic plates with cracks, Appl. Anal., 88 (2009), 961-975. doi: 10.1080/00036810903042224.
[9]	F. D. Gakhov, Boundary value problems, Pergamon Press, Oxford-New York-Paris, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1966.
[10]	A. S. Fokas, The Davey-Stewartson equation on the half-plane, Comm. Math. Phys., 289 (2009), 957-993. doi: 10.1007/s00220-009-0809-1.
[11]	Yiping Fu and Yongsheng Li, Initial boundary value problem for generalized 2D complex Ginzburg-Landau equation, J. Partial Differential Equations, 20 (2007), 65-70.
[12]	Helmut Friedrich, Initial boundary value problems for Einstein's field equations and geometric uniqueness, Gen. Relativity Gravitation, 41 (2009), 1947-1966. doi: 10.1007/s10714-009-0800-3.
[13]	N. Hayashi, E. I. Kaikina, P. I. Naumkin and I. A. Shishmarev, "Asymptotics for Dissipative Nonlinear Equations," Lecture Notes in Mathematics, 1884, Springer-Verlag, Berlin, 2006.
[14]	Nakao Hayashi and Elena Kaikina, "Nonlinear Theory of Pseudodifferential Equations on a Half-line," North-Holland Mathematics Studies, 194, Elsevier Science B.V., Amsterdam, 2004.
[15]	Elena I. Kaikina, Subcritical pseudodifferential equation on a half-line with nonanalytic symbol, Differential Integral Equations, 18 (2005), 1341-1370.
[16]	Elena I. Kaikina, Pseudodifferential operator with a nonanalytic symbol on a half-line, J. of Mathematical Physics, 48 (2007), 113509, 20 pp. doi: 10.1063/1.2804860.
[17]	Elena I. Kaikina, Critical Ostrovskiy-type equation on a half-line, Differential Integral Equations, 22 (2009), 69-98.
[18]	Elena I. Kaikina, Ott-Sudan-Ostrovskiy type equations on a segment with large initial data, Z. Angew. Math. Phys., 59 (2008), 647-675. doi: 10.1007/s00033-007-6075-1.
[19]	Elena I. Kaikina, Nonlinear pseudoparabolic type equations on a half-line with large initial data, Nonlinear Anal., 67 (2007), 2839-2858. doi: 10.1016/j.na.2006.09.044.
[20]	L. A. Ostrovsky, Short-wave asymptotics for weak shock waves and solitons in mechanics, Int. J. Non-Linear Mechanics, 11 (1976), 401-416. doi: 10.1016/0020-7462(76)90026-3.
[21]	E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic waves with Landau damping, Phys. Fluids, 12 (1969), 2388-2394. doi: 10.1063/1.1692358.
[22]	S. G. Samko, A. A. Kilbas and O. I. Marichev, "Fractional Integrals and Derivatives. Theory and Applications," Gordon and Breach Science Publishers, Yverdon, 1993.
[23]	Zhi-Qiang Shao, Global existence of classical solutions to the mixed initial-boundary value problem for quasilinear hyperbolic systems of diagonal form with large BV data, J. Math. Anal. Appl., 360 (2009), 398-411. doi: 10.1016/j.jmaa.2009.06.066.
[24]	Sheng Zhang, A domain embedding method for mixed boundary value problems, C. R. Math. Acad. Sci. Paris, 343 (2006), 287-290.

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