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 & Continuous Dynamical Systems - A, 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. Google Scholar
[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. doi: 10.2478/s11533-009-0049-9. Google Scholar
[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. doi: 10.1007/s00020-009-1694-x. Google Scholar
[4]	Mouffak Benchohra and Samira Hamani, Nonlinear boundary value problems for differential inclusions with Caputo fractional derivative,, Topol. Methods Nonlinear Anal., 32 (2008), 115. Google Scholar
[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. doi: 10.1090/S0002-9947-09-04774-6. Google Scholar
[6]	Jean-François Coulombel, Stability of finite difference schemes for hyperbolic initial boundary value problems,, SIAM J. Numer. Anal., 47 (2009), 2844. doi: 10.1137/080728342. Google Scholar
[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. doi: 10.1142/S0219891609001939. Google Scholar
[8]	I. Chudinovich and C. Constanda, The traction initial-boundary value problem for bending of thermoelastic plates with cracks,, Appl. Anal., 88 (2009), 961. doi: 10.1080/00036810903042224. Google Scholar
[9]	F. D. Gakhov, Boundary value problems,, Pergamon Press, (1966). Google Scholar
[10]	A. S. Fokas, The Davey-Stewartson equation on the half-plane,, Comm. Math. Phys., 289 (2009), 957. doi: 10.1007/s00220-009-0809-1. Google Scholar
[11]	Yiping Fu and Yongsheng Li, Initial boundary value problem for generalized 2D complex Ginzburg-Landau equation,, J. Partial Differential Equations, 20 (2007), 65. Google Scholar
[12]	Helmut Friedrich, Initial boundary value problems for Einstein's field equations and geometric uniqueness,, Gen. Relativity Gravitation, 41 (2009), 1947. doi: 10.1007/s10714-009-0800-3. Google Scholar
[13]	N. Hayashi, E. I. Kaikina, P. I. Naumkin and I. A. Shishmarev, "Asymptotics for Dissipative Nonlinear Equations,", Lecture Notes in Mathematics, 1884 (2006). Google Scholar
[14]	Nakao Hayashi and Elena Kaikina, "Nonlinear Theory of Pseudodifferential Equations on a Half-line,", North-Holland Mathematics Studies, 194 (2004). Google Scholar
[15]	Elena I. Kaikina, Subcritical pseudodifferential equation on a half-line with nonanalytic symbol,, Differential Integral Equations, 18 (2005), 1341. Google Scholar
[16]	Elena I. Kaikina, Pseudodifferential operator with a nonanalytic symbol on a half-line,, J. of Mathematical Physics, 48 (2007). doi: 10.1063/1.2804860. Google Scholar
[17]	Elena I. Kaikina, Critical Ostrovskiy-type equation on a half-line,, Differential Integral Equations, 22 (2009), 69. Google Scholar
[18]	Elena I. Kaikina, Ott-Sudan-Ostrovskiy type equations on a segment with large initial data,, Z. Angew. Math. Phys., 59 (2008), 647. doi: 10.1007/s00033-007-6075-1. Google Scholar
[19]	Elena I. Kaikina, Nonlinear pseudoparabolic type equations on a half-line with large initial data,, Nonlinear Anal., 67 (2007), 2839. doi: 10.1016/j.na.2006.09.044. Google Scholar
[20]	L. A. Ostrovsky, Short-wave asymptotics for weak shock waves and solitons in mechanics,, Int. J. Non-Linear Mechanics, 11 (1976), 401. doi: 10.1016/0020-7462(76)90026-3. Google Scholar
[21]	E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic waves with Landau damping,, Phys. Fluids, 12 (1969), 2388. doi: 10.1063/1.1692358. Google Scholar
[22]	S. G. Samko, A. A. Kilbas and O. I. Marichev, "Fractional Integrals and Derivatives. Theory and Applications,", Gordon and Breach Science Publishers, (1993). Google Scholar
[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. doi: 10.1016/j.jmaa.2009.06.066. Google Scholar
[24]	Sheng Zhang, A domain embedding method for mixed boundary value problems,, C. R. Math. Acad. Sci. Paris, 343 (2006), 287. Google Scholar

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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. Google Scholar
[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. doi: 10.2478/s11533-009-0049-9. Google Scholar
[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. doi: 10.1007/s00020-009-1694-x. Google Scholar
[4]	Mouffak Benchohra and Samira Hamani, Nonlinear boundary value problems for differential inclusions with Caputo fractional derivative,, Topol. Methods Nonlinear Anal., 32 (2008), 115. Google Scholar
[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. doi: 10.1090/S0002-9947-09-04774-6. Google Scholar
[6]	Jean-François Coulombel, Stability of finite difference schemes for hyperbolic initial boundary value problems,, SIAM J. Numer. Anal., 47 (2009), 2844. doi: 10.1137/080728342. Google Scholar
[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. doi: 10.1142/S0219891609001939. Google Scholar
[8]	I. Chudinovich and C. Constanda, The traction initial-boundary value problem for bending of thermoelastic plates with cracks,, Appl. Anal., 88 (2009), 961. doi: 10.1080/00036810903042224. Google Scholar
[9]	F. D. Gakhov, Boundary value problems,, Pergamon Press, (1966). Google Scholar
[10]	A. S. Fokas, The Davey-Stewartson equation on the half-plane,, Comm. Math. Phys., 289 (2009), 957. doi: 10.1007/s00220-009-0809-1. Google Scholar
[11]	Yiping Fu and Yongsheng Li, Initial boundary value problem for generalized 2D complex Ginzburg-Landau equation,, J. Partial Differential Equations, 20 (2007), 65. Google Scholar
[12]	Helmut Friedrich, Initial boundary value problems for Einstein's field equations and geometric uniqueness,, Gen. Relativity Gravitation, 41 (2009), 1947. doi: 10.1007/s10714-009-0800-3. Google Scholar
[13]	N. Hayashi, E. I. Kaikina, P. I. Naumkin and I. A. Shishmarev, "Asymptotics for Dissipative Nonlinear Equations,", Lecture Notes in Mathematics, 1884 (2006). Google Scholar
[14]	Nakao Hayashi and Elena Kaikina, "Nonlinear Theory of Pseudodifferential Equations on a Half-line,", North-Holland Mathematics Studies, 194 (2004). Google Scholar
[15]	Elena I. Kaikina, Subcritical pseudodifferential equation on a half-line with nonanalytic symbol,, Differential Integral Equations, 18 (2005), 1341. Google Scholar
[16]	Elena I. Kaikina, Pseudodifferential operator with a nonanalytic symbol on a half-line,, J. of Mathematical Physics, 48 (2007). doi: 10.1063/1.2804860. Google Scholar
[17]	Elena I. Kaikina, Critical Ostrovskiy-type equation on a half-line,, Differential Integral Equations, 22 (2009), 69. Google Scholar
[18]	Elena I. Kaikina, Ott-Sudan-Ostrovskiy type equations on a segment with large initial data,, Z. Angew. Math. Phys., 59 (2008), 647. doi: 10.1007/s00033-007-6075-1. Google Scholar
[19]	Elena I. Kaikina, Nonlinear pseudoparabolic type equations on a half-line with large initial data,, Nonlinear Anal., 67 (2007), 2839. doi: 10.1016/j.na.2006.09.044. Google Scholar
[20]	L. A. Ostrovsky, Short-wave asymptotics for weak shock waves and solitons in mechanics,, Int. J. Non-Linear Mechanics, 11 (1976), 401. doi: 10.1016/0020-7462(76)90026-3. Google Scholar
[21]	E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic waves with Landau damping,, Phys. Fluids, 12 (1969), 2388. doi: 10.1063/1.1692358. Google Scholar
[22]	S. G. Samko, A. A. Kilbas and O. I. Marichev, "Fractional Integrals and Derivatives. Theory and Applications,", Gordon and Breach Science Publishers, (1993). Google Scholar
[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. doi: 10.1016/j.jmaa.2009.06.066. Google Scholar
[24]	Sheng Zhang, A domain embedding method for mixed boundary value problems,, C. R. Math. Acad. Sci. Paris, 343 (2006), 287. Google Scholar

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