American Institute of Mathematical Sciences

Advanced
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

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, 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.  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), 694-716. 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), 495-537. 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), 115-130.  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), 1211-1230. 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), 2844-2871. 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), 577-614. 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), 961-975. doi: 10.1080/00036810903042224.  Google Scholar

[9]

F. D. Gakhov, Boundary value problems, Pergamon Press, Oxford-New York-Paris, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1966.  Google Scholar

[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.  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), 65-70.  Google Scholar

[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.  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, Springer-Verlag, Berlin, 2006.  Google Scholar

[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.  Google Scholar

[15]

Elena I. Kaikina, Subcritical pseudodifferential equation on a half-line with nonanalytic symbol, Differential Integral Equations, 18 (2005), 1341-1370.  Google Scholar

[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.  Google Scholar

[17]

Elena I. Kaikina, Critical Ostrovskiy-type equation on a half-line, Differential Integral Equations, 22 (2009), 69-98.  Google Scholar

[18]

Elena I. Kaikina, Ott-Sudan-Ostrovskiy type equations on a segment with large initial data, Z. Angew. Math. Phys., 59 (2008), 647-675. 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), 2839-2858. 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), 401-416. 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), 2388-2394. 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, Yverdon, 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), 398-411. 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), 287-290.  Google Scholar

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.  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), 694-716. 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), 495-537. 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), 115-130.  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), 1211-1230. 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), 2844-2871. 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), 577-614. 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), 961-975. doi: 10.1080/00036810903042224.  Google Scholar

[9]

F. D. Gakhov, Boundary value problems, Pergamon Press, Oxford-New York-Paris, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1966.  Google Scholar

[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.  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), 65-70.  Google Scholar

[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.  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, Springer-Verlag, Berlin, 2006.  Google Scholar

[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.  Google Scholar

[15]

Elena I. Kaikina, Subcritical pseudodifferential equation on a half-line with nonanalytic symbol, Differential Integral Equations, 18 (2005), 1341-1370.  Google Scholar

[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.  Google Scholar

[17]

Elena I. Kaikina, Critical Ostrovskiy-type equation on a half-line, Differential Integral Equations, 22 (2009), 69-98.  Google Scholar

[18]

Elena I. Kaikina, Ott-Sudan-Ostrovskiy type equations on a segment with large initial data, Z. Angew. Math. Phys., 59 (2008), 647-675. 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), 2839-2858. 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), 401-416. 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), 2388-2394. 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, Yverdon, 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), 398-411. 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), 287-290.  Google Scholar

[1]

Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93
[2]

Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1447-1478. doi: 10.3934/cpaa.2021028
[3]

Jerry L. Bona, Laihan Luo. Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 15-50. doi: 10.3934/dcdss.2011.4.15
[4]

Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 383-400. doi: 10.3934/dcds.1997.3.383
[5]

Yalçin Sarol, Frederi Viens. Time regularity of the evolution solution to fractional stochastic heat equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 895-910. doi: 10.3934/dcdsb.2006.6.895
[6]

Kaouther Bouchama, Yacine Arioua, Abdelkrim Merzougui. The Numerical Solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021026
[7]

Hiroshi Takeda. Large time behavior of solutions for a nonlinear damped wave equation. Communications on Pure & Applied Analysis, 2016, 15 (1) : 41-55. doi: 10.3934/cpaa.2016.15.41
[8]

Yajing Li, Yejuan Wang. The existence and exponential behavior of solutions to time fractional stochastic delay evolution inclusions with nonlinear multiplicative noise and fractional noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2665-2697. doi: 10.3934/dcdsb.2020027
[9]

Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic & Related Models, 2011, 4 (3) : 701-716. doi: 10.3934/krm.2011.4.701
[10]

Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations & Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007
[11]

Walter Allegretto, Yanping Lin, Zhiyong Zhang. Convergence to convection-diffusion waves for solutions to dissipative nonlinear evolution equations. Conference Publications, 2009, 2009 (Special) : 11-23. doi: 10.3934/proc.2009.2009.11
[12]

Gaku Hoshino. Dissipative nonlinear schrödinger equations for large data in one space dimension. Communications on Pure & Applied Analysis, 2020, 19 (2) : 967-981. doi: 10.3934/cpaa.2020044
[13]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137
[14]

Tran Bao Ngoc, Nguyen Huy Tuan, R. Sakthivel, Donal O'Regan. Analysis of nonlinear fractional diffusion equations with a Riemann-liouville derivative. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021007
[15]

Olivier Ley, Erwin Topp, Miguel Yangari. Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3555-3577. doi: 10.3934/dcds.2021007
[16]

Tomasz Komorowski. Long time asymptotics of a degenerate linear kinetic transport equation. Kinetic & Related Models, 2014, 7 (1) : 79-108. doi: 10.3934/krm.2014.7.79
[17]

Vladimir Varlamov. Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 675-702. doi: 10.3934/dcds.2001.7.675
[18]

Gaohang Yu. A derivative-free method for solving large-scale nonlinear systems of equations. Journal of Industrial & Management Optimization, 2010, 6 (1) : 149-160. doi: 10.3934/jimo.2010.6.149
[19]

Yigui Ou, Wenjie Xu. A unified derivative-free projection method model for large-scale nonlinear equations with convex constraints. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021125
[20]

Chun Wang, Tian-Zhou Xu. Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 505-521. doi: 10.3934/dcdss.2017025

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy