American Institute of Mathematical Sciences

Advanced
July  2015, 14(4): 1377-1393. doi: 10.3934/cpaa.2015.14.1377

Nonlinear dispersive wave equations in two space dimensions

Nakao Hayashi 1, , Seishirou Kobayashi 2, and  Pavel I. Naumkin 3,

1. 

Department of Mathematics, Graduate School of Science, Osaka University, Osaka Toyonaka 560-0043
2. 

Department of Mathematics, Graduate School of Science, Osaka University, Osaka, Toyonaka, 560-0043, Japan
3. 

Instituto de Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, Mexico

Received  February 2013 Revised  July 2013 Published  April 2015

We study the global existence and time decay of solutions to nonlinear dispersive wave equations $ \partial_t^2 u+\frac{1}{\rho^2}( -\Delta) ^{\rho }u=F ( \partial _t u )$ in two space dimensions, where $F( \partial _t u) =\lambda \vert \partial _t u\vert ^{p-1}\partial _t u$ or $\lambda \vert \partial _t u \vert ^p$, $\lambda \in \mathbf{C,}$ with $ p > 2 $ for $0 < \rho <1,$ $p > 3$ for $\rho =1,$ and $p > 1+\rho $ for $1 < \rho <2.$ If $\rho =1,$ then the equation converts into the well-known nonlinear wave equation.
Keywords: Dispersive nonlinear wave, global in time of solutions, asymptotic behavior, system of equations, vector field method..
Mathematics Subject Classification: Primary: 35Q55, 35P25; Secondary: 35B4.
Citation: Nakao Hayashi, Seishirou Kobayashi, Pavel I. Naumkin. Nonlinear dispersive wave equations in two space dimensions. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1377-1393. doi: 10.3934/cpaa.2015.14.1377
References:
[1]

P. Brenner, On $L^p-L^q$ estimate of the wave equation,, \emph{Math. Z.}, 145 (1975), 251.   Google Scholar

[2]

Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates,, \emph{Comm. Pure Appl. Anal.}, 10 (2011), 1121.  doi: 10.3934/cpaa.2011.10.1121.  Google Scholar

[3]

J. Ginibre and G. Velo, Generalized Strichartz inequality for the wave equation,, \emph{J. Funct. Anal.}, 133 (1995), 50.  doi: 10.1006/jfan.1995.1119.  Google Scholar

[4]

N. Hayashi, Global existence of small solutions to quadratic nonlinear Schrödinger equations,, \emph{Commun. P.D.E.}, 18 (1993), 1109.  doi: 10.1080/03605309308820965.  Google Scholar

[5]

N. Hayashi, S. Kobayashi and P. Naumkin, Global existence of solutions to nonlinear dispersive wave equations,, \emph{Differential and Integral Equations}, 25 (2012), 685.   Google Scholar

[6]

N. Hayashi, C. Li and P. Naumkin, Non existence of asymptotically free solution of systems of nolinear Schrödinger equations,, \emph{Electron. J. Diff. Equ.}, 162 (2012), 1.   Google Scholar

[7]

K. Hidano and K. Tsutaya, Global existence and asymptotic behavior of solutions for nonlinear wave equations,, \emph{Indiana Univ. Math. J.}, 44 (1995), 1273.  doi: 10.1512/iumj.1995.44.2028.  Google Scholar

[8]

C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations,, \emph{Indiana Univ. Math. J.}, 40 (1991), 33.  doi: 10.1512/iumj.1991.40.40003.  Google Scholar

[9]

S. Klainerman, The null condition and global existence to nonlinear wave equations,, \emph{Lect. Appl. Math.}, 23 (1986), 293.   Google Scholar

[10]

M. Nakamura, Remarks on Keel-Smith-Sogge estimates and some applications to nonlinear higher order wave equations,, \emph{Differential and Integral Equations}, 24 (2011), 519.   Google Scholar

[11]

T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations,, \emph{Nonlinear Analysis, 14 (1990), 765.  doi: 10.1016/0362-546X(90)90104-O.  Google Scholar

[12]

J-Q. Yao, Comportment à l'infini des solutions d'une équation de Schrödinger non linéaire dans un domaine extérier,, \emph{C. R. Acad, 294 (1982), 163.   Google Scholar

show all references

References:
[1]

P. Brenner, On $L^p-L^q$ estimate of the wave equation,, \emph{Math. Z.}, 145 (1975), 251.   Google Scholar

[2]

Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates,, \emph{Comm. Pure Appl. Anal.}, 10 (2011), 1121.  doi: 10.3934/cpaa.2011.10.1121.  Google Scholar

[3]

J. Ginibre and G. Velo, Generalized Strichartz inequality for the wave equation,, \emph{J. Funct. Anal.}, 133 (1995), 50.  doi: 10.1006/jfan.1995.1119.  Google Scholar

[4]

N. Hayashi, Global existence of small solutions to quadratic nonlinear Schrödinger equations,, \emph{Commun. P.D.E.}, 18 (1993), 1109.  doi: 10.1080/03605309308820965.  Google Scholar

[5]

N. Hayashi, S. Kobayashi and P. Naumkin, Global existence of solutions to nonlinear dispersive wave equations,, \emph{Differential and Integral Equations}, 25 (2012), 685.   Google Scholar

[6]

N. Hayashi, C. Li and P. Naumkin, Non existence of asymptotically free solution of systems of nolinear Schrödinger equations,, \emph{Electron. J. Diff. Equ.}, 162 (2012), 1.   Google Scholar

[7]

K. Hidano and K. Tsutaya, Global existence and asymptotic behavior of solutions for nonlinear wave equations,, \emph{Indiana Univ. Math. J.}, 44 (1995), 1273.  doi: 10.1512/iumj.1995.44.2028.  Google Scholar

[8]

C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations,, \emph{Indiana Univ. Math. J.}, 40 (1991), 33.  doi: 10.1512/iumj.1991.40.40003.  Google Scholar

[9]

S. Klainerman, The null condition and global existence to nonlinear wave equations,, \emph{Lect. Appl. Math.}, 23 (1986), 293.   Google Scholar

[10]

M. Nakamura, Remarks on Keel-Smith-Sogge estimates and some applications to nonlinear higher order wave equations,, \emph{Differential and Integral Equations}, 24 (2011), 519.   Google Scholar

[11]

T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear Schrödinger equations,, \emph{Nonlinear Analysis, 14 (1990), 765.  doi: 10.1016/0362-546X(90)90104-O.  Google Scholar

[12]

J-Q. Yao, Comportment à l'infini des solutions d'une équation de Schrödinger non linéaire dans un domaine extérier,, \emph{C. R. Acad, 294 (1982), 163.   Google Scholar

[1]

Kosuke Ono. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 651-662. doi: 10.3934/dcds.2003.9.651
[2]

C. I. Christov, M. D. Todorov. Investigation of the long-time evolution of localized solutions of a dispersive wave system. Conference Publications, 2013, 2013 (special) : 139-148. doi: 10.3934/proc.2013.2013.139
[3]

Hongqiu Chen, Jerry L. Bona. Periodic traveling--wave solutions of nonlinear dispersive evolution equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4841-4873. doi: 10.3934/dcds.2013.33.4841
[4]

Jerry L. Bona, Laihan Luo. More results on the decay of solutions to nonlinear, dispersive wave equations. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 151-193. doi: 10.3934/dcds.1995.1.151
[5]

Honglv Ma, Jin Zhang, Chengkui Zhong. Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4721-4737. doi: 10.3934/dcdsb.2019027
[6]

Lie Zheng. Asymptotic behavior of solutions to the nonlinear breakage equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 463-473. doi: 10.3934/cpaa.2005.4.463
[7]

Zhenhua Zhang. Asymptotic behavior of solutions to the phase-field equations with neumann boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 683-693. doi: 10.3934/cpaa.2005.4.683
[8]

Hideo Kubo. Asymptotic behavior of solutions to semilinear wave equations with dissipative structure. Conference Publications, 2007, 2007 (Special) : 602-613. doi: 10.3934/proc.2007.2007.602
[9]

Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 383-400. doi: 10.3934/dcds.1997.3.383
[10]

Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 609-622. doi: 10.3934/dcdsb.2010.13.609
[11]

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
[12]

Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707
[13]

P. R. Zingano. Asymptotic behavior of the $L^1$ norm of solutions to nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 151-159. doi: 10.3934/cpaa.2004.3.151
[14]

Luc Molinet, Francis Ribaud. On global well-posedness for a class of nonlocal dispersive wave equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 657-668. doi: 10.3934/dcds.2006.15.657
[15]

Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic & Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481
[16]

Jerry Bona, Hongqiu Chen. Well-posedness for regularized nonlinear dispersive wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1253-1275. doi: 10.3934/dcds.2009.23.1253
[17]

Francesca Bucci, Igor Chueshov, Irena Lasiecka. Global attractor for a composite system of nonlinear wave and plate equations. Communications on Pure & Applied Analysis, 2007, 6 (1) : 113-140. doi: 10.3934/cpaa.2007.6.113
[18]

Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-15. doi: 10.3934/dcds.2019229
[19]

Takashi Narazaki. Global solutions to the Cauchy problem for the weakly coupled system of damped wave equations. Conference Publications, 2009, 2009 (Special) : 592-601. doi: 10.3934/proc.2009.2009.592
[20]

Makoto Nakamura. Remarks on global solutions of dissipative wave equations with exponential nonlinear terms. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1533-1545. doi: 10.3934/cpaa.2015.14.1533

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences