Nonlinear dispersive wave equations in two space dimensions

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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

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

show all references

References:

[1]	P. Brenner, On $L^p-L^q$ estimate of the wave equation,, \emph{Math. Z.}, 145 (1975), 251.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	S. Klainerman, The null condition and global existence to nonlinear wave equations,, \emph{Lect. Appl. Math.}, 23 (1986), 293.
[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.
[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.
[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.

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