Nonlinear dispersive wave equations in two space dimensions

Nakao Hayashi; Seishirou Kobayashi; Pavel I. Naumkin

doi:10.3934/cpaa.2015.14.1377

Article Contents

2015, Volume 14, Issue 4: 1377-1393. Doi: 10.3934/cpaa.2015.14.1377

This issue Previous Article Modified wave operators without loss of regularity for some long range Hartree equations. II Next Article Ill-posedness for the quadratic nonlinear Schrödinger equation with nonlinearity $|u|^2$

Nonlinear dispersive wave equations in two space dimensions

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

Received: January 31, 2013

Revised: June 30, 2013

Published: June 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q55, 35P25; Secondary: 35B40.

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