Global existence for nonlinear Klein-Gordon equations in infinite homogeneous waveguides in two dimensions
type: Item
instance of: scholarly article
title: Global existence for nonlinear Klein-Gordon equations in infinite homogeneous waveguides in two dimensions
zbMATH Open document ID: 1120.35070
author: Daoyuan Fang
author: Sijia Zhong
author: Changqing Tong
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-05-04T00:00:00Z
review text: This paper deals with the Cauchy problem for the semilinear 2D Klein-Gordon equation \[ (\partial^2_t- \Delta_x- \Delta_g+ m^2)u= f(y, u,\partial_t u,\partial_x u,\nabla_g u)\tag{1} \] in infinite homogeneous waveguides \(\mathbb{R}^2_x\times\mathbb{M}\), \(m= \text{const}> 0\), where \(\mathbb{M}= (M,g)\) is a Zoll manifold, i.e. compact Riemannian manifold all of whose geodesics are periodic with the same period, \(\Delta_g\) is the Laplace-Beltrami operator on \(\mathbb{M}\) and \(f\) is a homogeneous polynomial of second degree. The initial data are ``sufficiently small'' and belong to \(C^\infty_0(\mathbb{R}^2\times \mathbb{M})\). Then it is proved that for almost all values of \(m\) the Cauchy problem for (1) admits a global smooth solution \(u\in C^\infty(\mathbb{R}_+\times \mathbb{R}^2\times\mathbb{M})\).
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 78A50
Mathematics Subject Classification ID: 58J45
Mathematics Subject Classification ID: 58J47
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 5149543
zbMATH Keywords: nonlinear Klein-Gordon equation in two dimensions
zbMATH Keywords: global existence
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Zoll manifold
zbMATH Keywords: infinite homogeneous waveguides
reviewed by: Peter R. Popivanov
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.08.065
OpenAlex ID: W2014802815
cites work: Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions
cites work: Long-time existence for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds
cites work: Global solutions for nonlinear Klein-Gordon equations in infinite homogeneous waveguides
cites work: Global existence of small amplitude solutions to nonlinear klein-gordon equations in four space-time dimensions
cites work: Nonlinear Hyperbolic Equations in Infinite Homogeneous Waveguides
cites work: Large time asymptotics of solutions to nonlinear Klein-Gordon systems
DOI: 10.1016/J.JMAA.2006.08.065
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description: scientific article; zbMATH DE number 5149543