Advances in Differential Equations

Scattering states for the nonlinear wave equation with small data

Tokio Matsuyama and Minoru Tanaka

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We investigate the energy nondecay and existence of scattering states for solutions to the initial-boundary-value problem for the nonlinear wave equation in exterior domains. When the space dimension is odd, the domain meets no geometrical condition. Otherwise, we assume that the obstacle is convex. For odd-dimensional general domains, taking into account the effective dissipation in trapping regions, we can derive the existence of scattering states. In particular, we can obtain also an $L^2$ bound of solutions. The method in deriving the energy nondecay is to utilize Huyghens' principle. For even-dimensional domains outside the convex obstacle, the asymptotics stated in the odd-dimensional case are also valid.

Article information

Adv. Differential Equations, Volume 9, Number 7-8 (2004), 721-744.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L70: Nonlinear second-order hyperbolic equations
Secondary: 35L20: Initial-boundary value problems for second-order hyperbolic equations 35P25: Scattering theory [See also 47A40]


Matsuyama, Tokio; Tanaka, Minoru. Scattering states for the nonlinear wave equation with small data. Adv. Differential Equations 9 (2004), no. 7-8, 721--744.

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