Journal of the Mathematical Society of Japan

Global well-posedness for the exterior initial-boundary value problem to the Kirchhoff equation

Tokio MATSUYAMA

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Abstract

The aim of this paper is to find a general class of data in which the global well-posedness for the exterior initial-boundary value problem to the Kirchhoff equation is assured. The result obtained in the present paper will be applied to the existence of scattering states. A class of weighted Sobolev spaces will be also presented in which the global well-posedness is assured. For this purpose, the method of generalized Fourier transforms is developed for some oscillatory integral associated with this equation. The crucial point is to obtain the resolvent expansion of the minus Laplacian around the origin in C, and the differentiability of the generalized Fourier transforms.

Article information

Source
J. Math. Soc. Japan, Volume 62, Number 4 (2010), 1167-1204.

Dates
First available in Project Euclid: 2 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1288703101

Digital Object Identifier
doi:10.2969/jmsj/06241167

Mathematical Reviews number (MathSciNet)
MR2761918

Zentralblatt MATH identifier
1223.35238

Subjects
Primary: 35L05: Wave equation
Secondary: 35L10: Second-order hyperbolic equations

Keywords
Kirchhoff equation generalized Fourier transform scattering theory

Citation

MATSUYAMA, Tokio. Global well-posedness for the exterior initial-boundary value problem to the Kirchhoff equation. J. Math. Soc. Japan 62 (2010), no. 4, 1167--1204. doi:10.2969/jmsj/06241167. https://projecteuclid.org/euclid.jmsj/1288703101


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