Advances in Differential Equations

Asymptotic behavior for quasilinear hyperbolic equations of Kirchhoff type with perturbation having integrable coefficient

Taeko Yamazaki

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Abstract

This paper is concerned with the abstract quasilinear hyperbolic equations of Kirchhoff type with perturbation whose coefficient is integrable in time. We show the unique existence of global solutions for small data in some class and that the solution has the same asymptotic behavior of a function obtained by a transformation of time variable from a solution of the free wave equation with an appropriate wave speed. Conversely, we show that there exists a solution of the Kirchhoff equation which has the same asymptotic behavior of a function obtained by a transformation of the time variable from the solution of the Cauchy problem of the free wave equation with an appropriate wave speed.

Article information

Source
Adv. Differential Equations Volume 20, Number 1/2 (2015), 143-192.

Dates
First available in Project Euclid: 11 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.ade/1418310445

Mathematical Reviews number (MathSciNet)
MR3297782

Zentralblatt MATH identifier
1316.35208

Subjects
Primary: 35L72: Quasilinear second-order hyperbolic equations 35L90: Abstract hyperbolic equations

Citation

Yamazaki, Taeko. Asymptotic behavior for quasilinear hyperbolic equations of Kirchhoff type with perturbation having integrable coefficient. Adv. Differential Equations 20 (2015), no. 1/2, 143--192. https://projecteuclid.org/euclid.ade/1418310445.


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