Differential and Integral Equations

Asymptotic profiles to the solutions for a nonlinear damped wave equation

Tatsuki Kawakami and Yoshihiro Ueda

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We consider the Cauchy problem for a nonlinear damped wave equation. Under suitable assumptions of the nonlinear term and the initial functions, the Cauchy problem has a global-in-time solution $u$ behaving like the Gauss kernel as time tends to infinity. In this paper we show the asymptotic profiles to the solutions and give precise decay estimates on the difference between the solutions and their asymptotic profiles. Our results are based on the $L^p$--$L^q$-type decomposition of the fundamental solutions of the linearized damped wave equation and asymptotic expansion of the solution of a nonlinear heat equation.

Article information

Differential Integral Equations, Volume 26, Number 7/8 (2013), 781-814.

First available in Project Euclid: 20 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L71: Semilinear second-order hyperbolic equations 35B40: Asymptotic behavior of solutions 35L15: Initial value problems for second-order hyperbolic equations


Kawakami, Tatsuki; Ueda, Yoshihiro. Asymptotic profiles to the solutions for a nonlinear damped wave equation. Differential Integral Equations 26 (2013), no. 7/8, 781--814. https://projecteuclid.org/euclid.die/1369057817

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