Large time behavior of solutions for a nonlinear damped wave equation

Hiroshi Takeda

doi:10.3934/cpaa.2016.15.41

Article Contents

2016, Volume 15, Issue 1: 41-55. Doi: 10.3934/cpaa.2016.15.41

This issue Previous Article Average error for spectral asymptotics on surfaces Next Article Existence and nonuniqueness of homoclinic solutions for second-order Hamiltonian systems with mixed nonlinearities

Large time behavior of solutions for a nonlinear damped wave equation

Hiroshi Takeda^1,

1.
Fukuoka Institute of Technology, Wajiro-higashi, Higashi-ku, Fukuoka, 811-0295

Received: October 2014

Revised: September 2015

Published: January 2016

Abstract / Introduction Related Papers Cited by

Abstract

We study the large time behavior of small solutions to the Cauchy problem for a nonlinear damped wave equation. We proved that the solution is approximated by the Gauss kernel with suitable choice of the coefficients and powers of $t$ for $N+1$ th order for all $N \in \mathbb{N}$. Our analysis is based on the approximation theorem of the linear solution by the solution of the heat equation [37]. In particular, as pointed out by Galley-Raugel [4], we explicitly observe that from third order expansion, the asymptotic behavior of the solutions of a nonlinear damped wave equation is different from that of a nonlinear heat equation.

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Mathematics Subject Classification: Primary: 35B40; Secondary: 35B33.

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