## Differential and Integral Equations

### On spatially periodic solutions of the damped Boussinesq equation

Vladimir V. Varlamov

#### Abstract

A classical solution of the damped Boussinesq equation $$u_{tt}-2bu_{txx}=-\alpha u_{xxxx}+u_{xx}+\beta (u^2)_{xx},\quad x\in {\Bbb R}^1,t>0,$$ with $\alpha ,b=\text{const}>0$, $\beta =\text{const}\in{\Bbb R}^1$, $\alpha >b^2$, and small initial data is constructed by means of the successive application of the spectral theory and the perturbation one. Its long-time asymptotic representation is obtained which shows that the major term increases linearly with time and the second term contains a combination of the Airy functions of a negative argument. A uniform-in-space estimate of the remainder is given.

#### Article information

Source
Differential Integral Equations, Volume 10, Number 6 (1997), 1197-1211.

Dates
First available in Project Euclid: 1 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1367438229

Mathematical Reviews number (MathSciNet)
MR1608069

Zentralblatt MATH identifier
0940.35162

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35C20: Asymptotic expansions 76D33: Waves

#### Citation

Varlamov, Vladimir V. On spatially periodic solutions of the damped Boussinesq equation. Differential Integral Equations 10 (1997), no. 6, 1197--1211. https://projecteuclid.org/euclid.die/1367438229