## Differential and Integral Equations

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

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

#### 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