## Advances in Differential Equations

- Adv. Differential Equations
- Volume 23, Number 3/4 (2018), 239-294.

### Asymptotics for the modified Boussinesq equation in one space dimension

Nakao Hayashi and Pavel I. Naumkin

#### Abstract

We consider the Cauchy problem for the modified Boussinesq equation in one space dimension \begin{equation*} \begin {cases} w_{tt}=a^{2}\partial _{x}^{2}w-\partial _{x}^{4}w+\partial _{x}^{2} ( w^{3} ) ,\text{ } ( t,x ) \in \mathbb{R}^{2}, \\ w ( 0,x ) =w_{0} ( x ) ,\text{ }w_{t} ( 0,x ) =w_{1} ( x ) ,\text{ }x\in \mathbb{R}\text{,} \end {cases} \end{equation*} where $a > 0.$ We study the large time asymptotics of solutions to the Cauchy problem for the modified Boussinesq equation. We apply the factorization technique developed recently in papers [5], [6], [7], [8].

#### Article information

**Source**

Adv. Differential Equations Volume 23, Number 3/4 (2018), 239-294.

**Dates**

First available in Project Euclid: 19 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1513652447

**Subjects**

Primary: 35B40: Asymptotic behavior of solutions 35Q35: PDEs in connection with fluid mechanics

#### Citation

Hayashi, Nakao; Naumkin, Pavel I. Asymptotics for the modified Boussinesq equation in one space dimension. Adv. Differential Equations 23 (2018), no. 3/4, 239--294. https://projecteuclid.org/euclid.ade/1513652447