## Advances in Differential Equations

- Adv. Differential Equations
- Volume 7, Number 5 (2002), 581-616.

### Periodic problem for a model nonlinear evolution equation

Elena I. Kaikina, Pavel I. Naumkin, and Ilya A. Shishmarev

#### Abstract

We study the large time asymptotic behavior of solutions to the periodic problem for a model nonlinear evolution equation. This equation has a general structure, it contains many well-known equations of mathematical physics, such as: the Korteweg-de Vries equation and nonlinear Schrödinger equation. We find the asymptotic representation of solution. Depending on the linear part of equation and the structure of the nonlinearity the solution can exponentially decay with time, oscillate or grow exponentially with time. Taking into account the symmetry of the nonlinear term we consider the case of large initial data.

#### Article information

**Source**

Adv. Differential Equations Volume 7, Number 5 (2002), 581-616.

**Dates**

First available in Project Euclid: 27 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1356651751

**Mathematical Reviews number (MathSciNet)**

MR1895033

**Zentralblatt MATH identifier**

1057.35055

**Subjects**

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]

Secondary: 34G20: Nonlinear equations [See also 47Hxx, 47Jxx] 35B10: Periodic solutions 35B40: Asymptotic behavior of solutions 35K57: Reaction-diffusion equations 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

#### Citation

Kaikina, Elena I.; Naumkin, Pavel I.; Shishmarev, Ilya A. Periodic problem for a model nonlinear evolution equation. Adv. Differential Equations 7 (2002), no. 5, 581--616. https://projecteuclid.org/euclid.ade/1356651751