## Differential and Integral Equations

- Differential Integral Equations
- Volume 9, Number 5 (1996), 879-890.

### Local existence in time of solutions to higher-order nonlinear dispersive equations

#### Abstract

We study a system of nonlinear dispersive equations of the form $$ \partial_t u_k + \partial_x^{2j+1}u_k + F_k(u_1,\ldots ,u_n,\ldots , \partial_x^{2j}u_1, \ldots , \partial_x^{2j} u_n)=0, (t,x) \in \mathbf{R}\times h{\mathbf{R}}, $$ where $k=1,\ldots ,n$, $j\in \mathbf{N}$ and $F_k(\cdot)$ is a polynomial having no constant or linear terms. Local existence of solutions to the associated initial value problem is shown without a smallness condition on the data.

#### Article information

**Source**

Differential Integral Equations, Volume 9, Number 5 (1996), 879-890.

**Dates**

First available in Project Euclid: 6 May 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1367871521

**Mathematical Reviews number (MathSciNet)**

MR1392085

**Zentralblatt MATH identifier**

0849.35129

**Subjects**

Primary: 35G25: Initial value problems for nonlinear higher-order equations

Secondary: 35A07 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]

#### Citation

Hayashi, Nakao. Local existence in time of solutions to higher-order nonlinear dispersive equations. Differential Integral Equations 9 (1996), no. 5, 879--890. https://projecteuclid.org/euclid.die/1367871521