Differential and Integral Equations

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

Nakao Hayashi

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