## Differential and Integral Equations

### A remark on the Cauchy problem for the generalized Benney-Luke equation

José Raúl Quintero

#### Abstract

In this article, we address the well posedness of the Cauchy problem associated with the generalized Benney--Luke equation in $\mathbb R^{1+2}:$ \begin{multline*} \Phi_{tt} - \Delta \Phi + a \Delta^2 \Phi - b \Delta \Phi_{tt} + \theta\Big ( \Phi_t \big [\partial_{x} \big [ \big (\partial_{x} \Phi \big )^{p} \big ]+ \partial_{y} \big [ \big (\partial_{y}\Phi \big )^{p} \big ] \big ] \\ + 2 \big [ \big (\partial_{x} \Phi \big )^{p}\Phi_{xt}+ \big (\partial_{y} \Phi \big )^{p}\Phi_{yt} \big ] \Big ) + \beta \nabla \cdot \big (|\nabla \Phi|^m \nabla \Phi \big )=0, \end{multline*} under a reasonable physical" initial condition, which is imposed from the formal derivation of the Benney-Luke water wave model.

#### Article information

Source
Differential Integral Equations, Volume 21, Number 9-10 (2008), 859-890.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356038590

Mathematical Reviews number (MathSciNet)
MR2483339

Zentralblatt MATH identifier
1224.35263

#### Citation

Quintero, José Raúl. A remark on the Cauchy problem for the generalized Benney-Luke equation. Differential Integral Equations 21 (2008), no. 9-10, 859--890. https://projecteuclid.org/euclid.die/1356038590