## Differential and Integral Equations

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

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

#### 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

**Subjects**

Primary: 35L75: Nonlinear higher-order hyperbolic equations

Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35L30: Initial value problems for higher-order hyperbolic equations 35Q35: PDEs in connection with fluid mechanics 76B03: Existence, uniqueness, and regularity theory [See also 35Q35] 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30]

#### 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