Differential and Integral Equations

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

José Raúl Quintero

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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


Export citation