## Differential and Integral Equations

### The Cauchy problem for the (generalized) Kadomtsev-Petviashvili-Burgers equation

Luc Molinet

#### Abstract

We investigate the Cauchy problem for the generalized Kadomtsev-Petviashvili-Burgers (KP-Burgers) equation in Sobolev spaces. This nonlinear wave equation has both dispersive and dissipative parts which makes it quite particular. After showing local existence by contraction principle for initial data $\varphi\in H^s(\mathbb R^2)$ such that ${\mathcal F}^{-1} (\frac{k_2}{k_1} \widehat{\varphi})\in H^r(\mathbb R^2)$, $0{\leqslant} r {\leqslant} s- 1$, we try to extend the solutions for all positive times. Whereas for ${\varepsilon}=-1$ and $1{\leqslant} p < 4/3$ this will be done without any assumption on the the initial data, we will require a smallness condition on the initial data otherwise. In a last part we prove a local smoothing effect in the transverse direction, which enables us to establish the existence of weak global solutions in $L^2(\mathbb R^2)$ when ${\varepsilon}=-1$ and $1{\leqslant} p < 4/3$.

#### Article information

Source
Differential Integral Equations, Volume 13, Number 1-3 (2000), 189-216.

Dates
First available in Project Euclid: 21 December 2012