## Differential and Integral Equations

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

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

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356124296

**Mathematical Reviews number (MathSciNet)**

MR1811955

**Zentralblatt MATH identifier**

0974.35109

**Subjects**

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]

Secondary: 35A05 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35B65: Smoothness and regularity of solutions

#### Citation

Molinet, Luc. The Cauchy problem for the (generalized) Kadomtsev-Petviashvili-Burgers equation. Differential Integral Equations 13 (2000), no. 1-3, 189--216. https://projecteuclid.org/euclid.die/1356124296