Differential and Integral Equations

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

Luc Molinet

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

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.


Export citation