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

Abstract

We investigate the Cauchy problem for the generalized Kadomtsev-Petviashvili-Burgers equation
u t +u xxx +u p u x +εv y -νu xx =0,v x =u y ,u(0)=ϕ
in Sobolev spaces. This nonlinear wave equation has both dispersive and dissipative parts. After showing local existence by the contraction principle for initial data ϕ∈H s (ℝ 2 ) such that ℱ -1 (k 2 k 1 ϕ ^)∈H r (ℝ 2 ), 0≤r≤s-1, we extend the solutions for all positive times. Whereas for ε=-1 and 1≤p<4/3 this is done without any assumption on the initial data, we 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 (ℝ 2 ) when ε=-1 and 1≤p<4/3.