Asymptotic profile of solutions to the Cauchy problem for the generalized Kadomtsev-Petviashvili equations with anisotropic dissipation in 2D

Abstract

We consider the Cauchy problem for the generalized Kadomtsev-Petviashvili equations with the dissipation term $-\nu u_{xx}$ in 2D. This is one of the nonlinear dispersive-dissipative type equations, which has a spatial anisotropy. In this paper, we investigate the large time behavior of the solution to this problem. Especially, we show that the $L^{\infty }$-norm of the solution decays at the rate of $t^{-7/4}$ if the initial data $u_0(x,y)$ satisfies $(1+|x|)u_0\in L^1({\mathbb{R}}^2)$ with the zero-mass condition and some appropriate regularity assumptions. Moreover, combining techniques used for parabolic equations and the Schrödinger equation, we also derive the detailed asymptotic profile of the solution.

arXiv comment24 pages