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

TL;DR

This paper analyzes the long-term behavior of solutions to a 2D anisotropic dissipative Kadomtsev-Petviashvili equation, showing decay rates and detailed asymptotic profiles under specific initial conditions.

Contribution

It provides the first detailed asymptotic profile and decay rate analysis for solutions to the generalized KP equations with anisotropic dissipation in 2D.

Findings

Solution decays at rate t^{-7/4} in L^{ } norm.

Derived detailed asymptotic profile of solutions.

Established decay and profile results under zero-mass and regularity conditions.

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 Schr\"{o}dinger equation, we also derive the detailed asymptotic profile of the solution.