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