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

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 $L^{\mathrm{\infty }}$-norm of the solution decays at the rate of $t^{-7\mathrm{/}4}$ if the initial data $u_0(x,y)$ satisfies $(1+\mathrm{\mid }x\mathrm{\mid })u_0\in L^1({\mathbb{R}}^2)$ 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.