Optimal Decay Estimate and Asymptotic Profile for Solutions to the Generalized Zakharov-Kuznetsov-Burgers Equation in 2D

TL;DR

This paper analyzes the decay rates and asymptotic behavior of solutions to a 2D generalized Zakharov-Kuznetsov-Burgers equation, establishing optimal decay rates and explicit profiles using combined techniques from parabolic and Schrödinger equations.

Contribution

It provides the first rigorous proof of the optimal decay rate and explicit asymptotic profile for solutions to this nonlinear dispersive-dissipative equation in 2D.

Findings

Solution decays at rate t^{-3/4} in L^{ ext{infinity}}

Lower bound confirms decay rate is optimal

Explicit asymptotic profile derived using combined techniques

Abstract

We consider the Cauchy problem for the generalized Zakharov-Kuznetsov-Burgers equation in 2D. This is one of the nonlinear dispersive-dissipative equations, which has a spatial anisotropic dissipative term $-\mu u_{xx}$. In this paper, we prove that the solution to this problem decays at the rate of $t^{-\frac{3}{4}}$ in the $L^{\infty}$-sense, provided that the initial data $u_{0}(x, y)$ satisfies $u_{0}\in L^{1}(\mathbb{R}^{2})$ and some appropriate regularity assumptions. Moreover, we investigate the more detailed large time behavior and obtain a lower bound of the $L^{\infty}$-norm of the solution. As a result, we prove that the given decay rate $t^{-\frac{3}{4}}$ of the solution to be optimal. Furthermore, combining the techniques used for the parabolic equations and for the Schr$\ddot{\mathrm{o}}$dinger equation, we derive the explicit asymptotic profile for the solution.