Large time behavior and optimal decay estimate for solutions to the generalized Kadomtsev--Petviashvili--Burgers equation in 2D

TL;DR

This paper analyzes the long-term behavior of solutions to a 2D generalized Kadomtsev--Petviashvili--Burgers equation, establishing the optimal decay rate and providing an approximate formula for large times.

Contribution

It offers a detailed description of the large time asymptotics and confirms the optimality of the known decay rate for solutions.

Findings

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

Constructs an approximate formula for large t

Proves the decay rate is optimal

Abstract

We consider the Cauchy problem for the generalized Kadomtsev--Petviashvili--Burgers equation in 2D. This is one of the nonlinear dispersive-dissipative type equations, which has a spatial anisotropic dissipative term. Under some suitable regularity assumptions on the initial data $u_{0}$, especially the condition $\partial_{x}^{-1}u_{0} \in L^{1}(\mathbb{R}^{2})$, it is known that the solution to this problem decays at the rate of $t^{-\frac{7}{4}}$ in the $L^{\infty}$-sense. In this paper, we investigate the more detailed large time behavior of the solution and construct the approximate formula for the solution at $t\to \infty$. Moreover, we obtain a lower bound of the $L^{\infty}$-norm of the solution and prove that the decay rate $t^{-\frac{7}{4}}$ of the solution given in the previous work to be optimal.