Asymptotic behavior of solutions to a higher-order KdV-type equation with critical nonlinearity

TL;DR

This paper analyzes the long-time behavior of solutions to a higher-order KdV-type equation with critical nonlinearity, proving global existence and decay estimates using wave packet methods.

Contribution

It establishes the global existence and decay properties of solutions for a higher-order KdV-type equation with critical nonlinearity, employing the testing by wave packets method.

Findings

Existence of unique global solutions with linear decay rates

Long-time behavior divided into three distinct regions

Method of testing by wave packets applied successfully

Abstract

We consider the Cauchy problem of the higher-order KdV-type equation: \[ \partial_t u + \frac{1}{\mathfrak{m}} |\partial_x|^{\mathfrak{m}-1} \partial_x u = \partial_x (u^{\mathfrak{m}}) \] where $\mathfrak{m} \ge 4$. The nonlinearity is critical in the sense of long-time behavior. Using the method of testing by wave packets, we prove that there exists a unique global solution of the Cauchy problem satisfying the same time decay estimate as that of linear solutions. Moreover, we divide the long-time behavior of the solution into three distinct regions.