Global Well-posedness and Asymptotic Behavior of a Class of Initial-Boundary-Value Problem of the Korteweg-de Vries Equation on a Finite Domain

TL;DR

This paper proves global existence and exponential decay of solutions for a class of Korteweg-de Vries IBVP on a finite domain with small initial and boundary data, addressing a key open problem.

Contribution

It establishes global well-posedness and decay results for the KdV IBVP with nonhomogeneous boundary conditions, which was previously unresolved.

Findings

Solutions exist globally for small initial and boundary data.

Solutions decay exponentially if boundary data decay exponentially.

Provides a priori estimates for the IBVP of the KdV equation.

Abstract

In this paper, we study a class of initial boundary value problem (IBVP) of the Korteweg- de Vries equation posed on a finite interval with nonhomogeneous boundary conditions. The IBVP is known to be locally well-posed, but its global $L^2 $ a priori estimate is not available and therefore it is not clear whether its solutions exist globally or blow up in finite time. It is shown in this paper that the solutions exist globally as long as their initial value and the associated boundary data are small, and moreover, those solutions decay exponentially if their boundary data decay exponentially