Local well-posedness for the Sixth-Order Boussinesq Equation

TL;DR

This paper establishes local well-posedness results for a nonlinear sixth-order Boussinesq equation in Sobolev spaces with negative indices, expanding understanding of its initial-value problem.

Contribution

It proves local well-posedness for the sixth-order Boussinesq equation in non-homogeneous Sobolev spaces with negative regularity indices, which was previously unaddressed.

Findings

Well-posedness holds for initial data in $H^s(\R)$ with $s<0$

Results apply to both cases $eta=\pm1$

Advances the theory of higher-order nonlinear wave equations

Abstract

This work studies the local well-posedness of the initial-value problem for the nonlinear sixth-order Boussinesq equation $u_{tt}=u_{xx}+\beta u_{xxxx}+u_{xxxxxx}+(u^2)_{xx}$, where $\beta=\pm1$. We prove local well-posedness with initial data in non-homogeneous Sobolev spaces $H^s(\R)$ for negative indices of $s \in \R$.