Well-posedness and analyticity of solutions for the sixth-order Boussinesq equation

TL;DR

This paper investigates the sixth-order Boussinesq equation, establishing local well-posedness, dispersive estimates, and persistence of analyticity, thereby advancing understanding of solutions in high dimensions with nonlinearities.

Contribution

It extends the local well-posedness theory to high dimensions and improves results for cubic nonlinearities despite challenging terms.

Findings

Established local solutions for the sixth-order Boussinesq equation

Derived dispersive estimates despite the 'bad' fourth term

Proved persistence of spatial analyticity for cubic nonlinearities

Abstract

Studied in this paper is the sixth-order Boussinesq equation. We extend the local well-posedness theory for this equation with quadratic and cubic nonlinearities to the high dimensional case. In spite of having the ``bad'' fourth term $\Delta u$ in the equation, we derive some dispersive estimates leading to the existence of local solutions which also improves the previous results in the cubic case. In addition, we show persistence of spatial analyticity of solutions for the cubic nonlinearity.