Well-posedness and Ill-posedness for the Nonlinear Beam Equation

TL;DR

This paper analyzes the well-posedness and ill-posedness of the nonlinear beam equation, extending existing estimates to determine minimal regularity for solutions and demonstrating ill-posedness in certain cases.

Contribution

It extends Strichartz estimates to the nonlinear beam equation and identifies the minimal regularity needed for well-posedness and scattering, also proving ill-posedness in the defocusing case.

Findings

Extended Strichartz estimates for nonlinear beam equations.

Determined minimal regularity for well-posedness and scattering.

Proved ill-posedness in the defocusing case for certain regularities.

Abstract

We investigate Strichartz estimates for the nonlinear beam equation with initial data $f\in\dot{H}^s, g\in\dot{H}^{s-2}$ and $f\in H^s, g\in H^{s-2}$. We extend results of H. Lindblad and C. D.Sogge [10] and T. Cazenave and F. B. Weissler [4] to nonlinear beam equations to determine the minimal regularity that is needed to prove well-posedness and scattering results with low regularity data. Finally, we also use small dispersion analysis of M. Christ, J. Colliander and T. Tao [2] to prove the nonlinear beam equation is ill-posed in defocusing case $\omega=-1$ when $ 0<s<s_c=\frac{n}{2}-\frac{4}{\kappa-1}$.