On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces

TL;DR

This paper establishes local well-posedness and ill-posedness results for the nonlinear semi-relativistic (half-wave) equation in Sobolev spaces, using contraction mapping and Strichartz estimates, advancing understanding of its mathematical properties.

Contribution

It provides the first rigorous proof of local well-posedness and ill-posedness for the nonlinear semi-relativistic equation in Sobolev spaces, employing novel analytical techniques.

Findings

Proved local well-posedness in Sobolev spaces for (NLHW).

Identified conditions leading to ill-posedness in super-critical cases.

Applied Strichartz estimates and Christ-Colliander-Tao techniques effectively.

Abstract

We proved the local well-posedness for the power-type nonlinear semi-relativistic or half-wave equation (NLHW) in Sobolev spaces. Our proofs mainly bases on the contraction mapping argument using Strichartz estimate. We also apply the technique of Christ-Colliander- Tao in \cite{ChristCollianderTao} to prove the ill-posedness for (NLHW) in some cases of the super-critical range.