On global well-posedness for nonlinear semirelativistic equations in some scaling subcritical and critical cases

TL;DR

This paper investigates the conditions under which semirelativistic equations with power nonlinearities are globally well-posed in various scaling subcritical and critical cases, using Strichartz estimates and a priori bounds.

Contribution

It establishes local and global well-posedness results for semirelativistic equations in multiple dimensions, including radial solutions, by employing advanced Strichartz estimates and uniform bounds.

Findings

Global well-posedness in 2D for 1 ≤ s ≤ 2

Local well-posedness in higher dimensions for radial solutions

Extension of local solutions via a priori estimates

Abstract

In this paper, the global well-posedness of semirelativistic equations with a power type nonlinearity on Euclidean spaces is studied. In two dimensional $H^s$ scaling subcritical case with $1 \leq s \leq 2$, the local well-posedness follows from a Strichartz estimate. In higher dimensional $H^1$ scaling subcritical case, the local well-posedness for radial solutions follows from a weighted Strichartz estimate. Moreover, in three dimensional $H^1$ scaling critical case, the local well-posedness for radial solutions follows from a uniform bound of solutions which may be derived by the corresponding one dimensional problem. Local solutions may be extended by a priori estimates.