Remark on the global non-existence of semirelativistic equations with non-gauge invariant power type nonlinearity with mass

TL;DR

This paper investigates the non-existence of global solutions for certain semirelativistic equations with non-gauge invariant nonlinearities, demonstrating small data blowup even with small mass using weighted integral estimates.

Contribution

It introduces a novel approach using weighted integral estimates and fractional derivatives to establish non-existence results for these equations.

Findings

Proves non-existence of global solutions under specific conditions.

Establishes small data blowup with small mass.

Develops a new a priori control method for weighted solutions.

Abstract

The non-existence of global solutions for semirelativistic equations with non-gauge invariant power type nonlinearity with mass is studied in the frame work of weighted $L^1$. In particular, a priori control of weighted integral of solutions is obtained by introducing a pointwise estimate of fractional derivative of some weight functions. Especially, small data blowup with small mass is obtained.