Note on the lifespan estimate of solutions for non-gauge invariant semilinear massless semirelativistic equations with some scaling critical nonlinearity

TL;DR

This paper derives lifespan estimates for solutions to non-gauge invariant semilinear semirelativistic equations at the critical scaling, using a modified test function method to handle fractional Laplacian non-locality.

Contribution

It introduces a novel approach with special test functions to estimate solution lifespan in the critical case involving fractional Laplacians.

Findings

Lifespan estimates are established for the critical case.

The method effectively handles non-local fractional Laplacian.

Results contribute to understanding solution behavior in semirelativistic equations.

Abstract

In this manuscript, in the $L^1$ scaling critical case, a lifespan estimate of solutions to the Cauchy problem for non-gauge invariant semilinear semirelativistic equations is considered. The lifespan estimate is given by the modified test function method with a fractional Laplace operator. The main obstacle to obtaining the lifespan estimate is the non-locality of the fractional Laplace operator. To treat the non-locality, special test functions are introduced.