Lifespan of solutions to nonlinear Schr\"odinger equations with general homogeneous nonlinearity of the critical order

TL;DR

This paper refines the upper bound of solution lifespan for nonlinear Schrödinger equations with critical homogeneous nonlinearity by employing a unified test function, advancing understanding of blow-up behavior in such equations.

Contribution

It introduces a refined upper bound for solution lifespan using a unified test function, improving previous estimates for equations with critical nonlinearity.

Findings

Refined upper bound of solution lifespan for critical nonlinear Schrödinger equations.

Application of a unified test function to improve lifespan estimates.

Enhanced understanding of blow-up phenomena in critical cases.

Abstract

This paper is concerned with the upper bound of the lifespan of solutions to nonlinear Schr\"odinger equations with general homogeneous nonlinearity of the critical order. In [8], Masaki and the first author obtain the upper bound of the lifespan of solutions to our equation via a test function method introduced by [16, 17]. Their nonlinearity contains a non-oscillating term $|u|^{1+2/d}$ which causes difficultly for constructing an even small data global solution. The non-oscillating term corresponds to the $L^1$-scaling critical. In this paper, it turns out that the upper bound can be refined by employing an unified test function by Ikeda and the second author [5].