2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/222596In this paper, we establish the linear profile decomposition for the Airy equation with complex or real initial data in $L^2$, respectively. As an application, we obtain a dichotomy result on the existence of maximizers for the symmetric Airy-Strichartz inequality.34 pages.A sign error is corrected throught the paper; the statement of Theorem 1.9 is slightly modified since version 3; add a new reference in remark 1.7; some improvement in exposition. To appear Analysis and PDEAnalysis of PDEs35Q53text