On profile decomposition for Airy type equation

TL;DR

This paper investigates the linear profile decomposition for Airy type equations, focusing on associated Strichartz inequalities and phenomena observed in odd curve cases, including inhomogeneous scenarios.

Contribution

It extends the understanding of profile decompositions to Airy type equations and explores the two-profile phenomenon in odd curve cases, including inhomogeneous situations.

Findings

Identified profile decomposition for Airy type equations.

Analyzed the two-profile phenomenon in odd curve cases.

Extended results to inhomogeneous cases with combined odd curves.

Abstract

We study the linear profile decomposition for the Airy type equation, where the associated Strichartz inequality corresponds to the Fourier extension inequality on the odd curve $\xi^{\ell}$. We also investigate an inhomogeneous case, modeled by the odd curve $\xi^3+\xi^5$ case. We note that, as observed by Frank and Sabin [Math. Ann., 2018], there is a two-profile phenomenon in the profile decomposition associated with odd curves.