Maximizers for the Strichartz Inequalities for the Wave Equation

TL;DR

This paper proves the existence of maximizers for Strichartz inequalities related to the wave equation in dimensions three and higher, extending previous profile decomposition results to higher dimensions.

Contribution

It introduces a linear profile decomposition for the wave equation in dimensions d≥3, enabling the proof of maximizer existence for Strichartz inequalities.

Findings

Existence of maximizers for wave equation Strichartz inequalities in d≥3

Extension of profile decomposition from 3D to higher dimensions

Method adapted from Schrödinger equation to wave equation

Abstract

We prove the existence of maximizers for Strichartz inequalities for the wave equation in dimensions $d\geq 3$. Our approach follows the scheme given by Shao, which obtains the existence of maximizers in the context of the Schr\"odinger equation. The main tool that we use is the linear profile decomposition for the wave equation which we prove in $\mathbb{R}^d$, $d\geq 3$, extending the profile decomposition result of Bahouri and Gerard, previously obtained in $\mathbb{R}^3$.