## Differential and Integral Equations

### A sharp bilinear estimate for the Klein--Gordon equation in arbitrary space-time dimensions

Chris Jeavons

#### Abstract

We prove a sharp bilinear inequality for the Klein--Gordon equation on $\mathbb{R}^{d+1}$, for any $d \geq 2$. This extends work of Ozawa--Rogers and Quilodrán for the Klein--Gordon equation and generalizes work of Bez--Rogers for the wave equation. As a consequence, we obtain a sharp Strichartz estimate for the solution of the Klein--Gordon equation in five spatial dimensions for data belonging to $H^1$. We show that maximizers for this estimate do not exist and that any maximizing sequence of initial data concentrates at spatial infinity.

#### Article information

Source
Differential Integral Equations, Volume 27, Number 1/2 (2014), 137-156.

Dates
First available in Project Euclid: 12 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1384282857

Mathematical Reviews number (MathSciNet)
MR3161599

Zentralblatt MATH identifier
1313.35036

#### Citation

Jeavons, Chris. A sharp bilinear estimate for the Klein--Gordon equation in arbitrary space-time dimensions. Differential Integral Equations 27 (2014), no. 1/2, 137--156. https://projecteuclid.org/euclid.die/1384282857