A sharp bilinear estimate for the Klein-Gordon equation in arbitrary space-time dimensions. (English) Zbl 1313.35036

Jeavons, Chris

Differ. Integral Equ. 27, No. 1-2, 137-156 (2014).

Summary: 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.

MSC:

35B45	A priori estimates in context of PDEs
35L10	Second-order hyperbolic equations

Keywords:

Klein-Gordon equation; sharp bilinear estimate; Strichartz estimate