A semilinear wave equation on hyperbolic spaces.

Abstract

The aim of this article is to study whether such a critical exponent exists on IH{sup n}, which is the most approachable rank one symmetric space G/K of the noncompact type. Notice that, though linear wave equations have ben well studied on symmetric spaces, few authors were interested in nonlinear wave equations on such spaces. We are only concerned here in low dimensions (n = 2 or 3) and our results may be summarized by Theorems A and B below: these are based on assumptions similar to Asakura`s, the comparison scale being defined here by functions related to the areas of spheres, namely (sinh r){sup n-1} on IH{sup n}.