Wave equations with logarithmic nonlinearity on hyperbolic spaces

TL;DR

This paper studies nonlinear wave equations with logarithmic nonlinearity on hyperbolic spaces, identifying a critical power that determines global existence or blow-up of solutions.

Contribution

It establishes the critical power for global existence versus blow-up in logarithmic nonlinear wave equations on hyperbolic spaces, specifically proving the critical power is 3 when n=3.

Findings

Global existence for p > 3 in 3D hyperbolic space

Generic blow-up for p in (1,3) in 3D hyperbolic space

Identification of the critical power p_c(n) > 1

Abstract

In light of the exponential decay of solutions of linear wave equations on hyperbolic spaces $\mathbb{H}^n$, to illustrate the critical nature, we investigate nonlinear wave equations with logarithmic nonlinearity, which behaves like $\left(\ln {1}/{|u|}\right)^{1-p}|u|$ near $u=0$, on hyperbolic spaces. Concerning the global existence vs blow up with small data, we expect that the problem admits a critical power $p_c(n)>1$. When $n=3$, we prove that the critical power is $3$, by proving global existence for $p>3$, as well as generically blow up for $p\in (1,3)$.