The Critical Conjecture of Wave Equations with Logarithmic Nonlinearity on $\mathbb{H}^2$

TL;DR

This paper confirms the critical exponent for nonlinear wave equations with logarithmic nonlinearity on hyperbolic space, establishing global existence for powers above 3 and blow-up below 3, thus resolving a conjecture in the field.

Contribution

It verifies the critical conjecture for wave equations with logarithmic nonlinearity on hyperbolic space, identifying the critical power as p_c(2)=3.

Findings

Global existence for p>3

Blow-up for p in (1,3)

Critical power p_c(2)=3 confirmed

Abstract

In this paper, we verified the critical conjecture in our previous work \cite{wang2023wave} on two-dimensional hyperbolic space, that is, concerning nonlinear wave equations with logarithmic nonlinearity, which behaves like $\left(\ln {1}/{|u|}\right)^{1-p}|u|$ near $u=0$, on hyperbolic spaces, we demonstrate that the critical power is $p_c(2)=3$, by proving global existence for $p>3$, as well as blow up for $p\in (1,3)$.