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

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 ${(\ln 1\mathrm{/}\mathrm{\mid }u\mathrm{\mid })}^{1-p}\mathrm{\mid }u\mathrm{\mid }$ 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 $p\in (1,3)$.