Wave equations with logarithmic nonlinearity on hyperbolic spaces

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 $p\in (1,3)$.

17 pages, 4 figures. All comments are welcome