The blow-up rate for a log non-scaling invariant semilinear wave equation in the conformal regime

Abstract

We consider the blow-up behavior of solutions to the semilinear wave equation $${\partial }_t^{2}u-\Delta u=|u{|}^{p-1}u{\ln }^a(u^2+2),(x,t)\in {\mathbb{R}}^n\times [0,T),$$ in the conformal case $p=p_c=1+\frac{4}{n-1}$. Previous results in \cite{HZjmaa2020, HZ2022} show that for $a\in \mathbb{R}$, solutions in the subconformal regime $p<p_c$ blow up with a Type~I rate at any non-characteristic point. The objective of this work is to extend this blow-up rate to the conformal regime under the assumption $a<0$. We establish an a priori upper bound for any blow-up solution and construct a Lyapunov functional in similarity variables. The resulting functional exhibits only weak dissipation, which necessitates delicate energy arguments to obtain the sharp blow-up rate in the conformal case. To the best of our knowledge, this provides the first result for the blow-up rate in a critical framework for an evolution problem where the scaling symmetry is broken.

57 pages