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

TL;DR

This paper investigates the blow-up behavior of solutions to a semilinear wave equation with a logarithmic nonlinearity in the critical conformal regime, establishing sharp blow-up rates despite broken scaling symmetry.

Contribution

It extends blow-up rate results to the conformal case with a logarithmic nonlinearity, introducing a Lyapunov functional and energy methods for critical regimes.

Findings

Established an a priori upper bound for blow-up solutions.

Constructed a Lyapunov functional with weak dissipation.

Obtained the first sharp blow-up rate result in a critical framework with broken scaling symmetry.

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.