The blow-up rate for a loglog non-scaling invariant semilinear wave equation

TL;DR

This paper investigates the blow-up rate of solutions to a semilinear wave equation with a loglog perturbation, establishing bounds that relate to an associated ODE, despite the PDE lacking scaling invariance.

Contribution

It provides the first precise bounds on the blow-up rate for a non-scaling invariant semilinear wave equation with loglog perturbation.

Findings

Blow-up rate bounds are proportional to the associated ODE solution.

The PDE's non-scaling invariance presents significant analytical challenges.

The results extend understanding of blow-up behavior in perturbed wave equations.

Abstract

We consider blow-up solutions of a semilinear wave equation with a loglog perturbation of the power nonlinearity in the subconformal case, and show that the blow-up rate is given by the solution of the associated ODE which has the same blow-up time. In fact, our result shows an upper bound and a lower bound of the blow-up rate, both proportional to the blow-up solution of the associated ODE. The main difficulty comes from the fact that the PDE is not scaling invariant.