The blow-up rate for strongly perturbed semilinear wave equations in the conformal regime without a radial assumption

TL;DR

This paper establishes the blow-up rate for a broad class of non-radial, strongly perturbed semilinear wave equations with critical nonlinearity, using a novel Lyapunov functional in similarity variables.

Contribution

It extends previous radial results by constructing a new Lyapunov functional applicable to non-radial cases, enabling analysis of blow-up rates in more general settings.

Findings

Blow-up rate matches the non-perturbed ODE solution.

Constructed a Lyapunov functional valid in non-radial scenarios.

Introduced a new Pohozaev-type identity for this analysis.

Abstract

We consider in this paper a large class of perturbed semilinear wave equations with critical (in the conformal transform sense) power nonlinearity. We will show that the blow-up rate of any singular solution is given by the solution of the non-perturbed associated ODE. The result in the radial case has been proved in [13]. The same approach will be followed here, but the main difference is to construct a Lyapunov functional in similarity variables valid in the non-radial case, which is far from being trivial. That functional is obtained by combining some classical estimates and a new identity of the Pohozaev type.