Blow-up results for semilinear wave equations in the super-conformal case

TL;DR

This paper investigates blow-up phenomena for semilinear wave equations with super-conformal power nonlinearities, improving existing bounds by employing similarity variables and perturbation techniques in higher dimensions.

Contribution

It advances the understanding of blow-up solutions in super-conformal wave equations by refining upper bounds using a novel perturbation approach in similarity variables.

Findings

Improved upper bounds on blow-up solutions.

Effective handling of lower order perturbations.

Application of similarity variables to super-conformal cases.

Abstract

We consider the semilinear wave equation in higher dimensions with power nonlinearity in the super-conformal range, and its perturbations with lower order terms, including the Klein-Gordon equation. We improve the upper bounds on blow-up solutions previously obtained by Killip, Stovall and Vi\c{s}an [6]. Our proof uses the similarity variables' setting. We consider the equation in that setting as a perturbation of the conformal case, and we handle the extra terms thanks to the ideas we already developed in [5] for perturbations of the pure power case with lower order terms.