Refined estimates of the blow-up profile for a strongly perturbed semilinear wave equations in one space dimension

TL;DR

This paper refines the understanding of blow-up profiles in strongly perturbed semilinear wave equations in one dimension, achieving exponential convergence to stationary solutions, improving upon previous polynomial estimates.

Contribution

It introduces an implicit solution approach that demonstrates exponential convergence to stationary solutions in strongly perturbed wave equations, refining prior polynomial convergence results.

Findings

Established exponential convergence to stationary solutions

Constructed an implicit solution approaching unperturbed problem solutions

Improved convergence estimates from polynomial to exponential

Abstract

We consider in this paper a class of strongly perturbed semilinear wave equations with a non-characteristic point in one space dimension, for general initial data. Working in the framework of similarity variables, in \cite {MZ} Merle and Zaag constructed an explicit stationary solution of the unperturbed problem and proved an exponential convergence to this family of solutions. If we follow the same strategy under our strongly perturbed equation we just obtain a polynomial convergence which is a rough estimate compared to the one obtained in the unperturbed problem. In order to refine this approximation, we constructed an implicit solution to the perturbed problem which approaches the stationary solutions of the unperturbed problem and we prove the exponential convergence to this prescribed blow-up profile.