Construction of a stable periodic solution to a semilinear heat equation with a prescribed profile. (January 2016)

• Record Type: Journal Article
• Authors: Mahmoudi, Fethi; Nouaili, Nejla; Zaag, Hatem
• Is Part Of: Nonlinear analysis. Volume 131(2016)
• Page Start: 300
• Page End: 324
• Publication Date: 2016-01
• Subjects: 35K57 -- 35K40 -- 35B44; Blow-up profile -- Periodic semilinear heat equation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.09.002
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We construct a periodic solution to the semilinear heat equation with power nonlinearity, in one space dimension, which blows up in finite time T only at one blow-up point. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude. Thanks to the geometrical interpretation of the finite-dimensional parameters in terms of the blow-up time and blow-up point, we derive the stability of the constructed solution with respect to initial data.