Separate variable blow-up patterns for a reaction–diffusion equation with critical weighted reaction. (April 2022)

• Record Type: Journal Article
• Authors: Iagar, Razvan Gabriel; Sánchez, Ariel
• Is Part Of: Nonlinear analysis. Volume 217(2022)
• Publication Date: 2022-04
• Subjects: 35B33 -- 35B40 -- 35K10 -- 35K67 -- 35Q79; Reaction–diffusion equations -- Weighted reaction -- Blow-up -- Separate variable solutions -- Phase space analysis -- Critical exponents; 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.2021.112740
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the separate variable blow-up patterns associated to the following second order reaction–diffusion equation: ∂ t u = Δ u m + | x | σ u m, posed for x ∈ R N, t ≥ 0, where m > 1, dimension N ≥ 2 and σ > 0 . A new and explicit critical exponent σ c = 2 ( m − 1 ) ( N − 1 ) 3 m + 1 is introduced and a classification of the blow-up profiles is given. The most interesting contribution of the paper is showing that existence and behavior of the blow-up patterns is split into different regimes by the critical exponent σ c and also depends strongly on whether the dimension N ≥ 4 or N ∈ { 2, 3 } . These results extend previous works of the authors in dimension N = 1 .