Eigenvalues and bifurcation for problems with positively homogeneous operators and reaction–diffusion systems with unilateral terms. (January 2018)

• Record Type: Journal Article
• Authors: Kučera, Milan; Navrátil, Josef
• Is Part Of: Nonlinear analysis. Volume 166(2018)
• Page Start: 154
• Page End: 180
• Publication Date: 2018-01
• Subjects: Positively homogeneous operators -- Maximal eigenvalue -- Variational characterization -- Global bifurcation -- Reaction–diffusion systems -- Unilateral sources; 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.2017.10.004
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Reaction–diffusion systems satisfying assumptions guaranteeing Turing's instability and supplemented by unilateral terms of type v − and v + are studied. Existence of critical points and sometimes also bifurcation of stationary spatially non-homogeneous solutions are proved for rates of diffusions for which it is excluded without any unilateral term. The main tool is a general result giving a variational characterization of the largest eigenvalue for positively homogeneous operators in a Hilbert space satisfying a condition related to potentiality, and existence of bifurcation for equations with such operators. The originally non-variational (non-symmetric) system is reduced to a single equation with a positively homogeneous potential operator and the abstract results mentioned are used.