Bifurcation for a reaction–diffusion system with unilateral obstacles with pointwise and integral conditions

• Published in: Nonlinear analysis: real world applications Vol. 12; no. 2; pp. 817 - 836
• Main Author: Vaeth, Martin
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.04.2011
• Subjects: Averaging on obstacle atoms; Bifurcations; Degree; Diffusion; Global bifurcation; Inclusion; Instability; Integral condition; Integrals; Laplace operator; Mathematical models; Nonlinearity; Nonlocal condition; Obstacle problem; Obstacles; Reaction–diffusion system; Signorini boundary condition; Stability; Stationary solutions; Steady state; Variational inequality; Laplace operator; Integral condition; Inclusion; Degree; Variational inequality; Reaction–diffusion system; Nonlocal condition; Averaging on obstacle atoms; Stationary solutions; Global bifurcation; Signorini boundary condition; Obstacle problem
• ISSN: 1468-1218; 1878-5719
• DOI: 10.1016/j.nonrwa.2010.08.009

A reaction–diffusion system of activator–inhibitor or substrate-depletion type is considered which is subject to diffusion driven instability. It is shown that obstacles (e.g. a unilateral membrane) for one or both quantities introduce a new bifurcation of spatially non-homogeneous steady states in a parameter domain where the trivial branch is exponentially stable without obstacles. The obstacles are modeled in terms of inclusions. Moreover, simultaneously some of the obstacles can be modeled also using nonlocal integral conditions.