Unilateral sources and sinks of an activator in reaction-diffusion systems exhibiting diffusion-driven instability

0504264 - MÚ 2020 RIV GB eng J - Journal Article
Fencl, M. - Kučera, Milan
Unilateral sources and sinks of an activator in reaction-diffusion systems exhibiting diffusion-driven instability.
Nonlinear Analysis: Theory, Methods & Applications. Roč. 187, October (2019), s. 71-92. ISSN 0362-546X. E-ISSN 1873-5215
Institutional support: RVO:67985840
Keywords : maximal eigenvalue * positively homogeneous operators * reaction–diffusion systems * unilateral terms * Turing's patterns
OECD category: Pure mathematics
Impact factor: 1.587, year: 2019
Method of publishing: Limited access
http://dx.doi.org/10.1016/j.na.2019.04.001

A reaction–diffusion system exhibiting Turing's diffusion driven instability is considered. The equation for an activator is supplemented by unilateral terms of the type s − (x)u − , s + (x)u + describing sources and sinks active only if the concentration decreases below and increases above, respectively, the value of the basic spatially constant solution which is shifted to zero. We show that the domain of diffusion parameters in which spatially non-homogeneous stationary solutions can bifurcate from that constant solution is smaller than in the classical case without unilateral terms. It is a dual information to previous results stating that analogous terms in the equation for an inhibitor imply the existence of bifurcation points even in diffusion parameters for which bifurcation is excluded without unilateral sources. The case of mixed (Dirichlet–Neumann) boundary conditions as well as that of pure Neumann conditions is described.
Permanent Link: http://hdl.handle.net/11104/0295933