Geodesic
A variational approach to bifurcation points of a reaction-diffusion system with obstacles and Neumann boundary conditions
Applications of Mathematics, Tome 61 (2016) no. 1, pp. 1-25.

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Given a reaction-diffusion system which exhibits Turing's diffusion-driven instability, the influence of unilateral obstacles of opposite sign (source and sink) on bifurcation and critical points is studied. In particular, in some cases it is shown that spatially nonhomogeneous stationary solutions (spatial patterns) bifurcate from a basic spatially homogeneous steady state for an arbitrarily small ratio of diffusions of inhibitor and activator, while a sufficiently large ratio is necessary in the classical case without unilateral obstacles. The study is based on a variational approach to a non-variational problem which even after transformation to a variational one has an unusual structure for which usual variational methods do not apply.
DOI : 10.1007/s10492-016-0119-9
Classification : 35B32, 35J50, 35J57, 35K57, 47J20
Keywords: reaction-diffusion system; unilateral condition; variational inequality; local bifurcation; variational approach; spatial patterns; Turing instability 
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Eisner, Jan; Kučera, Milan; Väth, Martin. A variational approach to bifurcation points of a reaction-diffusion system with obstacles and Neumann boundary conditions. Applications of Mathematics, Tome 61 (2016) no. 1, pp. 1-25. doi : 10.1007/s10492-016-0119-9. http://geodesic.mathdoc.fr/articles/10.1007/s10492-016-0119-9/

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