A variational approach to bifurcation points of a reaction-diffusion system with obstacles and Neumann boundary conditions

Jan Eisner; Milan Kučera; Martin Väth

Applications of Mathematics (2016)

  • Volume: 61, Issue: 1, page 1-25
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Eisner, Jan, Kučera, Milan, and Väth, Martin. "A variational approach to bifurcation points of a reaction-diffusion system with obstacles and Neumann boundary conditions." Applications of Mathematics 61.1 (2016): 1-25. <http://eudml.org/doc/276057>.

@article{Eisner2016,
abstract = {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.},
author = {Eisner, Jan, Kučera, Milan, Väth, Martin},
journal = {Applications of Mathematics},
keywords = {reaction-diffusion system; unilateral condition; variational inequality; local bifurcation; variational approach; spatial patterns; Turing instability; reaction-diffusion system; unilateral condition; variational inequality; local bifurcation; variational approach; spatial patterns; Turing instability},
language = {eng},
number = {1},
pages = {1-25},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A variational approach to bifurcation points of a reaction-diffusion system with obstacles and Neumann boundary conditions},
url = {http://eudml.org/doc/276057},
volume = {61},
year = {2016},
}

TY - JOUR
AU - Eisner, Jan
AU - Kučera, Milan
AU - Väth, Martin
TI - A variational approach to bifurcation points of a reaction-diffusion system with obstacles and Neumann boundary conditions
JO - Applications of Mathematics
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 1
SP - 1
EP - 25
AB - 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.
LA - eng
KW - reaction-diffusion system; unilateral condition; variational inequality; local bifurcation; variational approach; spatial patterns; Turing instability; reaction-diffusion system; unilateral condition; variational inequality; local bifurcation; variational approach; spatial patterns; Turing instability
UR - http://eudml.org/doc/276057
ER -

References

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