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

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• 作者：Jan Eisner ; Milan Kučera ; Martin Väth
• 关键词：reaction ; diffusion system ; unilateral condition ; variational inequality ; local bifurcation ; variational approach ; spatial patterns ; Turing instability ; 35B32 ; 35K57 ; 35J50 ; 35J57 ; 47J20
• 刊名：Applications of Mathematics
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：61
• 期：1
• 页码：1-25
• 全文大小：266 KB
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• 作者单位：Jan Eisner (1) (2)
  Milan Kučera (3) (4)
  Martin Väth (3)
  
  1. Institute of Mathematics and Biomathematics, Faculty of Science, University of South Bohemia, Branišovská 1760, 370 05, České Budějovice, Czech Republic
  2. Laboratory of Fish Genetics, Institute of Animal Physiology and Genetics, Academy of Sciences of the Czech Republic, Rumburská 89, 277 21, Liběchov, Czech Republic
  3. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Liběchov, Czech Republic
  4. Department of Mathematics, Faculty of Applied Sciences, University of West Bohemia in Pilsen, Univerzitní 8, 306 14, Plzeň, Czech Republic
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Applications of Mathematics
  Mechanics, Fluids and Thermodynamics
  Analysis
  Mathematical and Computational Physics
  Applied Mathematics and Computational Methods of Engineering
  Optimization
  
• 出版者：Springer Netherlands
• ISSN：1572-9109

文摘

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. Keywords reaction-diffusion system unilateral condition variational inequality local bifurcation variational approach spatial patterns Turing instability