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A Description of the Global Attractor for a Class of Reaction – Diffusion Systems with Periodic Solutions
Author(s) -
Büger Matthias
Publication year - 2001
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200102)222:1<31::aid-mana31>3.0.co;2-m
Subject(s) - mathematics , attractor , class (philosophy) , pure mathematics , reaction–diffusion system , boundary value problem , construct (python library) , diffusion , zero (linguistics) , orbit (dynamics) , order (exchange) , mathematical analysis , finite set , boundary (topology) , physics , engineering , economics , linguistics , philosophy , finance , artificial intelligence , computer science , thermodynamics , programming language , aerospace engineering
We examine the autonomous reaction–diffusion system$${{d} \over {dt}} \pmatrix {u \cr v \cr} = \pmatrix {\alpha & -\beta \cr \beta & \alpha \cr} \Delta \pmatrix {u \cr v \cr} +g \pmatrix {\left \Vert \pmatrix {u \cr v \cr} \right \Vert ^2 _{L^2}} \pmatrix {u \cr v \cr} \leqno (RD)$$with Dirichlet boundary conditions on (0, 1), where α , β are real, α > 0, and g is C 1 and satisfies some conditions which we need in order to prove the existence of solutions. We construct a solution of ( RD ) for every initial value in L 2 ((0, 1)) × L 2 ((0, 1)), we show that this solution is uniquely determined and that the solution has C ∞ –smooth representatives for all positive t . We determine the long time behaviour of each solution. In particular, we show that each solution of ( RD ) tends either to the zero solution or to a periodic orbit. We construct all periodic orbits and show that their number is always finite. It turns out that the global attractor is a finite union of subsets of L 2 × L 2 , which are finite–dimensional manifolds, and the dynamics in these sets can be described completely.