Existence of attractors in L ∞ (Ω) for a class of reaction-diffusion systems

Philippe Benilan and Halima Labani
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Philippe Benilan: Université de Franche-Comté, Faculté des Siences et Techniques Laboratoire de Mathématiques
Halima Labani: Université Chouaib Doukkali, Faculté des Siences Departement de Mathématiques et Informatique

A chapter in Nonlinear Evolution Equations and Related Topics, 2004, pp 771-784 from Springer

Abstract: Abstract Let us consider as an example, the reaction-diffusion system named “Brusselator”: 0.1 $$ {u_t} - {d_1}\Delta u = {u^2}v - \left( {B + 1} \right)u + A in \left( {0,T} \right) \times \Omega$$ 0.2 $$ {v_t} - {d_2}\Delta v = - {u^2}v + Bu in \left( {0,T} \right) \times \Omega$$ where Ω is smooth bounded open subset of IRn andT >0, with boundary conditions 0.3 $$ {{\lambda }_{1}}\frac{{\partial u}}{{\partial n}} + \left( {1 - {{\lambda }_{1}}} \right)u = {{\alpha }_{1}} on \partial \Omega $$ 0.4 $$ {\lambda _2}\frac{{\partial v}}{{\partial n}} + \left( {1 - {\lambda _2}} \right)u = {\alpha _2} on \partial \Omega $$ where d 1 d 2 B, Aare positive constants 0 ≤ λ 1, λ 2 ≤ 1 and α1, α2≥. Here u, v are functions of (t, x) with x ∈ Ω.

Date: 2004
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DOI: 10.1007/978-3-0348-7924-8_36

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