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

Abstract

Let us consider as an example, the reaction-diffusion system named “Brusselator”:

$$ {u_t} - {d_1}\Delta u = {u^2}v - \left( {B + 1} \right)u + A in \left( {0,T} \right) \times \Omega$$

(0.1)

$$ {v_t} - {d_2}\Delta v = - {u^2}v + Bu in \left( {0,T} \right) \times \Omega$$

(0.2)

where Ω is smooth bounded open subset of IRⁿ andT >0, with boundary conditions

$$ {{\lambda }_{1}}\frac{{\partial u}}{{\partial n}} + \left( {1 - {{\lambda }_{1}}} \right)u = {{\alpha }_{1}} on \partial \Omega $$

(0.3)

$$ {\lambda _2}\frac{{\partial v}}{{\partial n}} + \left( {1 - {\lambda _2}} \right)u = {\alpha _2} on \partial \Omega $$

(0.4)

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.