Approximate Controllability of a Reaction-Diffusion System with a Cross-Diffusion Matrix and Fractional Derivatives on Bounded Domains. (19th January 2010) - Badraoui Badraoui, Salah Salah
Abstract : We study the following reaction-diffusion system with a cross-diffusion matrix and fractional derivatives u t = a 1 Δ u + a 2 Δ v - c 1 ( - Δ ) α 1 u - c 2 ( - Δ ) α 2 v + 1 ω f 1 ( x, t ) in Ω × ] 0, t * [, v t = b 1 Δ u + b 2 Δ v - d 1 ( - Δ ) β 1 u - d 2 ( - Δ ) β 2 v + 1 ω f 2 ( x, t ) in Ω × ] 0, t * [, u = v = 0 on ∂ Ω × ] 0, t * [, u ( x, 0 ) = u 0 ( x ), v ( x, 0 ) = v 0 ( x ) in x ∈ Ω, where Ω ⊂ ℝ N ( N ≥ 1 ) is a smooth bounded domain, u 0, v 0 ∈ L 2 ( Ω ), the diffusion matrix M = ( a 1 a 2 b 1 b 2 ) has semisimple and positive eigenvalues 0 < ρ 1 ≤ ρ 2, 0 < α 1, α 2, β 1, β 2 < 1, ω ⊂ Ω is an open nonempty set, and 1 ω is the characteristic function of ω . Specifically, we prove that under some conditions over the coefficients a i, b i, c i, d i ( i = 1, 2 ), the semigroup generated by the linear operator of the system is exponentially stable, and under other conditions we prove that for all t * > 0 the system is approximately controllable on [ 0, t * ] .