Copyright © 2010 Salah Badraoui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the following reaction-diffusion system with a cross-diffusion matrix and fractional derivatives ut=a1Δu+a2Δv-c1(-Δ)α1u-c2(-Δ)α2v+1ωf1(x,t) in Ω×]0,t*[, vt=b1Δu+b2Δv-d1(-Δ)β1u-d2(-Δ)β2v+1ωf2(x,t) in Ω×]0,t*[, u=v=0 on ∂Ω×]0,t*[, u(x,0)=u0(x), v(x,0)=v0(x) in x∈Ω, where Ω⊂RN (N≥1) is a smooth bounded domain, u0,v0∈L2(Ω), the diffusion matrix M=(a1a2b1b2) has semisimple and positive eigenvalues 0<ρ1≤ρ2, 0<α1,α2,β1,β2<1, ω⊂Ω is an open nonempty set, and 1ω is the characteristic function of ω. Specifically, we prove that under some conditions over the coefficients ai,bi,ci,di(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*].