Boundary Value Problems
Volume 2009 (2009), Article ID 560407, 9 pages
Research Article

Interior Controllability of a 2×2 Reaction-Diffusion System with Cross-Diffusion Matrix

Departamento de Matemáticas, Universidad de Los Andes, Mérida 5101, Venezuela

Received 30 December 2008; Accepted 27 May 2009

Academic Editor: Gary Lieberman

Copyright © 2009 Hanzel Larez and Hugo Leiva. 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.


We prove the interior approximate controllability for the following 2×2 reaction-diffusion system with cross-diffusion matrix ut=aΔuβ(Δ)1/2u+bΔv+1ωf1(t,x) in (0,τ)×Ω, vt=cΔudΔvβ(Δ)1/2v+1ωf2(t,x) in (0,τ)×Ω, u=v=0, on (0,T)×Ω, u(0,x)=u0(x), v(0,x)=v0(x), xΩ, where Ω is a bounded domain in N(N1), u0,v0L2(Ω), the 2×2 diffusion matrix D=[abcd] has semisimple and positive eigenvalues 0<ρ1ρ2, β is an arbitrary constant, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, and the distributed controls f1,f2L2([0,τ];L2(Ω)). Specifically, we prove the following statement: if λ11/2ρ1+β>0 (where λ1 is the first eigenvalue of Δ), then for all τ>0 and all open nonempty subset ω of Ω the system is approximately controllable on [0,τ].