Unique Continuation Principle for Systems of Parabolic Equations

In this paper we prove a unique continuation result for a cascade system of parabolic equations, in which the solution of the first equation is (partially) used as a forcing term for the second equation. As a consequence we prove the existence of ε-insensitizing controls for some parabolic equations when the control region and the observability region do not intersect. Mathematics Subject Classification. 35B37, 35B60, 93C20. Received December 10, 2007. Revised October 10, 2008. Published online February 10, 2009. 1. Statement of the problem and main results This paper is devoted to the study of unique continuation properties for cascade systems of parabolic equations. This kind of problems has been studied in particular by Bodart and Fabre [1] in the context of the so called ε-insensitizing control problems for the heat equation, and has been solved only in the particular case in which the control domain and the observability domain have non empty intersection (see Sect. 5 or [1] for a complete description of the problem). To begin with a simple example, as far as the unique continuation property is concerned, let Ω ⊂ R be a Lipschitz bounded domain and, for p0 ∈ L(Ω), let p be the solution of ⎪⎨⎪⎩ ∂tp− div(a∇p) = 0 p(0, x) = p0(x) p(t, σ) = 0 in (0, T )× Ω in Ω on (0, T )× ∂Ω. (1.1) Here a := (aij)1≤i,j≤N is a self-adjoint matrix such that for some positive constant c0 > 0 and all ξ ∈ R , and all x ∈ Ω ∑ 1≤i,j≤N aij(x)ξiξj ≥ c0|ξ|, aij ∈ W 1,∞(Ω). (1.2) The first kind of unique continuation result which we are interested in, can be illustrated with the following: