Null Exact Controllability of the Parabolic Equations with Equivalued Surfaceboundary Condition

Let T > 0, Ω ⊂ Rn (n ∈ N) be a given bounded domain, ∂Ω = Γ0 ∪ Γ1 (Γ1 = ∅) (where Γ0 is the interior boundary and Γ1 the outer boundary), Γ0 ∩ Γ1 =∅. For simplicity, we assume that Γ0,Γ1 ∈ C∞ and ω = ∅ is a given subdomain of Ω. Denote the characteristic function of ω by χω, and the unit outward normal vector of Ω by (n1, . . . ,nn). Put Q = Ω× (0,T), Qω = ω× (0,T), and Σ= ∂Ω× (0,T). Let ai j(x)∈ C2(Ω) satisfy ai j = aji, and for some Λ > 0, it holds that