Tamtam Proceedings - lamsin

126 A. Guezane-Lakoud et al.3. Functional spacesNow we start the description of functional spaces. We construct the Hilbert spaceW (t) on D ( A 1/2 (t) ) for all t ∈ D, equipped with the norm∣|u| t= ∣A 1/2 (t)u∣ .We define by L the operator (L, l 1 , l 2 ) = (f, ϕ, ψ) generated by the GBVP (1)-(2) withthe domain{D(L) = u ∈ L 2 (D, H), u(t) ∈ D(A(t)), ∂u , ∂u}, A(t)u ∈ L 2 (D, H) .∂t 1 ∂t 2The operator L acts from E into F, where E is the completion of D(L) according tothe norm( ∫ s1∣‖u‖ 2 1 =u(t 1 , s 2 ) ∣∣∣2 ∫ s2∣ ∣ dt 1 +u(s 1 , t 2 ) ∣∣∣2 ∫ ∫0 ∂t 1∣ dt 2)+2 |u| 2 t0 ∂t dt 1dt 2 ;2 Dsup(s 1,s 2)∈Dand F is the Hilbert space L 2 (D, H) × L 2 (]0, T 1 [ , H) × L 2 (]0, T 2 [ , H), whose elementsF = (f, ϕ, ψ) are such that ϕ(0) = ψ(0) and the norm‖F ‖ 2 2 = ‖f‖2 L 2(D,H) + ‖ϕ‖2 L 2(]0,T 1[,H) + ‖ψ‖2 L 2(]0,T 2[,H)is finite.For the operator L we present the following Lemma:Lemma 1. Assume that conditions (a) and (b) hold, then D(L) is dense in L 2 (I, H).Proof. It follows from condition (a).Remark 1. When the domain of definition is depending on the variable t, then the regularizingoperators are introduced for the uniqueness theorem.4. A priori estimateTheorem 1. Under the conditions of Lemma 1, we get for all u ∈ D(L) the followingestimate‖u‖ 2 1 ≤ C ‖Lu‖2 2 . (3)Where the positive constant C is independent on t and on u.Proof. Since the operators A(t) are not bounded, we approximate them by a bounded andstrongly differentiable operators A(t)A −1ε (t) whereA −1ε (t) = (I + ε A(t)) −1 ; ε ≥ 0.The regularizing operators A −1ε (t) have the following properties:TAMTAM –Tunis– 2005