Weak and Strong Solutions for a Strongly Damped Quasilinear Membrane Equation

and Applied Analysis 3 Theorem2. Assume that (y0, y1, f) ∈ H1 0 ×L2×L2(0, T;H−1), and U ∈ L∞(Q). Then problem (2) has a unique weak solution y ∈ W(0, T). Moreover, the solution mapping p = (y0, y1, f,U) → y(p) ofP ≡ H1 0 ×L2×L2(0, T;H−1)×L∞(Q) into W(0, T) is locally Lipschitz continuous. Indeed, let p1 = (y1 0 , y1 1 , f1,U1) ∈ P and p2 = (y2 0 , y2 1 , f2,U2) ∈ P. We prove Theorem 2 by showing the inequality 󵄩󵄩󵄩󵄩y (p1) − y (p2)󵄩󵄩󵄩󵄩W(0,T) ≤ C(󵄩󵄩󵄩󵄩󵄩∇ (y1 0 − y2 0)󵄩󵄩󵄩󵄩󵄩22 + 󵄩󵄩󵄩󵄩󵄩y1 1 − y2 1󵄩󵄩󵄩󵄩󵄩22 + 󵄩󵄩󵄩󵄩f1 − f2󵄩󵄩󵄩󵄩2L2(0,T;H−1) + 󵄩󵄩󵄩󵄩U1 − U2󵄩󵄩󵄩󵄩2L∞(Q))1/2 ≡ C 󵄩󵄩󵄩󵄩p1 − p2󵄩󵄩󵄩󵄩P , (13) where C > 0 is a constant depending on data. Next we introduce the solution space S(0, T) for strong solutions of (2) defined by S (0, T) = {g | g ∈ L2 (0, T;D (Δ)) , g󸀠 ∈ L2 (0, T;D (Δ)) , g󸀠󸀠 ∈ L2 (Q)} (14) endowed with a norm 󵄩󵄩󵄩󵄩g󵄩󵄩󵄩󵄩S(0,T) = (󵄩󵄩󵄩󵄩g󵄩󵄩󵄩󵄩2L2(0,T;D(Δ)) + 󵄩󵄩󵄩󵄩󵄩g󸀠󵄩󵄩󵄩󵄩󵄩2L2(0,T;D(Δ)) + 󵄩󵄩󵄩󵄩󵄩g󸀠󸀠󵄩󵄩󵄩󵄩󵄩2L2(Q))1/2 , (15) where g󸀠 and g󸀠󸀠 denote the firstand second-order distributive derivatives of g. We remark also from Dautray and Lions [3, p. 555] that S(0, T) is continuously embedded in C([0, T]; D(Δ)) ∩ C1([0, T];H1 0 ). Definition 3. A function y is said to be a strong solution of (2) if y ∈ S(0, T) and y satisfies y󸀠󸀠 (t) − ∇ ⋅ G (∇y (t)) − μΔy󸀠 (t) = U (t) y (t) + f (t) , a.e. t ∈ [0, T] , y (0) = y0 ∈ D (Δ) , y󸀠 (0) = y1 ∈ H1 0 . (16) The next theorem gives a well-posedness result for strong solutions of (2). Theorem4. Assume that (y0, y1, f) ∈ D(Δ)×H1 0×L2(Q), and U ∈ L∞(Q). Then (2) has a unique strong solution y ∈ S(0, T) and it satisfies 󵄩󵄩󵄩󵄩y󵄩󵄩󵄩󵄩S(0,T) ≤ C(󵄩󵄩󵄩󵄩y0󵄩󵄩󵄩󵄩2D(Δ) + 󵄩󵄩󵄩󵄩y1󵄩󵄩󵄩󵄩2H1 0 + 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩2L2(Q) + ‖U‖2L∞(Q)) , (17) where C is a constant depending on data. Now we give the result on the continuous dependence of strong solutions of (2) on p = (y0, y1, f,U). Let F be a product space defined by F = D (Δ) × H1 0 × L2 (Q) × L∞ (Q) (18) endowed with a norm 󵄩󵄩󵄩󵄩p󵄩󵄩󵄩󵄩F = (󵄩󵄩󵄩󵄩y0󵄩󵄩󵄩󵄩2D(Δ) + 󵄩󵄩󵄩󵄩y1󵄩󵄩󵄩󵄩2H1 0 + 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩2L2(Q) + ‖U‖2L∞(Q)) . (19) For each p = (y0, y1, f,U) ∈ F we have a strong solution y = y(p) ∈ S(0, T) of (2) by Theorem 4. Thus, we can define the solution mapping p = (y0, y1, f,U) → y(p) of F into S(0, T). Theorem 5. The nonlinear solution mapping p = (y0, y1, f,U) → y(p) ofF into S(0, T) of (2) is continuous. Throughout this paper, we will use C as a generic constant and omit writing the integral variables in any definite integrals without confusion. 3. Proof of Main Results Proof of Theorem 2. Since Ui ∈ L∞(Q) (i = 1, 2), by the results in [1], we can deduce that the weak solutions y(pi) of (2) corresponding to pi (i = 1, 2) exist in W(0, T) such that 󵄩󵄩󵄩󵄩y (pi)󵄩󵄩󵄩󵄩W(0,T) ≤ C(󵄩󵄩󵄩󵄩󵄩∇yi 0󵄩󵄩󵄩󵄩󵄩22 + 󵄩󵄩󵄩󵄩󵄩yi 1󵄩󵄩󵄩󵄩󵄩22 + 󵄩󵄩󵄩󵄩fi󵄩󵄩󵄩󵄩2L2(0,T;H−1))1/2 ≤ C 󵄩󵄩󵄩󵄩pi󵄩󵄩󵄩󵄩P (i = 1, 2) . (20) We denote y1 − y2 ≡ y(p1) − y(p2) by ψ. Then, we can get from (2) thatψ satisfies the following equation in weak sense: ψ󸀠󸀠 − μΔψ󸀠 = ∇ ⋅ G (∇y1) − ∇ ⋅ G (∇y2) + U1ψ + Uy2 + f in Q, ψ = 0 on Σ, ψ (0) = y1 0 − y2 0 , ψ󸀠 (0) = y1 1 − y2 1 in Ω, (21) where U = U1 − U2 and f = f1 − f2. Wemultiply (21) by ψ󸀠 + ψ to have 12 d dt (󵄩󵄩󵄩󵄩󵄩ψ󸀠 (t)󵄩󵄩󵄩󵄩󵄩22 + μ 󵄩󵄩󵄩󵄩∇ψ (t)󵄩󵄩󵄩󵄩22) + μ 󵄩󵄩󵄩󵄩󵄩∇ψ󸀠 (t)󵄩󵄩󵄩󵄩󵄩22 = − (G (∇y1 (t)) − G (∇y2 (t)) , ∇ (ψ󸀠 (t) + ψ (t)))2 4 Abstract and Applied Analysis + (U1 (t) ψ (t) + U (t) y2 (t) , ψ󸀠 (t) + ψ (t))2 − ⟨ψ󸀠󸀠 (t) , ψ (t)⟩ −1,1 + ⟨f (t) , ψ󸀠 (t) − ψ (t)⟩ −1,1 . (22) By integrating (22) over [0, t], we obtain 󵄩󵄩󵄩󵄩󵄩ψ󸀠 (t)󵄩󵄩󵄩󵄩󵄩22 + 2μ∫ 0 󵄩󵄩󵄩󵄩󵄩∇ψ󸀠󵄩󵄩󵄩󵄩󵄩22 ds + μ 󵄩󵄩󵄩󵄩∇ψ (t)󵄩󵄩󵄩󵄩22 = 󵄩󵄩󵄩󵄩󵄩ψ󸀠 (0)󵄩󵄩󵄩󵄩󵄩22 + μ 󵄩󵄩󵄩󵄩∇ψ (0)󵄩󵄩󵄩󵄩22 − 2∫ 0 (G (∇y1) − G (∇y2) , ∇ (ψ󸀠 + ψ))2 ds + 2∫ 0 (U1ψ + Uy2, ψ󸀠 + ψ)2 ds − 2 (ψ󸀠 (t) , ψ (t)) 2 + 2 (ψ󸀠 (0) , ψ (0)) 2 + 2∫ 0 󵄩󵄩󵄩󵄩󵄩ψ󸀠󵄩󵄩󵄩󵄩󵄩22 ds + 2∫ 0 ⟨f , ψ󸀠 + ψ⟩ −1,1 ds. (23) Let ε > 0 be an arbitrary real number. Then, by (9), (20), and the Schwartz inequality we can obtain the following: 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨2 ∫ 0 (G (∇y1) − G (∇y2) , ∇ (ψ󸀠 + ψ))2 ds󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 4∫ 0 󵄩󵄩󵄩󵄩∇ψ󵄩󵄩󵄩󵄩2 󵄩󵄩󵄩󵄩󵄩∇ψ󸀠󵄩󵄩󵄩󵄩󵄩2 ds + 4∫ 0 󵄩󵄩󵄩󵄩∇ψ󵄩󵄩󵄩󵄩22 ds ≤ (4ε + 4)∫ 0 󵄩󵄩󵄩󵄩∇ψ󵄩󵄩󵄩󵄩22 ds + ε∫ 0 󵄩󵄩󵄩󵄩󵄩∇ψ󸀠󵄩󵄩󵄩󵄩󵄩22 ds; 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨2 ∫ 0 (U1ψ + Uy2, ψ󸀠 + ψ)2 ds󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 2∫ 0 (󵄩󵄩󵄩󵄩U1ψ󵄩󵄩󵄩󵄩2 + 󵄩󵄩󵄩󵄩Uy2󵄩󵄩󵄩󵄩2) (󵄩󵄩󵄩󵄩󵄩ψ󸀠󵄩󵄩󵄩󵄩󵄩2 + 󵄩󵄩󵄩󵄩ψ󵄩󵄩󵄩󵄩2) ds ≤ 2 󵄩󵄩󵄩󵄩U1󵄩󵄩󵄩󵄩L∞(Q) ∫ 0 (󵄩󵄩󵄩󵄩ψ󵄩󵄩󵄩󵄩2 󵄩󵄩󵄩󵄩󵄩ψ󸀠󵄩󵄩󵄩󵄩󵄩2 + 󵄩󵄩󵄩󵄩ψ󵄩󵄩󵄩󵄩22) ds + ∫ 0 󵄩󵄩󵄩󵄩Uy2󵄩󵄩󵄩󵄩22 ds + 2∫ 0 (󵄩󵄩󵄩󵄩ψ󵄩󵄩󵄩󵄩22 + 󵄩󵄩󵄩󵄩󵄩ψ󸀠󵄩󵄩󵄩󵄩󵄩22) ds ≤ C∫ 0 (󵄩󵄩󵄩󵄩∇ψ󵄩󵄩󵄩󵄩22 + 󵄩󵄩󵄩󵄩󵄩ψ󸀠󵄩󵄩󵄩󵄩󵄩22) ds + ‖U‖2L∞(Q) 󵄩󵄩󵄩󵄩y2󵄩󵄩󵄩󵄩2L2(Q) ≤ C(∫ 0 (󵄩󵄩󵄩󵄩∇ψ󵄩󵄩󵄩󵄩22 + 󵄩󵄩󵄩󵄩󵄩ψ󸀠󵄩󵄩󵄩󵄩󵄩22) ds + ‖U‖2L∞(Q) 󵄩󵄩󵄩󵄩y2󵄩󵄩󵄩󵄩2W(0,T)) ≤ C(∫ 0 (󵄩󵄩󵄩󵄩∇ψ󵄩󵄩󵄩󵄩22 + 󵄩󵄩󵄩󵄩󵄩ψ󸀠󵄩󵄩󵄩󵄩󵄩22) ds + ‖U‖2L∞(Q) 󵄩󵄩󵄩󵄩p2󵄩󵄩󵄩󵄩2P) ; 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨2 ∫ 0 ⟨f, ψ󸀠 + ψ⟩−1,1 ds󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 2∫ 0 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩H−1 (󵄩󵄩󵄩󵄩󵄩∇ψ󸀠󵄩󵄩󵄩󵄩󵄩2 + 󵄩󵄩󵄩󵄩∇ψ󵄩󵄩󵄩󵄩2) ds ≤ (1ε + 1) 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩2L2(0,T;H−1) + ε∫ 0 󵄩󵄩󵄩󵄩󵄩∇ψ󸀠󵄩󵄩󵄩󵄩󵄩22 ds + ∫ 0 󵄩󵄩󵄩󵄩∇ψ󵄩󵄩󵄩󵄩22 ds; 2 󵄨󵄨󵄨󵄨󵄨(ψ󸀠 (t) , ψ (t))2󵄨󵄨󵄨󵄨󵄨 ≤ 2 󵄩󵄩󵄩󵄩󵄩ψ󸀠 (t)󵄩󵄩󵄩󵄩󵄩2 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩ψ (0) + ∫ 0 ψ󸀠ds󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩2 ≤ ε 󵄩󵄩󵄩󵄩󵄩ψ󸀠 (t)󵄩󵄩󵄩󵄩󵄩22 + 1ε 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩ψ (0) + ∫ 0 ψ󸀠ds󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 2 2 ≤ ε 󵄩󵄩󵄩󵄩󵄩ψ󸀠 (t)󵄩󵄩󵄩󵄩󵄩22 + 2ε 󵄩󵄩󵄩󵄩ψ (0)󵄩󵄩󵄩󵄩22 + 2T ε ∫ 0 󵄩󵄩󵄩󵄩󵄩ψ󸀠󵄩󵄩󵄩󵄩󵄩22 ds. (24) We also note that |2(ψ󸀠(0), ψ(0))2| ≤ C(‖∇ψ(0)‖22 + ‖ψ󸀠(0)‖22). Therefore, from (23) and (24), we can obtain the following inequality: (1 − ε) 󵄩󵄩󵄩󵄩󵄩ψ󸀠 (t)󵄩󵄩󵄩󵄩󵄩22 + μ 󵄩󵄩󵄩󵄩∇ψ (t)󵄩󵄩󵄩󵄩22 + (2μ − 2ε) ⋅ ∫ 0 󵄩󵄩󵄩󵄩󵄩∇ψ󸀠󵄩󵄩󵄩󵄩󵄩22 ds ≤ C(1 + 1 + T ε + 󵄩󵄩󵄩󵄩p2󵄩󵄩󵄩󵄩2P) ⋅ (󵄩󵄩󵄩󵄩∇ψ (0)󵄩󵄩󵄩󵄩22 + 󵄩󵄩󵄩󵄩󵄩ψ󸀠 (0)󵄩󵄩󵄩󵄩󵄩22 + ‖f‖2L2(0,T;H−1) + ‖U‖2L∞(Q) + ∫ 0 (󵄩󵄩󵄩󵄩󵄩ψ󸀠󵄩󵄩󵄩󵄩󵄩22 + 󵄩󵄩󵄩󵄩∇ψ󵄩󵄩󵄩󵄩22) ds) . (25) If we choose ε = min{1/2, μ/2}, then by Bellman-Gronwall’s inequality it follows that 󵄩󵄩󵄩󵄩󵄩ψ󸀠 (t)󵄩󵄩󵄩󵄩󵄩22 + 󵄩󵄩󵄩󵄩∇ψ (t)󵄩󵄩󵄩󵄩22 + ∫ 0 󵄩󵄩󵄩󵄩󵄩∇ψ󸀠󵄩󵄩󵄩󵄩󵄩22 ds ≤ C (T, p2) ⋅ (󵄩󵄩󵄩󵄩∇ψ (0)󵄩󵄩󵄩󵄩22 + 󵄩󵄩󵄩󵄩󵄩ψ󸀠 (0)󵄩󵄩󵄩󵄩󵄩22 + ‖f‖2L2(0,T;H−1) + ‖U‖2L∞(Q)) = C (T, p2) 󵄩󵄩󵄩󵄩p1 − p2󵄩󵄩󵄩󵄩2P . (26) By (21) and (26) we have 󵄩󵄩󵄩󵄩󵄩ψ󸀠󸀠󵄩󵄩󵄩󵄩󵄩L2(0,T;H−1) ≤ C(󵄩󵄩󵄩󵄩󵄩∇ψ󸀠󵄩󵄩󵄩󵄩󵄩L2(Q) + 󵄩󵄩󵄩󵄩G (∇y1) − G (∇y2)󵄩󵄩󵄩󵄩L2(Q) + 󵄩󵄩󵄩󵄩ψ󵄩󵄩󵄩󵄩L2(Q) + ‖U‖L∞(Q) 󵄩󵄩󵄩󵄩y2󵄩󵄩󵄩󵄩L2(Q) + ‖f‖L2(0,T;H−1)) ≤ C (󵄩󵄩󵄩󵄩󵄩∇ψ󸀠󵄩󵄩󵄩󵄩󵄩L2(Q) + 󵄩󵄩󵄩󵄩∇ψ󵄩󵄩󵄩󵄩L2(Q) + ‖U‖L∞(Q) 󵄩󵄩󵄩󵄩y2󵄩󵄩󵄩󵄩W(0,T) + ‖f‖L2(0,T;H−1)) ≤ C1 (T, p2) (󵄩󵄩󵄩󵄩p1 − p2󵄩󵄩󵄩󵄩P + ‖U‖L∞(Q) 󵄩󵄩󵄩󵄩p2󵄩󵄩󵄩󵄩P + ‖f‖L2(0,T;H−1)) ≤ C2 (T, p2) 󵄩󵄩󵄩󵄩p1 − p2󵄩󵄩󵄩󵄩P . (27) Finally, by combining (26) and (27) we obtain (13). This completes the proof. Abstract and Applied Analysis 5and Applied Analysis 5 Lemma 6. Let X,Y, and Z be Banach spaces such that the imbeddings X ⊂ Y ⊂ Z are continuous and the imbedding X ⊂ Y is compact. Then a bounded set of W1,∞(0, T;X, Z) = {g | g ∈ L∞(0, T;X), g󸀠 ∈ L∞(0, T; Z)} is relatively compact in C([0, T]; Y). Proof. See Simon [13]. Proof of Theorem 4. We divide the proof into three steps. Step 1 (approximate solutions and a priori estimates). We construct approximate solutions of (2) by a Faedo–Galerkin’s procedure. Since D(Δ) is separable, there exists a complete orthonormal system {wm}∞m=1 in L2 such that {wm}∞m=1 is free and total in D(Δ). For each m ∈ N we can define an approximate solution of (2) by ym (t) = m ∑ j=1 gjm (t) wj, (28) where ym(t) satisfies (2). Then (2) can be written as m vector differential equations E d2 dt2 ⃗ gm + G (∇ym) ⃗ gm + μΔ̃ d dt ⃗ gm = Ũ ⃗ gm + ⃗ fm (29) with initial values ⃗ gm (0) = [(y0, w1)2 , (y0, w2)2 , . . . , (y0, wm)2]t , d dt ⃗ gm (0) = [(y1, w1)2 , (y1, w2)2 , . . . , (y1, wm)2]t . (30) Notations of (29) can be explained as follows: ⃗ gm = [g1m, . . . , gmm]t , E = ((wi, wj)2 = δij : i = 1, . . . , m, j = 1, . . . , m) ≡ Im, G (∇ym) = ( 1 √1 + 󵄨󵄨󵄨󵄨∇ym󵄨󵄨󵄨󵄨2 (∇wi, ∇wj)2 : i = 1, . . . , m, j = 1, . . . , m) , μΔ̃ = (μ (∇wi, ∇wj)2 : i = 1, . . . , m, j = 1, . . . , m) , Ũ = ((Uwi, wj)2 : i = 1, . . . , m, j = 1, . . . , m) , ⃗ fm = [(f, w1)2 , . . . , (f, wm)2]t , (31) where [⋅ ⋅ ⋅ ]t denotes the transpose of [⋅ ⋅ ⋅ ]. Since G(∇⋅) : H1 0 → [L∞]n is Lipschitz continuous and U ∈ L∞(Q), we can deduce by Carathéodory type existence theorem that the nonlinear vector differential equation (29) admits a unique solution [g1m, g2m, . . . , gmm]t on [0, T]. Hence, we can construct the approximate solutionym(t) of (2). Next we shall derive a priori estimates of ym(t). By analogy with (22), we take L2 product of the equations for approximate solutions ym(t) with −Δy󸀠 m(t) − Δym(t) to have 1 2 d dt (󵄩󵄩󵄩󵄩󵄩∇y󸀠 m (t)󵄩󵄩󵄩󵄩󵄩22 + μ 󵄩󵄩󵄩󵄩Δym (t)󵄩󵄩󵄩󵄩22) + μ 󵄩󵄩󵄩󵄩󵄩Δy󸀠 m (t)󵄩󵄩󵄩󵄩󵄩22 = − (∇ ⋅ G (∇ym (t)) , Δy󸀠 m (t) + Δym (t))2 − (U (t) ym (t) , Δy󸀠 m (t) + Δym (t))2 + (y󸀠󸀠 m (t) , Δym (t))2 − (f (t) , Δy󸀠 m (t) + Δym (t))2 . (32) By integrating (32) over [0, t], we obtain 󵄩󵄩󵄩󵄩󵄩∇y󸀠 m (t)󵄩󵄩󵄩󵄩󵄩22 + 2μ∫ 0 󵄩󵄩󵄩󵄩󵄩Δy󸀠 m󵄩󵄩󵄩󵄩󵄩22 ds + μ 󵄩󵄩󵄩󵄩Δym (t)󵄩󵄩󵄩󵄩22 = 󵄩󵄩󵄩󵄩∇y1m󵄩󵄩󵄩󵄩22 + μ 󵄩󵄩󵄩󵄩Δy0m󵄩󵄩󵄩󵄩22 − 2∫ 0 (∇ ⋅ G (∇ym) , Δy󸀠 m + Δym)2 ds − 2∫ 0 (Uym, Δy󸀠 m + Δym)2 ds + 2 (y󸀠 m (t) , Δym (t))2 − 2 (y1m, Δy0m)2 + 2∫ 0 󵄩󵄩󵄩󵄩󵄩∇y󸀠 m󵄩󵄩󵄩󵄩󵄩22 ds − 2∫ 0 (f, Δy󸀠 m + Δym)2 ds. (33) Here we note from the elliptic regularity theory that 󵄩󵄩󵄩󵄩∇ ⋅ G (∇φ)󵄩󵄩󵄩󵄩2 = 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 Δφ √1 + 󵄨󵄨󵄨󵄨∇φ󵄨󵄨󵄨󵄨2 − n ∑ i,j=1 φxiφxjφxixj (1 + 󵄨󵄨󵄨󵄨∇φ󵄨󵄨󵄨󵄨2)3/2 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩2 ≤ 󵄩󵄩󵄩󵄩Δφ󵄩󵄩󵄩󵄩2 + n ∑ i,j=1 󵄩󵄩󵄩󵄩󵄩󵄩φxixj󵄩󵄩󵄩󵄩󵄩󵄩2 ≤ C 󵄩󵄩󵄩󵄩Δφ󵄩󵄩󵄩󵄩2 ∀φ ∈ D (Δ) . (34) Thus, by (34) we can deduce for ε > 0 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨2 ∫ 0 (∇ ⋅ G (∇ym) , Δy󸀠 m + Δym)2 ds󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≤ 2∫ 0 󵄩󵄩󵄩󵄩∇ ⋅ G (∇ym)󵄩󵄩󵄩󵄩2 󵄩󵄩󵄩󵄩󵄩Δy󸀠 m + Δym󵄩󵄩󵄩󵄩󵄩2 ds ≤ C∫ 0 󵄩󵄩󵄩󵄩Δym?


Introduction
Let Ω be an open bounded set of R  with the smooth boundary Γ.We set  = (0, ) × Ω, Σ = (0, ) × Γ for  > 0. The nonlinear equation of the longitudinal motion of vibrating membrane surrounding Ω with clamped boundary is described by the following Dirichlet boundary value problem: where  is the height of a membrane,  > 0,  is a forcing function, and | ⋅ | denotes the Euclidean norm on R  .A brief physical background of ( 1) is given in our previous paper [1].
For damped linear or semilinear systems, there are many books and articles about the well-posedness with applications to various dynamic system's topics (cf.[2][3][4], etc.) with semigroup or unified variational treatments.However, the quasilinear cases like (1) require more manipulations in the analysis of systems, because the systems like (1) are very much model-dependent due to the strong nonlinearity.
Equation (1) is proposed in Kobayashi et al. [5] and the well-posedness of strongly regular solutions is studied by using the resolvent estimates of linearized operators in a modified Banach space.Besides, the well-posedness of less regular solutions is proved in [1], called weak solutions in the framework of the variational method in Dautray and Lions [3].Based on these results, we have treated the associated optimal control and identification problems in [6] and [7], respectively.Furthermore, in [8] we have extended the results in [1] to more general quasilinear nonautonomous wave equation with strong damping term.
where  > 0 is a constant depending on data.
Next we introduce the solution space (0, ) for strong solutions of (2) defined by The next theorem gives a well-posedness result for strong solutions of (2).Theorem 4. Assume that ( 0 ,  1 , ) ∈ (Δ)× where  is a constant depending on data.
Throughout this paper, we will use  as a generic constant and omit writing the integral variables in any definite integrals without confusion.
Proof of Theorem 4. We divide the proof into three steps.
By the standard argument of Dautray and Lions [3, pp.564-566], it can be verified that the limit  of {   } is a strong solution of the linear problem Step 3 (strong convergence of approximate solutions).In order to prove that  is a strong solution of (2), it is sufficient to prove G = ∇ ⋅ (∇).For this, we shall show   () → () strongly in (Δ) for all  ∈ [0, ].To prove the strong convergence, we use the modified arguments in Dautray and Lions [3, pp. 579-581] and the classical compact imbedding theorem.
First as in (33), we take   (47) For simplicity we set It is verified by direct computations that where as  → ∞.Thus, we readily have Therefore, we have proved the existence of a strong solution of (2).By similar estimations as in (37) and (38), we can show (17).
The uniqueness of strong solutions is evident from the uniqueness of weak solutions.