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J. Korean Math. Soc. 44 (2007), No. 2, pp. 443–453

WEAK SOLUTIONS OF THE EQUATION OF MOTION OF MEMBRANE WITH STRONG VISCOSITY Jin-soo Hwang and Shin-ichi Nakagiri

Reprinted from the Journal of the Korean Mathematical Society Vol. 44, No. 2, March 2007 c °2007 The Korean Mathematical Society

J. Korean Math. Soc. 44 (2007), No. 2, pp. 443–453

WEAK SOLUTIONS OF THE EQUATION OF MOTION OF MEMBRANE WITH STRONG VISCOSITY Jin-soo Hwang and Shin-ichi Nakagiri Abstract. We study the equation of a membrane with strong viscosity. Based on the variational formulation corresponding to the suitable function space setting, we have proved the fundamental results on existence, uniqueness and continuous dependence on data of weak solutions.

1. Introduction A freely flexible stretched film is called a membrane. Let Ω be an open bounded set of Rn with the smooth boundary Γ. We set Q = (0, T ) × Ω, Σ = (0, T ) × Γ for T > 0. The nonlinear equation of the longitudinal motion of a vibrating membrane surrounding Ω is described by the following Dirichlet boundary value problem: ³ ´ ∂2y ∇y (1.1) − div p =0 in Q, 2 ∂t 1 + |∇y|2 with (1.2)

y=0

on Σ,

∂y (0, x) = y1 (x) in Ω, ∂t where y is the height of the membrane. Then the reasonable physical candidate for the potential energy is the surface area of h = y(x), x ∈ Ω, since energy is stored in the membrane when it is stretched. Equation (1.1) is derived as the Euler-Lagrange equation of the action integral Z T ³Z ´ 1 ¯¯ ∂y ¯¯2 ¯ ¯ dx − J(y) dt, 0 Ω 2 ∂t (1.3)

y(0, x) = y0 (x),

where J(y) is the surface area of the graph y. And it is well known that the nonlinear term in (1.1) appears in the minimal surface problems as a nonlinear elliptic operator. For the base of that we refer to Gilbarg and Trudinger [3]. Received September 12, 2005. 2000 Mathematics Subject Classification. 35B30, 35L70. Key words and phrases. equation of membrane with strong viscosity, weak solution, variational method. c °2007 The Korean Mathematical Society

443

444

JIN-SOO HWANG AND SHIN-ICHI NAKAGIRI

Recently, there are several authors related to this problem (1.1)-(1.3). Kikuchi [8] has treated this problem in the space of functions having bounded variation and constructed approximate solutions by Rothe’s method. But it seems to be difficult to construct a solution in a Hilbert or reflexive Banach spaces not only for theoretical construction but also for any other applications. So we consider some modified but more realistic model given by the following problem with viscosity terms:  2 ³ ´ ∇y ∂y ∂ y   p − div − µ∆ = f in Q,  2  ∂t2 ∂t 1 + |∇y| (1.4) y = 0 on Σ,    ∂y  y(0, x) = y (x), (0, x) = y1 (x) in Ω, 0 ∂t where µ > 0 and f is a forcing function. Damping mechanism appears extendly and naturally in physical situations and there are many factors of it. We classify it largely by air and structural factors. Among them the modified problem (1.4) seems to be structually damped case. In the linear and semilinear cases, for the research of damped systems, there are a lot of books and articles about the well posedness and the practical applications (cf. [2], [10], [1], etc.) with semigroup or unified variational treatments. However the quasilinear cases like as this case require more manipulations in the analysis of systems. Because the damped systems are very much model-dependent due to the strong nonlinearity. In fact, the proposal of this problem can be found in [4] and [5] as a model of quasilinear wave equations (see also Temam [10]). Especially, it is given in Kobayashi, Pecher and Shibata [9] the proof of the existence of solutions of (1.4). Using some regular data conditions, they used resolvent estimates to construct regular solutions in a modified Banach space. However, it seems that there are a little researches on the variational treatment of (1.4) and the related control problems. Our aim of this paper is to prove the basic results on existence and uniqueness of weak solutions of (1.4) in the framework of variational method in Dautray and Lions [2]. The most difficult part of the existence proof is to show the strong convergence of nonlinear terms, and the part is completed by using the argument in [6] (see also [2, p.569] for the linear case). Finally we note that the quadratic optimal control problems associated with the equation (1.4) are studied in Hwang and Nakagiri [7]. 2. Main results We study the following Dirichlet boundary value problem for the equation of motion of a membrane with strong viscosity  2 ³ ´ ∂y ∂ y ∇y   p − div − µ∆ = f in Q,  ∂t2 2  ∂t 1 + |∇y| (2.1) y=0 on Σ,     y(0, x) = y (x), ∂y (0, x) = y (x) in Ω, 0 1 ∂t

EQUATION OF MOTION OF MEMBRANE WITH STRONG VISCOSITY

445

where f is a forcing function, y0 and y1 are initial data and µ > 0 is a constant. In (2.1) we suppose f ∈ L2 (0, T ; H −1 (Ω)), y0 ∈ H01 (Ω) and y1 ∈ L2 (Ω). The solution space W (0, T ) of (2.1) is defined by {g|g ∈ L2 (0, T ; H01 (Ω)), g 0 ∈ L2 (0, T ; H01 (Ω)), g 00 ∈ L2 (0, T ; H −1 (Ω))} endowed with the norm ³ ´ 12 kgkW (0,T ) = kgk2L2 (0,T ;H 1 (Ω)) + kg 0 k2L2 (0,T ;H 1 (Ω)) + kg 00 k2L2 (0,T ;H −1 (Ω)) , 0

0

dg dt

00

0

d2 g dt2

where g = and g = (cf. Dautray and Lions [2, p.471]). We remark that W (0, T ) is continuously imbedded in C([0, T ]; H01 (Ω)) ∩ C 1 ([0, T ]; L2 (Ω)) (cf. [2, p.555]). The scalar products and norms on L2 (Ω) and H01 (Ω) are denoted by (φ, ψ), |φ| and (φ, ψ)H01 (Ω) , kφk, respectively. The scalar product and norm on [L2 (Ω)]n are also denoted by (φ, ψ) and |φ|. Then the scalar product (φ, ψ)H01 (Ω) and the norm kφk of H01 (Ω) are given by (∇φ, ∇ψ) and 1 kφk = |(∇φ, ∇φ)| 2 , respectively. The duality pairing between H01 (Ω) and −1 H (Ω) is denoted by hφ, ψi. Related to the nonlinear term in (2.1), we define x the function G : Rn → Rn by G(x) = p , x ∈ Rn . Then it is easily 1 + |x|2 verified that (2.2)

|G(x) − G(y)| ≤ 2|x − y|,

∀x, y ∈ Rn .

The nonlinear operator G(∇·) : H01 (Ω) → [L2 (Ω)]n is introduced by (2.3)

G(∇φ)(x) = p

∇φ(x) 1+

|∇φ(x)|2

,

a.e. x ∈ Ω,

∀φ ∈ H01 (Ω).

By the definition of G(∇·) in (2.3), we have the following useful property on G(∇·): (2.4) |G(∇φ)| ≤ |∇φ|, |G(∇φ) − G(∇ψ)| ≤ 2|∇φ − ∇ψ|, ∀φ, ψ ∈ H01 (Ω). Definition 2.1. A function y is said to be a weak solution of (2.1) if y ∈ W (0, T ) and y satisfies  00  hy (·), φi + (G(∇y(·)), ∇φ) + µ(∇y 0 (·), ∇φ) = hf (·), φi for all φ ∈ H01 (Ω) in the sense of D0 (0, T ), (2.5)  1 y(0) = y0 ∈ H0 (Ω), y 0 (0) = y1 ∈ L2 (Ω). The following theorem gives the result on existence, uniqueness and regularity of the weak solution of (2.1). Theorem 2.2. Assume that µ > 0, f ∈ L2 (0, T ; H −1 (Ω)) and y0 ∈ H01 (Ω), y1 ∈ L2 (Ω). Then the problem (2.1) has a unique weak solution y in W (0, T ). Furthermore, y has the following estimate Z t (2.6) |y 0 (t)|2 + |∇y(t)|2 + |∇y 0 (s)|2 ds 0

≤ C(ky0 k2 + |y1 |2 + kf k2L2 (0,T ;H −1 (Ω)) ),

∀t ∈ [0, T ],

446


where C is a constant depending only on µ > 0. Next we give the result on the continuous dependence of weak solutions of (2.1) on initial values y0 , y1 and forcing terms f . Let P be a product space defined by P = H01 (Ω) × L2 (Ω) × L2 (0, T ; H −1 (Ω)).

(2.7)

For each p = (y0 , y1 , f ) ∈ P we have a unique weak solution y = y(p) ∈ W (0, T ) of (2.1) by Theorem 2.2. Hence we can define the solution mapping p = (y0 , y1 , f ) → y(q) of P into W (0, T ). Theorem 2.3. The solution mapping p = (y0 , y1 , f ) → y(q) of P into W (0, T ) is strongly continuous. Further, for each p1 = (y01 , y11 , f1 ) ∈ P and p2 = (y02 , y12 , f2 ) ∈ P we have the inequality |y 0 (p1 ; t) − y 0 (p2 ; t)|2 + |∇y(p1 ; t) − ∇y(p2 ; t)|2 Z t + |∇y 0 (p1 ; s) − ∇y 0 (p2 ; s)|2 ds

(2.8)

0

≤

C(ky01 − y02 k2 + |y11 − y12 |2 + kf1 − f2 k2L2 (0,T ;H −1 (Ω)) ), ∀t ∈ [0, T ]. 3. Proof of main results

We will omit writing the integral variables in the definite integral withRt out any confusion. For example, in (2.6) we will write 0 |∇y 0 |2 ds instead of Rt |∇y 0 (s)|2 ds. 0 Proof of Theorem 2.2. We construct approximate solutions of (2.1) by the Galerkin’s procedure. Since H01 (Ω) is separable, there exists a complete orthonor1 2 ∞ mal system {wm }∞ m=1 in L (Ω) such that {wm }m=1 is free and total in H0 (Ω). For each m = 1, 2, . . . , we define an approximate solution ym (t) of the equation (2.1) m X ym (t) = gjm (t)wj , j=1

where ym (t) satisfies  00 0 (ym (t), wj ) + (G(∇ym (t)), ∇wj ) + µ(∇ym (t), ∇wj )     = hf (t), wj i, t ∈ [0, T ], 1 ≤ j ≤ m, m m (3.1) X X  0  y (0) = (y , w )w , y (0) = (y1 , wi )wi .  m 0 i i m  i=1

i=1

Let Vm be m dimensional space spanned by {w1 , . . . , wm }. Then we can see that m X (3.2) y0m = (y0 , wi )wi ∈ Vm → y0 in H01 (Ω) as m → ∞, i=1


(3.3)

y1m =

447

m X (y1 , wi )wi ∈ Vm → y1 in L2 (Ω) as m → ∞. i=1

Hence the equation (3.1) in Vm induces the system of nonlinear second or0 (0) = der equations for gjm (t) with initial conditions gjm (0) = (y0 , wj ), gjm (y1 , wj )j = 1, . . . , m. Since the nonlinear term G(∇ym (t)) in (3.1) is Lipschitz continuous by (2.4), i.e., for fixed j ∈ {1, . . . , m} 1 2 1 2 |(G(∇ym (t)) − G(∇ym (t)), ∇wj )| ≤ 2|∇wj ||∇ym (t) − ∇ym (t)|,

the system is also Lipschitz continuous in gim . Hence, it is verified that the system admits a unique solution gjm (t), j = 1, . . . , m over [0, T ]. This allows us to construct the approximate solution ym (t) of (3.1). Now we will derive a priori estimates of ym (t). At first, we multiply both sides of the equation (3.1) 0 by gjm (t) and sum over j to have (3.4)

00 0 0 0 0 (ym (t), ym (t)) + (G(∇ym (t)), ∇ym (t)) + µ|∇ym (t)|2 = hf (t), ym (t)i.

Secondly, we also multiply both sides of the equation (3.1) by gjm (t) and sum over j to have (3.5)

00 0 (ym (t), ym (t)) + (G(∇ym (t)), ∇ym (t)) + µ(∇ym (t), ∇ym (t)) = hf (t), ym (t)i.

We sum (3.4) and (3.5) to have (3.6)

00 0 0 0 (ym (t), ym (t)) + µ|∇ym (t)|2 + µ(∇ym (t), ∇ym (t)) 00 0 = − (ym (t), ym (t)) + hf (t), ym (t) + ym (t)i 0 −(G(∇ym (t)), ∇ym (t) + ∇ym (t)).

Since 1 d 0 µ d 0 |ym (t)|2 , µ(∇ym (t), ∇ym (t)) = |∇ym (t)|2 , 2 dt 2 dt (3.6) can be rewritten as 1 d 0 µ d 0 (3.7) |y (t)|2 + µ|∇ym (t)|2 + |∇ym (t)|2 2 dt m 2 dt 00 0 = − (ym (t), ym (t)) + hf (t), ym (t) + ym (t)i 0 −(G(∇ym (t)), ∇ym (t) + ∇ym (t)). 00 0 (ym (t), ym (t)) =

By integrating (3.7) over [0, t], we obtain Z t 0 2 0 2 (3.8) |∇ym | ds + µ|∇ym (t)|2 |ym (t)| + 2µ 0

=

0 |y1m |2 + µ|∇y0m |2 − 2(ym (t), ym (t)) + 2(y1m , y0m ) Z t Z t 0 2 0 +2 |ym | ds + 2 hf, ym + ym ids 0 0 Z t 0 −2 (G(∇ym ), ∇ym + ∇ym )ds. 0

448


Let ² > 0 be an arbitrary real number. Then, we have by (2.4) and the Schwarz inequality Z t ¯ Z t ¯ 1Z t ¯ ¯ 0 2 0 2 |∇ym | ds + ² |∇ym | ds, (3.9) (G(∇ym ), ∇ym )ds¯ ≤ ¯2 ² 0 0 0 Z t ¯ ¯ Z t ¯ ¯ (G(∇ym ), ∇ym )ds¯ ≤ 2 |∇ym |2 ds, ¯2

(3.10)

0

(3.11)

0

Z t ¯ 1 ¯ Z t ¯ ¯ 0 2 0 2 hf, ym ids¯ ≤ kf kL2 (0,T ;H −1 (Ω)) + ² |∇ym | ds, ¯2 ² 0 0 Z t ¯ ¯ Z t ¯ ¯ 2 hf, ym ids¯ ≤ kf kL2 (0,T ;H −1 (Ω)) + |∇ym |2 ds. ¯2

(3.12)

0

0

Also we have (3.13)

0 (t), ym (t))| 2|(ym

Z t ¯2 1 ¯¯ ¯ 0 0 (t)|2 + ¯y0m + ²|ym (s)ds¯ ym ² 0 Z 2 2T t 0 2 0 (t)|2 + |y0m |2 + ²|ym |y | ds. ² ² 0 m

≤ ≤

We note by (3.2), (3.3) that |y1m | ≤ C 0 |y1 |, |∇y0m | ≤ C 0 ky0 k for some C 0 > 0 independent of m. Therefore from (3.8) to (3.13), we can obtain the following inequality Z t 0 0 2 (1 − ²)|ym (t)|2 + µ|∇ym (t)|2 + (2µ − 2²) (3.14) |∇ym | ds 0

≤

2

2

kf k2L2 (0,T ;H −1 (Ω)) )

C² (ky0 k + |y1 | + Z t 1 + 2T 0 2 ) (|ym | + |∇ym |2 )ds, + (5 + ² 0

for some C² > 0. If we choose ² = min{ 12 , µ2 }, then by Gronwall’s inequality it follows that Z t 0 0 2 |ym (t)|2 + |∇ym (t)|2 + (3.15) |∇ym | ds 0

≤ C(ky0 k2 + |y1 |2 + kf k2L2 (0,T ;H −1 (Ω)) ), 0 where C is some positive constant independent of m. Therefore ym and ym ∞ 1 ∞ 2 2 remain in a bounded sets of L (0,T ;H0 (Ω)) and L (0,T ;L (Ω)) ∩ L (0, T ;H01 (Ω)), respectively. And the nonlinear term G(∇ym ) is uniformly bounded. Hence by the extraction theorem of Rellich’s, we can extract a subsequence {ymk } of {ym } and find z ∈ L∞ (0, T ; H01 (Ω)), z 0 ∈ L∞ (0, T ; L2 (Ω)) ∩ L2 (0, T ;


449

H01 (Ω)) and also F (·) ∈ L2 (0, T ; L2 (Ω)) such that ymk → z weakly-star in L∞ (0, T ; H01 (Ω)),

i)

(3.16)

and weakly in L2 (0, T ; H01 (Ω)), 0 ym → z 0 weakly-star in L∞ (0, T ; L2 (Ω)), k

ii)

and weakly in L2 (0, T ; H01 (Ω)), iii)

0 → µ∆z 0 weakly in L2 (0, T ; H −1 (Ω)), µ∆ym k

iv)

G(∇ym ) → F (·) weakly in L2 (0, T ; L2 (Ω)),

as k → ∞. By the standard argument of Dautray and Lions [2, pp. 564-566], in view of (3.16), it can be verified that the limit z of {ymk } belong to W (0, T ) and is a weak solution of the linear problem  2 ∂z ∂ z     ∂t2 − divF (t) − µ∆ ∂t = f in Q, (3.17) z=0 on Σ,     z(0, x) = y0 , ∂z (0, x) = y1 in Ω. ∂t Thus, to prove that z is a weak solution of (2.1), we need to show that F (·) = G(∇z(·)). For the purpose we shall show ∇ym (t) → ∇z(t) strongly in [L2 (Ω)]n . To this end we use the strong convergence arguments in [6]. For notational simplicity, we denote ymk by ym again. The approximate solution ym (t) satisfies (3.8). For the weak solution z of (3.17), we can obtain the following equality similarly as for ym (t) as in [2, p.567]. Z t 0 2 |z (t)| + 2µ (3.18) |∇z 0 |2 ds + µ|∇z(t)|2 0

=

|y1 |2 + µ|∇y0 |2 − 2(z 0 (t), z(t)) + 2(y1 , y0 ) Z t Z t Z t +2 |z 0 |2 ds + 2 hf, z 0 + zids − 2 (F, ∇z 0 + ∇z)ds. 0

0

0

Moreover the following equalities hold:

=

0 (ym (t), ym (t)) + (z 0 (t), z(t)) 0 0 (ym (t) − z 0 (t), ym (t) − z(t)) + (ym (t), z(t)) + (z 0 (t), ym (t));

|φm (t)|2 + |ψ(t)|2 = |φm (t) − ψ(t)|2 + 2(φm (t), ψ(t)); |∇φm (t)|2 + |∇ψ(t)|2 = |∇(φm (t) − ψ(t))|2 + 2(∇φm (t), ∇ψ(t)); (G(∇ym (t)), ∇φm (t)) + (F (t), ∇ψ(t)) = (G(∇ym (t)) − G(∇ψ(t)), ∇(φm (t) − ψ(t))) + (G(∇ym (t)), ∇ψ(t)) +(G(∇ψ(t)), ∇(φm (t) − ψ(t))) + (F (t), ∇ψ(t)),

450


0 where φm (t) = ym (t) or ym (t) and ψ(t) = z(t) or z 0 (t). Adding (3.8) to (3.18) and using the above equalities, we have Z t 0 0 2 0 (3.19) |ym (t) − z (t)| + 2µ |∇(ym − z 0 )|2 ds + µ|∇(ym (t) − z(t))|2 0

=

Φ0m + 2 Z 0

Z

0

0 Φim (t) − 2(ym (t) − z 0 (t), ym (t) − z(t))

i=1 t

+2 −2

3 X

t

Z

0 − z 0 |2 ds − 2 |ym

t

(G(∇ym ) − G(∇z), ∇ym − ∇z)ds 0

0 − ∇z 0 )ds, (G(∇ym ) − G(∇z), ∇ym

where Φ0m

= |y1m |2 + |y1 |2 + µ(|∇y0m |2 + |∇y0 |2 ) +2((y1m , y0m ) + (y1 , y0 )),

Z t 0 0 , ∇z 0 )ds, (t), z 0 (t)) − µ(∇ym (t), ∇z(t)) − 2µ (∇ym = −(ym 0 Z t Z t 0 , z 0 )ds + hf, z + z 0 ids (ym Φ2m (t) = 2 0 0 Z t 0 0 + hf, ym + ym ids − (ym (t), z(t)) − (z 0 (t), ym (t)), 0 Z t Z t 3 0 Φm (t) = − (G(∇ym ), ∇(z + z ))ds − (F, ∇(z + z 0 ))ds 0 0 Z t 0 − (G(∇z), ∇((ym − z) + (ym − z 0 )))ds.

Φ1m (t)

0

For simplicity we set Sm (t) = Φ0m + 2

3 X

Φim (t).

i=1

It is clear from (3.2) and (3.3) that (3.20)

Φ0m → 2|y1 |2 + 2µ|∇y0 |2 + 4(y1 , y0 ).

0 (t) → z 0 (t) weakly in By (3.16) we have ∇ym (t) → ∇z(t) weakly in [L2 (Ω)]n , ym 2 0 0 2 2 n L (Ω) and ∇ym → ∇z weakly in L (0, T ; [L (Ω)] ), and moreover G(∇ym ) → F weakly in L2 (0, T ; [L2 (Ω)]n ), so that Z t 1 0 2 2 (3.21) Φm (t) → −|z (t)| − µ|∇z(t)| − 2µ |∇z 0 |2 ds, 0 Z t Z t 2 0 2 (3.22) Φm (t) → 2 |z | ds + 2 hf, z + z 0 ids − 2(z 0 (t), z(t)), 0

0


Z Φ3m (t)

(3.23)

→ −2

t

451

(F, ∇(z + z 0 ))ds.

0

Whence by (3.20)-(3.23) and by the equality (3.18) for z, we have (3.24)

Sm (t) → 0

as m → ∞.

The right hand side of (3.19) is estimated as follows: (3.25)

(r.h.s. of (3.19)) 1 0 (t) − z 0 (t)|2 + 4|y0m − y0 |2 ≤ Sm (t) + |ym 2 Z t Z t 0 +4 |∇(ym − z)|2 ds + (4T + 2) |ym − z 0 |2 ds 0 0 Z t 0 − z 0 )||∇(ym − z)|ds +4 |∇(ym 0 Z t 1 0 0 2 2 0 − z 0 )|2 ds ≤ Sm (t) + |ym (t) − z (t)| + 4|y0m − y0 | + µ |∇(ym 2 0 Z t³ ´ 0 +K |∇(ym − z)|2 + |ym − z 0 |2 ds, 0

where K = (2 + 4T + 4 + 16 µ ). By (3.19) and (3.25) we can obtain the following inequality Z t 1 0 0 2 2 0 (3.26) |y (t) − z (t)| + µ|∇(ym (t) − z(t))| + µ |∇(ym − z 0 )|2 ds 2 m 0 Z t³ ´ 2 0 ≤ Sm (t) + 4|y0m − y0 | + K |∇(ym − z)|2 + |ym − z 0 |2 ds. 0

We divide (3.26) by α =

min{ 12 , µ}

> 0 and if we set C =

(3.28) then we can have

Z

t

Mm (t) ≤ Ψm (t) + C

Mm (s)ds. 0

Here we note, thanks to (3.2) and (3.24) that (3.30)

and

0 Mm (t) = |∇(ym (t) − z(t))|2 + |ym (t) − z 0 (t)|2 , 1 Ψm (t) = (Sm (t) + 4|y0m − y0 |2 ), α

(3.27)

(3.29)

1 αK

Ψm (t) → 0 when m → ∞, for all t ∈ [0, T ].

We apply Gronwall’s inequality to (3.29), then we have Z T (3.31) Mm (t) ≤ Ψm (t) + C exp(CT ) Ψm (s)ds. 0

452


By the equality (3.18), we see that Ψm (t) is uniformly bounded. Then it follows from (3.30) and (3.31) that (3.32)

lim Mm (t) = 0.

m→∞

Therefore we can verify that (3.33)

ym (t) → z(t) strongly in H01 (Ω),

(3.34)

0 (t) → z 0 (t) strongly in L2 (Ω). ym

Finally, by (3.33) and Lipschitz property of G(∇z) in (2.4) we can deduce (3.35)

F (t) = G(∇z(t))

in [L2 (Ω)]n

∀t ∈ [0, T ].

The energy inequality (2.6) follows from (3.15) and the strong convergence of each terms in (3.15) by (3.33), (3.34) and (3.26). ¤ The uniqueness of weak solutions follows directly from the continuous dependence (2.8) given in Theorem 2.3. Proof of Theorem 2.3. Let y(p1 ) and y(p2 ) be the weak solutions of (2.1) corresponding to p1 ∈ P and p2 ∈ P , respectively. Set ϕ = y(q1 ) − y(q2 ). Then ϕ satisfies  2 ³ ´ ∂ ϕ   − div G(∇y(p )) − G(∇y(p )) 1 2    ∂t2   ∂ϕ −µ∆ = f1 − f2 in Q, (3.36) ∂t   ϕ=0 on Σ,      ϕ(0, x) = y 1 (x) − y 2 (x), ∂ϕ (0, x) = y 1 (x) − y 2 (x) in Ω 0 0 1 1 ∂t in the weak sense. Since |G(∇y(p1 )) − G(∇y(p2 ))| ≤ 2|∇ϕ|, we can repeat the same calculations as for (2.6) to have the estimates Z t 2 0 2 |∇ϕ(t)| + |ϕ (t)| + |∇ϕ0 |2 ds 0

≤

C(ky01 − y02 k2 + |y11 − y12 |2 + kf1 − f2 k2L2 (0,T ;H −1 (Ω)) ).

This proves the strong Lipshitz continuity of solution mapping p = (y0 , y1 , f ) → y(q) of P into W (0, T ). ¤ References [1] H. T. Banks, R. C. Smith, and Y. Wang, Smart Material Structures, Modeling, Estimation and Control, RAM, John Wiley and Sons, Masson, 1996. [2] R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5, Evolution Problems I, Springer-Verlag, 1992. [3] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New York, 1977.


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[4] J. Greenberg, On the existence, uniqueness and stability of equation ρ0 Xtt = E(Xx ) Xxx + Xxxt , J. Math. Anal. Appl. 25 (1969), 575–591. [5] J. Greenberg, R. MacCamy, and V. Mizel, On the existence, uniqueness and stability of equation σ(ux )utt + λuxxt = ρ0 utt , J. Math. Mech. 17 (1967/68), 707–728. [6] J. Ha and S. Nakagiri, Existence and regularity of weak solutions for semilinear second order evolution equations, Funcial. Ekvac. 41 (1998), no. 1, 1–24. [7] J. S. Hwang and S. Nakagiri, Optimal control problems for the equation of motion of membrane with strong viscosity, J. Math. Anal. Appl. 321 (2006), no. 1, 327–342. [8] K. Kikuchi, An analysis of the nonlinear equation of motion of a vibrating membrane in the space of BV functions, J. Math. Soc. Japan 52 (2000), no. 4, 741–766. [9] T. Kobayashi, H. Pecher, and Y. Shibata, On a global in time existence theorem of smooth solutions to nonlinear wave equation with viscosity, Math. Ann. 296 (1993), no. 2, 215–234. [10] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physis, Second Edition, Applied Mathematical Sciences. Vol. 68, Springer-Verlag, Berlin-HeidelbergNew York, 1997. Jin-soo Hwang Department of Mathematics Dong-A University Pusan 604-714, Korea E-mail address: [email protected] Shin-ichi Nakagiri Department of Applied Mathematics Faculty of Engineering Kobe University Kobe 657-8501, Japan E-mail address: [email protected]