PERIODIC STABILITY OF ELLIPTIC-PARABOLIC ... - CiteSeerX

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS SUPPLEMENT 2007

Website: www.AIMSciences.org pp. 614–623

PERIODIC STABILITY OF ELLIPTIC-PARABOLIC VARIATIONAL INEQUALITIES WITH TIME-DEPENDENT BOUNDARY DOUBLE OBSTACLES

Masahiro Kubo Department of Mathematics Nagoya Institute of Technology Gokiso-cho, Showa-ku, Nagoya Aichi, 466-8555, Japan

Noriaki Yamazaki Department of Mathematical Science Common Subject Division Muroran Institute of Technology 27-1 Mizumoto-ch¯ o, Muroran Hokkaido, 050-8585, Japan

Abstract. We consider periodic problems of elliptic-parabolic variational inequalities with time-dependent boundary double obstacles. In this paper we assume that the given boundary obstacles change periodically in time. Then, we prove the existence, uniqueness and asymptotic stability of a periodic solution to our problem.

1. Introduction. In this paper, for the given obstacle functions g1 , g2 , we study the time-dependent boundary double obstacle problem of the following form: Problem (P) u(t) ∈ K(t) := {z ∈ H 1 (Ω); z = g1 (t) on Γ1 , g1 (t) ≤ z ≤ g2 (t) on Γ2 }, Z (b(u)t , u − v) + a(x, b(u), ∇u) · ∇(u − v)dx ≤ (f, u − v)

(1.1) (1.2)

Ω

for all v ∈ K(t) and a.e. t ∈ J, b(u(0, ·)) = b0

in Ω.

(1.3)

Here, J is a time interval in R (e.g. J = R, [t0 , ∞), [t0 , t1 ]). Ω is a bounded domain in RN (N ≥ 1) with a smooth boundary Γ that admits a mutually disjoint decomposition such as Γ = Γ1 ∪ Γ2 . The given function b : R → R is bounded, nondecreasing and Lipschitz continuous, and f (t, x) is a given function on R × Ω. The term a(x, s, p) is a quasi-linear elliptic vector field satisfying some structure conditions, in particular we assume a(x, s, p) = ∂p A(x, s, p) for a potential function A : Ω × R × RN → R. The inequality (1.2) is called an elliptic-parabolic variational inequality, since it is elliptic in the region {b0 (u) = 0}, and is parabolic in {b0 (u) > 0}, respectively. 2000 Mathematics Subject Classification. 35B10, 35B40, 35J60, 35K55, 35M10. Key words and phrases. Periodic stability, elliptic-parabolic variational inequality, boundary constraints.

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ELLIPTIC-PARABOLIC VARIATIONAL INEQUALITIES

615

Clearly, the problem (P) is a weak (variational) formulation for the following elliptic-parabolic equation with boundary double obstacles (cf. Remark 2.2): b(u)t − ∇ · a(x, b(u), ∇u) = f (t, x)

in J × Ω.

(1.4)

Note that when Γ1 = Γ, this problem (P) is the usual Dirichlet boundary value problem with Dirichlet data g1 (t). Many studies have been made of elliptic-parabolic equations (cf. (1.4)). In mathematical analyses of elliptic-parabolic equations, two kinds of solutions have been studied: weak and strong. Strong solutions admit strong time derivatives of b(u) in L2 (Ω) (or L1 (Ω)), while weak ones do not. The notion of weak solutions was introduced by Alt–Luckhaus [1]. Other kinds of weak solutions of ellipticparabolic equations without a strong time-derivative in L1 (Ω): mild solutions and renormalized solutions, have also been studied (cf. [2, 4, 5, 14]). Time-periodic problems to elliptic-parabolic equations have been studied by Kenmochi–Kr¨ oner–Kubo [6] and Kenmochi–Kubo [7, 8] with various boundary constraints. Note that to date the large-time behavior of a solution to elliptic-parabolic equations has only been studied within the frame of a strong solution (cf. [6, 7, 8, 10, 11]). The main object of this paper is to discuss the periodic problem of a strong solution for (P) assuming that the given obstacle functions g1 and g2 are periodic in time with period T0 > 0: gi (t + T0 ) = gi (t) for any t ∈ R and i = 1, 2. In fact, by applying the abstract theory, we show the existence of a T0 -periodic (strong) solution ω satisfying the periodic condition ω(t + T0 ) = ω(t)

for any t ∈ R.

Moreover, we consider the special case when a(x, s, p) = a(x)[p + k(s)] in (1.2). Then, using the ideas of Kenmochi–Kubo [8], we prove the uniqueness and asymptotic stability of the time-periodic (strong) solution to (P). The plan of this paper is as follows. In the next Section 2, after defining the solution to (P), we state the main results (Theorem 2.1, 2.2) concerning the existence, uniqueness and asymptotic stability of a time-periodic solution to (P). In Section 3, we prove Theorem 2.1 by applying the abstract results in [12, 13]. In Section 4, we prepare some order properties of boundary fluxes of solutions to (P). In Section 5, we prove Theorem 2.2 by an argument similar to [8, Section 5]. Notation and basic assumptions. Throughout this paper, we put H := L2 (Ω) with the usual real Hilbert space structures. The inner product and norm in H are denoted by (·, ·) and by | · |H , respectively. We put V := H 1 (Ω) with the usual norm 1 1 | · |V . Also, we denote by h·, ·iΓ the duality pairing between H − 2 (Γ) and H 2 (Γ). ∞ ∞ ∞ Various L -norms, e.g., norms in L (Ω), L (R), etc., are all denoted by the same symbol | · |∞ . Also, for the sake of order relationships, we use the following notation: u ∨ v := max{u, v},

u ∧ v := min{u, v},

[u]+ := u ∨ 0.

Let us now give the assumption of the elliptic vector field a(·, ·, ·): (A) a(x, s, p) = ∂p A(x, s, p) for some potential function A(x, s, p). There exist constants µ1 > 0, C1 = C1 (a) > 0 and C2 = C2 (a) > 0 such that [a(x, s, p) − a(x, s, pˆ)] · (p − pˆ) ≥ µ1 |p − pˆ|2 ,

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M. KUBO AND N. YAMAZAKI

|a(x, s, p)|2 + |A(x, s, p)| + |∂s A(x, s, p)|2 ≤ C1 (1 + |s|2 + |p|2 ), |a(x, s, p) − a(x, sˆ, p)| ≤ C2 (1 + |p|)|s − sˆ| for all x ∈ Ω, s, sˆ ∈ R, p, pˆ ∈ RN . Moreover, a(·, ·, ·) and A(·, ·, ·) satisfy the Carath´eodory condition. As for the function b(·) and the data {b0 , T0 , f, g1 , g2 , Γ}, the following conditions (B), (C), (D), (E) and (F) are always assumed in this paper. (B) b : R → R is bounded, nondecreasing and Lipschitz continuous. (C) b0 ∈ H and T0 > 0. 1,2 (D) f ∈ Wloc (R; H), f (t + T0 ) = f (t) for all t ∈ R. (E) g1 , g2 are T0 -periodic obstacle functions on R × Ω so that for i = 1, 2 ∞ gi ∈ L∞ loc (R; V ) ∩ L (R × Ω),

sup |gi0 |L2 (t,t+1;V ) + sup |gi0 |L2 (t,t+1;L∞ (Ω)) < ∞, t∈R

t∈R

gi (t, x) = gi (t + T0 , x), g2 − g1 ≥ C3

a.e. on R × Ω

∀x ∈ Ω,

∀t ∈ R,

(1.5)

for some constant C3 > 0.

(F) The boundary Γ of the domain Ω is smooth and admits a mutually disjoint decomposition such as Γ = Γ1 ∪ Γ2 , where Γ1 and Γ2 are measurable subsets of Γ, and Γ1 has a positive surface measure. 2. Main theorem. At first we give the definition of solution for (P). Definition 2.1. Let J := [t0 , t1 ] be an interval in R. A function u : J → V is a solution of (P) on J, or (P;f, g1 , g2 ) on J when the data are specified, if the following items (a)–(c) are satisfied. 1,2 ∗ ∗ (a) u ∈ L∞ loc ((t0 , t1 ]; V ) and there exists u ∈ Wloc ((t0 , t1 ]; H) such that b(u) = u for a.e. t ∈ J (cf. Remark 2.1). (b) u ∈ K(t) = {z ∈ V ; z = g1 (t) on Γ1 , g1 (t) ≤ z ≤ g2 (t) on Γ2 } for a.e. t ∈ J. (c) For a.e. t ∈ J, the following inequality holds: Z ∗ (ut , u − v) + a(x, b(u), ∇u) · ∇(u − v)dx ≤ (f, u − v), ∀v ∈ K(t). Ω

Further, if the following (d) is satisfied, we call u a solution of the Cauchy problem (P;b0 ), or (P;b0 , f, g1 , g2 ), on J = [t0 , t1 ]: (d) u∗ (t0 ) = b0 in H. Remark 2.1. In what follows, we identify the function u∗ with b(u) in the condition (a) of Definition 2.1, and always write b(u) for u∗ . Remark 2.2. From Green’s formula and the same argument in [9, Section 3] it follows that (P;b0 , f, g1 , g2 ) on J is equivalent to the following system: b(u)t − ∇ · a(x, b(u), ∇u) = f hν · a(x, b(u), ∇u), u − viΓ ≤ 0, u(t) ∈ K(t)

in J × Ω,

∀v ∈ K(t) and a.e. t ∈ J,

(2.1) (2.2)

for a.e. t ∈ J,

b(u(t0 , ·)) = b0

in Ω,

where ν is the outward normal vector on the boundary Γ. Now, we mention the following main result:

(2.3)


617

Theorem 2.1. Suppose (A) and (B)–(F). H

(i) Let J := [t0 , t1 ] ⊂ R. Put b0 := b(u0 ) for some u0 ∈ K(t0 ) (= H), where H

K(t0 ) denotes the closure of K(t0 ) in H. Then, there is a unique solution u of (P;b0 , f, g1 , g2 ) on J. H

(ii) Let J := [t0 , t1 ] ⊂ R. Let u0 , u0,n ∈ K(t0 ) such that {u0,n } is bounded in H. Let u and un be solutions to (P;b0 , f, g1 , g2 ) and (P;b0,n , f, g1 , g2 ) on J, respectively, where b0 = b(u0 ) and b0,n = b(u0,n ). If b0,n → b0 in H as n → ∞, then un → u b(un ) → b(u)

weakly in L2 (J; V ), strongly in C(J; Lq (Ω)) for all 1 ≤ q < ∞

as n → ∞. (iii) There exists at least one solution (called a T0 -periodic solution) ω of (P;f, g1 , g2 ) on R satisfying the T0 -periodic condition ω(t + T0 ) = ω(t) for all t ∈ R. We can prove the uniqueness and the asymptotic stability of the T0 -periodic solution to (P) under the additional assumption that the elliptic vector field is of a special form: a(x, s, p) = a(x)[p + k(s)]. We assume that the following condition (A)’ is satisfied: (A)’ a(x) = (ai,j (x)) is a symmetric and positive definite matrix with ai,j ∈ C 1 (Ω), and k : R → RN is Lipschitz continuous. There is a constant µ2 > 0 such that a(x)[p − pˆ] · (p − pˆ) ≥ µ2 |p − pˆ|2

for all x ∈ Ω, p, pˆ ∈ RN .

Note that (A)’ implies (A) for a(x, s, p) = a(x)[p + k(s)]. In fact, we define 1 A(x, s, p) := a(x)[ p + k(s)] · p 2 for (x, s, p) ∈ Ω × R × RN . Then, we have ∂p A(x, s, p) = a(x)[p + k(s)]. We now state this paper’s second main result, which is concerned with the uniqueness and stability of T0 -periodic solution to (P). Theorem 2.2. Suppose (A)’, (B)–(F) and put a(x, s, p) := a(x)[p + k(s)]. Assume f∗ ≤ f (t) ≤ f ∗ gi,∗ ≤ gi (t) ≤ gi∗

for some f ∗ , f∗ ∈ H, for some gi∗ , gi,∗ ∈ V,

∀t ∈ R, ∀t ∈ R

for i = 1, 2. Then, there exists a unique T0 -periodic solution ω of (P;f, g1 , g2 ) on R satisfying ω(t + T0 ) = ω(t) for all t ∈ R. Moreover, the T0 -periodic solution ω is asymptotically stable in the sense that for any solution u of (P;f, g1 , g2 ) on [t0 , ∞) (t0 ∈ R), we have b(ω)(t) − b(u)(t) → 0

in Lq (Ω) as t → ∞

for all 1 ≤ q < ∞.

3. Proof of Theorem 2.1. In this section, we prove Theorem 2.1 by applying the abstract theorem obtained in [12, 13]. To do so, we prepare some key lemmas. Lemma 3.1 ((K1) of [12, 13]). Put Z t 0 α(t) = k |g1 |L∞ (Ω) + |g20 |L∞ (Ω) + |g10 |V + |g20 |V dτ, 0

where the constant k > 0 is dependent only on |gi |L∞ (R;V ) , |gi |L∞ (R×Ω) and C3 . Then, we have the following property (∗):

618


(∗) For any −∞ < s < t < ∞, w ∈ V with |w(x)| ≤ |b|∞ a.e. in Ω and z ∈ K(s), there exists z˜ ∈ K(t) such that

Z

|˜ z − z|H ≤ |α(t) − α(s)|(1 + |z|V ), Z A(x, w(x), ∇˜ z (x))dx− A(x, w(x), ∇z(x))dx ≤ |α(t)−α(s)|(1+|z|2V +|w|V |z|V ).

Ω

Ω

Proof. Put g2 (t) − g1 (t) + g1 (t). (3.1) g2 (s) − g1 (s) Then, we easily see that z˜ ∈ K(t) if z ∈ K(s). Moreover, from the same calculation of [15, Lemma 5.1] we observe that (∗) holds. z˜ := (z − g1 (s))

Also, we have the other properties of the time-dependent convex set K(t) in V . Lemma 3.2 ((K2)–(K4) of [12, 13], (E) of [13]). We have the following (k1)–(k4). (k1) There is a constant C4 = C4 (K) > 0 such that |z|V ≤ C4 (1 + |∇z|H )

for all z ∈ K(t) and t ∈ R.

(k2) For any z, z ∈ K(t) and w, w ∈ V with w ≤ z, z ≤ w, we have w ∨ z,

z ∧ w ∈ K(t).

+

(k3) If z, z ∈ K(t) and ∇[z − z] ≡ 0, then z ≤ z. (k4) K(t + T0 ) = K(t) for all t ∈ R. Proof. We can prove (k1) by (F) and Poincar´e inequality. By the definition of K(t) in (1.1), we can show the assertions (k2) and (k3). Also, we easily observe from the periodicity condition (1.5) that (k4) holds. Proof of Theorem 2.1. The authors [12, 13] considered the Cauchy problem and time-periodic problem of elliptic-parabolic variational inequalities with the more general time-dependent constraint K(t). By applying the abstract results in [12, 13] to (P), we prove Theorem 2.1. Indeed, by taking Lemmas 3.1 and 3.2 into account, we can apply the results in [12, 13] to (P). Therefore, we observe from [12, Theorems 2.1, 2.2] that there is a unique solution u of (P) on J with initial data b0 = b(u0 ) for u0 ∈ K(t0 ). In the H

case when b0 = b(u0 ) for u0 ∈ K(t0 ) , using a quite standard argument, we can obtain a unique solution of (P) on J. Hence, Theorem 2.1 (i) holds, and similarly so does Theorem 2.1 (ii). Also, by applying [13, Theorem 2.5] to (P), we get the solution ω to (P) on R satisfying the T0 -periodic condition b(ω)(t + T0 ) = b(ω)(t) for all t ∈ R. Hence, by the uniqueness of the solution to the Cauchy problem for (P) (cf. (k3) in Lemma 3.2), we see that the solution ω satisfies the T0 -periodic condition: ω(t + T0 ) = ω(t) for all t ∈ R. Hence, we get Theorem 2.1 (iii). 4. Order properties of boundary fluxes. In the previous Section 3, we got the T0 -periodic solution ω to (P) on R. However, it is very difficult to prove the generality of the uniqueness of the T0 -periodic solution to (P). So, to show the uniqueness and stability of T0 -periodic solution to (P), we consider the special case when a(x, s, p) = a(x)[p + k(s)] in (1.2), and assume (A’). Note that we obtain the results of the comparison of solutions to (P) with respect to given data b0 , f, g1 and g2 by using the L1 -technique (cf. [12, Theorem 2.2]):


619

Proposition 4.1 (cf. [12, Theorem 2.2]). Let u and u be solutions to (P;b0 , f, g1 , g2 ) and (P;b0 , f , g 1 , g 2 ) on J := [t0 , t1 ], respectively for two sets of data {b0 , f, g1 , g2 } and {b0 , f , g 1 , g 2 }. Assume that g1 , g2 and g 1 , g 2 satisfy the condition (E) and g1 (t) ≤ g 1 (t) on Γ

and

g2 (t) ≤ g 2 (t) on Γ2 ,

∀t ∈ R.

If b(u(t0 )) ≤ b(u(t0 )) on Ω, and f ≤ f on J × Ω, then, we have u ≤ u on J × Ω. By using the above property, we can prove the order properties of boundary fluxes of solutions as follows. Proposition 4.2. Let u1 and u2 be solutions to (P;f, g1 , g2 ) on J := [t0 , t1 ]. If u1 ≤ u2 on J × Ω, then ν · a(x)[∇u1 + k(b(u1 ))] ≥ ν · a(x)[∇u2 + k(b(u2 ))]

on Γ

for a.e. t ∈ J, i.e., hν · a(x)[∇u1 + k(b(u1 ))], ziΓ ≥ hν · a(x)[∇u2 + k(b(u2 ))], ziΓ

(4.1)

1 2

for all z ∈ H (Γ) with z ≥ 0 and a.e. t ∈ J. By using the idea of Kenmochi–Kubo [8, Section 4], we prove Proposition 4.2. To do so, we consider the approximating problem of (P) for λ > 0 as follows: Problem (P)λ 1,2 uλ ∈ L∞ loc ((t0 , t1 ]; V ) and b(uλ ) ∈ Wloc ((t0 , t1 ]; H) ∩ C([t0 , t1 ]; H), Z (b(uλ )t , uλ − v) + a(x)[∇uλ + k(b(uλ ))] · ∇(uλ − v)dx Z ZΩ 1 1 + (uλ − g1 )(uλ − v)dΓ + ([uλ − g2 ]+ − [g1 − uλ ]+ )(uλ − v)dΓ = (f, uλ − v) λ Γ1 λ Γ2 for any v ∈ V and a.e. t ∈ J := [t0 , t1 ], b(uλ (t0 , ·)) = b(u(t0 , ·)) in Ω.

Remark 4.1. From Green’s formulation and the same argument in [9, Section 3] it follows that Problem (P)λ on J is equivalent to the following system: b(uλ )t − ∇ · a(x)[∇uλ + k(b(uλ ))] = f

in J × Ω,

χ1 −ν · a(x)[∇uλ + k(b(uλ ))] = (uλ − g1 ) λ χ2 + [uλ − g2 ]+ − [g1 − uλ ]+ on J × Γ, λ b(uλ (t0 , ·)) = b(u(t0 , ·)) in Ω, where χi are the characteristic functions of the sets J × Γi (i = 1, 2).

(4.2)

(4.3)

By the same argument in [12, Theorem 2.1] (or Theorem 2.1 (i)), we get the unique solution of (P)λ as follows: Lemma 4.1 ([12, 13]). Let J := [t0 , t1 ] and u(t0 ) ∈ K(t0 ). Then, for each λ > 0 there is a unique solution uλ of (P)λ on J such that uλ ∈ L∞ (J; V ) and b(uλ ) ∈ W 1,2 (J; H). Moreover, there is a function u : J → V such that u is a solution to (P) with initial data b(u(t0 )) and uλ * u weakly-∗ in L∞ (J; V ), as λ → 0.

b(uλ ) * b(u) weakly in W 1,2 (J; H)

(4.4)

620


We now prove Proposition 4.2, which is concerned with the order properties of boundary fluxes of solutions for (P). Proof of Proposition 4.2. Let ui,λ be the solution to (P)λ on J = [t0 , t1 ] with initial data b(ui,λ (t0 , ·)) = b(ui (t0 , ·)) for i = 1, 2. Without loss of generality, we may assume that ui (t0 ) ∈ K(t0 ), because of ui (t) ∈ K(t) for a.e. t ∈ J. From the assumption u1 ≤ u2 on J × Ω, it follows that b(u1 (t0 )) ≤ b(u2 (t0 )). χ2 χ1 (· − g1 )(t, x) and [· − g2 ]+ − [g1 − ·]+ (t, x), Then, by the monotonicity of λ λ and applying Proposition 4.1 to (P)λ , we observe that u1,λ ≤ u2,λ

on J × Ω.

From Remark 4.1, more precisely, the boundary condition (4.3), we have ν · a(x)[∇u1,λ + k(b(u1,λ ))] ≥ ν · a(x)[∇u2,λ + k(b(u2,λ ))]

a.e. on J × Γ. (4.5)

We observe from (4.4) that a(x)[∇ui,λ + k(b(ui,λ ))] * a(x)[∇ui + k(b(ui ))] weakly in L2 (J; H),

(4.6)

b(ui,λ )t * b(ui )t weakly in L2 (J; H), hence b(ui,λ )t − f = ∇ · a(x)[∇ui,λ + k(b(ui,λ ))] *

b(ui )t − f = ∇ · a(x)[∇ui + k(b(ui ))]

weakly in L2 (J; H)

(4.7)

as λ → 0 for i = 1, 2. Here, we recall Green’s formula: Z Z ∇ · a(x)[∇ui,λ + k(b(ui,λ ))]zdx + a(x)[∇ui,λ + k(b(ui,λ ))] · ∇zdx Ω

Ω

= hν · a(x)[∇ui,λ + k(b(ui,λ ))], ziΓ

(4.8)

for any z ∈ V , λ > 0, i = 1, 2 and a.e. t ∈ J. Hence, from (4.6), (4.7) and (4.8) we observe ν · a(x)[∇ui,λ + k(b(ui,λ ))] *

ν · a(x)[∇ui + k(b(ui ))]

1

weakly in L2 (J; H − 2 (Γ))

(4.9)

as λ → 0 for i = 1, 2. Thus, from the result of the convergence (4.9) with (4.5) we conclude that the assertion (4.1) holds. 5. Proof of Theorem 2.2. In this section we prove Theorem 2.2 concerning the existence, uniqueness and asymptotic stability of the time-periodic solution to (P). Throughout this section, we assume that all the conditions of Theorem 2.2 hold. Now, let us consider the stationary problem (SP; F, G1 , G2 ) to (P) as follows: Problem (SP; F, G1 , G2 ): Find a function U ∈ V such that U ∈ K := {z ∈ V ; z = G1 on Γ1 and G1 ≤ z ≤ G2 on Γ2 }, Z a(x)[∇U + k(b(U ))] · ∇(U − v)dx ≤ (F, U − v), ∀v ∈ K. Ω

By a standard fixed point and a comparison argument (cf. Gilbarg–Trudinger [3]), we can easily show Proposition 5.1 as follows. We omit the detailed proof.


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Proposition 5.1. For any F ∗ , F∗ ∈ H, the stationary problems (SP; F ∗ , g1∗ , g2∗ ) and (SP; F∗ , g1,∗ , g2,∗ ) have one and only one solution u∗ and u∗ , respectively. Moreover if F∗ ≤ F ∗ on Ω, then, we have u∗ ≤ u∗ on Ω. The next Lemma 5.1 is the key to proving the uniqueness of the periodic solution. Lemma 5.1. Let ω1 and ω2 be two T0 -periodic solutions of (P; f, g1 , g2 ) such that ω1 ≤ ω2 on R × Ω. Then, we have ω1 = ω2 on R × Ω. Proof. At first, we show ν · a(x)[∇ω1 + k(b(ω1 ))] = ν · a(x)[∇ω2 + k(b(ω2 ))]

1

in H − 2 (Γ)

(5.1)

for a.e. t ∈ R. From (2.1) in Remark 2.2 it follows that [b(ω1 ) − b(ω2 )]t − {∇ · a(x)[∇ω1 + k(b(ω1 ))] − ∇ · a(x)[∇ω2 + k(b(ω2 ))]} = 0 (5.2) in R × Ω. By integrating (5.2) over [0, T0 ] × Ω and using the T0 -periodicity of ωi , we get Z T0 Z {∇ · a(x)[∇ω1 + k(b(ω1 ))] − ∇ · a(x)[∇ω2 + k(b(ω2 ))]} dxdt = 0. (5.3) 0

Ω

Note here that the above integration makes sense since the integrand belongs to L2 (0, T0 ; H) because of (2.1) and b(ωi )t ∈ L2 (0, T0 ; H) (cf. Definition 2.1 (a)). On the other hand, Proposition 4.2 implies that Z {∇ · a(x)[∇ω1 + k(b(ω1 ))] − ∇ · a(x)[∇ω2 + k(b(ω2 ))]} dx Ω

= hν · a(x)[∇ω1 + k(b(ω1 ))] − ν · a(x)[∇ω2 + k(b(ω2 ))], 1iΓ ≥ 0 for a.e. t ∈ R. Therefore from the above inequality and (5.3) it follows that hν · a(x)[∇ω1 + k(b(ω1 ))] − ν · a(x)[∇ω2 + k(b(ω2 ))], 1iΓ = 0, which implies that (5.1) holds. Note that the above calculation is formal one. In fact, we prove (5.1) by using the smooth approximating function of 1. Here, we define the functions ηδ and ω i by [r − δ]+ (r ∈ R) for each δ > 0, [r − δ]+ + δ Z T0 ω i (x) := ωi (t, x)dt for x ∈ Ω (i = 1, 2). ηδ (r) :=

0

By multiplying (5.2) by the smooth approximating function of ηδ (·), and using (5.1) and the T0 -periodicity of ωi , we conclude that Z a(x)[∇(ω 1 − ω 2 )] · ∇ηδ (ω 1 − ω 2 )dx Ω

Z

T0

Z a(x)[k(b(ω1 )) − k(b(ω2 ))] · ∇ηδ (ω 1 − ω 2 )dxdt = 0.

+ 0

(5.4)

Ω

Note that the assumption (F) implies ω 1 − ω 2 = 0 on Γ1 . Then, by the same argument in [6, Lemma 4.1], we infer that ω1 = ω2 on R × Ω. Using Propositions 4.2, 5.1 and Lemma 5.1, we prove Theorem 2.2.

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Proof of Theorem 2.2. Let u1 and u2 be any two solutions of (P; f, g1 , g2 ) on [t0 , ∞) (t0 ∈ R). Note that we can regard the function ui (s) as the solution of the stationary problem (SP; f (s) − b(u)t (s), g1 (s), g2 (s)) for a.e. s ∈ [t0 , ∞) (i = 1, 2). For the fixed s(≥ t0 ), we put F ∗ := f ∗ + |b(u1 )t (s)| + |b(u2 )t (s)|,

F∗ := f∗ − |b(u1 )t (s)| − |b(u2 )t (s)|.

Then, by Proposition 5.1, we see that the stationary problems (SP; F ∗ , g1∗ , g2∗ ) and (SP; F∗ , g1,∗ , g2,∗ ) have one and only one solution u∗ and u∗ , respectively, satisfying u∗ ≤ inf{u1 (s), u2 (s)} ≤ sup{u1 (s), u2 (s)} ≤ u∗ .

(5.5) ∗

From Theorem 2.1 (i), we see that for the stationary solutions u and u∗ there are unique solutions ω ∗ and ω∗ on [s, ∞) to the problem (P;b(u∗ ), f, g1 , g2 ) and (P;b(u∗ ), f, g1 , g2 ), respectively. Note that we can regard the functions u∗ and u∗ as the solution to the problems (P; b(u∗ ), F ∗ , g1∗ , g2∗ ) and (P; b(u∗ ), F∗ , g1,∗ , g2,∗ ) on [s, ∞), respectively. So, applying Proposition 4.1 with (B) and (5.5) we see that u∗ ≤ ω∗ ≤ inf{u1 , u2 } ≤ sup{u1 , u2 } ≤ ω ∗ ≤ u∗ on [s, ∞) × Ω.

(5.6)

By repeating the above arguments, we have u∗ (s) = u∗ (s + nT0 ) ≤ ≤ ≤ ≤ ≤ ≤ ≤

ω∗ (s + nT0 ) ω∗ (s + nT0 + T0 ) inf{u1 (s + nT0 + T0 ), u2 (s + nT0 + T0 )} sup{u1 (s + nT0 + T0 ), u2 (s + nT0 + T0 )} ω ∗ (s + nT0 + T0 ) ω ∗ (s + nT0 ) u∗ (s + nT0 ) = u∗ (s) on Ω

(5.7)

for any n = 0, 1, 2, · · · . Therefore, we see that {ω∗ (s + nT0 )} (resp. {ω ∗ (s + nT0 )} is a non-decreasing (resp. non-increasing) sequence in H with respect to n. So, there are elements ω 0 , ω 0 ∈ H such that ω 0 = lim ω∗ (s + nT0 ), n→∞

ω 0 = lim ω ∗ (s + nT0 ) n→∞ ∗

in H,

(5.8)

in H. (5.9) u∗ ≤ ω 0 ≤ ω 0 ≤ u Hence, by Theorem 2.1 (ii), there are solutions ω and ω to (P;b(ω 0 ), f, g1 , g2 ) and (P;b(ω 0 ), f, g1 , g2 ) on [s, ∞), respectively, such that for all t > s ω∗ (· + nT0 ) → ω

and

ω ∗ (· + nT0 ) → ω

weakly in L2 (s, t; H),

b(ω∗ )(· + nT0 ) → b(ω) and b(ω ∗ )(· + nT0 ) → b(ω) as n → ∞. Therefore we easily see that for all t ∈ R b(ω)(t)

= =

strongly in C([s, t]; H)

lim b(ω∗ )(t + nT0 )

n→∞

lim b(ω∗ )(t + T0 + nT0 ) = b(ω)(t + T0 ).

n→∞

(5.10)

Hence, it follows from (5.10) and the T0 -periodic conditions (D), (E) of f , g1 , g2 that ω is a T0 -periodic solution of (P;f, g1 , g2 ) on R satisfying ω(t + T0 ) = ω(t) in H,

∀t ∈ R.

Similarly, we see that ω is a T0 -periodic solution of (P;f, g1 , g2 ) on R. Moreover we observe from (5.9) that ω ≤ ω (cf. Proposition 4.1). Thus, by Lemma 5.1 we have ω = ω, which implies the uniqueness of the T0 -periodic solution to (P;f, g1 , g2 ) on R.


623

Finally, we show the T0 -periodic stability to (P;f, g1 , g2 ). From (5.10), the boundedness of b and the T0 -periodicity of ω, we observe that for all t ∈ R b(ω∗ )(t + nT0 ) − b(ω)(t + nT0 ) = b(ω∗ )(t + nT0 ) − b(ω)(t) −→ 0

in Lq (Ω)

as n → ∞

(5.11)

for all 1 ≤ q < ∞. Similarly, we have b(ω ∗ )(t + nT0 ) − b(ω)(t) −→ 0

in Lq (Ω)

as n → ∞

(5.12)

for all 1 ≤ q < ∞. Therefore, we can infer from (5.6), (5.11), (5.12) and the uniqueness of T0 -periodic solution (i.e. ω = ω) that b(u1 )(t) − b(u2 )(t) → 0


for all 1 ≤ q < ∞.

Hence, we observe that the T0 -periodic solution ω := ω = ω is asymptotically stable in the following sense: for any solution u of (P;f, g1 , g2 ) on [t0 , ∞), we have b(ω)(t) − b(u)(t) → 0


for all 1 ≤ q < ∞.

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Received September 2006; revised January 2007. E-mail address: [email protected] E-mail address: [email protected]