Convergence of solutions of a nonlocal biharmonic MEMS equation with the fringing field Tosiya MIYASITA
∗
Abstract We study a nonlocal biharmonic MEMS equation with the fringing field. It arises in the Micro-Electro Mechanical System(MEMS) devices. First, we establish the local solution and extend it globally in time. Next, we discuss the dynamical properties of omega limit set. Especially, we show that the omega limit set is included in the set of the stationary solution. Finally, we consider the convergence rate to the stationary solution.
1
Introduction
In this paper, we study the following nonlocal biharmonic equation with the fringing field 1+δ|∇u|2 2 u + κu + ∆ u = G (β, γ, ∇u) ∆u + λ I (σ, χ, u) x ∈ Ω, t > 0, tt t (1−u)σ u = ∆u = 0 x ∈ ∂Ω, t > 0, u(x, 0) = u (x) x ∈ Ω, 0 u (x, 0) = u (x) x∈Ω t 1 (1) and the corresponding stationary problem { 2 ∆2 η = G (β, γ, ∇η) ∆η + λ 1+δ|∇η| x ∈ Ω, σ I (σ, χ, η) (1−η) (2) η = ∆η = 0 x ∈ ∂Ω, Mathematics subject classification(2010): 35G30, 35A01, 35B45, 37L25, 74K15. Keywords and Phrases: nonlocal biharmonic equation, MEMS, fringing field, global solution, omega limit set, convergence rate. ∗ “ This is the submitted version of the following article: J. Math. Anal. Appl., 454, (2017), 265–284, which has been published in final form at http://authors.elsevier.com/sd/article/S0022247X17304365
1
where κ ≥ 0, λ > 0, δ > 0, β > 0, γ > 0, χ > 0, σ ≥ 2, Ω ⊂ Rn for n ∈ N is a bounded domain with smooth boundary ∂Ω, ∫ G (β, γ, ∇u) = β |∇u|2 dx + γ, Ω
1 I (σ, χ, u) = (H (σ, χ, u))σ
∫ and
H (σ, χ, u) = 1 + χ Ω
dx . (1 − u)σ−1
This kind of equation arises in the study of the Micro-Electro Mechanical System(MEMS) devices. For more details, see [9, 10, 24, 31]. Here λ denotes the voltage and δ |∇u|2 is called the fringing field [24, 32]. I (σ, χ, u) represents a nonlocal term due to the capacitor in the circuit. Thus, if the solution u(x, t) of MEMS equations reaches 1 at some point in Ω in finite time t = Tq , the right-hand side becomes infinite, which leads to the singularity. In this case, the solution u(x, t) is said to quench in finite time t = Tq and Tq is called the quenching time of the solution. We have several results for the equation with the nonlocal term or fringing field. In fact, for δ = 0 in (1), we have the solution under the periodic boundary condition [4] and the Navier boundary condition [28]. For δ = 0 in (2), we have a stationary solution [2] under Steklov and Dirichlet boundary conditions. For the problem with the fringing field, the equation 1 + δ |∇u|2 εutt + κut = ∆u + λ (1 − u)σ is studied under the Dirichlet boundary condition. There are results of stationary problem (ε = κ = 0) [8, 11, 37], the global solution of parabolic problem (ε = 0) [25, 35], the global solution of damping hyperbolic problem [27]. Nowadays, it seems that there are no studies for the fourth order problem with the fringing field. The aim of this paper is to investigate (1) and (2), respectively. We refer to the studies for the second order problem εutt + κut = ∆u + λ
1 I (σ, χ, u) . (1 − u)σ
For parabolic (ε = 0), hyperbolic (κ = 0), damped hyperbolic or elliptic problem (ε = κ = 0), we have many results about stationary solution, global solution and quenching problem. For the stationary and parabolic problems with χ = 0, see [9]. For the hyperbolic and damped hyperbolic problems with χ = 0, see [5, 6, 14, 20, 21, 22, 29, 34]. A nonlocal problem χ ̸= 0 is studied intensively. Although we can not apply the comparison principle 2
owing to the nonlocal term, there are similar results to those without nonlocal term. For example, see [13, 15, 18, 19, 30, 33] for the stationary solution, [13, 14, 15, 18, 19, 23, 26] for the global existence and [13, 15, 18, 19] for the quenching. To state the theorems, let us give some definitions and notations. Throughout this paper, the definition of the function spaces and their norms is presented in Section 2. We denote by |C| the Lebesgue measure in Rn for the set C ⊂ Rn . X represents the space X ≡ {u ∈ Y, ∆u ∈ Y } ⊂ H 4 (Ω) and Y ≡ H 2 (Ω) ∩ H01 (Ω), respectively. The first two theorems are concerned with the local and global existence of the solution. Theorem 1 Let Ω ⊂ Rn be a bounded domain with smooth boundary ∂Ω with n ≤ 5. We denote D ≡ X × Y and H ≡ Y × L2 (Ω). For any κ ≥ 0, λ > 0, δ > 0, β > 0, γ > 0, χ > 0, σ ≥ 2 and ϕ0 ≡ (u0 , u1 ) ∈ D with ∥u0 ∥X ≤
1−θ CS
for some 0 < θ < 1,
(3)
there exists a unique solution of (1) with ϕ ≡ (u, ut ) ∈ C ([0, T ); D) ∩ C 1 ([0, T ); H) for sufficiently small T > 0, where T depends only on κ, λ, δ, β, γ, χ, σ, Ω, ϕ0 and θ. The solution u can be continued as long as ∥u( · , t)∥X < CS−1 . Here, CS > 0 is an embedding constant which depends only on Ω determined by ∥u∥C + ∥∇u∥C ≤ CS ∥u∥X .
Theorem 2 Let Ω ⊂ Rn be a bounded domain with smooth boundary ∂Ω with n ≤ 5. For any κ > 1/2, β > 0, γ > 0, χ > 0, σ ≥ 2 and ϕ0 ∈ D with (3), there exist two sets } { 0 , Λ0 ≡ (λ, δ) ∈ R2 | 0 < λ < Cλ0 , 0 < λδ < Cλδ 0 where Cλ0 and Cλδ depend only on κ, β, γ, χ, σ, Ω and θ and
D0 ≡ {ϕ0 ∈ D | ∥u0 ∥X < Cu0 , ∥u1 ∥Y < Cu1 } ,
3
where Cu0 and Cu1 depend only on κ, β, γ, Ω and θ such that (1) has a unique global solution with ϕ = (u, ut ) ∈ C ([0, ∞); D) ∩ C 1 ([0, ∞); H) and ∥u( · , t)∥X ≤
1−θ CS
provided that (λ, δ) ∈ Λ0 and ϕ0 ∈ D0 . λ,δ,β To argue the behaviour as t → ∞, we introduce the set Sγ,χ,σ of stationary solution by λ,δ,β Sγ,χ,σ = {η ∈ X | η = η(x) is a solution of (2) for λ, δ, β, γ, χ and σ} .
To state the condition of theorem, we denote by Cβ0 the positive constant which is determined in Proposition 2 and depends only on κ, γ and Ω. Next, we consider the dynamical properties of global solution. Theorem 3 Let Ω ⊂ Rn be a bounded domain with smooth boundary ∂Ω with n ≤ 5. For any κ > 1/2, γ > 0, χ > 0, σ ≥ 2 and ϕ0 ∈ D with (3), we take β ∈ (0, Cβ0 ). Then for any (λ, δ) ∈ Λ0 and ϕ0 ∈ D0 , ω(ϕ0 ) is invariant, λ,δ,β non-empty, compact and connected in D. Moreover ω(ϕ0 ) ⊂ Sγ,χ,σ × {0}. In particular, λ,δ,β Sγ,χ,σ ̸= ∅ for (λ, δ) ∈ Λ0 . λ,δ,β Theorem 3 implies that (η, 0) ∈ ω(ϕ0 ) for some η ∈ Sγ,χ,σ . In other words, λ,δ,β there exist η ∈ Sγ,χ,σ and tn → +∞ such that ( ) lim ∥u( · , tn ) − η∥X + ∥ut ( · , tn )∥Y = 0. (4) n→+∞
We have to fix smaller parameters than those in Theorem 3 to derive the exponential convergence rate to the stationary solution in H. We introduce some positive constant Cβ1 which is determined in the proof of the theorem and depends only on κ, γ and Ω with 0 < Cβ1 ≤ Cβ0 . The main result is on the convergence rate. Theorem 4 Under the same hypotheses as Theorem 3, we take β ∈ (0, Cβ1 ). Then there exists a set } { 1 Λ1 ≡ (λ, δ) ∈ Λ0 | 0 < λ < Cλ1 , 0 < λδ < Cλδ 4
1 0 for some 0 < Cλ1 ≤ Cλ0 and 0 < Cλδ ≤ Cλδ which depend only on κ, γ, χ, σ, Ω and θ such that ( ) 1 lim ∥u( · , t) − η∥Y + ∥ut ( · , t)∥2 = O(e− 4 C2 t ) t→+∞
holds as t → +∞ provided that (λ, δ) ∈ Λ1 and ϕ0 ∈ D0 , where C2 ∈ (0, 2] depends only on κ, γ and Ω and is defined in Section 4. This paper is organized as follows: In Section 2, we recall the facts about Sobolev space. For the notations and definitions, see this section. In Section 3, we establish the local solution. This procedure can be found in many papers. We introduce the method and sketch the proof. In Section 4, we extend the local solution globally in time as long as the parameters and initial values are sufficiently small. In section 5, we discuss the dynamical property. The decaying property and compactness play an important role in the proof. In Section 6, we derive the exponential convergence rate in the same manner as in Section 4. In Section 7, we introduce the results with δ = 0 in (1). We discuss the difference between δ ̸= 0 and δ = 0.
2
Sobolev space
We introduce the notations of function spaces and the Sobolev embedding theorems. In this paper, C(Ω) denotes the space of all continuous functions in Ω with the norm ∥u∥C = sup |u(x)| x∈Ω
for u ∈ C(Ω). W
s,p
(Ω) denotes the usual Sobolev space in Ω with the norm p1 s ∑ ∥Dα u∥p ∥u∥ s,p = p
W
|α|=0
for u ∈ W s,p (Ω) with s ∈ N for 1 ≤ p < ∞ and ∥u∥W s,∞ =
s ∑
∥Dα u∥∞
|α|=0
for u ∈ W s,∞ (Ω) with s ∈ N, respectively. Here, ∥ · ∥p denotes the standard Lp norm in Ω with p ∈ [1, ∞], α is a multi index α = (α1 , α2 , . . . , αn ), |α| = α1 + α2 + · · · + αn and Dα =
∂ |α| . ∂xα1 1 ∂xα2 2 · · · ∂xαnn 5
In particular, we denote W s,2 (Ω) = H s (Ω). H0s (Ω) is defined as the closure of the set D(Ω) in the space H s (Ω), where we denote by D(Ω) the space of all infinitely differentiable functions on Ω with compact supports. H −s (Ω) is defined as the dual space of H0s (Ω). By Proposition A.1.2 in [3], we adopt the norm in H01 (Ω), Y and X as
∥u∥H01 = ∥∇u∥2 , ∥u∥Y = ∥∆u∥2 and ∥u∥X = ∆2 u 2 , respectively. Thus D and H are supposed to be equipped with ( )1 ∥ϕ∥D = ∥u∥2X + ∥v∥2Y 2
and
( )1 ∥ϕ∥H = ∥u∥2Y + ∥v∥22 2
for ϕ = (u, v). By CS and CP , we denote the embedding constants which depend only on Ω. Henceforth we shall adopt universal notations CS and CP to denote various constants. Under these notations, we have the following embedding inequalities. For details, see Chapitre IX in [1]. Lemma 1 We have the following embedding inequalities: ∥u∥2 ≤ CP ∥u∥H01
for u ∈ H01 ,
∥u∥H01 ≤ CP ∥u∥Y
for u ∈ Y ,
∥u∥C + ∥∇u∥C ≤ CS ∥u∥X
3
for u ∈ X with n ≤ 5.
Local existence
First we consider the linear wave equation utt + κut + Au = 0 x ∈ Ω, t > 0, u = ∆u = 0 x ∈ ∂Ω, t > 0, u(x, 0) = u (x) x ∈ Ω, 0 ut (x, 0) = u1 (x) x ∈ Ω,
(5)
where Au = ∆2 u − γ∆u and derive the decay estimate of the solution. Next we construct the time local solution of (1) by the contraction mapping theorem. We omit the detail of the computations. See [16, 21, 26, 27].
6
Lemma 2 (Cf. Proposition 4.3.4 in [17]) For any ϕ0 = (u0 , u1 ) ∈ D, there exists a unique solution ϕ = (u, ut ) ∈ C ([0, ∞); D) ∩ C 1 ([0, ∞); H) of (5). Moreover, we have ∥ϕ∥D ≤ K2 ∥ϕ0 ∥D e−K1 t , where K1 > 0 and K2 > 0 depend only on κ, γ and Ω. Proof of Theorem 1. To deal with the nonlinear term with the singularity, we modify (1 − p)−1 by { 1 for p ≤ 1 − θ, 1−p Fθ (p) = 2 for p ≥ 1 − θ/2 θ for 0 < θ < 1 and extend Fθ suitably in the range (1 − θ, 1 − θ/2) so that Fθ ∈ W 3,∞ (R). Moreover we define aθ (q) = aθ (q1 , q2 , . . . , qn ) by { 1 for |q| ≤ 1 − θ, aθ (q) = 0 for |q| ≥ 1 − θ/2 for 0 < θ < 1 and extend aθ suitably in the range 1 − θ < |q| < 1 − θ/2 so that aθ ∈ W 3,∞ (Rn ). We suppose that ∥Fθ ∥W 3,∞ + ∥aθ ∥W 3,∞ ≤ Lθ , where a constant Lθ > 0 depends only on θ. Note that ( ) α k Dq aθ (q) |q| ≤ ∥aθ ∥W 3,∞ ≤ Lθ
(6)
α for q ∈ Rn , α = 0, 1, 2, 3 and k ∈ {0} ∪ N, where Dq is a multi index with respect to q defined in Section 2. Under the abstract setting
ϕ = (u, ut ) ,
ϕ0 = (u0 , u1 ) ,
Bϕ = (−ut , κut + Au) , ∫ Hθ (σ, χ, u) = 1 + χ Fθ (u)σ−1 dx,
Iθ (σ, χ, u) = (Hθ (σ, χ, u))−σ , σ
Jθ (δ, σ, u) = Fθ (u) and Kθ (λ, δ, σ, β, χ, u) =
(
(∫
(
Ω
1 + δaθ (∇u) |∇u|
) ) |∇u| dx ∆u + λJθ (δ, σ, u) Iθ (σ, χ, u) , 2
0, β
2)
Ω
7
we consider the modified problem ϕt + Bϕ = Kθ (λ, δ, σ, β, χ, u) and convert it to the corresponding integral equation ∫ t −Bt ϕ = e ϕ0 + e−B(t−s) Kθ (λ, δ, σ, β, χ, u(s)) ds. 0
We can construct the local solution ϕ ∈ C ([0, T ); D) ∩ C 1 ([0, T ); H) by the contraction mapping theorem. If the solution of modified equation begins with (3) and satisfies ∥u( · , t)∥X ≤ (1 − θ)/CS for all t > 0, then we obtain ∥u∥C + ∥∇u∥C ≤ CS ∥u∥X ≤ 1 − θ by Lemma 1, which implies that u is also a solution of (1). Thus the solution u of (1) can be continued as long as ∥u( · , t)∥X < CS−1 . 2 Remark 1 In the case of n = 1, the Sobolev embedding H 2 (Ω) ⊂ C 1 (Ω) holds. Hence it suffices to construct the local solution in the class of ( ) ϕ = (u, ut ) ∈ C ([0, T ); H) ∩ C 1 [0, T ); L2 (Ω) × H −2 (Ω) .
4
Global existence
Originally, in [23], the authors establish necessary estimates for the hyperbolic MEMS equation with a nonlocal term. Next, by use of their idea, in [27], the author deals with the hyperbolic problem with the fringing field. Also in this paper, their method is applicable to the modified problem utt + κut + ∆2 u = G (β, γ, ∇u) ∆u + λJθ (δ, σ, u) Iθ (σ, χ, u)
(7)
for x ∈ Ω, t > 0 with the same boundary and initial conditions as (1). In this section, we introduce the necessary inequalities and prove that ∥u( · , t)∥X < CS−1 holds for all t > 0 if (λ, δ) ∈ Λ0 and ϕ0 ∈ D0 . Lemma 3 Let a, b ∈ R and κ > 1/2. Then there exists a constant Mi > 8 which depends only on κ for i = 1,2,3,4 such that the following inequalities
8
hold: κ 2 a + ab + b2 . 2 (κ ) ( ) |a + 2b|2 ≤ 2 a2 + 4b2 ≤ M1 a2 + ab + b2 . 2 (κ ) 2 2 2 a ≤ M2 a + ab + b . 2 (κ ) b2 ≤ M3 a2 + ab + b2 . 2 ) (κ ( ) 2 a2 + ab + b2 . |κa + b| ≤ 2 κ2 a2 + b2 ≤ M4 2 0≤
Henceforth let M = max (M1 , M2 , M3 , M4 ). Proof. h(a, b) = (κ/2)a2 +ab+b2 achieves a local minimum 0 at (a, b) = (0, 0) provided that κ > 1/2. Similarly, we prove that a local minimum is achieved at (a, b) = (0, 0) for the following Mi : ( ) 20 4 (2κ + 1 + α) 4 (2κ + 1 + α) M1 > max , = > 8, κ+2 2κ − 1 2κ − 1 ( ) 4 (κ2 + 1) 2κ (2κ + 1 + α) 2κ (2κ + 1 + α) M4 > max , = > 4κ, κ+2 2κ − 1 2κ − 1 where α=
√ 4κ2 − 4κ + 5 > 2κ − 1.
The rest inequalities follow easily. In fact, we may take M2 = M1 /2 and M3 = M1 /8. 2 Setting v = ut , we define
∫ ( ) γ κ 2 u + ut u + u2t + |∇u|2 + (∆u)2 dx, Φ(t) = 2 2 ∫Ω ( ) κ 2 Ψ(t) = v + vt v + vt2 + (∆v)2 dx Ω 2
for u ∈ X, v = ut ∈ Y and vt ∈ L2 (Ω), respectively. We note that ∫ ( ) (2κ − 1) u2t + (∆u)2 dx, 0 ≤ Φ(t) ≤ C1 ∫Ω ( ) (2κ − 1) vt2 + (∆v)2 dx 0 ≤ Ψ(t) ≤ C1 Ω
9
(8)
by Lemmas 1 and 3, where C1 > 0 depends only on κ, γ and Ω. Let ( ) 1 C2 = min ,2 . C1 In order to derive the global existence, we estimate Φ and Ψ. Henceforth we omit the subscript of constants δ, β, γ, χ and σ if there is no confsion. For instance, we’ll denote G (∇u) = G (β, γ, ∇u), Iθ (u) = Iθ (σ, χ, u), Jθ (u) = Jθ (δ, σ, u) and so on. Lemma 4 We have ( ) ) 1 2λ2 C3 ( 2 − 21 C2 t Φ(t) ≤ e Φ(0) + G (β, γ, ∇u0 ) ∥u0 ∥H01 + 1 + δ 2 L2θ , 2 C2 where C3 ≡
L2σ θ M |Ω| . C2
Proof. Multiplying the modified equation (7) by u and ut , we have ∫ ( ∫ ∫ ( 2 d κ 2) 2) ut u + u dx + −ut + (∆u) dx + G (∇u) |∇u|2 dx dt Ω 2 Ω Ω ∫ = λIθ (u) uJθ (u) dx Ω
and d dt
∫
(
2)
u2t
∫
+ (∆u) dx + 2κ u2t dx Ω Ω ( ) ∫ ∫ 1d 2 2 G (∇u) |∇u| dx + γ |∇u| dx + 2 dt Ω Ω ∫ = 2λIθ (u) ut Jθ (u) dx, Ω
respectively. Adding these inequalities together, we have ) ( d 1 1 2 Φ + G (∇u) ∥u∥H01 + Φ + G (∇u) ∥u∥2H01 dt 2 C1 ∫ ≤ λIθ (u) |u + 2ut | Jθ (u) dx ∫ Ω ∫ C2 λ2 M 2 ≤ |u + 2ut | dx + (Jθ (u))2 dx 2M Ω 2C2 Ω 2 2σ λ Lθ M C2 Φ+ (1 + δLθ )2 |Ω| ≤ 2 2C2 ( ) C2 ≤ Φ + C3 1 + δ 2 L2θ λ2 2 10
by (6), (8) and Lemma 3, replace the left-hand side by ( ) d 1 C2 2 Φ + G (∇u) ∥u∥H01 + C2 Φ + G (∇u) ∥u∥2H01 dt 2 4 from below and solve the differential inequality to obtain the conclusion. 2 Next, we compute Ψ and v in the same way as Φ and u in Lemma 4. Lemma 5 If we choose sufficiently small (λ, δ) and ϕ0 ∈ D so that λ2 ≤
C2 , 10C7
λ2 δ 2 ≤
C2 , 10C8
and G (β, γ, ∇u0 ) ∥u0 ∥2H01 ≤
Φ(0) ≤
C2 60βM
C2 30βM
hold, then we have
( ) 1 Ψ(t) ≤ e− 2 C2 t Ψ(0) + G (β, γ, ∇u0 ) ∥u1 ∥2H01 .
Here C7 > 0 and C8 > 0 are constants to be determined in the proof and depend only on κ, β, γ, χ, σ, Ω and θ. Proof. We have ∫ ( ∫ ∫ ( 2 d κ 2) 2) vt v + v dx + −vt + (∆v) dx + G (∇u) |∇v|2 dx dt Ω 2 Ω Ω ∫ ≤ λ v {Jθ (u) Iθ (u)}t dx Ω
and
( ) ∫ d 2 + (∆v) dx + 2κ dx + G (∇u) |∇v| dx dt Ω Ω Ω ∫ ∫ ∫ = 4β ∇u · ∇ut dx vt ∆u dx + 2λ vt {Jθ (u) Iθ (u)}t dx Ω Ω Ω ∫ ∫ +2β ∇u · ∇ut dx |∇v|2 dx,
d dt
∫
(
2)
vt2
Ω
∫
vt2
Ω
respectively. Hence we have ) 1 d ( Ψ + G (∇u) ∥v∥2H01 + Ψ + G (∇u) ∥v∥2H01 dt C 1 ∫ ≤ λ |v + 2vt | |{Jθ (u) Iθ (u)}t | dx Ω
+4β ∥u∥2Y ∥v∥2 ∥vt ∥2 + 2β ∥u∥Y ∥ut ∥2 ∥v∥Y ∥v∥2 . 11
Since |{Jθ (u) Iθ (u)}t | ≤ |{Jθ (u)}t | + |Jθ (u) {Iθ (u)}t | ≤ σ (1 +
δLθ ) Lσθ
|v| +
3δLσ+1 θ
∫
|∇v| + σ (σ −
1) χL2σ−1 θ
|v| dx
(1 + δLθ ) Ω
≤ σ (1 + δLθ ) Lσθ |v| + 3δLσ+1 |∇v| + C4 (1 + δLθ ) ∥v∥2 θ by (6), where C4 = σ (σ − 1) χL2σ−1 θ
√ |Ω|,
we have I1 ≡ ∥{Jθ (u) I (u)}t ∥22 { √ }2 σ+1 σ ≤ σ (1 + δLθ ) Lθ ∥v∥2 + 3δLθ ∥v∥H01 + C4 (1 + δLθ ) ∥v∥2 |Ω| { ( ) ( ) } 2 2 2(σ+1) 2 2 2 ≤ 3 2σ 2 M 1 + δ 2 L2θ L2σ + 9δ C L + 2M C 1 + δ L θ P θ 4 θ |Ω| Ψ ( ) ≤ C5 + C6 δ 2 Ψ by Lemmas 1 and 3, where ( ) 2 C5 = 6M σ 2 L2σ θ + C4 |Ω|
2(σ+1)
and C6 = C5 L2θ + 27CP2 Lθ
.
Hence we have ) C2 d ( Ψ + G (∇u) ∥v∥2H01 + C2 Ψ + G (∇u) ∥v∥2H01 dt 2 ∫ C2 5λ2 M ≤ |v + 2vt |2 dx + I1 + 6βM ΦΨ 10M Ω 2C2 ( ) C2 1 2 2 2 2 ≤ Ψ + λ C7 Ψ + λ δ C8 Ψ + 6βM Φ(0) + G (∇u0 ) ∥u0 ∥H01 Ψ 10 2 C2 ≤ Ψ 2 by means of Lemma 4, where C7 =
5C5 M 12βC3 M + 2C2 C2
and C8 =
5C6 M 12βC3 L2θ M + . 2C2 C2
It suffices to solve the integral inequality ) C ( ) d ( 2 2 2 Ψ + G (∇u) ∥v∥H01 + Ψ + G (∇u) ∥v∥H01 ≤ 0. dt 2 12
In this estimate, for instance, we choose ϕ0 ∈ D with Φ(0) ≤ (2κ − 1) C1 CP4 ∥u1 ∥2Y + C1 CP4 ∥u0 ∥2X ≤
C2 60βM
and ) ( G (β, γ, ∇u0 ) ∥u0 ∥2H01 ≤ CP6 βCP6 ∥u0 ∥2X + γ ∥u0 ∥2X ≤
C2 . 30βM 2
Lemma 6 Let C9 =
1−θ √ . CS (2κ − 1) C1 M
Suppose that λ≤
C9 √ , 50Lσθ |Ω|
λδ ≤
C9 √ , 50Lσ+1 |Ω| θ
∥u0 ∥X ≤
and that G (β, γ, ∇u0 ) ∥u0 ∥Y ≤ Then we have ∥(u1 )t ∥2 ≤
C9 , 50
∥u1 ∥2 ≤
C9 50κ
C9 . 50
C9 . 10
Proof. The following inequality leads us to the conclusion: ∥(u1 )t ∥2 ≤ κ ∥u1 ∥2 + ∥u0 ∥X + G (∇u0 ) ∥u0 ∥Y + λLσθ (1 + δLθ )
√ |Ω|. 2
Lemma 7 We suppose that the hypotheses in Lemma 6 hold. If we choose sufficiently small u1 ∈ Y so that ∥u1 ∥Y ≤
1−θ √ , 10CS C1 M
then we have Ψ(0) ≤
(1 − θ)2 . 50CS2 M
13
Proof. Note that Ψ(0) ≤ (2κ − 1) C1 ∥(u1 )t ∥22 + C1 ∥u1 ∥2Y 2
along with Lemma 6.
Lemma 8 We suppose that the hypotheses in Lemmas 5, 6 and 7 hold. If we choose sufficiently small ϕ0 ∈ D so that G (β, γ, ∇u0 ) ∥u1 ∥2H01 then we have Ψ(t) ≤ e− 2 C2 t 1
(1 − θ)2 ≤ , 50CS2 M
(1 − θ)2 (1 − θ)2 ≤ . 25CS2 M 25CS2 M 2
Proof. It follows from Lemmas 5 and 7 right away. Lemma 9 Let C10
(1 − θ)2 = 100γ 2 CS2
and C11
1 = 4
(
γ (1 − θ) 10βCS
) 23 .
Suppose that λ2 ≤
C2 Ci , 2C3
λ2 δ 2 ≤
C2 Ci , 2C3 L2θ
Φ(0) ≤ Ci
and that G (β, γ, ∇u0 ) ∥u0 ∥2H01 ≤ 2Ci for i = 10,11. Then we have Φ(t) ≤ 4Ci .
2
Proof. It follows from Lemma 4.
Proof of Theorem 2. First, we take sufficiently small (λ, δ) and ϕ0 ∈ D so that the hypotheses in all of the lemmas are satisfied. Moreover we assume that 1−θ 1−θ √ . √ λ≤ and that λδ ≤ σ 5CS Lθ |Ω| 5CS Lσ+1 |Ω| θ 14
Then we conclude that ∥u∥X ≤ ∥vt + κv∥2 + G (∇u) ∥u∥Y + λ ∥Jθ (u)∥2 ( ) √ √ √ 2β ≤ M Ψ(t) + Φ(t) + γ Φ(t) + λLσθ (1 + δLθ ) |Ω| γ 1−θ ≤ CS for all t ≥ 0.
2
Remark 2 From Lemma 8, we can deduce the decay estimate of ut and utt , that is, lim ∥ut ∥Y = lim ∥utt ∥2 = 0. t→+∞
t→+∞
Moreover, we derive ϕt = (ut , utt ) ∈ Lp ((0, ∞); H) and ∫ t2 t1
∥ϕt ∥pH
ds ≤ M
1 p 2
∫
t2
Ψ(s)
1 p 2
t1
4 ds ≤ pC2
(
1−θ 5CS
)p (
e− 4 pC2 t1 − e− 4 pC2 t2 1
1
)
for p ≥ 1 and 0 ≤ t1 ≤ t2 . Remark 3 In the case of n = 1, the Sobolev embedding H 2 (Ω) ⊂ C 1 (Ω) holds. Hence we estimate only Φ(t) and reach ( ) ϕ = (u, ut ) ∈ C ([0, ∞); H) ∩ C 1 [0, ∞); L2 (Ω) × H −2 (Ω) .
5
Omega limit set
In this section, we consider the behaviour of the solution obtained in Section 4. First, we prepare the lemma. Lemma 10 Ω ⊂ Rn be a bounded domain with smooth boundary ∂Ω with n ≤ 5. Then we have 1
∥u1 u2 ∥2 ≤ |Ω| 10 ∥u1 ∥ 10 ∥u2 ∥10 ≤ E1 ∥u1 ∥H01 ∥u2 ∥Y , 3
∥u2 u3 ∥2 ≤ |Ω|
3 10
∥u2 ∥10 ∥u3 ∥10 ≤ E1 ∥u2 ∥Y ∥u3 ∥Y
for u1 ∈ H01 (Ω) and u2 , u3 ∈ Y , where a universal constant E1 > 0 depends only on Ω. 15
Next, we introduce the proposition concerned with the omega limit set. Proposition 1 (Cf. Corollary 7.2.1 in [17]) Let ϕ0 ∈ H and the orbit ∪t≥0 ϕ(t) be relatively compact in H. Assume that 1,1 ϕ ∈ Wloc ((0, ∞); H) .
Then if there exists α > 0 such that ∫ t+α ∥ϕt (s)∥H ds → 0 t λ,δ,β as t → +∞, then the omega limit set ω(ϕ0 ) is contained in Sγ,χ,σ × {0} in H.
In order to apply this proposition, we decompose u into w and z as mentioned below. For a solution u of (1) obtained in Theorem 2, let w be a solution of (∫ ) 2 w + κw + Aw = β |∇u| dx ∆w x ∈ Ω, t > 0, tt t Ω w = ∆w = 0 x ∈ ∂Ω, t > 0, (9) w(x, 0) = u0 (x) x ∈ Ω, wt (x, 0) = u1 (x) x ∈ Ω. Then we have the decay property of w. Proposition 2 Under the same hypotheses as Theorem 2, we assume that β
0, ztt + κzt + A1 z = λ 1+δ|∇u| (1−u)σ z = ∆z = 0 x ∈ ∂Ω, t > 0, z(x, 0) = z (x, 0) = 0 x ∈ Ω, t )
(∫
where
|∇u| dx ∆z.
A1 z = Az − β
2
Ω
For ξ = (z, zt ), setting B1 ξ = (−zt , κzt + A1 z) ,
( ) J (δ, σ, u) = (1 − u)−σ 1 + δ |∇u|2 (
and
)
L (δ, σ, χ, u) = 0, I (σ, χ, u) J (δ, σ, u) , ∫
then we have ξ=λ
t
e−B1 (t−s) L (δ, σ, χ, u(s)) ds.
0
We find ϕ=ψ+ξ and cite the compactness result of the orbit, which is proven originally in [36]. 17
Proposition 3 (Cf. Lemma 7.6.2 in [17]) We assume that exp(−tB1 ) is the contractive semi-group on D for all t ≥ 0. If there exists a compact set K ⊂ D such that L (δ, σ, χ, u(t)) ∈ L∞ ((0, ∞); D) and L (δ, σ, χ, u(t)) ⊂ K for almost every t ∈ (0, ∞), then the orbit ∪t≥0 ϕ(t) is relatively compact in D. Lemma 11 Under the same hypotheses as Theorem 2 and Proposition 2, L (δ, σ, χ, u(t)) ∈ L∞ ((0, ∞); H 5 (Ω) × H 3 (Ω)) and ∥L (δ, σ, χ, u(t))∥H 5 ×H 3 ≤ E2 for all t ≥ 0, where E2 > 0 depends only on δ, σ, Ω and θ. Proof. It follows from (10), Lemma 10 and ∥u(t)∥X ≤
1−θ CS
for all t ≥ 0.
2
Proof of Theorem 3. First, by Proposition 2, exp(−tB1 ) is contractive. Next, by Lemma 11, L(u(t)) is bounded in H 5 (Ω) × H 3 (Ω) for all t > 0. Since the embedding H 5 (Ω) × H 3 (Ω) ⊂ H 4 (Ω) × H 2 (Ω) is compact, Proposition 3 shows that the orbit ∪t≥0 ϕ(t) is relatively compact in H 4 (Ω) × H 2 (Ω) and finally D. Hence, the omega limit set ω(ϕ0 ) is invariant, non-empty, compact and connected in D by Theorem 5.1.8 in [17]. By Lemma 4 and Remark 2, Proposition 1 is applicable for the topology H. The precompactness of the orbit and elliptic regularity of the stationary solution guarantee λ,δ,β the convergence in D. Hence we conclude that ω(ϕ0 ) ⊂ Sγ,χ,σ × {0} in D. 2
6
Convergence rate
λ,δ,β In the proof of Theorem 3 in Section 5, we show that we have some η ∈ Sγ,χ,σ and tn → +∞ such that (4) holds. For the case δ = 0 [28], we conclude that the omega limit set is composed of a single point by the Lyapunov function and Lojasiewicz-Simon inequality [7, 12, 17]. In our problem, we do not have any appropriate Lyapunov function. Therefore, in order to deal with the convergence problem in H instead of D, we have to restrict the parameters on the smaller set Λ1 than Λ0 . Setting
p = p(x, t) ≡ u(x, t) − η(x)
18
and noting that u and η satisfy all of the estimates obtained in Section 4, we define ∫ ( ) κ 2 p + pt p + p2t + (∆p)2 dx, X(t) ≡ Ω 2 Y (t) ≡ I (σ, χ, u) J (δ, σ, u) − I (σ, χ, η) J (δ, σ, η) , respectively. Then we introduce the lemma. Lemma 12 We have ∥Y (t)∥2 ≤ (F4 + F5 δ) ∥p(t)∥Y for all t ≥ 0, where F4 > 0 and F5 > 0 depend only on χ, σ, Ω and θ. Proof. Since
( ) |aτ − bτ | ≤ τ |a|τ −1 + |b|τ −1 |a − b|
holds for a, b ∈ R and τ ≥ 1, it suffices to utilize (10) and Lemma 1.
2
Lemma 13 We suppose that the hypotheses in Theorem 3 hold. If there exists t0 > 0 such that X(t0 ) = 0, then X(t) = 0 holds for all t ≥ t0 . Proof. Lemma 3 implies that (p(x, t0 ), pt (x, t0 )) ≡ (0, 0), that is, u(x, t0 ) ≡ η(x)
and
ut (x, t0 ) ≡ 0.
Thus we consider (1) under the initial value ϕ(t0 ) = (η(x), 0). Then we have utt (x, t0 ) = −∆2 η + G (∇η) ∆η + λI (η) J (η) = 0 and moreover ∫ ( ) κ 2 2 2 ut (t0 ) + utt (t0 )ut (t0 ) + utt (t0 ) + (∆ut (t0 )) dx = 0. Ψ(t0 ) = Ω 2 Hence Lemma 5 yields − 12 C2 (t−t0 )
0 ≤ Ψ(t) ≤ e
( Ψ(t0 ) +
G (∇u(t0 )) ∥ut (t0 )∥2H01
) =0
for all t ≥ t0 . Since |X ′ (t)| ∫ ( ) 2 κppt + pt + pptt + 2pt ptt + 2∆p∆pt dx = Ω
≤ κ ∥p∥2 ∥v∥2 + ∥pt ∥2 ∥v∥2 + ∥p∥2 ∥vt ∥2 + 2 ∥pt ∥2 ∥vt ∥2 + 2 ∥p∥Y ∥v∥Y √ ≤ (κM + 4M + 2) XΨ 19
holds for all t ≥ t0 , finally we have X(t0 ) = 0 and X ′ (t) = 0 for t ≥ t0 , which implies X(t) = 0 for all t ≥ t0 . 2 Proof of Theorem 4. In the same manner as the proof of Lemma 4 in Section 4, we have ∫ ( ∫ ) ( 2 d κ 2) pt p + p dx + −pt + (∆p)2 dx dt Ω 2 Ω ∫ ∫ ∫ −G (∇u) ∆up dx + G (∇η) ∆ηp dx = λ pY dx Ω
and d dt
∫
(
Ω
∫
2)
p2t
Ω
+ (∆p) dx + 2κ p2t dx Ω Ω ∫ ∫ ∫ −2G (∇u) ∆upt dx + 2G (∇η) ∆ηpt dx = 2λ pt Y dx. Ω
Ω
Ω
Adding them together, we obtain X ′ + C2 X ∫ ( )∫ ≤ G (∇u) − G (∇η) ∆u (p + 2pt ) dx + G (∇η) ∆p (p + 2pt ) dx Ω Ω ∫ +λ (p + 2pt ) Y dx ∫ Ω ∫ ∫ ≤ β ∇p · ∇ (u + η) dx ∆u (p + 2pt ) dx + 2G (∇η) ∆ppt dx Ω Ω Ω √ +λ M (F4 + F5 δ) X √ √ √ ≤ 2βCP3 M |Ω| ∥u∥X X + 2 (β |Ω| + γ) M XΨ + λ M (F4 + F5 δ) X √ 2βCP3 √ 2 (β |Ω| + γ) − 1 C2 t √ ≤ M |Ω|X + e 4 X + λ M (F4 + F5 δ) X CS 5CS by (10), Lemmas 8 and 12. Let ρ ∈ (1/4, 1/2) fixed arbitrarily. For any β
0, x ∈ ∂Ω, t > 0, x ∈ Ω, x ∈ Ω. (11)
Different from (1), E(ϕ(t)) ≡ E(ϕ(t)) +
λ (H (σ, χ, u))1−σ 2 (σ − 1) χ
is a Lyapunov function for (11), where )2 (∫ ∫ ∫ ∫ 1 β γ 1 2 2 2 E(ϕ(t)) = ut dx + (∆u) dx + |∇u| dx + |∇u|2 dx. 2 Ω 2 Ω 4 2 Ω Ω In fact,
∫ d E(ϕ(t)) = −κ u2t dx ≤ 0 dt Ω holds. Hence for sufficiently small λ and ϕ0 , we can construct the global solution in the class of ( ) ϕ ∈ C ([0, ∞); H) ∩ C 1 [0, ∞); L2 (Ω) × H −2 (Ω) and establish the dynamical properties in H. Thanks to the Lyapunov function, we can prove that Lojasiewicz-Simon inequality holds, which implies that ( ) lim ∥u( · , t) − η∥Y + ∥ut ( · , t)∥2 = 0 t→+∞
21
without any extra restriction on the parameters such as the hypotheses in Theorem 4. However in [28], the convergence rate is not treated. In our problem, we truncate the nonlinear term by aθ (∇u) and utilize the embedding X ⊂ C 1 (Ω). Therefore we deal with the space X.
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Japan e-mail:
[email protected]
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