Remarks on the beam evolution equations in noncylindrical domains

Nonlinear Analysis 70 (2009) 693–710 www.elsevier.com/locate/na

Remarks on the beam evolution equations in noncylindrical domains C.S.Q. De Caldas ∗ , J. Limaco, R.K. Barreto Universidade Federal Fluminense, IM, Rua M´ario Santos Braga s/n., CEP: 24020-140, Niter´oi, RJ, Brazil Received 17 April 2007; accepted 8 January 2008

Abstract In this article, we present results concerning the existence, uniqueness and the asymptotic behavior of solutions for a beam evolution equation with variable coefficients in noncylindrical domains. c 2008 Elsevier Ltd. All rights reserved.

MSC: 39B52 Keywords: Beam evolution equation; Existence; Exponential decay; Uniqueness; Noncylindrical domains

1. Introduction The main purpose of this paper is to present results concerning the existence, uniqueness and the asymptotic behavior of solutions for the equation   Z 2 00 |∇u(x, t)| dx ∆u + δ u 0 = 0 (1.1) a(x) u + ∆(b(x)∆u) − M x, t, Ω

b Some mathematical aspects related with the initial-boundary value problem (1.1) have in a noncylindrical domain Q. been investigated during the last years by several authors in several contexts. In fact, in the cylindrical case, when the dimension is one (n = 1), for the function M(x, t, λ) depending only on λ, with M(λ) ≥ m 0 > 0, and the functions a = b = 1 and δ = 0, we mention the works of Ball [1], Dickey [4] and Woinwsky and Krieger [13] in which the existence of global solutions was investigated. In the precedent conditions and the two-dimensional case, we cite for instance, the paper of Eiesley [5] that also yields results of the existence of global solutions. In the n-dimensional case, still in the case of constant coefficients, with M depending only on λ, Eq. (1.1) in the presence of internal damping δu 0 was studied by Biler [2] and Pereira [11], proving the existence, uniqueness and decay estimate for the energy associated with the global solutions. Of course, there are several other good papers related with problem (1.1) in the literature: see for example the papers of Brito [3], Medeiros [10], Patcheu [12] and a number of other interesting references cited in the mentioned papers. The function still represented by M(x, t, λ) is a restriction of M(x, t, λ) b × [0, T ] × [0, ∞[, and Q b × [0, T ]. Of particular b and λ > 0, where M(x, t, λ) is defined in Ω b⊂Ω to (x, t) ∈ Q ∗ Corresponding address: Universidade Federal Fluminense, Departamento de Matematica Aplicada, Rua Mario Santos Braga s/n., CEP 24020140, Niter´oi, RJ, Brazil. Tel.: +55 21 2629 2074. E-mail address: [email protected] (C.S.Q. De Caldas).

c 2008 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2008.01.003

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relevance to our paper, in the cylindrical case, is the recent work due to Limaco et al. [7] in which the existence of a unique global weak and strong solution is established and a rate decay of energy for a problem similar to (1.1), but with hypotheses on the function M(x, t, λ) more restrictive, namely b, M(x, t, λ) is a C 1 -real function in the variables x ∈ Ω M(x, t, λ) ≥ 0, |∇x M| ≤ c1 |λ| , ∂M ≤ c2 |λ| p and ∂ M ≤ c3 |λ| p−1 ∂t ∂λ

t ≥ 0, λ ≥ 0,

p

(1.2) with p ≥ 1,

where | · | is the absolute value of R. Finally, in the noncylindrical domain, we cite the work of Limaco et al. [8] where the existence of global solutions and exponential decay of energy assuming for the function M the same hypotheses fixed in (1.2) in an increasing noncylindrical domain was obtained. In this paper we generalize the result of Limaco et al. [8] in the sense of considering hypotheses more generally on the function M and the not necessarily increasing noncylindrical domain: b = {x = K (t)y; y ∈ , 0 < t < T }, Q

(1.3)

where Ω ⊂ Rn is an open, bounded and regular subset and K : [0, ∞[→ R is a C 2 -real function. To solve the problem we will employ a change of variable. We also prove the uniqueness of solutions which was not proved in [8]. Our paper is divided into five sections. In the second section we fix notations and we prove the existence of weak solutions for Eq. (1.1). In the third section we prove the exponential decay of the energy, in the fourth section the uniqueness of solutions and in the fifth section we give an example of one M function satisfying (2.6). 2. Weak solutions We use the standard notations for all functional spaces to be encountered throughout R the paper. The scalar product and the norm in L 2 (Ω ) and H01 (Ω ) are denoted respectively by (u, v) = Ω u(x) v(x)dx, |u|2 = R R Pn R ∂u 2 2 2 2 i=1 Ω | ∂ xi (x)| dx. We denote the absolute value Ω |u(x)| dx, ((u, v)) = Ω ∇u(x) ∇v(x)dx, kuk = |∇u| = 2 of u(x) in R and the norm of u in L (Ω ) by |u(x)| and |u| respectively. We will study the problem a(x)u00 (x, t) + ∆(b(x)∆u(x,t)) Z −M x, t, |∇u(x, t)|2 dx ∆u(x, t) + δu 0 (x, t) = 0 Ω ∂u(x, t) u(x, t) = =0 ∂ν u(x, 0) = u (x), u 0 (x, 0) = u (x) 0

1

b in Q, (2.1) b, on Σ in Ω0 ,

b = {(x, t) ∈ Rn ×]0, T [; x = K (t) y, y ∈ Ω , 0 ≤ t < T }. where Q b K , a, b, and M given in Eq. (2.1): We consider the following hypotheses for Q, b ×]0, T [, b⊂Ω Q K ∈ C ([0, +∞[) 2

b is an open bounded set of Rn , 0 ∈ Ω b where Ω with 0 < K 0 ≤ K (t) ≤ K 1,

b1 |K (t)| ≤ K 0

b ) with 0 < a0 ≤ a(x) ≤ 1, 0 < b0 ≤ b(x) ≤ 1 a, b ∈ C 1 (Ω δ δ b0 P(|K 0 |, d(Ω )) < , S(|K 0 |, d(Ω )) < , 2 12K 14 where P and S are two variable real polynomials of degree 2, with positive coefficients, defined in (2.40) and (2.41) respectively and d(Ω ) = sup{|y|; y ∈ Ω }.

(2.2) (2.3) (2.4)

(2.5)

C.S.Q. De Caldas et al. / Nonlinear Analysis 70 (2009) 693–710

M(x, t, λ) is a C 1 function in the variables x ∈ Ω b , t > 0 and λ > 0, 0 ≤ M(x, t, λ) ≤ φ(λ), where φ is an increasing and continuous function in [0, +∞[, with lim φ(λ) = 0; |∇ M| ≤ c √λ f (λ), ∂ M ≤ c g(λ), x 2 1 ∂t λ→0 ∂ M ≤ c3 h(λ) where f, g, and h are C 1 increasing functions in [0, +∞[ with lim g(λ) = 0. λ→0 ∂λ

695

(2.6)

On the other hand, by the change of variable: v(y, t) = u(x, t)

with x = K (t) y

(2.7)

the problem (2.1) is transformed into 1 1 e ∆(e b(y, t) ∆v(y, t)) + 2 M(y, t, kv(y, t)k2 ) ∆v(y, t) + δ v 0 (y, t) K4 K ∂ 2v ∂v 0 ∂v (y, t) + di j (y, t) (y, t) + ei (y, t) (y, t) = 0 in Q = Ω × (0, T ) +ci (y, t) ∂ yi ∂ yi ∂ y j ∂ yi X ∂v v(y, t) = (y, t) = 0 in ∂ν v(y, 0) = v0 (y), v 0 (y, 0) = v1 (x) in Ω ,

e a (y, t) v 00 (y, t) +

(2.8)

where, e e a (y, t) = a(K (t)y), b(y, t) = b(K (t)y), 2 e t, kvk ) = M(K (t)y, t, k(t)n−2 kvk2 ) M(y, −yi K + 2yi (K 0 )2 δyi K 0 +e a (y, t) K K2 0 2 (K ) K0 di j (y, t) = e a (y, t) yi y j , e (y, t) = −2e a (y, t)y . i i K K2 We define b h0 = b h 0 (u 0 , u 1 ) as that transformed from h 0 = h 0 (v0 , v1 ), through the change of variable where ci (y, t) = −

(2.9)

h 0 = c10 [|v1 |2 + |∆v0 |2 + φ(kv0 k2 )kv0 k2 ], and

c10 is defined in (2.51).

(2.10)

b → R is said to be a weak solution of (2.1), if u ∈ L 2 (0, T ; H 2 (Ωt )), u 0 ∈ Definition 2.1. The function u : Q 0 2 2 L (0, T ; L (Ωt )) and satisfies: Z TZ Z TZ − a(x) u 0 (x, t)φ 0 (x, t)dxdt + b(x) ∆u(x, t) ∆φ(x, t)dxdt Ωt

0

0 T

Z



Z

|∇u(x, t)|2 dx ∇u(x, t) · ∇φ(x, t)dxdt

M x, t,

+ Ωt

0 T

Z

Ωt



Z

+ Ωt

0

+δ

Z 0

T

Ωt



Z

Z Ωt

 Z ∇x M x, t, Ωt

  |∇u(x, t)|2 dx · ∇u(x, t) φ(x, t)dxdt

u 0 (x, t) φ(x, t)dxdt = 0

for all φ ∈ L 2 (0, T ; H02 (Ωt )), φ 0 ∈ L 2 (0, T ; L 2 (Ωt )), with φ(x, 0) = φ(x, T ) = 0, u(x, 0) = u 0 (x), u 0 (x, 0) = u 1 (x) for all x ∈ Ω0. Theorem 2.1. Suppose u 0 ∈ H02 (Ω0 ), u 1 ∈ L 2 (Ω0 ), the hypotheses (2.2)–(2.6) and δ b0 F(b h0) < , 13 K 14

(2.11)

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where F defined in (2.49) is a positive increasing function that depends on φ, f, g and h. Then there exists at least one function u : Q → R which is a weak solution of the initial-boundary value problem (2.1) in the sense of Definition 2.1. Theorem 2.2. Suppose v0 ∈ H02 (Ω ), v1 ∈ L 2 (Ω ), the hypotheses (2.2)–(2.6) and F(h 0 ) <

δ b0 , 13 K 14

(2.12)

where F and h 0 are defined in (2.49) and (2.10) respectively. Then there exists at least one function v : Q → R weak solution of (2.8). From (2.7) we can conclude that problem (2.1) is equivalent to the problem (2.8). Therefore it is sufficient to prove Theorem 2.2. Proof P of Theorem 2.2. We employ the Faedo–Galerkin method. Let (wi )i∈N be a basis of H02 (Ω ). We look for m vm = i=1 gim (t)wi , solution of the nonlinear system of ordinary differential equations. The solution exists. 1 e 1 00 b ∆vm (t), ∆w) + 2 ( M(t, kvm (t)k2 ) ∇vm (t), ∇w) (av ˜ m (t), w) + 4 (e K K   1 ∂vm (t) 0 e kvm (t)k2 )∇vm (t), w) + (δvm + 2 (∇ y M(t, (t), w) + ci ,w ∂ yi K     0 (t) ∂ 2 vm (t) ∂vm + di j , w + ei ,w = 0 ∂ yi ∂ y j ∂ yi vm (0) = v0m → v0 in H02 (Ω ) 0 vm (0) = v1m → v1 in L 2 (Ω ).

(2.13)

0 (t) in (2.13) we have: First estimate. Set w = vm

1 e 1 e 0 0 (b ∆vm (t), ∆vm (t)) + 2 ( M(t, kvm (t)k2 ) ∇vm (t), ∇vm (t)) K4 K   ∂vm (t) 0 1 2 0 0 0 e + 2 (∇ y M(t, kvm (t)k )∇vm (t), vm (t)) + (δvm (t), vm (t)) + ci , vm (t) ∂ yi K     ∂ 2 vm (t) 0 ∂v 0 (t) 0 + di j , vm (t) + ei m , vm (t) = 0. ∂ yi ∂ y j ∂ yi

00 0 (av ˜ m (t), vm (t)) +

Therefore, we have the following identities: Z Z 0 2 0 2 1 d 1 00 0 • (a˜ vm (t), vm (t)) = a(y, ˜ t) vm (y, t) dy − a˜ 0 (y, t) vm (y, t) dy. 2 dt Ω 2 Ω Z 1 e 1 d 1 0 e • (b∆vm (t), ∆vm (t)) = b(y, t) |∆vm (y, t)|2 dy 2 dt Ω K 4 K4 0 Z  1 1 e − b(y, t) |∆vm (y, t)|2 dy. 2 Ω K4 Z 1 e 1 d 1 e 2 0 • ( M(t, kvm (t)k ) ∇vm (t), ∇vm (t)) = M(y, t, kvm (t)k2 ) |∇vm (y, t)|2 dy 2 dt Ω K 2 K2 0 Z  1 1 e 2 |∇vm (y, t)|2 dy. − ( M(y, t, kv (t)k )) m 2 Ω K2   Z Z 0 (t) 2 ∂vm K0 1 ∂ 0 K 0 0 0 2 • ei , vm (t) = −2yi (vm (y, t)) dy = n vm (y, t) dy. ∂ yi K 2 ∂ yi Ω Ω K

(2.14)

(2.15)

(2.16)

(2.17) (2.18)

C.S.Q. De Caldas et al. / Nonlinear Analysis 70 (2009) 693–710

697

And we have the following inequalities:   ci ∂vm (t) , v 0 (t) ≤ c0 m ∂y i

2 ! 0 δd(Ω ) K 0 + d(Ω ) 2d(Ω ) K 0 |∆vm (t)| vm (t) , + 2 K0 K0

(2.19)

where c0 is such that kvm (t)k ≤ c0 |∆vm (t)| .   2 di j ∂ vm (t) , v 0 (t) ≤ m ∂y ∂y i

cb0 (d(Ω ))2 (K 0 )2 K 02

j

! 0 , |∆vm | vm

(2.20)

where cb0 is such that |vm (t)| H 2 (Ω ) ≤ cb0 |∆vm (t)| . Since e t, kvm (t)k2 ) = M(K (t)y, t, K n−2 kvm (t)k2 ), M(y, then e ∂M ∂M (y, t, kvm (t)k2 ) = K (t) (K (t)y, t, K n−2 kvm k2 ), ∂ yi ∂ xi e = K (t)∇x M. Then, where ∇ y M 1 2 0 ≤ 1 ∇x M(K (t)y, t, K n−2 kvm (t)k2 ) kvm (t)k|v 0 (t)| e (∇ M(y, t, kv (t)k )∇v , v (t)) m m m m K K2 y 0 c1 n−2 0 K 2 kvm (t)k f (K n−2 kvm (t)k2 )kvm (t)k|vm ≤ (t)| K0 c1 12 0 ≤ c02 c f (K n−2 kvm (t)k2 )|∆vm (t)|2 |vm (t)|, K0 6 where c6 = max{1, K 1n−2 , K 02−n }. On the other hand (e a (y, t))0 = K 0 (t)∇a.y, then Z 1 2 c7 0 0 0 0 (t)|2 , (e a (y, t)) vm (y, t) dy ≤ |K |d(Ω )|vm 2 2 Ω where c7 = sup b |∇a(x)|. x∈Ω Similarly (e b(y, t))0 = K 0 (t)∇x b.y, then Z  Z  0  1 1 1 e −4K 0 e K0 2 2 |∆v |∆v b(y, t) (y, t)| dy = b(y, t) + ∇ b.y (y, t)| dy m x m 2 4 2 Ω K5 K4 Ω K ! 2b1 K 0 c8 |K 0 |d(Ω ) ≤ + |∆vm (t)|2 , K 04 K 05 where c8 = sup b |∇b(x)|. x∈Ω Since !0 e t, kvm (t)k2 ) M(y, −2K 0 e 1 ∂M = M(y, t, kvm (t)k2 ) + 2 (K (t)y, t, K n−2 kvm (t)k2 ) 2 3 K K K ∂t 1 0 0 K (t) ∇x M(K (t)y, t, K n−2 kvm (t)k2 ) · y − 2K n−2 (∆vm (t), vm (t)) K2 1 ∂M + 2 (K (t)y, t, K n−2 kvm (t)k2 )[(n − 2)K 0 K n−3 kvm (t)k2 ] K ∂λ +

(2.21)

(2.22)

(2.23)

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then, Z 1 2 Ω +

+

e t, kvm (t)k2 ) M(y, K2 c02 c2 2K 02

!0

c2 K 0 |∇vm (y, t)| dy ≤ 0 3 φ(K n−2 kvm (t)k2 )|∆vm (t)|2 K0 2

1

g(K

n−2

kvm (t)k )|4vm (t)| + 2

|n − 2|c03 c3 c6 |K 0 | 2K 03

2

c03 c1 c62 2K 02

|K 0 |d(Ω ) f (K n−2 kvm (t)k2 )|∆vm (t)|3

h(K n−2 kvm (t)k2 )|∆vm (t)|3 +

c02 c3 c6 K 02

0 h(K n−2 kvm (t)k2 )|∆vm (t)|3 |vm (t)|.

(2.24)

From (2.14)–(2.24), we have # Z " e t, kvm (t)k2 ) 0 2 0 2 1 ∂ 1 e M(y, 2 2 e a (y, t) vm (y, t) + 4 b(y, t) |∆vm (y, t)| + |∇vm (y, t)| dy + δ vm (t) 2 2 ∂t Ω K K " # 0 δc0 |K 0 |d(Ω ) + c0 d(Ω ) 2c0 d(Ω )(K 0 )2 + cb0 (d(Ω ))2 (K 0 )2 v (t) |∆v ≤ + (t)| m m K0 K 02 # " # " 0 2b1 K c8 |K 0 |d(Ω ) n K 0 c7 |K 0 |d(Ω ) 0 2 + |vm (t)|2 + + |∆vm (t)| + K0 2 K 04 K 05 1

c 2 c1 c 2 0 + 0 6 f (K n−2 kvm (t)k2 )|∆vm (t)|2 |vm (t)| K0 c02 K 0 c02 c2 n−2 2 2 + φ(K kv (t)k )|∆v (t)| + g(K n−2 kvm (t)k2 )|∆vm (t)|2 m m 2K 02 K 03 1 c03 c1 c62 K 0 d(Ω ) |n − 2|c03 c3 c6 |K 0 | 3 n−2 2 + f (K kv (t)k )|∆v (t)| + h(K n−2 kvm (t)k2 )|∆vm (t)|3 m m 2K 02 2K 03 +

c02 c3 c6 K 02

0 h(K n−2 kvm (t)k2 )|∆vm (t)|3 |vm (t)|.

(2.25)

Second estimate. Set w = vm (t) in (2.13), we get: 1 e 1 e (b∆vm (t), ∆vm (t)) + 2 ( M(t, kvm (t)k2 )∇vm (t), ∇vm (t)) K4 K   ∂vm (t) 1 0 2 e + 2 (∇ y M(t, kvm (t)k )∇vm (t), vm (t)) + δ(vm (t), vm (t)) + ci , vm (t) ∂ yi K     0 (t) ∂ 2 vm (t) ∂vm + di j , vm (t) + ei , vm (t) = 0. ∂ yi ∂ y j ∂ yi

00 (e a vm (t), vm (t)) +

(2.26)

On the other hand, we have 00 (e a vm (t), vm (t)) =

Z Z 0 2 d 0 a(y, ˜ t)vm (y, t)vm (y, t)dy − a˜ (y, t) vm (y, t) dy dt Ω Ω Z 0 − a˜ 0 (y, t)vm (y, t)vm (y, t)dy

(2.27)

Ω

and we get the following inequalities: Z (a˜ 0 (y, t))v 0 (y, t)vm (y, t)dy ≤ c0 c∼0 c7 |K 0 |d(Ω )|v 0 (t)||∆vm (t)|, m m Ω

(2.28)

C.S.Q. De Caldas et al. / Nonlinear Analysis 70 (2009) 693–710 ∼

699

∼

where c0 is such that |vm (t)| ≤ c0 kvm (t)k 1 ∼ 1 c03 c0 c1 c62 2 e f (K n−2 kvm (t)k2 )|∆vm (t)|3 K 2 (∇ y M(t, kvm (t)k ) · ∇vm (t), vm (t)) ≤ K0 !   δd(Ω ) K 0 + a1 d(Ω ) 2a1 d(Ω )(K 0 )2 ci ∂vm (t) , vm (t) ≤ c2 c∼0 |∆vm (t)|2 + 0 ∂ yi K0 K 02 !   ∼ 2 2 0 2 di j ∂ vm (t) , vm (t) ≤ c0 cb0 c0 a1 d(Ω ) (K ) |∆vm (t)|2 . ∂ yi ∂ y j K 02

(2.29) (2.30)

(2.31)

Since 

 Z Z 0 (t) ∂vm ∂vm (t) 0 ∂ 0 0 , vm (t) = − (ei vm (y, t))(vm (y, t))dy − ei v (y, t)dy ∂ yi ∂ yi m Ω ∂ yi Ω  Z  K0 ∂a K 0 0 − 2a(y, ˜ t) = − −2K yi vm (y, t)(vm (y, t))dy, ∂ xi K K Ω

ei

we obtain " #  ∼  0 c a K 0 + a1 d(Ω ) K 0 0 ei ∂vm (t) , vm (t) ≤ 2c0 c1 c7 |K 0 |d(Ω ) + 0 1 |vm (t)||∆vm (t)|. ∂ yi K0

(2.32)

From (2.26)–(2.32), we have,  Z  Z 0 2 d δ 0 a(y, ˜ t)vm (y, t)vm (y, t) + |vm (y, t)|2 dy − a˜ (y, t) vm (y, t) dy dt Ω 2 Ω  Z  Z 1 1 2 e e t, kvm (t)k2 )|∇vm (y, t)|2 dy + 4 b(y, t) |∆vm (y, t)| dy + 2 M(y, K K Ω Ω  ∼ 0  K + 2a1 d(Ω ) K 0 2c a ∼ 0 1 0  |vm ≤ c0  c0 c7 |K 0 |d(Ω ) + 2c1 c7 |K 0 |d(Ω ) + (t)||∆vm (t)| K0 # " δc0 d(Ω )|K 0 | + c0 a1 d(Ω ) 2c0 a1 d(Ω )(K 0 )2 + cb0 a1 d(Ω )2 (K 0 )2 ∼ |∆vm (t)|2 + + c0 c0 K0 K 02 ∼

1

+ c03 c0 c1 c62 f (K n−2 kvm (t)k2 )|∆vm (t)|3 .

(2.33)

Let, h m (t) = h m (u m (t)) =

Z  1 1 0 e a (y, t)|vm (y, t)|2 + 4 e b(y, t) |∆vm (y, t)|2 2 Ω K  1 e t, kvm (t)k2 )|∇vm (y, t)|2 dy + 2 M(y, K  Z  δ δ 0 2 e + a (y, t)vm (y, t)vm (y, t) + |vm (y, t)| dy. 4 Ω 2

(2.34)

Since e a (y, t) ≤ 1, from the Cauchy–Schwartz inequality we obtain Z Z Z δ 1 0 2 δ2 0 e e |vm (y, t)|2 dy. a (y, t)vm (y, t)vm (y, t)dy ≤ a (y, t) vm (y, t) dy + 2 2 Ω 8 Ω Ω Hence, Z Z Z 0 2 δ 1 δ2 0 e |vm (y, t)|2 dy. [e a (y, t)vm (y, t)vm (y, t)dy] ≥ − a (y, t) vm (y, t) dy − 2 Ω 2 Ω 8 Ω

(2.35)

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From (2.34) and (2.35), we have  Z  0 2 1 1 e 1 e 2 2 2 |∆v h m (t) ≥ a(y, ˜ t) vm (y, t) + . (2.36) b(y, t) (y, t)| + M(y, t, kv (t)k )|∇v (y, t)| m m m 2K 4 2K 2 Ω 4 From (2.3), (2.4) and (2.36) we have, # Z " 2 b 1 a0 0 0 e t, kvm (t)k2 )|∇vm (t)|2 dy |4vm (y, t)|2 + v (y, t) + M(y, h m (t) ≥ 4 m 2K 4 2K 12 Ω   Z 0 2 2 2 2 e ≥ c9 vm (t) + |4vm (y, t)| + M(y, t, kvm (t)k )|∇vm (t)| dy,

(2.37)

Ω

where, c9 = min{ a40 , Also, multiplying

b0 , 1 }. 2K 14 2K 12 (2.33) by 4δ

and adding to (2.25), we have, Z Z 2 2 δ 0 0 δ d e e (t) dy + (t) − a (y, t) vm b(y, t) |∆vm (t)|2 dy (h m (t)) + δ vm dt 4 Ω 4K 4 Ω Z δ e t, kvm (t)k2 )|∇vm (t)|2 dy + M(y, 4K 2 Ω 0 2 ≤ γm (t) |∆vm (t)|2 + P(|K 0 |, d(Ω )) vm (t) + S(|K 0 |, d(Ω )) |∆vm (t)|2 ,

(2.38)

where, 1

c02 K 0

c02 c1 c62 0 γm (t) = φ(K kv (t)k ) + f (K n−2 kvm (t)k2 )|vm (t)| m K0 K 03   1 1 ∼ c03 c1 c62 0 δc03 c0 c1 c62  f (K n−2 kvm (t)k2 )|∆vm (t)| + |K |d(Ω ) + 4K 0 2K 02 +

+

n−2

2

c02 c2

c02 c3 c6

2K 0

K 02

g(K n−2 kvm (t)k2 ) + 2

|n − 2|c03 c3 c6 2K 03

0 h(K n−2 kvm (t)k2 )|∆vm (t)||vm (t)|

K 0 h(K n−2 kvm (t)k2 )|∆vm (t)|,

(2.39)

and P and S are two variable polynomials. They are P and S polynomials in the variables α and β defined by: P(α, β) = p10 α + p01 β + p11 αβ + p21 α 2 β + p12 αβ 2 + p22 α 2 β 2

(2.40)

S(α, β) = s10 α + s01 β + s11 αβ + s21 α β + s12 αβ + s22 α β

(2.41)

2

2

2 2

with, ∼

p10 =

4n + δc0 c0 a , 4K 0

p01 =

c0 , 2K 0

p11 =

c7 δc0 c7 ∼ δc0 + ( c0 +2c1 ) + (2 + a1 ), 2 8 4K 0 ∼

2c0 + δc02 c0 a1 c0 cb0 p21 = 2 , p12 = 0, p22 = , s01 = , 2 4K 0 K0 c8 δc0 c7 ∼ δc0 ∼ s11 = 4 + ( c0 +2c1 ) + (2 + a1 + δc0 c0 ), 8 4K 0 K0 ∼

s21 =

2c0 + δc02 c0 a1 2K 02

∼

,

s12 = 0,

s22 =

2b c0 + δc0 cb0 a1 c0 . 4K 02

s10

∼

δc0 c0 a1 = 5 + , 4K 0 K0 2b1

C.S.Q. De Caldas et al. / Nonlinear Analysis 70 (2009) 693–710

701

We have, Z Z 0 2 δ 0 2 0 2 2 δ 3δ 0 e δ vm (t) − a (y, t) vm (y, t) dy = [4 − e a (y, t)] vm (y, t) dy ≥ vm (y, t) . 4 Ω 4 Ω 4 From (2.38) and (2.42), we get: Z 2 δb0 δ 3δ 0 d 2 e |∆v vm (t) + (t)| + b(y, t) |∆vm (t)|2 dy (h m (t)) + m dt 4 12K 4 Ω 6K 14 Z δ e t, kvm (t)k2 )|∇vm (t)|2 dy + M(y, 4K 2 Ω 0 2 ≤ γm (t) |∆vm (t)|2 + P(|K 0 |, d(Ω )) vm (t) + S(|K 0 |, d(Ω )) |∆vm (t)|2 .

(2.42)

(2.43)

From (2.5) and (2.43), we have: d (h m (t)) + dt

! δb0 − γ (t) |∆vm (t)|2 ≤ 0. 12K 14

(2.44)

From (2.37), we obtain: 0 v (t) ≤ √1 h m (t) 12 , m c9

1 1 |∆vm (t)| ≤ √ h m (t) 2 . c9

(2.45)

kvm (t)k ≤ c0 |∆vm (t)| ,

1 c0 then kvm (t)k ≤ √ h m (t) 2 . c9

(2.46)

Since

From the definition of c6 and (2.46) we have: K n−2 kvm (t)k2 ≤

c0 c6 h m (t). c9

(2.47)

Since φ, f, g and h are increasing functions, then from (2.3), (2.39) and (2.45)–(2.47), we have: ! ! r c1 c02 K c02 c6 c02 c1 c6 c02 c6 1 γm (t) ≤ φ h m (t) + f h m (t) h m (t) 2 3 c9 K 0 c9 c9 K0   ! 1 1 ∼ c02 c6 δc03 c0 c1 c62 1 1  c03 c1 c62 c f h m (t) h m (t) 2 +√ K 1 d(Ω ) + 2 c9 4K 0 c9 2K 0 ! ! c1 c02 c2 c02 c6 |n − 2|c03 c3 c6 K c02 c6 1 + g h h m (t) + h m (t) h m (t) 2 √ c9 c9 2K 02 2K 03 c9 ! c02 c3 c6 c02 c6 h h m (t) h m (t). + c9 c9 K 02

(2.48)

We define ! c02 c6 1 F(s) = φ s s2 3 c9 K0   ! 1 1 ∼ δc03 c0 c1 c62 c02 c6 1 1  c03 c1 c42 c f +√ K 1 d(Ω ) + s s2 c9 4K 0 c9 2K 02 ! ! c1 c02 c2 c02 c3 c6 |n − 2|c03 c3 c6 K c02 c6 c02 c6 + g s + h s s + h √ 2 2 3 c9 c9 2K 0 c9 K 0 2K 0 c9 c1 c02 K

! r c2 c1 c6 c02 c6 s + 0 f c9 K 0 c9

! c02 c6 1 s s2. c9

(2.49)

702

C.S.Q. De Caldas et al. / Nonlinear Analysis 70 (2009) 693–710

From (2.48) and (2.49), we obtain: γm (t) ≤ F(h m (t))

for every t ≥ 0.

(2.50)

On the other hand, from (2.3), (2.4), (2.6) and (2.34) we have:   ∼ ∼ ! 2 b1 δc0 c0 +δ 2 c0 c0 1 1 δ 0 |∆vm (t)|2 + vm (t) + + + φ(kvm (t)k2 )kvm (t)k2 h m (t) ≤ 4 2 8 8 2K 0 2K 02 h i 2 0 ≤ c10 vm (t) + |∆vm (t)|2 + φ(kvm (t)k2 )kvm (t)k2 , where c10 = max{ 21 + 8δ , We define

b1 2K 04

∼

+

∼

δc0 c0 +δ 2 c0 c0 8

,

(2.51)

1 }. 2K 02

h 0 = c10 [|v1 |2 + |∆v0 |2 + φ(kv0 k2 )kv0 k2 ].

(2.52)

From (2.51), we have: h m (0) ≤ c10 [|v1m |2 + |∆v0m |2 + φ(kv0m k2 )kv0m k2 ].

(2.53)

From (2.53) and the fact that F(s) is increasing, we obtain F(h m (0)) ≤ F[c10 [|v1m |2 + |∆v0m |2 + φ(kv0m k2 )kv0m k2 ]].

(2.54)

Taking limit in (2.54), as m −→ ∞, we have: δb0 . 13K 14

lim F(h m (0)) ≤ F(h 0 ) <

m−→∞

Then there exists m 0 ∈ Z+ , such that F(h m (0)) <

δb0 13K 14

for every m ≥ m 0 .

(2.55)

We assert that γm (t) ≤

δb0 12K 14

for every t ≥ 0 and m ≥ m 0 .

In fact, suppose that γm (t) >

δb0 12K 14

(2.56)

for some t > 0 and for every m ≥ m 0 .

From (2.50) and (2.55), we have γm (0) < F(h m (0)) <

δb0 . 13K 14

(2.57)

Since γm (t) is continuous, then there exists t ∗ > 0, t ∗ = min{t > 0; γm (t) = γm (t) <

δb0 12K 14

for every 0 ≤ t ≤ t ∗

and

γm (t ∗ ) =

δb0 . 12K 14

δb0 }, 12K 14

such that (2.58)

Integrating (2.44) from 0 to t ∗ , we have h m (t ∗ ) ≤ h m (0).

(2.59)

From (2.50), (2.57) and (2.59), we get γm (t ∗ ) ≤ F(h m (0)) <

δb0 δb0 < 13K 14 12K 14

for every m ≥ m 0

703

C.S.Q. De Caldas et al. / Nonlinear Analysis 70 (2009) 693–710

which is a contradiction to (2.58). Then 0 < γm (t) <

δb0 12K 14

for every t ≥ 0.

(2.60)

From (2.44), (2.52) and (2.56), it follows that h m (t) ≤ h m (0) for every t ≥ 0. Then, 0 v (t) 2 + |∆vm (t)|2 ≤ c for every t ≥ 0. m

(2.61)

From (2.61), we can obtain a subsequence, still represented by (vm )m∈N , such that vm * v

weak star in L ∞ (0, T ; H02 (Ω )), as m → ∞,

(2.62)

0 vm * v0

weak star in L ∞ (0, T ; L 2 (Ω )), as m → ∞.

(2.63)

From (2.61) and the Aubin–Lions Theorem, [9], we obtain a subsequence, still represented by (vεm )m∈N , such that vm → v

strongly in L 2 (0, T ; H01 (Ω )), as m → ∞.

From (2.13) and using (2.61)–(2.64), we can take the limit as m → ∞ and Theorem 2.2 is proved.

(2.64) 

3. Exponential decay b of system (2.1), satisfies Theorem 3.1. With the same hypothesis of Theorem 2.1, we have that the energy E(t) c b ≤c E(t) A0 e− A1 t for every t ≥ 0, where, Z b =1 E(t) [a(x)|u 0 (x, t)|2 + b(x)|∆u(x, t)|2 + M(x, t, ku(t)k2 )|∇u(x, t)|2 ]dx 2 Ω

and c A0 , c A1 are positive constants. b of system (2.8), satisfies Theorem 3.2. With the same hypothesis of Theorem 2.2, we have that the energy E(t) E(t) ≤ A0 e−A1 t for every t ≥ 0 where, Z n o 2 1 e t, kv(t)k2 )|∇v(y, t)|2 dy, e E(t) = a (y, t) v 0 (y, t) + e b(y, t)|∆v(y, t)|2 + M(y, 2 Ω where A0 and A1 be are positive constants given in (3.7). Proof of the Theorem 3.2. From (2.5) and (2.43), we have Z 2 d δ 0 δ e (h m (t)) + vm (t) + b(y, t) |∆vm (y, t)|2 dy dt 4 12K 14 Ω ! Z δ δb 0 2 2 e t, kvm (t)k )|∇vm (y, t)| dy + + 4 M(y, − γm (t) |∆vm (t)|2 ≤ 0. 4K 1 Ω 12K 14 From (2.60) and (3.1), we have Z n o 0 d 2 2 v (y, t) 2 + |∆vm (y, t)|2 + M(y, e (h m (t)) + c11 t, kv (t)k )|∇v (y, t)| dy ≤ 0, m m m dt Ω where c11 = min{ 4δ ,

δb0 , δ }. 12K 14 4K 12

On the other hand, from (2.34), we obtain ) Z ( 0 2 1 1 b 1 2 2 2 v (y, t) + e t, kvm (t)k )|∇vm (y, t)| dy |∆vm (y, t)| + 2 M(y, h m (t) ≤ m 2 Ω K 04 K0

(3.1)

(3.2)

704

C.S.Q. De Caldas et al. / Nonlinear Analysis 70 (2009) 693–710

 Z Z 0 2 δ δ δ 2 2 |vm (y, t)| dy + |vm (y, t)| dy + vm (y, t) dy + 8 Ω 8 Ω 2 Ω Z n o 0 2 2 v (y, t) 2 + |∆vm (y, t)|2 + M(y, e t, kv (t)k )|∇v (y, t)| dy, ≤ c12 m m m Z 

(3.3)

Ω

where c12 =

1 2

max{1 + 4δ ,

b1 K 04

∼

+ 54 δc0 c0 ,

1 }. K 02

From (3.2) and (3.3), we get d (h m (t)) + c13 h m (t) ≤ 0 dt

where c13 =

c11 . c12

(3.4)

From (3.4), we have h m (t) ≤ c14 e−c13 t

for every t ≥ 0.

(3.5)

From (2.36), we obtain h m (t) ≥ c15 E m (t), where c15 = min{ 21 , E m (t) =

(3.6)

1 , 1 } K 14 K 12

and

Z n o 0 2 1 e t, kvm (t)k2 )|∇vm (y, t)|2 dy. e a (y, t) vm (y, t) + e b(y, t) |∆vm (y, t)|2 + M(y, 2 Ω

From (3.5) and (3.6), we have: E m (t) ≤ A0 e−A1 t where A0 =

c14 c15

for every t ≥ 0,

and A1 = c13.

(3.7)



Proof of the Theorem 3.1. The proof is a consequence of Theorem 3.2, (2.5) and the Change of Variable Theorem.  4. Uniqueness of solutions In this section we use the following hypotheses: K 0 , K 00 ∈ L 1 (0, ∞) C5 d(Ω ) K 1 + 4 K 05 b0

|K 0 | L 1 (0,+∞) ≤

1 , 16 (K (0))4

(4.1)

where C5 is given in (4.18) b1 |K 0 (t)| + |K 00 (t)| ≤ K

(4.2)

|∆M| ≤ C4 β(λ),

(4.3)

where β ∈ C 0 [0, ∞[ and C4 is given in (4.17). Note that here we are using capital letters instead of lower case letters to differentiate from the given constants in the existence and exponential decay of solutions. Theorem 4.1 (Uniqueness of Solutions). Suppose that the hypotheses of Theorem 2.1 hold. Additionally assume that (4.1)–(4.3) hold. Then the solution to the initial-boundary value problem (2.1) is unique in [0, T ] for all T > 0 Proof. It is enough to show the uniqueness of solutions to (2.8), since the problems (2.1) and (2.8) are equivalent. Suppose v1 and v2 are two solutions to (2.8). Thus, w = v1 − v2 satisfies

705

C.S.Q. De Caldas et al. / Nonlinear Analysis 70 (2009) 693–710

1 a (y, t)w00 (y, t) + 4 ∆(e b(y, t)∆w(y, t)) + δw 0 (y, t) e K 2 0 +c (y, t) ∂w(y, t) + d (y, t) ∂ w(y, t) + e (y, t) ∂w (y, t) i ij i ∂ yi ∂ yi ∂ y j ∂ yi 1 e = − 1 M(y, 2 e t, kv1 (t)k ) ∆v1 (y, t) + 2 M(y, t, kv2 k2 )∆v2 K2 K ∂w w(y, t) = (y, t) = 0 ∂ν w(y, 0) = 0, w0 (y, 0) = 0

in Q (4.4) in

X

in Ω .

Eq. (4.4) is given in the sense of L 2 (0, T ; H −2 (Ω )) and w 0 ∈ L 2 (0, T ; L 2 (Ω )); then the duality (w00 (t), w0 (t)) H −2 (Ω )×L 2 (Ω ) does not make sense. Thus, the energy method cannot be used. To overcome this difficulty, the uniqueness will be obtained by the Ladyzhenskaya method [6]. In fact, for each s ∈ (0, T ) let ψ : (0, T ) → H02 (Ω ) be a function defined by Z s − w(y, r )dr if 0 < t < s, ψ(y, t) = t 0 if s ≤ t < T, where w is a solution of (4.4). Note that ψ 0 (y, t) = w(y, t)

and ψ(y, s) = 0.

(4.5)

Besides, if ψ1 (y, t) =

Z

t

w(y, r )dr

0

then ψ(y, t) = ψ1 (y, t) − ψ1 (y, s)

and

ψ(y, 0) = −ψ1 (y, s).

(4.6)

As w ∈ L ∞ (0, T ; H02 (Ω )) then ψ ∈ L ∞ (0, T ; H02 (Ω )). Considering the duality of Eq. (4.4), satisfied in L 2 (0, T ; H −2 (Ω )) with ψ ∈ L 2 (0, T ; H02 (Ω )), we have       1 e ∂w ∂ 2w 0 he a w 00 , ψi + b ∆w, ∆ψ + hδw , ψi + c , ψ + d , ψ i ij ∂ yi ∂ yi ∂ y j K4     0 ∂w 1 e 1 e 2 2 + ei M(kv k )∆v , ψ . (4.7) , ψ = − 2 M(kv 1 k ) ∆v1 + 2 2 ∂ yi K K2 Now, we analyze each term of (4.7). In these analyses we will use several times integration by parts, the properties (4.5) and (4.6), the null-initial condition, the hypotheses (4.1)–(4.3), and the usual inequalities. We have Z Z Z Z sZ 1 1 s e e e he a (y, t)w00 , ψi = − a 0 |w|2 dydt − a 0 w 0 ψdydt (4.8) a |w(s)|2 dy + 2 Ω 2 0 Ω 0 Ω Z sZ 0 hδw , ψi = −δ |w|2 dydt (4.9) 0

Ω

 Z Z Z 1 e 1 1 s 2 e b(y, t)∆w, ∆ψ = − b(y, 0)|∆ψ (s)| dy − 1 2 0 Ω K4 2K 4 (0) Ω   Z sZ ∂w ci w(∇ψ.c + ψdiv c)dydt, ,ψ = − ∂ yi 0 Ω



where c = c(y, t) = (ci (y, t))   Z sZ ∂ 2 (di j ψ) ∂ 2w di j ,ψ = w dydt ∂ yi ∂ y j ∂ yi ∂ y j 0 Ω

e b K4

!0 |∆ψ|2 dydt

(4.10) (4.11)

(4.12)

706

C.S.Q. De Caldas et al. / Nonlinear Analysis 70 (2009) 693–710

 Z sZ ∂w0 ∂w ei ,ψ =− (ei ψ)0 dydt ∂ yi ∂ y i 0 Ω   1 2 − e 1 k ) ∆v1 + 1 M(kv e 2 k2 )∆v2 , ψ M(kv K2 2 K Z s Z Z s Z 1 e 1 2 2 e e ≤ w∆( M ψ)dydt + ( M(y, t, kv2 k ) − M(y, t, kv1 k )) ∆v2 ψdydt . 2 2 0 Ω K 0 Ω K 

From (4.4) and (4.7)–(4.14) we obtain Z Z Z sZ 1 1 2 e e a |w(s)|2 dy + b(y, 0) |∆ψ (s)| dy + δ |w|2 dy dt 1 2 Ω 2K 4 (0) Ω 0 Ω !0 Z Z Z sZ Z Z e 1 s 1 s b 0 2 0 0 e e ≤ a |w| dydt − a w ψdydt − |∆ψ|2 dydt 2 0 Ω 2 0 Ω K4 0 Ω Z sZ Z sZ Z sZ ∂ 2 (di j ψ) ∂w − w(∇ψ.c + ψdiv c)dydt + w dydt − (ei ψ)0 dydt ∂ yi ∂ y j 0 Ω 0 Ω 0 Ω ∂ yi Z s Z Z s Z 1 1 e 2 2 . e e + + w∆( M ψ)dydt ( M(y, t, kv k ) − M(y, t, kv k ))∆v ψdydt 2 1 2 2 2 0 Ω K 0 Ω K

(4.13)

(4.14)

(4.15)

Now we use the hypotheses (4.1)–(4.3), the equalities (4.5) and (4.6) and also the usual inequalities to estimate the terms on the right-hand side of (4.15). We have the following inequalities: Z sZ b1 C1 d(Ω ) Z s Z 1 K 0 2 e e a |w| dydt ≤ a |w|2 dydt, 2 2a0 0 Ω 0 Ω where C1 = sup Z

s

Z

|∇a(x)|

e a w ψdydt ≤ 0

Ω

0

b x∈Ω

(4.16)

0

b1 C1 d(Ω ) + ( K b1 )2 (d(Ω ))2 C2 + 8 (K (0))4 C e2 C4 b−1 2K 0 0 2 a0

!Z

s 0

Z Ω

e a |w|2 dydt

Z

+

1 e b(y, 0)|∆ψ1 (s)|2 dydt, 16(K (0))4 Ω

e0 is such that |v| L 2 (Ω ) ≤ C e0 |∆v| L 2 (Ω ) , C4 = 2 (C1 d(Ω ) K b1 (d(Ω ))2 C2 )2 where C C2 = sup

∂ 2 a(x)

b ,1≤i, j≤n | ∂ x j ∂ xi x∈Ω

1Z sZ − 2 0 Ω

where C5 = sup

!0

Z

R∞ 0

(|K 0 | + |K 00 |)dt, and

|

! Z Z C5 d(Ω ) K 0 K 1 + 4K 0 1 s |∆ψ| dydt ≤ 2 (|∆(ψ1 (t))|2 + |∆(ψ1 (s))|2 )dydt 5 2 0 Ω K0 Z Z b b C5 d(Ω ) K 1 K 1 + 4 K 1 s e ≤ b(y, 0)|∆ψ1 (s)|2 dydt K 05 b0 Ω 0 ! Z C5 d(Ω ) K 1 + 4 0 e + |K | b(y, 0)|∆ψ1 (s)|2 dy, (4.18) 1 L (0,∞) K 05 b0 Ω 2

|∇ b(x)|

Z sZ 1 1 e t, kv1 k2 ) ψ)dydt ≤ C6 w ∆( M(y, |w||∆ψ|dydt 2 2 0 Ω K 0 Ω K Z sZ 1 ≤ C6 |w|(|∆ψ1 (t)| + |∆ψ1 (s)|)dydt 2 0 Ω K

Z

s

b x∈Ω

e b K4

(4.17)

707

C.S.Q. De Caldas et al. / Nonlinear Analysis 70 (2009) 693–710

4

≤

+

C62 (K (0))4 T

K 04 b0 a0 Z sZ C6 K 02 b0 0 Ω

C6 + 2 K 0 a0

!Z

s 0

Z Ω

e a |w|2 dydt Z 1 e b(y, 0)|∆ψ1 (s)|2 dy, 16 (K (0))4 Ω

e b(y, 0)|∆ψ1 (t)|2 dydt +

(4.19)

where C6 = (C4 K 12 + 2C1 K 1 + 1) max (|β(λ)| + 0≤λ≤θ1

√ e1 ), λ f (λ) + φ(λ))(2C

where θ1 = θ1 (v1 ) = (K 0n−2 + K 1n−2 )kv1 k2 ∞

L (0,T,H01 (Ω ))

s

e1 is such that |∇v| L 2 (Ω ) ≤ C e1 |∆v| L 2 (Ω ) and C

1 e 2 2 e ( M(y, t, kv2 k ) − M(y, t, kv1 k ))∆v2 ψdydt 2 K 0 ΩZ Z s e 1 ∂M 2 2 2 2 2 = (kv k + θ (kv k − kv k )) (kv k − kv k ) ∆v ψdydt 1 1 2 2 1 2 0 Ω K 2 ∂λ Z sZ 1 n−2 2 2 2 ≤ c K h (kv k + θ (kv k − kv k )) |w|(|∆v | + |∆v |)∆v ψ 1 1 2 2 1 2 dydt K2 3 0 Ω Z s Z s ≤ C7 |w| |ψ|dt ≤ C7 |w| (|ψ1 (t)| + |ψ1 (s)|)dt 0 0 ! e2 C 2 T Z s Z e1 Z s Z 4(K (0))4 C C7 C C7 2 1 7 e e a |w| dydt + ≤ + b |∆ψ1 |2 dydt 2a0 b0 a0 2b0 0 Ω 0 Ω Z 1 e b(y, 0) |∆ψ1 |2 dy, + 16(K (0))4 Ω

Z

Z

(4.20)

where C7 = c3 (K 0n−4 + K 1n−4 ) max |h(λ)|(kv1 k L ∞ (0,T,H 2 (Ω )) + kv2 k L ∞ (0,T,H 2 (Ω )) ) kv2 k L ∞ (0,T,H 2 (Ω )) 0

0≤λ≤θ2

and θ2 = θ2 (v1 , v2 ) = (kv1 k2 ∞

L (0,T,H01 (Ω ))

+ kv2 k2 ∞

L (0,T,H01 (Ω ))

0

0

).

We observe that kv2 k2 − kv1 k2 =

Z ZΩ

= Ω

Z

(∇v2 − ∇v1 ) · (∇v2 + ∇v1 )dy Z ∇(v2 − v1 ) · ∇(v2 + v1 )dy = (v2 − v1 )∆(v2 + v1 )dy.

(|∇v2 |2 − |∇v1 |2 )dy =

Ω

Ω

Hence |kv2 k2 − kv1 k2 | ≤ |v2 − v1 | L 2 (Ω ) |∆v2 + ∆v1 | L 2 (Ω ) Z sZ Z s Z  0  Z Z ∂ei 1 s ∂w2 ∂w 0 0 0 ∂ψ − (ei ψ + ei ψ )dydt = ψ + ei dydt − dydt w ∂ yi ∂ yi 2 0 Ω ∂ yi 0 Ω ∂ yi 0 Ω Z sZ ≤ C8 (|w| |ψ| + |w||∇ψ| + |w|2 )dydt 0 Ω Z sZ Z Z C8 s e1 e ≤ 2 C8 C |w| (|∆ψ1 (t)| + |∆ψ1 (s)|)dydt + a0 |w|2 dydt C 0 0 Ω 0 Ω ! Z sZ e2 (K (0))4 T Z s Z e1 e 16C82 C C8 C C C 8 1 1 e e ≤ + a |w|2 dydt + b(y, 0) (|∆ψ1 (t)|2 dydt) a0 b0 a0 b0 0 Ω 0 Ω Z 1 e + b(y, 0) (|∆ψ1 (s)|2 dy), 16 (K (0))4 Ω

(4.21)

708

C.S.Q. De Caldas et al. / Nonlinear Analysis 70 (2009) 693–710

where C8 = max

∂e0

b ×[0,T ], 1≤i≤n (y,t)∈Ω

{| ∂ yii (y, t)| + |ei0 (y, t)| +

1 ∂ei 2 | ∂ yi

(y, t)|}

Z Z Z Z 2 ! s ∂ 2d s ∂di j ∂ψ ∂di j ∂ψ ∂ 2 (di j ψ) ij + + |di j | ∂ ψ dydt w |w| dydt = |ψ|+ 0 Ω ∂ yi ∂ y j ∂ yi ∂ y j ∂ yi ∂ y j ∂ y j ∂ yi ∂ yi ∂ y j 0 Ω Z s Z s e1 e1 ≤ C9 C |w||∆ψ|dt ≤ C9 C |w|(|∆ψ1 (t)|dt + |∆ψ1 (s)|)dt 0 !0 Z Z s e2 (K (0))2 T e1 4C92 C C9 C 1 e ≤ + a |w|2 dydt 2a0 b0 0 Ω Z sZ e C9 C1 e + b(y, 0) (|∆ψ1 (t)|2 dydt) 2b0 0 Ω Z 1 e + b(y, 0)(|∆ψ1 (s)|2 dy), (4.22) 16 (K (0))4 Ω where C9 = max

∂d

∂d

{| ∂ yi ∂iyj j (y, t)| + | ∂ yiij (y, t)| + | ∂ yijj (y, t)| + |di j (y, t)|}

Z sZ w(∇ψ.c + ψdiv c)dydt ≤ |w|(|∇ψ| |c| + |ψ| |div c|)dydt 0 Ω 0 Ω Z sZ Z sZ e1 ≤ C10 |w|(|∇ψ| + |ψ|)dydt ≤ C10 C |w||∆ψ|dydt 0 Ω 0 Ω Z sZ e1 ≤ C10 C |w|(|∆ψ1 (t)| + |∆ψ1 (s)|)dydt 0 Ω ! 2 C e2 (K (0))4 T Z s Z e1 Z s Z e1 4C10 C10 C C10 C 1 2 e e + a |w| dydt + ≤ b(y, 0) (|∆ψ1 (t)|2 dydt) 2a0 b0 a 0 2b0 0 Ω 0 Ω Z 1 e + b(y, 0)(|∆ψ1 (s)|2 dy), 16 (K (0))4 Ω

Z −

s

∂2d

b ×[0,T ], 1≤i, j≤n (y,t)∈Ω

Z

n where C10 = max b ×[0,T ] (|c(y, t)|R + |div c(y, t)|R ). (y,t)∈Ω Taking into account (4.15)–(4.23) in (4.15) and using the hypotheses (4.1)–(4.3) we obtain Z Z sZ Z 1 1 2 2 e e b(y, 0) |∆ψ1 (s)| dy + δ |w|2 dy dt a |w(s)| dy + 2 Ω 8K 4 (0) Ω 0 Ω Z sZ Z sZ e e ≤ C11 a |w(t)|2 dydt + C12 b(y, 0) |∆ψ1 (t)|2 dydt,

0

Ω

0

(4.24)

Ω

where C11 =

1 b1 )2 (d(Ω ))2 C2 + 8(K (0))4 C e2 C4 b−1 ) b1 C1 d(Ω ) + ( K (3 K 0 0 2a0 e2 (K (0))4 T e2 C 2 T e1 16C82 C C 2 (K (0))4 T 4(K (0))4 C C6 C7 C8 C 1 1 7 +4 6 4 + 2 + + + + 2a0 b0 a0 a0 b0 a 0 K 0 b0 a 0 K 0 a0 2 C e2 (K (0))2 T e2 (K (0))4 T e1 e1 4C92 C 4C10 C9 C C10 C 1 1 + + + + 2a0 b0 2a0 b0 a 0

and C12 =

b1 K 1 + 4 K b1 C5 d(Ω ) K K 05 b0

+

e1 e1 e1 e1 C6 C7 C C8 C C9 C C10 C + + + + . 2 2 b0 b0 2 b0 2 b0 (K (0)) b0

Then from (4.24) and Gronwall’s inequality we obtain the uniqueness in [0, T ] for all T > 0.

(4.23)



C.S.Q. De Caldas et al. / Nonlinear Analysis 70 (2009) 693–710

709

Acknowledgment We want to thank the referee for a careful reading and helpful suggestions which led to an improvement of our original manuscript. Appendix Example of a function M satisfying the property (2.6). Let c(x, t) be a C 1 function in the variables x and t, such that ∂c ≤ c2 and |∇c(x, t)| ≤ c3 , for all x ∈ Ω , t ≥ 0. 0 ≤ c(x, t) ≤ c1 , ∂t Let M(λ) be a C 2 function in [0, +∞], such that √ 0 M(0) = 0 and 0 ≤ M(λ) ≤ c4 λ |M (λ)|

for all λ ≥ 0.

(A.1)

(A.2)

Let us define M(x, t, λ) = c(x, t) M(λ) for all x ∈ Ω , t ≥ 0 and λ ≥ 0. From (A.1) and (A.2), it follows that M is a C 1 function in the variables x, t, and λ.

(A.3)

Setting Φ(λ) = c1 M(λ), we get 0 ≤ M(x, t, λ) ≤ φ(λ).

(A.4)

From (A.1), we have ∂ M ∂c = M(λ) ≤ c2 |M(λ)|. ∂t ∂t Rλ Since M(0) = 0, we have M(λ) = 0 M 0 (λ)ds. Then Z λ |M(λ)| ≤ g(λ), where g(λ) = |M 0 (λ)|ds.

(A.5)

(A.6)

0

From (A.5) and (A.6), we obtain ∂M 1 lim g(λ) = 0. ∂t ≤ c2 g(λ), where g is an increasing C function, and λ→0 From (A.1), we have ∂M 0 0 ∂λ = |c(x, t) M (λ)| ≤ c1 |M (λ)|   Z λ ≤ c1 |M 0 (0)| + |M 00 (λ)|ds = c1 h(λ),

(A.7)

(A.8)

0

Rλ where h(λ) = |M 0 (0)| + 0 |M 00 (λ)|ds is an increasing C 1 function in [0, +∞]. On the other hand from (A.1), we get |∇x M| = |∇c(x, t)||M (λ)| √ ≤ c3 c4 λ |M 0 (λ)|  Z √ 0 ≤ c3 c4 λ |M (0)| + √ = c3 c4 λ f (λ)

λ

|M (λ)|ds 00



0

where f (λ) = h(λ).

From (A.3), (A.4) and (A.7)–(A.9) we obtain that M satisfies (2.6).

(A.9)

710

C.S.Q. De Caldas et al. / Nonlinear Analysis 70 (2009) 693–710

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