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Boundary stabilization for a coupled system
Nonlinear Analysis 74 (2011) 6988–7004
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Boundary stabilization for a coupled system Aldo T. Lourêdo a , Alexandro M. Oliveira b,∗ , Marcondes R. Clark c a
Universidade Estadual da Paraíba - DM, CEP 58109095, Campina Grande, PB, Brazil
b
Universidade Federal do Piauí - DM, CEP 64.202.020, Parnaíba, PI, Brazil Universidade Federal do Piauí - DM, CEP 21941-901, Teresina, PI, Brazil
c
article
abstract
info
Article history: Received 29 April 2011 Accepted 9 July 2011 Communicated by Enzo Mitidieri
We investigate in this work the global existence of weak solutions for a nonlinear coupled system with mixed type boundary conditions. More precisely, Dirichlet and feedback boundary conditions. Further, we also prove the exponential decay of the energy associated with these solutions. © 2011 Elsevier Ltd. All rights reserved.
MSC: 35605 35N10 35S15 Keywords: Coupled system Nonlinear boundary conditions Exponential stability
1. Introduction Let Ω be an open bounded set of Rn whose boundary Γ of class C 2 is constituted by two closed disjoint parts Γ0 and Γ1 . By ν(x) is represented the outer unit normal vector at x ∈ Γ1 . Consider a real function h(x, s) defined in Γ1 ×]0, ∞[ and a real number σ > 0. Under these conditions we have the following mixed problem for a nonlinear thermoelastic system: u′′ (x, t ) − △u(x, t ) +
θ ′ (x, t ) − △θ (x, t ) +
n − ∂θ (x, t ) + |u(x, t )|σ u(x, t ) = 0 in Ω ×]0, ∞[, ∂ xi i =1 n − ∂ u′ i =1
u(x, t ) = 0,
∂ xi
(x, t ) = 0 in Ω ×]0, ∞[,
θ (x, t ) = 0 on Γ0 ×]0, ∞[,
(P)
∂u (x, t ) + h(x, u′ (x, t )) = 0 on Γ1 ×]0, ∞[, ∂ν ∂θ (x, t ) + h(x, θ (x, t )) = 0 on Γ1 ×]0, ∞[, ∂ν u(x, 0) = u0 (x), u′ (x, 0) = u1 (x), θ (x, 0) = θ 0 (x) in Ω .
∗
Corresponding author. Tel.: +55 86 33235406. E-mail addresses: [email protected] (A.T. Lourêdo), [email protected], [email protected] (A.M. Oliveira), [email protected] (M.R. Clark). 0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.07.019
A.T. Lourêdo et al. / Nonlinear Analysis 74 (2011) 6988–7004
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The thermoelastic systems have been studied by a lot of mathematicians. Among then we mention Hansen [1], Henry et al. [2], Kin [3] and Marinho et al. [4,5]. However the question of boundary stabilization with nonlinear conditions (P)4 and (P)5 has still not been discussed. This motivated us to develop this work. The existence of solutions of the wave equation with boundary damping can be obtained by applying the Galerkin method with a special basis. In this direction, we mention the pioneer work of Medeiros and Milla Miranda [6], and also, Araruna and Maciel [7] and Araújo et al. [8] for the linear boundary damping case, and Louredo and Milla Miranda [9] for the nonlinear boundary damping case. The Kirchhoff equation with linear boundary damping can be seen in [10]. In this paper we study the existence of solutions of Problem (P) by applying the Galerkin method with a special basis and using the Strauss approximations of a real continuous function. We study first the case h(x, s) and then the case α(x)h(s). In the second case the initial data are more general than in the first one. The exponential decay of the energy associated with (P) is obtained for a special α(x)h(s) and using a Lyapunov functional (see [11]). The papers [11,9,6] were our source of inspiration. Our work improves the results obtained by Marinho et al. [4,5]. 2. Notation and main results In this section we fix the notation and state the main results. We represent by H m (Ω ) the Sobolev space of order m and by L2 (Ω ) the class of square Lebesgue integrable functions. The scalar product and norm of L2 (Ω ) are denoted by (u, v) and |u|, respectively. By V we represent the Hilbert space V = {v ∈ H 1 (Ω ) : γ0 v = 0
on Γ0 },
where γ0 denotes the trace operator of order zero on Γ . V is equipped with the scalar product
((u, v)) =
∫ Ω
∇ u(x)∇v(x)dx, 1
and norm ‖u‖ = ((u, u)) 2 . All scalar functions considered in the paper will be real valued. We consider the operator A = −∆ defined by the triplet {V , L2 (Ω ), ((u, v))}. Then the domain D(−∆) of −∆ is given by D(−∆) =
u ∈ V ∩ H 2 (Ω );
∂u = 0 on Γ1 . ∂ν
We note that D(−∆) is dense in V .
(1)
See [12]. In order to state the first result we introduce the following hypotheses on the function h and the real number σ :
h ∈ C 0 (R; L∞ (Γ 1)) such that h(x, 0) = 0, a.e. x in Γ1 ; (h(x, s) − h(x, r ))(s − r ) ≥ d0 (s − r )2 , a.e. x in Γ1 , ∀s, r ∈ R(d > 0 constant);
(H1)
2d20
(H2)
0
2
≥n ; 1 2 <σ ≤ if n ≥ 3, n n−2 1 if n = 1, 2. σ >
(H3)
n
We have: Theorem 1. Assume hypotheses (H1)–(H3). Consider u0 ∈ D(−△), u0 ∈ H01 (Ω ) and
θ 0 ∈ H01 (Ω ) ∩ D(−∆).
(2)
Then there exist a pair of functions {u, θ} in the class
u ∈ L∞ (0, ∞; V ), u′ ∈ L∞ (0, ∞; L2 (Ω )) ∩ L∞ loc (0, ∞; V ) ′′ u ∈ L∞ (0, ∞; L2 (Ω )) loc θ ∈ L∞ (0, ∞; L2 (Ω )) ∩ L2 (0, ∞; V ) ′ θ ∈ L∞ (0, ∞; L2 (Ω )) ∩ L2 (0, ∞; V ) loc loc
(3)
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satisfying
n − ′′ ∂θ u − ∆u + ∂ xi i=1 n − ′ ∂ u′ θ − ∆θ + ∂ xi i=1 ∂u ′ ∂ν + h(·, u ) = 0 ∂θ + h(·, θ ) = 0 ∂ν
2 + |u|σ u = 0 in L∞ loc (0, ∞; L (Ω )),
(4) 2 = 0 in L∞ loc (0, ∞; L (Ω )),
in L1loc (0, ∞; L1 (Γ1 )),
(5)
in L1loc (0, ∞; L1 (Γ1 )),
and u(0) = u0 ,
u′ (0) = u1 ,
θ (0) = θ 0 .
(6)
Remark 1. We note that the initial data u0 , u1 and θ 0 given by (2) satisfy
∂ u0 ∂θ 0 + h(·, u1 ) = 0 on Γ1 and + h(·, θ 0 ) = 0 on Γ1 . ∂ν ∂ν We introduce a supplementary hypothesis on h:
|h(x, s)| ≤ c |s|,
∀s ∈ R, a.e. x in Γ1 .
(H4)
Corollary 1. Under hypotheses (H1)–(H4) the solution {u, θ} given by Theorem 1 satisfies 3 2 u ∈ L∞ loc (0, ∞; V ∩ H (Ω )),
∂u ′ ∂ν + h(·, u ) = 0 ∂θ + h(·, u) = 0 ∂ν
3
θ ∈ L2loc (0, ∞; V ∩ H 2 (Ω ))
2 in L∞ loc (0, ∞; L (Γ1 ))
in L2 (0, ∞; L2 (Γ1 ))
(7)
(8)
and this solution is unique in class (3) and (7). In the next result, we consider initial data a little more general than in Theorem 1; in compensation, the function h will depend on a single variable. More precisely, we introduce the following hypotheses on the functions α , β and h:
α ∈ W 1,∞ (Γ1 ), α(x) ≥ α0 > 0,
∀x ∈ Γ1 , (α0 constant)
1,∞
(H5)
β ∈ W (Γ1 ), β(x) ≥ β0 > 0, ∀x ∈ Γ1 , (β0 constant) h ∈ C 0 (R) such that h(0) = 0, (h(s) − h(r ))(s − r ) ≥ d0 (s − r )2 , ∀s, r ∈ R (d0 > 0 constant)
(H6)
2α0 β0 d20 ≥ n2 .
(H8)
(H7)
We have: Theorem 2. Assume hypotheses (H3), (H5)–(H8). Consider u0 ∈ V ∩ H 2 (Ω ), u1 ∈ V ,
θ 0 ∈ H01 (Ω ) ∩ D(−∆)
(9)
verifying
∂ u0 ∂θ 0 + α h(u1 ) = 0 on Γ1 and + β h(θ 0 ) = 0 on Γ1 . ∂ν ∂ν Then there exist a pair of functions {u, θ} in the class u ∈ L∞ (0, ∞; V ), u′ ∈ L∞ (0, ∞; L2 (Ω )) ∩ L∞ loc (0, ∞; V ) ′′ loc∞ u ∈ L (0, ∞; L2 (Ω )) loc θ ∈ L∞ (0, ∞; L2 (Ω )) ∩ L2 (0, ∞; V ) ′ θ ∈ L∞ (0, ∞; L2 (Ω )) ∩ L2 (0, ∞; V ) loc loc
(10)
(11)
A.T. Lourêdo et al. / Nonlinear Analysis 74 (2011) 6988–7004
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satisfying
n − ′′ ∂θ 2 u − △ u + + |u|σ u = 0 in L∞ loc (0, ∞; L (Ω )), ∂ xi i=1 n − ′ ∂ u′ 2 θ − △θ + = 0 in L∞ loc (0, ∞; L (Ω )), ∂ x i i=1 ∂u ′ 1 1 ∂ν + α h(u ) = 0 in Lloc (0, ∞; L (Γ1 )) ∂θ + β h(θ ) = 0 in L1loc (0, ∞; L1 (Γ1 )) ∂ν
(12)
(13)
and u(0) = u0 ,
u′ (0) = u1 ,
θ (0) = θ 0 .
(14)
In order to state the result on the asymptotic behavior of the energy associated with system (12), we introduce some hypotheses. Assume that there exists x0 ∈ Rn such that
Γ0 = {x ∈ Γ ; m(x) · ν(x) ≤ 0} and Γ1 = {x ∈ Γ ; m(x) · ν(x) > 0}
(H9)
where m(x) = x − x0 , x ∈ Rn , and ξ · η denotes the scalar product of Rn of ξ , η ∈ Rn . We use the notation R(x0 ) = max ‖m(x)‖Rn ,
τ0 = max[m(x) · ν(x)] > 0.
(15)
x∈Γ1
x∈Ω
Consider the following supplementary hypotheses on h and σ :
|h(s)| ≤ k∗ |s|,
∀s ∈ R(k∗ > 0 constant)
(H10)
2 2 <σ ≤ if n ≥ 3; n − 1 n−2 σ > 2 if n = 1, 2.
(H11)
Introduce the following notation:
|u|2 ≤ k1 ‖u‖2 ,
‖u‖L2 (Γ1 ) ≤ k2 ‖u‖2 ,
∀u ∈ V ;
∀u ∈ V .
(16)
Also, d∗1 = β0 d0 −
n2 2α0 d0
≥ 0 (see hypotheses (H8))
(17)
C ∗ = 4R(x0 ) + (n − 1)(1 + 2k1 ) k=
1
τ0
(18)
(k∗ )2 ‖α‖2L∞ (Γ1 ) R2 (x0 )
(19)
L = (n − 1)2 k2 (k∗ )2 R2 (x0 )
(20)
M = σ (n − 1) − 2 > 0
(21)
(see hypothesis (H11))
N = 4nR (x ) + n(n − 1) k2
(22)
P = K + L + R(x ).
(23)
2
0
2
0
Introduce the energy E (t ) associated with (12): E (t ) =
1 2
|u (t )| + ‖u(t )‖ + |θ (t )| + ′
2
2
2
∫
2
σ +2
Ω
|u(t )|
σ +2
dx ,
t ≥ 0.
If we multiply (12)1 by u′ and (12)2 by θ and integrate over Ω , formally we obtain d dt
E (t ) + ‖θ (t )‖2 +
1 2
α0 d0
∫ Γ1
|u′ (t )|2 dΓ + d∗1
∫ Γ1
|θ (t )|2 dΓ ≤ 0,
where d∗1 was defined in (17), i.e., the energy E (t ) is not increasing.
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We have: Theorem 3. Let {u, θ} be the solution given by Theorem 2. Consider hypotheses (H9)–(H11). Then 2
E (t ) ≤ 3E (0)e− 3 ηt ,
∀t ≥ 0 ,
where 2
η = min{ε1 , ε3 }, ε1 = ∗ , C 1 α0 d0 1 , , . ε2 = min 2N
2P
ε3 = min
ε
2
2
, ε2 M
k1
3. Proof of the results with h(x, s) First of all, we note that hypothesis (H3)1 on σ implies q∗ = n2n ≥ 2σ + 2 and q∗ ≥ σ n. Then the Sobolev Embedding −2 Theorem gives
V ↩→ Lq∗ (Ω ) ↩→ L2σ +2 (Ω ), if n ≥ 3 V ↩→ L2σ +2 (Ω ), if n = 1, 2
(24)
V ↩→ Lq∗ (Ω ) ↩→ Lσ n (Ω ), if n ≥ 3 V ↩→ Lσ n (Ω ), if n = 1, 2.
(25)
and
Here, the notation X ↩→ Y indicates that the space X is continuously embedded in the space Y . Next, following the ideas of Strauss [13], we approximate the function h by Lipschitz continuous functions hl . Proposition 1. Assume that the function h satisfies hypothesis (H1) Then there exists a sequence (hl ) of functions of C 0 (R; L∞ (Γ1 )) satisfying: (i) hl (x, 0) = 0, a.e. x in Γ1 ; (ii) [hl (x, s) − hl (x, r )](s − r ) ≥ d(s − r )2 , ∀s, r ∈ R; a.e. x in Γ1 ; (iii) there exists a function cl ∈ L∞ (Γ1 ) such that
|hl (x, s) − hl (x, r )| ≤ cl (x)|s − r |,
∀ s, r ∈ R, a.e. x in Γ1 ;
and (iv) (hl ) converges to h uniformly on bounded sets of R, a.e. x in Γ1 . Proof. For each l ∈ N, we define
C (x)s, if 0 ≤ s ≤ 1 , 1l l ∫ 1 s+ l 1 l h(τ )dτ , if ≤ s ≤ l, l s C2l (x)s, if s > l, hl (x, s) = C3l (x)s, if − 1 ≤ s ≤ 0, ∫ l s 1 h(τ )dτ , if − l ≤ s ≤ − , −l 1 l s− l C4l (x)s, if s < −l, where
∫ 2 C1l (x) = l2 l h(x, τ )dτ , 1 ∫ l+l 1 l C2l (x) = h(x, τ )dτ , l ∫ − 1l 2 C ( x ) = − l h(x, τ )dτ , 3l − 2l ∫ −l C4l (x) = − h(x, τ )dτ . −l− 1l
The sequence (hl ) satisfies the conditions of the proposition.
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3.1. Proof of Theorem 1 We consider a sequence (u1l ) of vectors of D (Ω ) such that in H01 (Ω ).
u1l → u1
(26)
Note that
∂ u0 + hl (·, u1l ) = 0 on Γ1 , ∂ν
∀l,
∂θ 0 + hl (·, θ 0 ) = 0 on Γ1 , ∂ν
and
∀l.
(27)
Fix l ∈ N. Consider a special basis β = {w1l , w2l , w3l , . . .} of V ∩ H 2 (Ω ) such that u0 , u1l , θ 0 belong to the subspace [w , w2l , w3l ], generated by w1l , w2l , w3l . Denote by Vml = [w1l , w2l , . . . , wml ] the subspace generated by the first m vectors of the basis β . Consider the approximate solutions l 1
ulm (t ) =
m −
gjml (t )wjl ,
θlm (t ) =
j=1
m −
hjml (t )wjl
j =1
of system (4) defined by the following system of ordinary differential equations:
(u′′ℓm (t ), v) + ((uℓm (t ), v)) + n − ∂θℓm
+
Γ1
(θℓ′ m (t ), w) + ((θℓm (t ), w)) + uℓm (0) = u0ℓ ,
hl (·, u′ℓm (t ))v dΓ
(t ), v + (|um (t )|σ um (t ), v) = 0,
∂ xi
i =1
∫
∫ Γ1
u′ℓm (0) = u1ℓ
∀v ∈ Vml ;
hl (·, θℓm (t ))w dΓ +
n − ∂ u′
ℓm
∂ xi
i =1
(28)
(t ), w = 0,
∀w ∈ Vml ;
and θℓm (0) = θ 0 .
(29) (30)
This system has a local solution {ulm (t ), θlm (t )} defined over [0, tlm [ for all v, w ∈ extend this solution to all [0, ∞[.
Vml .
The next estimate will permit us to
The first estimate. Choosing v = u′lm (t ) in (28) and w = θlm (t ) in (29), we get 1 d 2 dt
|ulm (t )| + ′
+
2
1 d 2 dt
n − ∂θlm (t )
∂ xi
i =1
‖ulm (t )‖ + 2
, u′lm (t ) +
∫ Γ1
hl (·, u′lm (t ))u′lm (t )dΓ d
1
σ + 2 dt
∫ Ω
|ulm (t )|σ +2 dx = 0
and
n − ∂ u′lm (t ) hl (·, θlm (t ))θlm (t )dΓ + |θlm (t )| + ‖θlm (t )‖ + , θlm (t ) = 0. 2 dt ∂ xi Γ1 i=1 1 d
2
2
∫
Applying the identity n − ∂ u′
− n n ∫ − ∂θℓm (t ), θℓm (t ) = − u′ℓm (t ), (t ) + u′ℓm (t )θℓm (t )νi dΓ , ∂ xi ∂ x i Γ 1 i=1 i=1 ℓm
i =1
adding the last two equalities and using the hypothesis (H1) on h, we get d dt
E1lm (t ) + ‖θlm (t )‖ + d0 2
∫
|ulm (t )| dΓ + d0 ′
Γ1
2
∫
|θlm (t )| dΓ + 2
Γ1
n ∫ −
Γ1
i=1
u′lm (t )θlm (t )νi dΓ ≤ 0,
(31)
where E1lm (t ) =
1 2
|u′lm (t )|2 + ‖ulm (t )‖2 + |θlm (t )|2 +
2
σ +2
∫ Ω
|ulm (t )|
σ +2
dx .
We have, for ε > 0,
∫ ∫ n ∫ − ε n ′ ulm (t )θlm (t )νi dΓ ≤ |u′lm (t )|2 dΓ + |θlm (t )|2 dΓ . i=1 Γ1 2n Γ1 2ε Γ1
(32)
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Taking into account the last inequality with ε = d0 in (31) we obtain d dt
E1lm (t ) + ‖θlm (t )‖2 +
where d1 = d0 − obtain E1lm (t ) +
n2 2d0
d0 2
Γ1
|u′lm (t )|2 dΓ + d1
∫ Γ1
|θlm (t )|2 dΓ ≤ 0,
(33)
≥ 0 (see hypothesis (H2)). Integrating this expression over [0, t ] and using the embedding (24), we
t
∫
∫
‖θlm (s)‖2 ds +
d0 2
0
∫ t∫ Γ1
0
|u′lm (s)|2 dΓ ds + d1
∫ t∫ Γ1
0
≤ (|u1 |2 + η) + ‖u0 ‖2 + |θ 0 |2 + c1 ‖u0 ‖σ +2 ,
|θlm (s)|2 dΓ ds
∀l ≥ l0 , ∀m(η positive constant),
where ‖v‖Lσ +2 (Ω ) ≤ c1 ‖v‖ for all v ∈ V . This gives
(ulm ) is bounded in L∞ (0, ∞; V ), ′ (ulm )is bounded in L∞ (0, ∞; L2 (Ω )), (θlm ) is bounded in L∞ (0, ∞; L2 (Ω )) ∩ L2 (0, ∞; V ), ′ (ulm ) is bounded in L2 (0, ∞; L2 (Γ1 )).
(34)
The second estimate. Differentiate with respect to t the approximate equations (28) and (29) and then we choose v = u′′lm (t ) ′ and w = θlm (t ), respectively. We obtain 1 d 2 dt
|ulm (t )| + ′′
2
1 d 2 dt
‖ulm (t )‖ + ′
2
∫ Γ1
h′l (., u′lm (t ))|u′′lm (t )|2 dΓ
n ′ − ∂θlm (t ) ′′ , ulm (t ) + (σ + 1)(|ulm (t )|σ u′lm (t ), u′′lm (t )) = 0, ∂ xi i=1 ∫ n − 1 d ′ ∂ u′′lm (t ) ′ 2 ′ 2 ′ ′ 2 |θlm (t )| + ‖θlm (t )‖ + , θlm (t ) = 0. hl (., θlm (t ))|θlm (t )| dΓ + 2 dt ∂ xi Γ1 i=1 +
We observe that n − ∂ u′′
− n ∫ n − ∂θ ′ (t ), θℓ′ m (t ) = − u′′ℓm (t )θℓ′ m (t )νi dΓ . u′′ℓm (t ), ℓm (t ) + ∂ xi ∂ x i Γ 1 i=1 i=1 ℓm
i=1
Using this equality in the preceding equation, adding the last two equations and considering hypothesis (H1)3 on h(x, s), we get d dt
′ E2lm (t ) + ‖θlm (t )‖2 + d0
+
n ∫ − i=1
Γ1
∫ Γ1
|u′′lm (t )|2 dΓ + d0
∫ Γ1
′ |θlm (t )|2 dΓ
′ u′′lm (t )θlm (t )νi dΓ + (σ + 1)(|ulm (t )|σ u′lm (t ), u′′lm (t )) = 0,
(35)
where E2lm (t ) =
1 ′′ ′ |ulm (t )|2 + ‖u′lm (t )‖2 + |θlm (t )|2 . 2
(36)
We have, for ε > 0,
∫ ∫ ∫ n ′′ ′ ′′ 2 ≤ ε u ( t )θ ( t )ν d Γ | u ( t )| + |θ ′ (t )|2 dΓ . i lm lm lm 2n 2ε Γ1 Γ1 Γ1 Also, by the Hölder inequality with
|(|ulm (t ) |σ u′lm (t ), u′′lm (t ))| ≤
1 q∗ 1
+
∫ Ω
1 n
+
1 2
= 1 and by the embedding (25) and first estimate (34), we find
|ulm (t )|σ n σ
1n ∫ Ω ′′
∗
|u′lm (t )|q1
1∗ ∫ q
1
Ω ′′
|u′′lm (t )|σ 2
12
≤ c ‖ulm (t )‖ ‖ulm (t )‖ulm (t )| ≤ c ‖ulm (t )‖ulm (t )| ≤ C (‖u′lm (t )‖2 + |u′′lm (t )|2 ) ′
where the constant C is independent of l, m and t ≥ 0.
′
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Taking into account the last two inequalities (with ε = d0 ) in (35), we get d dt
′ E2lm (t ) + ‖θlm (t )‖2 +
d0
∫
2
Γ1
≤ C [‖u′lm (t )‖2 + |u′′lm (t )|2 ],
|u′′lm (t )|2 dΓ + d1
∫ Γ1
′ |θlm (t )|2 dΓ
∀l, m and t ≥ 0,
(37)
where d1 was introduced in (33). ′ So that (37) provides an estimate it is necessary to bound (u′′lm (0)) and (θlm (0)). We obtain this boundedness. We set ′′ ′ t = 0 in the approximate equations (28) and (29) and then choose v = ulm (0) and w = θlm (0), respectively. We find
|u′′lm (0)|2 + ((u0 , u′′lm (0))) +
∫
|θlm (0)| + ((θ , θlm (0))) +
∫
′
2
0
′
Γ1
hl (·, u1l )u′′lm (0)dΓ +
i =1
hl (·, θ )θlm (0)dΓ + 0
Γ1
n − ∂θ 0
′
∂ xi
n − ∂ u1 l
i =1
∂ xi
, u′′lm (0) + (|u0 |σ u0 , u′′lm (0)) = 0; , θlm (0) = 0. ′
The Green’s formula implies
] ∫ [ 0 ∂u 1 |u′′ (0)|2 − (△u0 , u′′ (0)) + + h (·, u ) u′′lm (0)dΓ l lm lm l ∂ν Γ1 n − ∂θ 0 ′′ , u (0) + (|u0 |σ u0 , u′′lm (0)) = 0, + ∂ xi lm i=1 ] ∫ [ 0 ∂θ 0 ′ |θ ′ (0)|2 − (∆θ 0 , θ ′ (0)) + + hl (·, θ ) θlm (0)dΓ lm lm ∂ν Γ1 n 1 − ∂u l ′ , θlm (0) = 0. + ∂ x i i=1
(38)
(39)
Thanks to (27), the integrals on Γ1 in (38) and (39) are null. Then convergence (26) and embedding (24) applied in (38) and (39) give
′′ |u (0)|2 ≤ c , ∀l, m lm |θ ′ (0)|2 ≤ c , ∀l, m. lm
(40)
By applying Gronwall’s inequality in (37) and using estimate (40), we have
′ uℓm ′′ u ℓ′ m θ ℓ′′m u ℓm
is bounded in L∞ loc (0, ∞; V ); 2 is bounded in L∞ loc (0, ∞; L (Ω )); ∞ 2 is bounded in Lloc (0, ∞; L (Ω )) ∩ L2loc (0, ∞; V ); is bounded in L2loc (0, ∞; L2 (Γ1 )).
(41)
We observe that one has estimate (41) thanks to the special basis β of V ∩ H 2 (Ω ). This is the key point in the proof of Theorem 1. By estimates (34), (41), induction and the diagonal process, we obtain subsequences of (uℓm ) and (θℓm ) still denoted by (uℓm ) and (θℓm ) respectively, and we also get the functions uℓ : Ω ×]0, ∞[−→ R and θℓ : Ω ×]0, ∞[−→ R such that
u → u weak star in L∞ (0, ∞; V ); ℓm l u′ → u′ weak star in L∞ (0, ∞; L2 (Ω )) ∩ L∞ (0, ∞; V ); ℓm l loc ′′ ′′ ∞ 2 uℓm → ul weak star in Lloc (0, ∞; L (Ω )); ′ ′ 2 2 uℓm → ul weak in L (0, ∞; L (Γ1 )); ′′ ′′ 2 uℓm → ul weak in Lloc (0, ∞; L2 (Γ1 )); θℓm → θl weak star in L∞ (0, ∞; L2 (Ω )); θℓm → θl weak in L2 (0, ∞; V ); ′ 2 θℓm → θℓ′ weak star in L∞ loc (0, ∞; L (Ω )); ′ ′ 2 θℓm → θℓ weak in Lloc (0, ∞; V ).
(42)
Analysis of the nonlinear terms. By estimate (34)1 , (34)2 , the compactness method (cf. [14]), embedding (24), induction and the diagonal process, we obtain a subsequence of (ulm ), which is denoted also by (ulm ), such that 2 |uℓm |σ ulm → |uℓ |σ uℓ weak star in L∞ loc (0, ∞; L (Ω )).
(43)
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1 ′′ 2 By (41)1 , we have that (u′lm ) is bounded in L∞ loc (0, ∞; H (Γ1 )). This, estimate (41)4 of (ulm ), and the compactness 1
embedding of H 2 (Γ1 ) in L2 (Γ1 ) give in L2loc (0, ∞; L2 (Γ1 )).
u′lm → u′l
Then Lipschitzian property (iii) and Proposition 1 for the function hl imply hl (·, u′lm ) → hl (·, u′l ) in L2loc (0, ∞; L2 (Γ1 )).
(44)
In a similar way, estimate (34)3 and (41)3 for (θlm ) and (θlm ), respectively, furnish ′
hl (·, θlm ) → hl (·, θl )
in L2loc (0, ∞; L2 (Γ1 )).
(45)
Passage to the limit in m. Convergences (42)–(45) permit us to pass to the limit in m in approximate equations (28) and (29). Then this and the density of Vml in V give ∞
∫
(u′′ℓ (t ), v)ψ(t ) dt +
0
∫
∞
∫
+ Γ1
0
∫
∞
+
∞
∫
((uℓ (t ), v))ψ(t ) dt 0 n ∫ −
h(·, u′ℓ (t )) v ψ(t ) dΓ dt +
0
i=1
((|uℓ (t )|σ uℓ (t )), v) ψ(t ) = 0,
∞
∂θℓ (t ), v ψ(t ) dt ∂ xi
∀ψ ∈ D (0, ∞), ∀v ∈ V
(46)
0
and ∞
∫
(θℓ′ , w)ψ(t ) dt +
0
+
n ∫ − i=1
∞ 0
∞
∫
((θℓ (t ), w))ψ(t ) dt + 0
∞
∫ 0
∂ u′ℓ (t ), w ψ(t ) dt = 0, ∂ xi
∫ Γ1
h(·, θℓ (t ))wψ(t )dΓ
∀ψ ∈ D (0, ∞), ∀w ∈ V .
(47)
Taking ψ ∈ D (0, ∞), v, w ∈ D (Ω ) in (46) and (47), respectively, and using the regularity of u and θ given by (42) and (43), we obtain u′′l − ∆ul +
n − ∂θl i =1
θl′ − ∆θl +
∂ xi
n − ∂ u′ l
i=1
∂ xi
2 + |ul |σ ul = 0 in L∞ loc (0, ∞; L (Ω )),
(48)
2 = 0 in L∞ loc (0, ∞; L (Ω )).
(49) ∂u
∞ 2 −1/2 l By (42)1 , we have ul ∈ L∞ (0, ∞; V ) and by (47), we have ∆ul ∈ L∞ (Γ1 )) loc (0, ∞; L (Ω )). Then ∂ν ∈ Lloc (0, ∞; H ∂θl (cf. Lions [12] and Medeiros and Milla Miranda [15]). In a similar way, ∂ν ∈ L2loc (0, ∞; H −1/2 (Γ1 )). Multiplying both sides ∂u
of (48) by vψ with v ∈ V and ψ ∈ D (0, ∞), and integrating over Ω and [0, ∞[, the preceding regularity ∂νl on Γ1 then gives ∞
∫
(ul , v)ψ dt + ′′
0
∞
∫
((ul , v))ψ dt −
∞
∫
0
0
∫ ∞ n ∫ ∞ − ∂ ul ∂θl σ , v ψ dt + (|ul | ul , v)ψ dt + , v ψ dt = 0, (50) ∂ν ∂ xi 0 i=1 0
where ⟨ , ⟩ denotes the duality pairing between H −1/2 (Γ1 ) and H 1/2 (Γ1 ). In similar way, from (48) it follows that ∞
∫
(θl′ , w)ψ dt + 0
∞
∫
((θl , w))ψ dt − 0
∞
∫
0
n ∫ ∞ − ∂θl ∂ u′l , w ψ dt + , w ψ dt = 0. ∂ν ∂ xi i=1 0
(51)
Comparing (46) and (50) and using the Lipschitz property of hl (x, s), we obtain
∂ ul + hl (·, u′l ) = 0 in L2loc (0, ∞; L2 (Γ1 )). ∂ν
(52)
In similar way, comparing (47) and (51), we find
∂θl + hl (·, θl ) = 0 in L2loc (0, ∞; L2 (Γ1 )). ∂ν
(53)
A.T. Lourêdo et al. / Nonlinear Analysis 74 (2011) 6988–7004
6997
Passage to the limit in l. As estimates (34) and (41) are independent of l and m, we obtain a subsequence of (ul ), still denoted by (ul ), and a function u such that all convergences (42) and (43) are valid. These convergences will be denoted by (42)l and (43)l , respectively. These results imply that {u, θ} belongs to class (3) and it is a solution of Eqs. (4). Convergence (42)l gives ul → u weak star in
L∞ (0, ∞; V )
and Eq. (48) gives
∆ul → ∆u weak star in
2 L∞ loc (0, ∞; L (Ω )).
∂ ul ∂u → weak star in ∂ν ∂ν
−2 L∞ (Γ1 )). loc (0, ∞; H
Then 1
(54)
Also convergence (42)l furnishes u′l → u′ in L2loc (0, ∞; L2 (Γ1 )). Fix T > 0. The preceding convergence implies u′l (x, t ) → u(x, t )
a.e in Σ1 = Γ1 ×]0, T [.
(55)
Fix (x, t ) ∈ Σ1 . Then by (55) the set {ul (x, t ) : l ∈ N} is bounded. Part (iv) of Proposition 1 says that (hl ) converges to h uniformly in bounded sets of R, a.e. x in Γ1 . These two results and (55) give ′
hl (x, u′l (x, t )) → h(x, u′ (x, t ))
a.e in Σ1 = Γ1 ×]0, T [.
(56)
On the other hand, by Eqs. (48) and (52), we obtain
∫ Γ1
hl (·, u′l (t ))u′l (t )dΓ = −
1 d 2 dt
|u′l (t )|2 −
1 d 2 dt
‖ul (t )‖2 −
n − ∂θl (t ) i=1
∂ xi
, u′l (t ) − (|ul (t )|σ ul (t ), u′l (t )).
(57)
We have
n − ∂θl (t ) ′ , ul (t ) ≤ C [‖θl (t )‖2 + |u′l (t )|2 ] i=1 ∂ xi and embedding (24) gives
|(|ul (t ) |σ ul (t ), u′l (t ))| ≤ C [‖ul (t )‖2(σ +2) + |u′l (t )|2 ], where C > 0 is a constant independent of l and t ∈ [0, T ]. Note that ul ∈ C 0 ([0, T ]; V ), u′l ∈ C 0 ([0, T ]; L2 (Ω )) and that (ul (T )) and (u′l (T )) are bounded in V and L2 (Ω ), respectively. By taking into account the preceding considerations, estimate (34) and convergence (26) in (57), we obtain T
∫ 0
∫ Γ1
hl (·, u′l (t ))u′l (t )dΓ dt ≤ C ,
∀l ≥ l0 , ∀t ∈ [0, T ],
(58)
where C > 0 is a constant independent of l ≥ l0 and t ∈ [0, T ]. Note that hl (·, u′l (t ))u′l (t ) ≥ 0. Expressions (56), (58), Strauss’s Theorem and the diagonal process provide hl (·, u′l ) → h(·, u′ ) in L1loc (0, ∞; L1 (Γ1 )). Convergences (54) and (59) imply
∂ ul ∂u → ∂ν ∂ν
in D ′ (Γ1 ×]0, ∞[)
and hl (·, u′l ) → h(·, u′ ) in D ′ (Γ1 ×]0, ∞[). We take the limit in Eq. (52). The last two convergences and the regularity of h(·, u′ ) imply
∂u + h(·, u′ ) = 0 in L1loc (0, ∞; L1 (Γ1 )). ∂ν In similar way, we obtain
∂θ + h(·, θ ) = 0 in L1loc (0, ∞; L1 (Γ1 )). ∂ν The verification of the initial conditions follows by convergences (42)l and initial conditions (32).
(59)
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3.2. Proof of Corollary 1 By (3)1 and (4) we have u ∈ L∞ (0, ∞; V ) and
2 △u ∈ L∞ loc (0, ∞; L (Ω )).
Hypothesis (H4), regularity (3)1 and Eq. (5)1 give
∂u 2 ∈ L∞ loc (0, ∞; L (Γ1 )). ∂ν 3 2 The above two results furnish u ∈ L∞ loc (0, ∞; V ∩ H (Ω )) (cf. Lions [14] and Medeiros and Milla Miranda [15]). 3
In similar way we obtain θ ∈ L2loc (0, ∞; V ∩ H 2 (Ω )). Expressions (8) are consequence of hypothesis (H4) and regularity (3). The uniqueness of the solutions follows by the energy method. 4. Proof the result with h(s) We begin by introducing results obtained previously. 1
Proposition 2. Let us consider f ∈ L2 (Ω ) and g ∈ H 2 (Γ1 ). Then the solution u of the boundary value problem in Ω on Γ0
−△u = f u = 0 ∂u =g ∂ν
(60)
on Γ1
belongs to V ∩ H 2 (Ω ) and
‖ u‖
2 H 2 (Ω )
[ ≤ c |f |2 + ‖g ‖2 1
]
H 2 (Γ1 )
,
where the constant c > 0 is independent of f , g and u. Proposition 3. In V ∩ H 2 (Ω ), the norms H 2 (Ω ) and the norm
12 2 ∂u u → |△u| + ∂ν H 21 (Γ1 )
2
are equivalent. The above two results can be found in Medeiros and Milla Miranda [6]. We equip V ∩ H 2 (Ω ) with the norm given by Proposition 3. 1
1
1
1
Proposition 4. Let h : R → R be a Lipschitz continuous function with h(0) = 0. Then h(u) ∈ H 2 (Γ1 ) for u ∈ H 2 (Γ1 ) and the mapping 1
1
h : H 2 (Γ1 ) → H 2 (Γ1 ),
u → h(u)
is continuous. The above result was obtained by Marcus and Mizael [16]. Proposition 5. Consider h as in Proposition 4 and α satisfying hypothesis (H5). Then α h(u) ∈ H 2 (Γ1 ) for u ∈ H 2 (Γ1 ) and
‖α h(u)‖
1
H 2 (Γ1 )
≤ c ‖h(u)‖
1
H 2 (Γ1 )
,
1
∀u ∈ H 2 (Γ1 )
where the constant c > 0 is independent of α, h and u. Proof. The linear mapping L2 (Γ1 ) → L2 (Γ1 ),
u → α u
and H 1 (Γ1 ) → H 1 (Γ1 ),
u → α u
A.T. Lourêdo et al. / Nonlinear Analysis 74 (2011) 6988–7004
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are continuous. Then by interpolation it follows that 1
1
H 2 (Γ1 ) → H 2 (Γ1 ),
u → α u
is continuous. Therefore
‖α u‖
1
H 2 (Γ1 )
≤ c ‖ u‖
1
H 2 (Γ1 )
1
,
∀u ∈ H 2 (Γ1 ).
This result and Proposition 4 give the proposition. Proposition 6. Assume that h is as in Proposition 4 and α satisfies hypothesis (H5). Suppose that u0 ∈ V ∩ H 2 (Ω ) and u1 ∈ V and that they satisfy
∂ u0 + α(x)h(u1 ) = 0 on Γ1 . ∂ν Then, for each ε > 0, there exist w and z in V ∩ H 2 (Ω ) such that
‖w − u0 ‖V ∩H 2 (Ω ) < ε,
‖ z − u1 ‖ < ε
and
∂w + α h(z ) = 0 on Γ1 . ∂ν Proof. Consider ε > 0. By the continuity of the mapping h (see Proposition 4) and the Trace Theorem there exist 0 < δ < ε and z ∈ V such that
‖z − u‖ < δ and ‖h(u) − h(u1 )‖
1
H 2 (Γ1 )
< ε.
By the density of V ∩ H 2 (Ω ) in V (see (2)) we can consider z ∈ V ∩ H 2 (Ω ). Let us consider w ∈ V ∩ H 2 (Ω ), a solution of the problem
−△w = −△u0 w = 0 ∂w = −α h(z ) ∂ν
in Ω on Γ0
(61)
on Γ1 .
By Propositions 3–5, we have
‖w −
u0 2V ∩H 2 (Ω )
‖
2 ∂w ∂ u0 = |△w − △u | + < c ε2 . − = ‖α h(z ) − α h(u1 )‖2 1 ∂ν ∂ν H 12 (Γ1 ) H 2 (Γ1 ) 0 2
Thus, we find w, z satisfying the conditions of the proposition.
The above results allow us to prove the result with h(s). 4.1. Proof of Theorem 2 We proceed as in the proof of Theorem 1. Proposition 2 gives a sequence (hl ) of functions of C 0 (R) that verifies (i)–(iv) of this proposition without x ∈ Γ1 . By hypothesis (9) we have θ 0 ∈ V ∩ H 2 (Ω ) and
∂θ 0 + β h(θ 0 ) = 0 on Γ1 . ∂ν
(62)
Proposition 6 provides two sequences (u0l ) and u1l of vectors of V ∩ H 2 (Ω ) such that u0l → u0
in V ∩ H 2 (Ω ),
u1l → u1
in V
(63)
and
∂ u0 + α h(u1 ) = 0 on Γ1 . ∂ν
(64)
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Fix l. Consider a basis β = {w1l , w2l , . . .} of V ∩ H 2 (Ω ) such that u0l , u1l and θ 0 belong to the subspace [w1l , w2l , w3l ] generated l by w1l , w2l , w3l . Denote by Vml = [w1l , w2l , . . . , wm ] the subspace generated by the first m vectors of the basis β. Consider the approximate solutions ulm (t ) =
m −
gjml (t )wjl ,
θlm (t ) =
m −
j =1
hjml (t )wjl
j =1
of system (12) defined by the following system of ordinary differential equations:
∫
n − ∂θℓm
(t ), v + (|um (t )|σ um (t ), v) = 0, ∂ xi ∫ n − ∂ u′ℓm β hl (θℓm (t ))w dΓ + (t ), w = 0, ∀w ∈ Vml ; (θℓ′ m (t ), w) + ((θℓm (t ), w)) + ∂ xi Γ1 i=1 (u′′ℓm (t ), v)
+ ((uℓm (t ), v)) +
uℓm (0) = u0ℓ ,
Γ1
u′ℓm (0) = u1ℓ
α hl (u′ℓm (t ))v dΓ
+
∀v ∈ Vml ;
(65)
i =1
and θℓm (0) = θ 0 .
(66) (67)
By arguments similar to those used to obtain the first and second estimates of Theorem 1, we find E1lm (t ) +
t
∫
‖θlm (s)‖ ds + 2
α0 d0 2
0
≤
1 2
∫ t∫
(|u1 |2 + ‖u0 ‖2 + |θ 0 |2 +
0
2c1
σ +2
|ulm (s)| dΓ ds + d1 ′
Γ1
2
∗
∫ t∫ 0
‖u0 ‖σ +2 ) + η,
Γ1
|θlm (s)|2 dΓ ds
∀l ≥ l0 , ∀m (η positive constant),
2
where d∗1 = β0 d0 − 2αn d (which is non-negative by hypothesis (H8)) and E1lm (t ) has a similar form to (32). Also 0 0 E2lm (t ) +
t
∫
′ ‖θlm (s)‖2 ds +
α0 d0
0
≤
1 2
∫ t∫
2
0
Γ1
|u′′lm (s)|2 dΓ ds + d∗1
′ [‖u′′lm (0)‖2 + ‖u1l ‖2 + |θlm (0)|2 ] + C
∫ t∫ 0
Γ1
′ |θlm (s)|2 dΓ ds
t
∫
[‖u′lm (s)‖2 + |u′′lm (s)|2 ]ds, 0
where E2ml (t ) has a similar form to (36). Thanks to (62) and (64), we obtain
′′ |u (0)| ≤ c , lm ′ |θlm (0)| ≤ c ,
∀l, m ∀l, m.
With the three preceding results we proceed as in the proof of Theorem 1 and obtain a solution {u, θ} for (12)–(14). 5. Proof of the decay of the energy We begin with some observations. Remark 2. Assume hypothesis (H10). Then the approximations (hl ) of h given in the proof of Proposition 2 satisfy
|hl (s)| ≤
3 ∗ k |s|, 2
∀s ∈ R, ∀l.
(68)
This is obtained by direct computations in the formula for hl given in the proof of Proposition 1. We consider the solution {ul , θl } given in the proof of Theorem 2 with hl and u0l , u1l , θ 0 . We have u′l ∈ L∞ loc (0, ∞; V ). We obtain
∂ ∂ u′ hl (u′l (x, t )) = h′l (u′l (x, t )) l (x, t ) ∂ xi ∂ xi (see Brezis and Cazenave [17]). Thus hl (u′l ) ∈ L∞ loc (0, ∞; V ). Therefore, 1
2 α hl (u′l ) ∈ L∞ loc (0, ∞; H (Γ1 )).
Then Eq. (52) provides
∂ ul 1 2 ∈ L∞ loc (0, ∞; H (Γ1 )). ∂ν
A.T. Lourêdo et al. / Nonlinear Analysis 74 (2011) 6988–7004
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The further regularity ul ∈ L ∞ loc (0, ∞; V ),
2 ∆ ul ∈ L ∞ loc (0, ∞; L (Ω ))
implies 2 ul ∈ L ∞ loc (0, ∞; V ∩ H (Ω )).
(69)
In similar way,
θl ∈ L2loc (0, ∞; V ∩ H 2 (Ω )).
(70)
5.1. Proof of Theorem 3 We consider the solution {ul , θl } given in the proof of Theorem 2 with hl and u0l , u1l , θ 0 . Thus ul , θl belong to class (11), and ul and θl verify (69) and (70) respectively. Also ul , θl satisfy Eqs. (12) and the equations
∂ ul 1 2 + α h(u′l ) = 0 in L∞ loc (0, ∞; H (Γ1 )) ∂ν ∂θl 1 + β h(θl ) = 0 in L2loc (0, ∞; H 2 (Γ1 )) ∂ν
(71) (72)
and verify the initial conditions ul (0) = u0l ,
u′l (0) = u1l ,
θl (0) = θ 0 .
We prove Theorem 3 for the solution {ul , θl }. Theorem 3 will follow by taking the lim inf of both sides of the inequality with l for the theorem. Introduce the energy 1
Eℓ ( t ) =
2
|u′ℓ (t )|2 + ‖uℓ (t )‖2 + |θℓ (t )|2 +
∫
2
σ +2
Ω
|uℓ (t )|σ +2 dx ,
t ≥ 0,
(73)
and the functional
ρ(t ) = 2(u′ℓ (t ), m · ∇ uℓ (t )) + (n − 1)(u′ℓ (t ), uℓ (t )),
t ≥0
(74)
and consider the perturbed energy Eεl (t ) = El (t ) + ερ(t ),
(ε > 0).
(75)
In order to ease the notation in what follows we do not write the sub-index l and the variable t; we have
|ρ(t )| ≤ c ∗ E (t ),
t ≥ 0,
∗
where c was defined in (18). The above inequality gives 1 2
E (t ) ≤ Eε (t ) ≤
3 2
E (t ),
∀ t ≥ 0 , ∀ 0 < ε ≤ ε1
(76)
where 2
ε1 =
c∗
.
(77)
We have Eε′ = E ′ + ερ ′ .
(78)
Multiplying Eq. (13)1 with u and Eq. (13)2 with θ and integrating over Ω , we obtain ′
′
E +
1 2k1
1
1
|θ| + ‖θ ‖ + α0 d0 2
2
2
2
∫
|u | dΓ + d1 ′ 2
Γ1
∗
∫ Γ1
|θ |2 dΓ ≤ 0
where k1 > 0 and d∗1 ≥ 0 were defined in (16) and (17), respectively. The idea is to obtain
|ρ ′ | ≤ −aE ,
a positive constant.
Eq. (12)1 provides u′′ = △u −
n − ∂θ 2 − |u|σ u in L∞ loc (0, ∞; L (Ω )). ∂ x i i=1
(79)
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We differentiate ρ(t ) with respect to t and use the preceding equality. We find
n − ∂θ , m · ∇ u − 2(|u|σ u, m · ∇ u) ρ (t ) = 2(∆u(t ), m · ∇ u(t )) − 2 ∂ x i i =1 n − ∂θ ′ ′ + 2(u (t ), m · ∇ u (t )) + (n − 1)(△u(t ), u(t )) − (n − 1) ,u ∂ xi i=1
′
− (n − 1)(|u|σ u, u) + (n − 1)|u′ (t )|2 = I1 + I2 + · · · + I8 . • Modifications of I1 . Integrating
∂u m ∂u ∂ xj k ∂ xk
∂ ∂ xj
∂ ∂ xk
and
I1 = −(2 − n)‖u‖2 −
[ mk
∂u ∂ xj
2 ]
over Ω , and applying Gauss’s Theorem, we find
n ∫ n ∫ − − ∂u ∂u 2 ∂u mk (mk νK )dΓ + 2 νj dΓ = −(2 − n)‖u‖2 + J1 + J2 . ∂ x ∂ x ∂ x j j k j,k=1 Γ j,k=1 Γ
We have
2 2 n ∫ n ∫ − − ∂u ∂u J1 = − (m · ν)dΓ − (m · ν)dΓ . ∂ x ∂ xj j j=1 Γ0 j=1 Γ1 Also
n ∫ n ∫ − − ∂u ∂u ∂u ∂u νj mk dΓ + 2 νj mk dΓ . J2 = 2 ∂ xj ∂ xk ∂ xj ∂ xk j,k=1 Γ1 j,k=1 Γ0 ∂u on Γ0 . Then Note that ∂∂xu = νj ∂ν j
∫ J2 = 2
Γ0
∂u ∂ν
2
(m · ν)dΓ + 2
∫ Γ1
∂u (m · ∇ u)dΓ . ∂ν
Using the boundary equation (13)1 and hypothesis (H10), and noting that m(x) · ν(x) ≥ τ0 > 0 on Γ1 , we get
∫ 2 Γ0
∫ ∂u (m · ∇ u)dΓ = −2 α h(u′ )(m · ∇ u)dΓ ∂ν Γ1 ∫ ≤2 ‖α‖L∞ (Γ1 ) k∗ |u′ |R(x0 )|∇ u|dΓ Γ1 ∫ ∫ 1 ∗ 2 2 2 0 ′ 2 ≤ (k ) ‖α‖L∞ (Γ1 ) R (x ) |u | dΓ + (m · ν)|∇ u|2 dΓ . τ0 Γ1 Γ1
Thus
∫ J2 ≤ 2
Γ0
∂u ∂ν
2
(m · ν)dΓ + K
∫ Γ1
|u′ |2 dΓ +
∫ Γ1
(m · ν)|∇ u|2 dΓ
where K was defined in (19). Adding J1 and J2 and noting that m(x) · ν(x) ≤ 0 on Γ0 , we find
∫ J1 + J2 ≤ K
Γ1
|u′ |2 dΓ .
Thus I1 ≤ −(2 − n)‖u‖ + K 2
∫ Γ1
|u′ |2 dΓ .
(80)
• Modification of I2 . By the Cauchy–Schwarz inequality and an elementary inequality, we find 1
|I2 | ≤ 4nR2 (x0 )‖θ ‖2 + ‖u‖2 . 4
(81)
A.T. Lourêdo et al. / Nonlinear Analysis 74 (2011) 6988–7004
7003
• Modification of I3 . In this and the two next modifications we use Gauss’s Theorem. Thus ∫ ∫ 2 2n |u|σ +2 dx − |u|σ +2 (m · ν)dΓ I3 = σ +2 Ω σ + 2 Γ1 which implies
∫
2n
I3 ≤
σ +2 • Modification of I4 .
Ω
I4 = −n|u′ |2 +
|u|σ +2 dx.
∫ Γ1
(82)
(m · ν)|u′ |2 dΓ ≤ −n|u′ |2 + R(x0 )
∫ Γ1
| u′ | 2 d Γ .
(83)
• Modification of I5 . I5 = −(n − 1)‖u‖2 + (n − 1)
∫ Γ1
(m · ν)
∂u udΓ . ∂ν
Using boundary equation (13)1 and by arguments similar to those applied to obtain the modification of J2 , we get
∫
(n − 1)
Γ1
(m · ν)
∂u udΓ = −(n − 1) ∂ν
∫ Γ1
(m · ν)h(u′ )udΓ ≤ (n − 1)2 R2 (x0 )k2 (k∗ )2
∫
1
Γ1
| u′ | 2 d Γ + ‖ u‖ 2 , 4
where k2 was introduced in (16). So I5 ≤ −(n − 1)‖u‖2 + L
∫
1
Γ1
|u′ |2 dΓ + ‖u‖2 ,
(84)
4
where L was introduced in (15).
• Modification of I6 . We have
I6 ≤ n(n − 1)2 k2 ‖θ ‖2 +
1 4
‖ u‖ 2 .
• Modifications of I7 and I8 . ∫ I7 = −(n − 1) |u|σ +2 dx,
(85)
I8 = (n − 1)|u′ |2 .
Ω
(86)
Finally, adding the two sides of (80)–(86) and simplifying similar terms, we obtain 1
2
4
σ +2
ρ ′ ≤ − ‖u‖2 − |u′ |2 − M
∫
|u|σ +2 dx + N ‖θ ‖2 + P Ω
∫ Γ1
| u′ | 2 d Γ
where M , N and P were defined in (21)–(23), respectively. The preceding inequality and (78) and (79) provide E′ ≤ −
1 2k1 ∗
∫
+ d1
1
1
2
2
|θ|2 − ‖θ ‖2 − α0 d0 ε
Γ1
|u′ |2 dΓ
|θ| dΓ − ‖u‖ − ε|u | − ε M 2
Γ1
∫
2
2
′ 2
4
σ +2
∫
σ +2
|u| Ω
dx + ε N ‖θ ‖ + ε P 2
∫ Γ1
| u′ | 2 d Γ .
Consider
ε2 = min
1 2N
,
α0 d0 1 , 2P
k1
,
ε3 = min
ε
2
2
, ε2 M .
We have Eε′ (t ) ≤ −ε E (t ),
∀ t ≥ 0 , ∀ 0 < ε ≤ ε3 .
Take η = min{ε1 , ε3 }. Then by (76) and (87), we obtain 2 Eη′ (t ) ≤ − ηEη (t ), 3 which implies
∀t ≥ 0,
2
Eη (t ) ≤ Eη (0)e− 3 ηt . This and (76) provide 2
El (t ) ≤ 3El (0)e− 3 ηt ,
∀t ≥ 0, ∀l ≥ l0 .
(87)
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Acknowledgments We thank the referees for their careful reading of our article and for the constructive suggestions and modifications that transformed the text into one that is more understandable. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Scott W. Hansen, Exponential energy decay in linear thermoelastic rod, J. Math. Anal. Appl. 167 (1992) 429–442. D. Henry, O. Lopes, A. Perisinotto, Linear thermoelasticity: asymptotic stability and essential spectrum, Nonlinear Anal. 21 (1) (1993) 65–75. J.U. Kin, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal. 23 (4) (1992) 889–899. A.O. Marinho, M.R. Clark, H.R. Clark, On a coupled linear system with homogeneous damping, Appl. Math. Sci. 02 (2008) 679–699. A.O. Marinho, M.R. Clark, H.R. Clark, Existence and boundary stabilization of solutions for the coupled semilinear system, Nonlinear Anal. 70 (2009) 4226–4244. L.A. Medeiros, M. Milla Miranda, On a boundary value problem for wave equations: existence, uniqueness–asymptotic behavior, Revista de Matemáticas Aplicadas, Univerdidade de Chile 17 (1996) 47–73. F.D. Araruna, A.B. Maciel, Existence and boundary stabilization of the semilinear wave equation, Nonlinear Anal. 67 (2007) 1288–1305. J.L.G. Araújo, M. Milla Miranda, L.A. Medeiros, Vibrations of beams by torsion or impact, Mat. Contemp. 36 (2009) 29–50. A.T. Louredo, M. Milla Miranda, Nonlinear boundary dissipation for a coupled system of Klein–Gordon equations, Electron. J. Differential Equations 120 (2010) 1–19. M. Milla Miranda, J.P. Jutuca, Existence and boundary stabilization of solutions for the Kirchhoff equation, Comm. Partial Differential Equations 24 (1999) 1759–1800. V. Komornik, E. Zuazua, A direct method for boundary stabilization of wave equations, J. Math. Pure Appl. 69 (1990) 33–54. J.L. Lions, Problèmes aux Limites das les Équations aux Derivées Partielles, in: Oeuvres Cholsis de Jacques-Louis Lions, V.I, EDP Sciences, Paris, France, 2003, pp. 431–588. W.A. Strauss, On weak solutions of semilinear hyperbolic equations, An. Acad. Brasil. Ciénc. 4 (2) (1970) 645–651. J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non-Linéaires, Dunod, Paris, 1969. L.A. Medeiros, M. Milla Miranda, Espaços de Sobolev (Iniciação aos Problemas Eliticos não Homogêneos), Ed. IM-UFRJ, Rio de Janeiro, 2006. M. Marcus, V.L. Mizel, Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal. 33 (1979) 217–229. H. Brezis, T. Cazenave, Nonlinear evolution equations, IM-UFRJ, Rio de Janeiro, 1994.
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Boundary stabilization for a coupled system Aldo T. Lourêdo a , Alexandro M. Oliveira b,∗ , Marcondes R. Clark c a
Universidade Estadual da Paraíba - DM, CEP 58109095, Campina Grande, PB, Brazil
b
Universidade Federal do Piauí - DM, CEP 64.202.020, Parnaíba, PI, Brazil Universidade Federal do Piauí - DM, CEP 21941-901, Teresina, PI, Brazil
c
article
abstract
info
Article history: Received 29 April 2011 Accepted 9 July 2011 Communicated by Enzo Mitidieri
We investigate in this work the global existence of weak solutions for a nonlinear coupled system with mixed type boundary conditions. More precisely, Dirichlet and feedback boundary conditions. Further, we also prove the exponential decay of the energy associated with these solutions. © 2011 Elsevier Ltd. All rights reserved.
MSC: 35605 35N10 35S15 Keywords: Coupled system Nonlinear boundary conditions Exponential stability
1. Introduction Let Ω be an open bounded set of Rn whose boundary Γ of class C 2 is constituted by two closed disjoint parts Γ0 and Γ1 . By ν(x) is represented the outer unit normal vector at x ∈ Γ1 . Consider a real function h(x, s) defined in Γ1 ×]0, ∞[ and a real number σ > 0. Under these conditions we have the following mixed problem for a nonlinear thermoelastic system: u′′ (x, t ) − △u(x, t ) +
θ ′ (x, t ) − △θ (x, t ) +
n − ∂θ (x, t ) + |u(x, t )|σ u(x, t ) = 0 in Ω ×]0, ∞[, ∂ xi i =1 n − ∂ u′ i =1
u(x, t ) = 0,
∂ xi
(x, t ) = 0 in Ω ×]0, ∞[,
θ (x, t ) = 0 on Γ0 ×]0, ∞[,
(P)
∂u (x, t ) + h(x, u′ (x, t )) = 0 on Γ1 ×]0, ∞[, ∂ν ∂θ (x, t ) + h(x, θ (x, t )) = 0 on Γ1 ×]0, ∞[, ∂ν u(x, 0) = u0 (x), u′ (x, 0) = u1 (x), θ (x, 0) = θ 0 (x) in Ω .
∗
Corresponding author. Tel.: +55 86 33235406. E-mail addresses: [email protected] (A.T. Lourêdo), [email protected], [email protected] (A.M. Oliveira), [email protected] (M.R. Clark). 0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.07.019
A.T. Lourêdo et al. / Nonlinear Analysis 74 (2011) 6988–7004
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The thermoelastic systems have been studied by a lot of mathematicians. Among then we mention Hansen [1], Henry et al. [2], Kin [3] and Marinho et al. [4,5]. However the question of boundary stabilization with nonlinear conditions (P)4 and (P)5 has still not been discussed. This motivated us to develop this work. The existence of solutions of the wave equation with boundary damping can be obtained by applying the Galerkin method with a special basis. In this direction, we mention the pioneer work of Medeiros and Milla Miranda [6], and also, Araruna and Maciel [7] and Araújo et al. [8] for the linear boundary damping case, and Louredo and Milla Miranda [9] for the nonlinear boundary damping case. The Kirchhoff equation with linear boundary damping can be seen in [10]. In this paper we study the existence of solutions of Problem (P) by applying the Galerkin method with a special basis and using the Strauss approximations of a real continuous function. We study first the case h(x, s) and then the case α(x)h(s). In the second case the initial data are more general than in the first one. The exponential decay of the energy associated with (P) is obtained for a special α(x)h(s) and using a Lyapunov functional (see [11]). The papers [11,9,6] were our source of inspiration. Our work improves the results obtained by Marinho et al. [4,5]. 2. Notation and main results In this section we fix the notation and state the main results. We represent by H m (Ω ) the Sobolev space of order m and by L2 (Ω ) the class of square Lebesgue integrable functions. The scalar product and norm of L2 (Ω ) are denoted by (u, v) and |u|, respectively. By V we represent the Hilbert space V = {v ∈ H 1 (Ω ) : γ0 v = 0
on Γ0 },
where γ0 denotes the trace operator of order zero on Γ . V is equipped with the scalar product
((u, v)) =
∫ Ω
∇ u(x)∇v(x)dx, 1
and norm ‖u‖ = ((u, u)) 2 . All scalar functions considered in the paper will be real valued. We consider the operator A = −∆ defined by the triplet {V , L2 (Ω ), ((u, v))}. Then the domain D(−∆) of −∆ is given by D(−∆) =
u ∈ V ∩ H 2 (Ω );
∂u = 0 on Γ1 . ∂ν
We note that D(−∆) is dense in V .
(1)
See [12]. In order to state the first result we introduce the following hypotheses on the function h and the real number σ :
h ∈ C 0 (R; L∞ (Γ 1)) such that h(x, 0) = 0, a.e. x in Γ1 ; (h(x, s) − h(x, r ))(s − r ) ≥ d0 (s − r )2 , a.e. x in Γ1 , ∀s, r ∈ R(d > 0 constant);
(H1)
2d20
(H2)
0
2
≥n ; 1 2 <σ ≤ if n ≥ 3, n n−2 1 if n = 1, 2. σ >
(H3)
n
We have: Theorem 1. Assume hypotheses (H1)–(H3). Consider u0 ∈ D(−△), u0 ∈ H01 (Ω ) and
θ 0 ∈ H01 (Ω ) ∩ D(−∆).
(2)
Then there exist a pair of functions {u, θ} in the class
u ∈ L∞ (0, ∞; V ), u′ ∈ L∞ (0, ∞; L2 (Ω )) ∩ L∞ loc (0, ∞; V ) ′′ u ∈ L∞ (0, ∞; L2 (Ω )) loc θ ∈ L∞ (0, ∞; L2 (Ω )) ∩ L2 (0, ∞; V ) ′ θ ∈ L∞ (0, ∞; L2 (Ω )) ∩ L2 (0, ∞; V ) loc loc
(3)
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A.T. Lourêdo et al. / Nonlinear Analysis 74 (2011) 6988–7004
satisfying
n − ′′ ∂θ u − ∆u + ∂ xi i=1 n − ′ ∂ u′ θ − ∆θ + ∂ xi i=1 ∂u ′ ∂ν + h(·, u ) = 0 ∂θ + h(·, θ ) = 0 ∂ν
2 + |u|σ u = 0 in L∞ loc (0, ∞; L (Ω )),
(4) 2 = 0 in L∞ loc (0, ∞; L (Ω )),
in L1loc (0, ∞; L1 (Γ1 )),
(5)
in L1loc (0, ∞; L1 (Γ1 )),
and u(0) = u0 ,
u′ (0) = u1 ,
θ (0) = θ 0 .
(6)
Remark 1. We note that the initial data u0 , u1 and θ 0 given by (2) satisfy
∂ u0 ∂θ 0 + h(·, u1 ) = 0 on Γ1 and + h(·, θ 0 ) = 0 on Γ1 . ∂ν ∂ν We introduce a supplementary hypothesis on h:
|h(x, s)| ≤ c |s|,
∀s ∈ R, a.e. x in Γ1 .
(H4)
Corollary 1. Under hypotheses (H1)–(H4) the solution {u, θ} given by Theorem 1 satisfies 3 2 u ∈ L∞ loc (0, ∞; V ∩ H (Ω )),
∂u ′ ∂ν + h(·, u ) = 0 ∂θ + h(·, u) = 0 ∂ν
3
θ ∈ L2loc (0, ∞; V ∩ H 2 (Ω ))
2 in L∞ loc (0, ∞; L (Γ1 ))
in L2 (0, ∞; L2 (Γ1 ))
(7)
(8)
and this solution is unique in class (3) and (7). In the next result, we consider initial data a little more general than in Theorem 1; in compensation, the function h will depend on a single variable. More precisely, we introduce the following hypotheses on the functions α , β and h:
α ∈ W 1,∞ (Γ1 ), α(x) ≥ α0 > 0,
∀x ∈ Γ1 , (α0 constant)
1,∞
(H5)
β ∈ W (Γ1 ), β(x) ≥ β0 > 0, ∀x ∈ Γ1 , (β0 constant) h ∈ C 0 (R) such that h(0) = 0, (h(s) − h(r ))(s − r ) ≥ d0 (s − r )2 , ∀s, r ∈ R (d0 > 0 constant)
(H6)
2α0 β0 d20 ≥ n2 .
(H8)
(H7)
We have: Theorem 2. Assume hypotheses (H3), (H5)–(H8). Consider u0 ∈ V ∩ H 2 (Ω ), u1 ∈ V ,
θ 0 ∈ H01 (Ω ) ∩ D(−∆)
(9)
verifying
∂ u0 ∂θ 0 + α h(u1 ) = 0 on Γ1 and + β h(θ 0 ) = 0 on Γ1 . ∂ν ∂ν Then there exist a pair of functions {u, θ} in the class u ∈ L∞ (0, ∞; V ), u′ ∈ L∞ (0, ∞; L2 (Ω )) ∩ L∞ loc (0, ∞; V ) ′′ loc∞ u ∈ L (0, ∞; L2 (Ω )) loc θ ∈ L∞ (0, ∞; L2 (Ω )) ∩ L2 (0, ∞; V ) ′ θ ∈ L∞ (0, ∞; L2 (Ω )) ∩ L2 (0, ∞; V ) loc loc
(10)
(11)
A.T. Lourêdo et al. / Nonlinear Analysis 74 (2011) 6988–7004
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satisfying
n − ′′ ∂θ 2 u − △ u + + |u|σ u = 0 in L∞ loc (0, ∞; L (Ω )), ∂ xi i=1 n − ′ ∂ u′ 2 θ − △θ + = 0 in L∞ loc (0, ∞; L (Ω )), ∂ x i i=1 ∂u ′ 1 1 ∂ν + α h(u ) = 0 in Lloc (0, ∞; L (Γ1 )) ∂θ + β h(θ ) = 0 in L1loc (0, ∞; L1 (Γ1 )) ∂ν
(12)
(13)
and u(0) = u0 ,
u′ (0) = u1 ,
θ (0) = θ 0 .
(14)
In order to state the result on the asymptotic behavior of the energy associated with system (12), we introduce some hypotheses. Assume that there exists x0 ∈ Rn such that
Γ0 = {x ∈ Γ ; m(x) · ν(x) ≤ 0} and Γ1 = {x ∈ Γ ; m(x) · ν(x) > 0}
(H9)
where m(x) = x − x0 , x ∈ Rn , and ξ · η denotes the scalar product of Rn of ξ , η ∈ Rn . We use the notation R(x0 ) = max ‖m(x)‖Rn ,
τ0 = max[m(x) · ν(x)] > 0.
(15)
x∈Γ1
x∈Ω
Consider the following supplementary hypotheses on h and σ :
|h(s)| ≤ k∗ |s|,
∀s ∈ R(k∗ > 0 constant)
(H10)
2 2 <σ ≤ if n ≥ 3; n − 1 n−2 σ > 2 if n = 1, 2.
(H11)
Introduce the following notation:
|u|2 ≤ k1 ‖u‖2 ,
‖u‖L2 (Γ1 ) ≤ k2 ‖u‖2 ,
∀u ∈ V ;
∀u ∈ V .
(16)
Also, d∗1 = β0 d0 −
n2 2α0 d0
≥ 0 (see hypotheses (H8))
(17)
C ∗ = 4R(x0 ) + (n − 1)(1 + 2k1 ) k=
1
τ0
(18)
(k∗ )2 ‖α‖2L∞ (Γ1 ) R2 (x0 )
(19)
L = (n − 1)2 k2 (k∗ )2 R2 (x0 )
(20)
M = σ (n − 1) − 2 > 0
(21)
(see hypothesis (H11))
N = 4nR (x ) + n(n − 1) k2
(22)
P = K + L + R(x ).
(23)
2
0
2
0
Introduce the energy E (t ) associated with (12): E (t ) =
1 2
|u (t )| + ‖u(t )‖ + |θ (t )| + ′
2
2
2
∫
2
σ +2
Ω
|u(t )|
σ +2
dx ,
t ≥ 0.
If we multiply (12)1 by u′ and (12)2 by θ and integrate over Ω , formally we obtain d dt
E (t ) + ‖θ (t )‖2 +
1 2
α0 d0
∫ Γ1
|u′ (t )|2 dΓ + d∗1
∫ Γ1
|θ (t )|2 dΓ ≤ 0,
where d∗1 was defined in (17), i.e., the energy E (t ) is not increasing.
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We have: Theorem 3. Let {u, θ} be the solution given by Theorem 2. Consider hypotheses (H9)–(H11). Then 2
E (t ) ≤ 3E (0)e− 3 ηt ,
∀t ≥ 0 ,
where 2
η = min{ε1 , ε3 }, ε1 = ∗ , C 1 α0 d0 1 , , . ε2 = min 2N
2P
ε3 = min
ε
2
2
, ε2 M
k1
3. Proof of the results with h(x, s) First of all, we note that hypothesis (H3)1 on σ implies q∗ = n2n ≥ 2σ + 2 and q∗ ≥ σ n. Then the Sobolev Embedding −2 Theorem gives
V ↩→ Lq∗ (Ω ) ↩→ L2σ +2 (Ω ), if n ≥ 3 V ↩→ L2σ +2 (Ω ), if n = 1, 2
(24)
V ↩→ Lq∗ (Ω ) ↩→ Lσ n (Ω ), if n ≥ 3 V ↩→ Lσ n (Ω ), if n = 1, 2.
(25)
and
Here, the notation X ↩→ Y indicates that the space X is continuously embedded in the space Y . Next, following the ideas of Strauss [13], we approximate the function h by Lipschitz continuous functions hl . Proposition 1. Assume that the function h satisfies hypothesis (H1) Then there exists a sequence (hl ) of functions of C 0 (R; L∞ (Γ1 )) satisfying: (i) hl (x, 0) = 0, a.e. x in Γ1 ; (ii) [hl (x, s) − hl (x, r )](s − r ) ≥ d(s − r )2 , ∀s, r ∈ R; a.e. x in Γ1 ; (iii) there exists a function cl ∈ L∞ (Γ1 ) such that
|hl (x, s) − hl (x, r )| ≤ cl (x)|s − r |,
∀ s, r ∈ R, a.e. x in Γ1 ;
and (iv) (hl ) converges to h uniformly on bounded sets of R, a.e. x in Γ1 . Proof. For each l ∈ N, we define
C (x)s, if 0 ≤ s ≤ 1 , 1l l ∫ 1 s+ l 1 l h(τ )dτ , if ≤ s ≤ l, l s C2l (x)s, if s > l, hl (x, s) = C3l (x)s, if − 1 ≤ s ≤ 0, ∫ l s 1 h(τ )dτ , if − l ≤ s ≤ − , −l 1 l s− l C4l (x)s, if s < −l, where
∫ 2 C1l (x) = l2 l h(x, τ )dτ , 1 ∫ l+l 1 l C2l (x) = h(x, τ )dτ , l ∫ − 1l 2 C ( x ) = − l h(x, τ )dτ , 3l − 2l ∫ −l C4l (x) = − h(x, τ )dτ . −l− 1l
The sequence (hl ) satisfies the conditions of the proposition.
A.T. Lourêdo et al. / Nonlinear Analysis 74 (2011) 6988–7004
6993
3.1. Proof of Theorem 1 We consider a sequence (u1l ) of vectors of D (Ω ) such that in H01 (Ω ).
u1l → u1
(26)
Note that
∂ u0 + hl (·, u1l ) = 0 on Γ1 , ∂ν
∀l,
∂θ 0 + hl (·, θ 0 ) = 0 on Γ1 , ∂ν
and
∀l.
(27)
Fix l ∈ N. Consider a special basis β = {w1l , w2l , w3l , . . .} of V ∩ H 2 (Ω ) such that u0 , u1l , θ 0 belong to the subspace [w , w2l , w3l ], generated by w1l , w2l , w3l . Denote by Vml = [w1l , w2l , . . . , wml ] the subspace generated by the first m vectors of the basis β . Consider the approximate solutions l 1
ulm (t ) =
m −
gjml (t )wjl ,
θlm (t ) =
j=1
m −
hjml (t )wjl
j =1
of system (4) defined by the following system of ordinary differential equations:
(u′′ℓm (t ), v) + ((uℓm (t ), v)) + n − ∂θℓm
+
Γ1
(θℓ′ m (t ), w) + ((θℓm (t ), w)) + uℓm (0) = u0ℓ ,
hl (·, u′ℓm (t ))v dΓ
(t ), v + (|um (t )|σ um (t ), v) = 0,
∂ xi
i =1
∫
∫ Γ1
u′ℓm (0) = u1ℓ
∀v ∈ Vml ;
hl (·, θℓm (t ))w dΓ +
n − ∂ u′
ℓm
∂ xi
i =1
(28)
(t ), w = 0,
∀w ∈ Vml ;
and θℓm (0) = θ 0 .
(29) (30)
This system has a local solution {ulm (t ), θlm (t )} defined over [0, tlm [ for all v, w ∈ extend this solution to all [0, ∞[.
Vml .
The next estimate will permit us to
The first estimate. Choosing v = u′lm (t ) in (28) and w = θlm (t ) in (29), we get 1 d 2 dt
|ulm (t )| + ′
+
2
1 d 2 dt
n − ∂θlm (t )
∂ xi
i =1
‖ulm (t )‖ + 2
, u′lm (t ) +
∫ Γ1
hl (·, u′lm (t ))u′lm (t )dΓ d
1
σ + 2 dt
∫ Ω
|ulm (t )|σ +2 dx = 0
and
n − ∂ u′lm (t ) hl (·, θlm (t ))θlm (t )dΓ + |θlm (t )| + ‖θlm (t )‖ + , θlm (t ) = 0. 2 dt ∂ xi Γ1 i=1 1 d
2
2
∫
Applying the identity n − ∂ u′
− n n ∫ − ∂θℓm (t ), θℓm (t ) = − u′ℓm (t ), (t ) + u′ℓm (t )θℓm (t )νi dΓ , ∂ xi ∂ x i Γ 1 i=1 i=1 ℓm
i =1
adding the last two equalities and using the hypothesis (H1) on h, we get d dt
E1lm (t ) + ‖θlm (t )‖ + d0 2
∫
|ulm (t )| dΓ + d0 ′
Γ1
2
∫
|θlm (t )| dΓ + 2
Γ1
n ∫ −
Γ1
i=1
u′lm (t )θlm (t )νi dΓ ≤ 0,
(31)
where E1lm (t ) =
1 2
|u′lm (t )|2 + ‖ulm (t )‖2 + |θlm (t )|2 +
2
σ +2
∫ Ω
|ulm (t )|
σ +2
dx .
We have, for ε > 0,
∫ ∫ n ∫ − ε n ′ ulm (t )θlm (t )νi dΓ ≤ |u′lm (t )|2 dΓ + |θlm (t )|2 dΓ . i=1 Γ1 2n Γ1 2ε Γ1
(32)
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A.T. Lourêdo et al. / Nonlinear Analysis 74 (2011) 6988–7004
Taking into account the last inequality with ε = d0 in (31) we obtain d dt
E1lm (t ) + ‖θlm (t )‖2 +
where d1 = d0 − obtain E1lm (t ) +
n2 2d0
d0 2
Γ1
|u′lm (t )|2 dΓ + d1
∫ Γ1
|θlm (t )|2 dΓ ≤ 0,
(33)
≥ 0 (see hypothesis (H2)). Integrating this expression over [0, t ] and using the embedding (24), we
t
∫
∫
‖θlm (s)‖2 ds +
d0 2
0
∫ t∫ Γ1
0
|u′lm (s)|2 dΓ ds + d1
∫ t∫ Γ1
0
≤ (|u1 |2 + η) + ‖u0 ‖2 + |θ 0 |2 + c1 ‖u0 ‖σ +2 ,
|θlm (s)|2 dΓ ds
∀l ≥ l0 , ∀m(η positive constant),
where ‖v‖Lσ +2 (Ω ) ≤ c1 ‖v‖ for all v ∈ V . This gives
(ulm ) is bounded in L∞ (0, ∞; V ), ′ (ulm )is bounded in L∞ (0, ∞; L2 (Ω )), (θlm ) is bounded in L∞ (0, ∞; L2 (Ω )) ∩ L2 (0, ∞; V ), ′ (ulm ) is bounded in L2 (0, ∞; L2 (Γ1 )).
(34)
The second estimate. Differentiate with respect to t the approximate equations (28) and (29) and then we choose v = u′′lm (t ) ′ and w = θlm (t ), respectively. We obtain 1 d 2 dt
|ulm (t )| + ′′
2
1 d 2 dt
‖ulm (t )‖ + ′
2
∫ Γ1
h′l (., u′lm (t ))|u′′lm (t )|2 dΓ
n ′ − ∂θlm (t ) ′′ , ulm (t ) + (σ + 1)(|ulm (t )|σ u′lm (t ), u′′lm (t )) = 0, ∂ xi i=1 ∫ n − 1 d ′ ∂ u′′lm (t ) ′ 2 ′ 2 ′ ′ 2 |θlm (t )| + ‖θlm (t )‖ + , θlm (t ) = 0. hl (., θlm (t ))|θlm (t )| dΓ + 2 dt ∂ xi Γ1 i=1 +
We observe that n − ∂ u′′
− n ∫ n − ∂θ ′ (t ), θℓ′ m (t ) = − u′′ℓm (t )θℓ′ m (t )νi dΓ . u′′ℓm (t ), ℓm (t ) + ∂ xi ∂ x i Γ 1 i=1 i=1 ℓm
i=1
Using this equality in the preceding equation, adding the last two equations and considering hypothesis (H1)3 on h(x, s), we get d dt
′ E2lm (t ) + ‖θlm (t )‖2 + d0
+
n ∫ − i=1
Γ1
∫ Γ1
|u′′lm (t )|2 dΓ + d0
∫ Γ1
′ |θlm (t )|2 dΓ
′ u′′lm (t )θlm (t )νi dΓ + (σ + 1)(|ulm (t )|σ u′lm (t ), u′′lm (t )) = 0,
(35)
where E2lm (t ) =
1 ′′ ′ |ulm (t )|2 + ‖u′lm (t )‖2 + |θlm (t )|2 . 2
(36)
We have, for ε > 0,
∫ ∫ ∫ n ′′ ′ ′′ 2 ≤ ε u ( t )θ ( t )ν d Γ | u ( t )| + |θ ′ (t )|2 dΓ . i lm lm lm 2n 2ε Γ1 Γ1 Γ1 Also, by the Hölder inequality with
|(|ulm (t ) |σ u′lm (t ), u′′lm (t ))| ≤
1 q∗ 1
+
∫ Ω
1 n
+
1 2
= 1 and by the embedding (25) and first estimate (34), we find
|ulm (t )|σ n σ
1n ∫ Ω ′′
∗
|u′lm (t )|q1
1∗ ∫ q
1
Ω ′′
|u′′lm (t )|σ 2
12
≤ c ‖ulm (t )‖ ‖ulm (t )‖ulm (t )| ≤ c ‖ulm (t )‖ulm (t )| ≤ C (‖u′lm (t )‖2 + |u′′lm (t )|2 ) ′
where the constant C is independent of l, m and t ≥ 0.
′
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6995
Taking into account the last two inequalities (with ε = d0 ) in (35), we get d dt
′ E2lm (t ) + ‖θlm (t )‖2 +
d0
∫
2
Γ1
≤ C [‖u′lm (t )‖2 + |u′′lm (t )|2 ],
|u′′lm (t )|2 dΓ + d1
∫ Γ1
′ |θlm (t )|2 dΓ
∀l, m and t ≥ 0,
(37)
where d1 was introduced in (33). ′ So that (37) provides an estimate it is necessary to bound (u′′lm (0)) and (θlm (0)). We obtain this boundedness. We set ′′ ′ t = 0 in the approximate equations (28) and (29) and then choose v = ulm (0) and w = θlm (0), respectively. We find
|u′′lm (0)|2 + ((u0 , u′′lm (0))) +
∫
|θlm (0)| + ((θ , θlm (0))) +
∫
′
2
0
′
Γ1
hl (·, u1l )u′′lm (0)dΓ +
i =1
hl (·, θ )θlm (0)dΓ + 0
Γ1
n − ∂θ 0
′
∂ xi
n − ∂ u1 l
i =1
∂ xi
, u′′lm (0) + (|u0 |σ u0 , u′′lm (0)) = 0; , θlm (0) = 0. ′
The Green’s formula implies
] ∫ [ 0 ∂u 1 |u′′ (0)|2 − (△u0 , u′′ (0)) + + h (·, u ) u′′lm (0)dΓ l lm lm l ∂ν Γ1 n − ∂θ 0 ′′ , u (0) + (|u0 |σ u0 , u′′lm (0)) = 0, + ∂ xi lm i=1 ] ∫ [ 0 ∂θ 0 ′ |θ ′ (0)|2 − (∆θ 0 , θ ′ (0)) + + hl (·, θ ) θlm (0)dΓ lm lm ∂ν Γ1 n 1 − ∂u l ′ , θlm (0) = 0. + ∂ x i i=1
(38)
(39)
Thanks to (27), the integrals on Γ1 in (38) and (39) are null. Then convergence (26) and embedding (24) applied in (38) and (39) give
′′ |u (0)|2 ≤ c , ∀l, m lm |θ ′ (0)|2 ≤ c , ∀l, m. lm
(40)
By applying Gronwall’s inequality in (37) and using estimate (40), we have
′ uℓm ′′ u ℓ′ m θ ℓ′′m u ℓm
is bounded in L∞ loc (0, ∞; V ); 2 is bounded in L∞ loc (0, ∞; L (Ω )); ∞ 2 is bounded in Lloc (0, ∞; L (Ω )) ∩ L2loc (0, ∞; V ); is bounded in L2loc (0, ∞; L2 (Γ1 )).
(41)
We observe that one has estimate (41) thanks to the special basis β of V ∩ H 2 (Ω ). This is the key point in the proof of Theorem 1. By estimates (34), (41), induction and the diagonal process, we obtain subsequences of (uℓm ) and (θℓm ) still denoted by (uℓm ) and (θℓm ) respectively, and we also get the functions uℓ : Ω ×]0, ∞[−→ R and θℓ : Ω ×]0, ∞[−→ R such that
u → u weak star in L∞ (0, ∞; V ); ℓm l u′ → u′ weak star in L∞ (0, ∞; L2 (Ω )) ∩ L∞ (0, ∞; V ); ℓm l loc ′′ ′′ ∞ 2 uℓm → ul weak star in Lloc (0, ∞; L (Ω )); ′ ′ 2 2 uℓm → ul weak in L (0, ∞; L (Γ1 )); ′′ ′′ 2 uℓm → ul weak in Lloc (0, ∞; L2 (Γ1 )); θℓm → θl weak star in L∞ (0, ∞; L2 (Ω )); θℓm → θl weak in L2 (0, ∞; V ); ′ 2 θℓm → θℓ′ weak star in L∞ loc (0, ∞; L (Ω )); ′ ′ 2 θℓm → θℓ weak in Lloc (0, ∞; V ).
(42)
Analysis of the nonlinear terms. By estimate (34)1 , (34)2 , the compactness method (cf. [14]), embedding (24), induction and the diagonal process, we obtain a subsequence of (ulm ), which is denoted also by (ulm ), such that 2 |uℓm |σ ulm → |uℓ |σ uℓ weak star in L∞ loc (0, ∞; L (Ω )).
(43)
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1 ′′ 2 By (41)1 , we have that (u′lm ) is bounded in L∞ loc (0, ∞; H (Γ1 )). This, estimate (41)4 of (ulm ), and the compactness 1
embedding of H 2 (Γ1 ) in L2 (Γ1 ) give in L2loc (0, ∞; L2 (Γ1 )).
u′lm → u′l
Then Lipschitzian property (iii) and Proposition 1 for the function hl imply hl (·, u′lm ) → hl (·, u′l ) in L2loc (0, ∞; L2 (Γ1 )).
(44)
In a similar way, estimate (34)3 and (41)3 for (θlm ) and (θlm ), respectively, furnish ′
hl (·, θlm ) → hl (·, θl )
in L2loc (0, ∞; L2 (Γ1 )).
(45)
Passage to the limit in m. Convergences (42)–(45) permit us to pass to the limit in m in approximate equations (28) and (29). Then this and the density of Vml in V give ∞
∫
(u′′ℓ (t ), v)ψ(t ) dt +
0
∫
∞
∫
+ Γ1
0
∫
∞
+
∞
∫
((uℓ (t ), v))ψ(t ) dt 0 n ∫ −
h(·, u′ℓ (t )) v ψ(t ) dΓ dt +
0
i=1
((|uℓ (t )|σ uℓ (t )), v) ψ(t ) = 0,
∞
∂θℓ (t ), v ψ(t ) dt ∂ xi
∀ψ ∈ D (0, ∞), ∀v ∈ V
(46)
0
and ∞
∫
(θℓ′ , w)ψ(t ) dt +
0
+
n ∫ − i=1
∞ 0
∞
∫
((θℓ (t ), w))ψ(t ) dt + 0
∞
∫ 0
∂ u′ℓ (t ), w ψ(t ) dt = 0, ∂ xi
∫ Γ1
h(·, θℓ (t ))wψ(t )dΓ
∀ψ ∈ D (0, ∞), ∀w ∈ V .
(47)
Taking ψ ∈ D (0, ∞), v, w ∈ D (Ω ) in (46) and (47), respectively, and using the regularity of u and θ given by (42) and (43), we obtain u′′l − ∆ul +
n − ∂θl i =1
θl′ − ∆θl +
∂ xi
n − ∂ u′ l
i=1
∂ xi
2 + |ul |σ ul = 0 in L∞ loc (0, ∞; L (Ω )),
(48)
2 = 0 in L∞ loc (0, ∞; L (Ω )).
(49) ∂u
∞ 2 −1/2 l By (42)1 , we have ul ∈ L∞ (0, ∞; V ) and by (47), we have ∆ul ∈ L∞ (Γ1 )) loc (0, ∞; L (Ω )). Then ∂ν ∈ Lloc (0, ∞; H ∂θl (cf. Lions [12] and Medeiros and Milla Miranda [15]). In a similar way, ∂ν ∈ L2loc (0, ∞; H −1/2 (Γ1 )). Multiplying both sides ∂u
of (48) by vψ with v ∈ V and ψ ∈ D (0, ∞), and integrating over Ω and [0, ∞[, the preceding regularity ∂νl on Γ1 then gives ∞
∫
(ul , v)ψ dt + ′′
0
∞
∫
((ul , v))ψ dt −
∞
∫
0
0
∫ ∞ n ∫ ∞ − ∂ ul ∂θl σ , v ψ dt + (|ul | ul , v)ψ dt + , v ψ dt = 0, (50) ∂ν ∂ xi 0 i=1 0
where ⟨ , ⟩ denotes the duality pairing between H −1/2 (Γ1 ) and H 1/2 (Γ1 ). In similar way, from (48) it follows that ∞
∫
(θl′ , w)ψ dt + 0
∞
∫
((θl , w))ψ dt − 0
∞
∫
0
n ∫ ∞ − ∂θl ∂ u′l , w ψ dt + , w ψ dt = 0. ∂ν ∂ xi i=1 0
(51)
Comparing (46) and (50) and using the Lipschitz property of hl (x, s), we obtain
∂ ul + hl (·, u′l ) = 0 in L2loc (0, ∞; L2 (Γ1 )). ∂ν
(52)
In similar way, comparing (47) and (51), we find
∂θl + hl (·, θl ) = 0 in L2loc (0, ∞; L2 (Γ1 )). ∂ν
(53)
A.T. Lourêdo et al. / Nonlinear Analysis 74 (2011) 6988–7004
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Passage to the limit in l. As estimates (34) and (41) are independent of l and m, we obtain a subsequence of (ul ), still denoted by (ul ), and a function u such that all convergences (42) and (43) are valid. These convergences will be denoted by (42)l and (43)l , respectively. These results imply that {u, θ} belongs to class (3) and it is a solution of Eqs. (4). Convergence (42)l gives ul → u weak star in
L∞ (0, ∞; V )
and Eq. (48) gives
∆ul → ∆u weak star in
2 L∞ loc (0, ∞; L (Ω )).
∂ ul ∂u → weak star in ∂ν ∂ν
−2 L∞ (Γ1 )). loc (0, ∞; H
Then 1
(54)
Also convergence (42)l furnishes u′l → u′ in L2loc (0, ∞; L2 (Γ1 )). Fix T > 0. The preceding convergence implies u′l (x, t ) → u(x, t )
a.e in Σ1 = Γ1 ×]0, T [.
(55)
Fix (x, t ) ∈ Σ1 . Then by (55) the set {ul (x, t ) : l ∈ N} is bounded. Part (iv) of Proposition 1 says that (hl ) converges to h uniformly in bounded sets of R, a.e. x in Γ1 . These two results and (55) give ′
hl (x, u′l (x, t )) → h(x, u′ (x, t ))
a.e in Σ1 = Γ1 ×]0, T [.
(56)
On the other hand, by Eqs. (48) and (52), we obtain
∫ Γ1
hl (·, u′l (t ))u′l (t )dΓ = −
1 d 2 dt
|u′l (t )|2 −
1 d 2 dt
‖ul (t )‖2 −
n − ∂θl (t ) i=1
∂ xi
, u′l (t ) − (|ul (t )|σ ul (t ), u′l (t )).
(57)
We have
n − ∂θl (t ) ′ , ul (t ) ≤ C [‖θl (t )‖2 + |u′l (t )|2 ] i=1 ∂ xi and embedding (24) gives
|(|ul (t ) |σ ul (t ), u′l (t ))| ≤ C [‖ul (t )‖2(σ +2) + |u′l (t )|2 ], where C > 0 is a constant independent of l and t ∈ [0, T ]. Note that ul ∈ C 0 ([0, T ]; V ), u′l ∈ C 0 ([0, T ]; L2 (Ω )) and that (ul (T )) and (u′l (T )) are bounded in V and L2 (Ω ), respectively. By taking into account the preceding considerations, estimate (34) and convergence (26) in (57), we obtain T
∫ 0
∫ Γ1
hl (·, u′l (t ))u′l (t )dΓ dt ≤ C ,
∀l ≥ l0 , ∀t ∈ [0, T ],
(58)
where C > 0 is a constant independent of l ≥ l0 and t ∈ [0, T ]. Note that hl (·, u′l (t ))u′l (t ) ≥ 0. Expressions (56), (58), Strauss’s Theorem and the diagonal process provide hl (·, u′l ) → h(·, u′ ) in L1loc (0, ∞; L1 (Γ1 )). Convergences (54) and (59) imply
∂ ul ∂u → ∂ν ∂ν
in D ′ (Γ1 ×]0, ∞[)
and hl (·, u′l ) → h(·, u′ ) in D ′ (Γ1 ×]0, ∞[). We take the limit in Eq. (52). The last two convergences and the regularity of h(·, u′ ) imply
∂u + h(·, u′ ) = 0 in L1loc (0, ∞; L1 (Γ1 )). ∂ν In similar way, we obtain
∂θ + h(·, θ ) = 0 in L1loc (0, ∞; L1 (Γ1 )). ∂ν The verification of the initial conditions follows by convergences (42)l and initial conditions (32).
(59)
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3.2. Proof of Corollary 1 By (3)1 and (4) we have u ∈ L∞ (0, ∞; V ) and
2 △u ∈ L∞ loc (0, ∞; L (Ω )).
Hypothesis (H4), regularity (3)1 and Eq. (5)1 give
∂u 2 ∈ L∞ loc (0, ∞; L (Γ1 )). ∂ν 3 2 The above two results furnish u ∈ L∞ loc (0, ∞; V ∩ H (Ω )) (cf. Lions [14] and Medeiros and Milla Miranda [15]). 3
In similar way we obtain θ ∈ L2loc (0, ∞; V ∩ H 2 (Ω )). Expressions (8) are consequence of hypothesis (H4) and regularity (3). The uniqueness of the solutions follows by the energy method. 4. Proof the result with h(s) We begin by introducing results obtained previously. 1
Proposition 2. Let us consider f ∈ L2 (Ω ) and g ∈ H 2 (Γ1 ). Then the solution u of the boundary value problem in Ω on Γ0
−△u = f u = 0 ∂u =g ∂ν
(60)
on Γ1
belongs to V ∩ H 2 (Ω ) and
‖ u‖
2 H 2 (Ω )
[ ≤ c |f |2 + ‖g ‖2 1
]
H 2 (Γ1 )
,
where the constant c > 0 is independent of f , g and u. Proposition 3. In V ∩ H 2 (Ω ), the norms H 2 (Ω ) and the norm
12 2 ∂u u → |△u| + ∂ν H 21 (Γ1 )
2
are equivalent. The above two results can be found in Medeiros and Milla Miranda [6]. We equip V ∩ H 2 (Ω ) with the norm given by Proposition 3. 1
1
1
1
Proposition 4. Let h : R → R be a Lipschitz continuous function with h(0) = 0. Then h(u) ∈ H 2 (Γ1 ) for u ∈ H 2 (Γ1 ) and the mapping 1
1
h : H 2 (Γ1 ) → H 2 (Γ1 ),
u → h(u)
is continuous. The above result was obtained by Marcus and Mizael [16]. Proposition 5. Consider h as in Proposition 4 and α satisfying hypothesis (H5). Then α h(u) ∈ H 2 (Γ1 ) for u ∈ H 2 (Γ1 ) and
‖α h(u)‖
1
H 2 (Γ1 )
≤ c ‖h(u)‖
1
H 2 (Γ1 )
,
1
∀u ∈ H 2 (Γ1 )
where the constant c > 0 is independent of α, h and u. Proof. The linear mapping L2 (Γ1 ) → L2 (Γ1 ),
u → α u
and H 1 (Γ1 ) → H 1 (Γ1 ),
u → α u
A.T. Lourêdo et al. / Nonlinear Analysis 74 (2011) 6988–7004
6999
are continuous. Then by interpolation it follows that 1
1
H 2 (Γ1 ) → H 2 (Γ1 ),
u → α u
is continuous. Therefore
‖α u‖
1
H 2 (Γ1 )
≤ c ‖ u‖
1
H 2 (Γ1 )
1
,
∀u ∈ H 2 (Γ1 ).
This result and Proposition 4 give the proposition. Proposition 6. Assume that h is as in Proposition 4 and α satisfies hypothesis (H5). Suppose that u0 ∈ V ∩ H 2 (Ω ) and u1 ∈ V and that they satisfy
∂ u0 + α(x)h(u1 ) = 0 on Γ1 . ∂ν Then, for each ε > 0, there exist w and z in V ∩ H 2 (Ω ) such that
‖w − u0 ‖V ∩H 2 (Ω ) < ε,
‖ z − u1 ‖ < ε
and
∂w + α h(z ) = 0 on Γ1 . ∂ν Proof. Consider ε > 0. By the continuity of the mapping h (see Proposition 4) and the Trace Theorem there exist 0 < δ < ε and z ∈ V such that
‖z − u‖ < δ and ‖h(u) − h(u1 )‖
1
H 2 (Γ1 )
< ε.
By the density of V ∩ H 2 (Ω ) in V (see (2)) we can consider z ∈ V ∩ H 2 (Ω ). Let us consider w ∈ V ∩ H 2 (Ω ), a solution of the problem
−△w = −△u0 w = 0 ∂w = −α h(z ) ∂ν
in Ω on Γ0
(61)
on Γ1 .
By Propositions 3–5, we have
‖w −
u0 2V ∩H 2 (Ω )
‖
2 ∂w ∂ u0 = |△w − △u | + < c ε2 . − = ‖α h(z ) − α h(u1 )‖2 1 ∂ν ∂ν H 12 (Γ1 ) H 2 (Γ1 ) 0 2
Thus, we find w, z satisfying the conditions of the proposition.
The above results allow us to prove the result with h(s). 4.1. Proof of Theorem 2 We proceed as in the proof of Theorem 1. Proposition 2 gives a sequence (hl ) of functions of C 0 (R) that verifies (i)–(iv) of this proposition without x ∈ Γ1 . By hypothesis (9) we have θ 0 ∈ V ∩ H 2 (Ω ) and
∂θ 0 + β h(θ 0 ) = 0 on Γ1 . ∂ν
(62)
Proposition 6 provides two sequences (u0l ) and u1l of vectors of V ∩ H 2 (Ω ) such that u0l → u0
in V ∩ H 2 (Ω ),
u1l → u1
in V
(63)
and
∂ u0 + α h(u1 ) = 0 on Γ1 . ∂ν
(64)
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Fix l. Consider a basis β = {w1l , w2l , . . .} of V ∩ H 2 (Ω ) such that u0l , u1l and θ 0 belong to the subspace [w1l , w2l , w3l ] generated l by w1l , w2l , w3l . Denote by Vml = [w1l , w2l , . . . , wm ] the subspace generated by the first m vectors of the basis β. Consider the approximate solutions ulm (t ) =
m −
gjml (t )wjl ,
θlm (t ) =
m −
j =1
hjml (t )wjl
j =1
of system (12) defined by the following system of ordinary differential equations:
∫
n − ∂θℓm
(t ), v + (|um (t )|σ um (t ), v) = 0, ∂ xi ∫ n − ∂ u′ℓm β hl (θℓm (t ))w dΓ + (t ), w = 0, ∀w ∈ Vml ; (θℓ′ m (t ), w) + ((θℓm (t ), w)) + ∂ xi Γ1 i=1 (u′′ℓm (t ), v)
+ ((uℓm (t ), v)) +
uℓm (0) = u0ℓ ,
Γ1
u′ℓm (0) = u1ℓ
α hl (u′ℓm (t ))v dΓ
+
∀v ∈ Vml ;
(65)
i =1
and θℓm (0) = θ 0 .
(66) (67)
By arguments similar to those used to obtain the first and second estimates of Theorem 1, we find E1lm (t ) +
t
∫
‖θlm (s)‖ ds + 2
α0 d0 2
0
≤
1 2
∫ t∫
(|u1 |2 + ‖u0 ‖2 + |θ 0 |2 +
0
2c1
σ +2
|ulm (s)| dΓ ds + d1 ′
Γ1
2
∗
∫ t∫ 0
‖u0 ‖σ +2 ) + η,
Γ1
|θlm (s)|2 dΓ ds
∀l ≥ l0 , ∀m (η positive constant),
2
where d∗1 = β0 d0 − 2αn d (which is non-negative by hypothesis (H8)) and E1lm (t ) has a similar form to (32). Also 0 0 E2lm (t ) +
t
∫
′ ‖θlm (s)‖2 ds +
α0 d0
0
≤
1 2
∫ t∫
2
0
Γ1
|u′′lm (s)|2 dΓ ds + d∗1
′ [‖u′′lm (0)‖2 + ‖u1l ‖2 + |θlm (0)|2 ] + C
∫ t∫ 0
Γ1
′ |θlm (s)|2 dΓ ds
t
∫
[‖u′lm (s)‖2 + |u′′lm (s)|2 ]ds, 0
where E2ml (t ) has a similar form to (36). Thanks to (62) and (64), we obtain
′′ |u (0)| ≤ c , lm ′ |θlm (0)| ≤ c ,
∀l, m ∀l, m.
With the three preceding results we proceed as in the proof of Theorem 1 and obtain a solution {u, θ} for (12)–(14). 5. Proof of the decay of the energy We begin with some observations. Remark 2. Assume hypothesis (H10). Then the approximations (hl ) of h given in the proof of Proposition 2 satisfy
|hl (s)| ≤
3 ∗ k |s|, 2
∀s ∈ R, ∀l.
(68)
This is obtained by direct computations in the formula for hl given in the proof of Proposition 1. We consider the solution {ul , θl } given in the proof of Theorem 2 with hl and u0l , u1l , θ 0 . We have u′l ∈ L∞ loc (0, ∞; V ). We obtain
∂ ∂ u′ hl (u′l (x, t )) = h′l (u′l (x, t )) l (x, t ) ∂ xi ∂ xi (see Brezis and Cazenave [17]). Thus hl (u′l ) ∈ L∞ loc (0, ∞; V ). Therefore, 1
2 α hl (u′l ) ∈ L∞ loc (0, ∞; H (Γ1 )).
Then Eq. (52) provides
∂ ul 1 2 ∈ L∞ loc (0, ∞; H (Γ1 )). ∂ν
A.T. Lourêdo et al. / Nonlinear Analysis 74 (2011) 6988–7004
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The further regularity ul ∈ L ∞ loc (0, ∞; V ),
2 ∆ ul ∈ L ∞ loc (0, ∞; L (Ω ))
implies 2 ul ∈ L ∞ loc (0, ∞; V ∩ H (Ω )).
(69)
In similar way,
θl ∈ L2loc (0, ∞; V ∩ H 2 (Ω )).
(70)
5.1. Proof of Theorem 3 We consider the solution {ul , θl } given in the proof of Theorem 2 with hl and u0l , u1l , θ 0 . Thus ul , θl belong to class (11), and ul and θl verify (69) and (70) respectively. Also ul , θl satisfy Eqs. (12) and the equations
∂ ul 1 2 + α h(u′l ) = 0 in L∞ loc (0, ∞; H (Γ1 )) ∂ν ∂θl 1 + β h(θl ) = 0 in L2loc (0, ∞; H 2 (Γ1 )) ∂ν
(71) (72)
and verify the initial conditions ul (0) = u0l ,
u′l (0) = u1l ,
θl (0) = θ 0 .
We prove Theorem 3 for the solution {ul , θl }. Theorem 3 will follow by taking the lim inf of both sides of the inequality with l for the theorem. Introduce the energy 1
Eℓ ( t ) =
2
|u′ℓ (t )|2 + ‖uℓ (t )‖2 + |θℓ (t )|2 +
∫
2
σ +2
Ω
|uℓ (t )|σ +2 dx ,
t ≥ 0,
(73)
and the functional
ρ(t ) = 2(u′ℓ (t ), m · ∇ uℓ (t )) + (n − 1)(u′ℓ (t ), uℓ (t )),
t ≥0
(74)
and consider the perturbed energy Eεl (t ) = El (t ) + ερ(t ),
(ε > 0).
(75)
In order to ease the notation in what follows we do not write the sub-index l and the variable t; we have
|ρ(t )| ≤ c ∗ E (t ),
t ≥ 0,
∗
where c was defined in (18). The above inequality gives 1 2
E (t ) ≤ Eε (t ) ≤
3 2
E (t ),
∀ t ≥ 0 , ∀ 0 < ε ≤ ε1
(76)
where 2
ε1 =
c∗
.
(77)
We have Eε′ = E ′ + ερ ′ .
(78)
Multiplying Eq. (13)1 with u and Eq. (13)2 with θ and integrating over Ω , we obtain ′
′
E +
1 2k1
1
1
|θ| + ‖θ ‖ + α0 d0 2
2
2
2
∫
|u | dΓ + d1 ′ 2
Γ1
∗
∫ Γ1
|θ |2 dΓ ≤ 0
where k1 > 0 and d∗1 ≥ 0 were defined in (16) and (17), respectively. The idea is to obtain
|ρ ′ | ≤ −aE ,
a positive constant.
Eq. (12)1 provides u′′ = △u −
n − ∂θ 2 − |u|σ u in L∞ loc (0, ∞; L (Ω )). ∂ x i i=1
(79)
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A.T. Lourêdo et al. / Nonlinear Analysis 74 (2011) 6988–7004
We differentiate ρ(t ) with respect to t and use the preceding equality. We find
n − ∂θ , m · ∇ u − 2(|u|σ u, m · ∇ u) ρ (t ) = 2(∆u(t ), m · ∇ u(t )) − 2 ∂ x i i =1 n − ∂θ ′ ′ + 2(u (t ), m · ∇ u (t )) + (n − 1)(△u(t ), u(t )) − (n − 1) ,u ∂ xi i=1
′
− (n − 1)(|u|σ u, u) + (n − 1)|u′ (t )|2 = I1 + I2 + · · · + I8 . • Modifications of I1 . Integrating
∂u m ∂u ∂ xj k ∂ xk
∂ ∂ xj
∂ ∂ xk
and
I1 = −(2 − n)‖u‖2 −
[ mk
∂u ∂ xj
2 ]
over Ω , and applying Gauss’s Theorem, we find
n ∫ n ∫ − − ∂u ∂u 2 ∂u mk (mk νK )dΓ + 2 νj dΓ = −(2 − n)‖u‖2 + J1 + J2 . ∂ x ∂ x ∂ x j j k j,k=1 Γ j,k=1 Γ
We have
2 2 n ∫ n ∫ − − ∂u ∂u J1 = − (m · ν)dΓ − (m · ν)dΓ . ∂ x ∂ xj j j=1 Γ0 j=1 Γ1 Also
n ∫ n ∫ − − ∂u ∂u ∂u ∂u νj mk dΓ + 2 νj mk dΓ . J2 = 2 ∂ xj ∂ xk ∂ xj ∂ xk j,k=1 Γ1 j,k=1 Γ0 ∂u on Γ0 . Then Note that ∂∂xu = νj ∂ν j
∫ J2 = 2
Γ0
∂u ∂ν
2
(m · ν)dΓ + 2
∫ Γ1
∂u (m · ∇ u)dΓ . ∂ν
Using the boundary equation (13)1 and hypothesis (H10), and noting that m(x) · ν(x) ≥ τ0 > 0 on Γ1 , we get
∫ 2 Γ0
∫ ∂u (m · ∇ u)dΓ = −2 α h(u′ )(m · ∇ u)dΓ ∂ν Γ1 ∫ ≤2 ‖α‖L∞ (Γ1 ) k∗ |u′ |R(x0 )|∇ u|dΓ Γ1 ∫ ∫ 1 ∗ 2 2 2 0 ′ 2 ≤ (k ) ‖α‖L∞ (Γ1 ) R (x ) |u | dΓ + (m · ν)|∇ u|2 dΓ . τ0 Γ1 Γ1
Thus
∫ J2 ≤ 2
Γ0
∂u ∂ν
2
(m · ν)dΓ + K
∫ Γ1
|u′ |2 dΓ +
∫ Γ1
(m · ν)|∇ u|2 dΓ
where K was defined in (19). Adding J1 and J2 and noting that m(x) · ν(x) ≤ 0 on Γ0 , we find
∫ J1 + J2 ≤ K
Γ1
|u′ |2 dΓ .
Thus I1 ≤ −(2 − n)‖u‖ + K 2
∫ Γ1
|u′ |2 dΓ .
(80)
• Modification of I2 . By the Cauchy–Schwarz inequality and an elementary inequality, we find 1
|I2 | ≤ 4nR2 (x0 )‖θ ‖2 + ‖u‖2 . 4
(81)
A.T. Lourêdo et al. / Nonlinear Analysis 74 (2011) 6988–7004
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• Modification of I3 . In this and the two next modifications we use Gauss’s Theorem. Thus ∫ ∫ 2 2n |u|σ +2 dx − |u|σ +2 (m · ν)dΓ I3 = σ +2 Ω σ + 2 Γ1 which implies
∫
2n
I3 ≤
σ +2 • Modification of I4 .
Ω
I4 = −n|u′ |2 +
|u|σ +2 dx.
∫ Γ1
(82)
(m · ν)|u′ |2 dΓ ≤ −n|u′ |2 + R(x0 )
∫ Γ1
| u′ | 2 d Γ .
(83)
• Modification of I5 . I5 = −(n − 1)‖u‖2 + (n − 1)
∫ Γ1
(m · ν)
∂u udΓ . ∂ν
Using boundary equation (13)1 and by arguments similar to those applied to obtain the modification of J2 , we get
∫
(n − 1)
Γ1
(m · ν)
∂u udΓ = −(n − 1) ∂ν
∫ Γ1
(m · ν)h(u′ )udΓ ≤ (n − 1)2 R2 (x0 )k2 (k∗ )2
∫
1
Γ1
| u′ | 2 d Γ + ‖ u‖ 2 , 4
where k2 was introduced in (16). So I5 ≤ −(n − 1)‖u‖2 + L
∫
1
Γ1
|u′ |2 dΓ + ‖u‖2 ,
(84)
4
where L was introduced in (15).
• Modification of I6 . We have
I6 ≤ n(n − 1)2 k2 ‖θ ‖2 +
1 4
‖ u‖ 2 .
• Modifications of I7 and I8 . ∫ I7 = −(n − 1) |u|σ +2 dx,
(85)
I8 = (n − 1)|u′ |2 .
Ω
(86)
Finally, adding the two sides of (80)–(86) and simplifying similar terms, we obtain 1
2
4
σ +2
ρ ′ ≤ − ‖u‖2 − |u′ |2 − M
∫
|u|σ +2 dx + N ‖θ ‖2 + P Ω
∫ Γ1
| u′ | 2 d Γ
where M , N and P were defined in (21)–(23), respectively. The preceding inequality and (78) and (79) provide E′ ≤ −
1 2k1 ∗
∫
+ d1
1
1
2
2
|θ|2 − ‖θ ‖2 − α0 d0 ε
Γ1
|u′ |2 dΓ
|θ| dΓ − ‖u‖ − ε|u | − ε M 2
Γ1
∫
2
2
′ 2
4
σ +2
∫
σ +2
|u| Ω
dx + ε N ‖θ ‖ + ε P 2
∫ Γ1
| u′ | 2 d Γ .
Consider
ε2 = min
1 2N
,
α0 d0 1 , 2P
k1
,
ε3 = min
ε
2
2
, ε2 M .
We have Eε′ (t ) ≤ −ε E (t ),
∀ t ≥ 0 , ∀ 0 < ε ≤ ε3 .
Take η = min{ε1 , ε3 }. Then by (76) and (87), we obtain 2 Eη′ (t ) ≤ − ηEη (t ), 3 which implies
∀t ≥ 0,
2
Eη (t ) ≤ Eη (0)e− 3 ηt . This and (76) provide 2
El (t ) ≤ 3El (0)e− 3 ηt ,
∀t ≥ 0, ∀l ≥ l0 .
(87)
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