On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback

Nonlinear Analysis 71 (2009) 5961–5978

Contents lists available at ScienceDirect

Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedbackI Bao-Zhu Guo a,b,∗ , Zhi-Chao Shao b,c a

Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, PR China

b

School of Information Technology and Management, University of International Business and Economics, Beijing 100029, PR China

c

School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa

article

info

Article history: Received 23 November 2007 Accepted 4 May 2009 MSC: 35L70 35B40 35B37 93D15 Keywords: Semilinear wave equation Nonlinear boundary feedback Riemannian geometry method Variable coefficients

abstract The uniform stabilization of an originally regarded nondissipative system described by a semilinear wave equation with variable coefficients under the nonlinear boundary feedback is considered. The existence of both weak and strong solutions to the system is proven by the Galerkin method. The exponential stability of the system is obtained by introducing an equivalent energy function and using the energy multiplier method on the Riemannian manifold. This equivalent energy function shows particularly that the system is essentially a dissipative system. This result not only generalizes the result from constant coefficients to variable coefficients for these kinds of semilinear wave equations but also simplifies significantly the proof for constant coefficients case considered in [A. Guesmia, A new approach of stabilization of nondissipative distributed systems, SIAM J. Control Optim. 42 (2003) 24–52] where the system is claimed to be nondissipative. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Many results concerning the boundary stabilization of classical wave equations are available in literature. We refer the reader to [1–4] for linear cases and [5–12] for nonlinear ones. The paper [5] is of special interest since the boundary feedback comprises both the nonlinear component and the memory source term that brings much difficulty to the problem. The earlier attempt using the classical analysis method for the wave equation with variable coefficients is probably [13]. The recent efforts on nonlinear boundary stabilization can be found in [14–16]. In [14], the stabilization of transmission problem for the wave equation with variable coefficients was investigated. [17] considered the stabilization of an Euler–Bernoulli plate equation with variable coefficients subject to nonlinear boundary feedback. The uniform stabilization of the damped Cauchy–Ventcel problem with variable coefficients and dynamic boundary conditions was obtained in [18]. In all these recent works mentioned above, the Riemannian geometry method is adopted in investigations. This method was first introduced in [19] to obtain the observability inequality, in terms of certain geometry conditions, for the wave equation with variable coefficients. The method was then applied to establish the exact controllability for the second-order

I This work was supported by the National Natural Science Foundation of China and the National Research Foundation of South Africa. Zhi-Chao Shao acknowledges the postdoctoral fellowship of the Claude Leon Foundation in South Africa and also acknowledges the support of Project 211, Phase 3 of University of International Business and Economics (project No. 31042). ∗ Corresponding author at: Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, PR China. Tel.: +86 1062651443; fax: +86 1062587343. E-mail address: [email protected] (B.-Z. Guo).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.05.018

5962

B.-Z. Guo, Z.-C. Shao / Nonlinear Analysis 71 (2009) 5961–5978

hyperbolic equations with variable coefficients and first-order terms [20], and the observability estimates for the secondorder hyperbolic equations with variable coefficients [21]. In [22], the following classical semilinear wave equation was considered:

 00 u − ∆u + h(∇ u) + f (u) = 0    u = 0 ∂u + l(u0 ) = 0     ∂ν u(x, 0) = u0 (x), u0 (x, 0) = u1 (x)

in Ω × (0, ∞), on Γ0 × (0, ∞), (1.1)

on Γ1 × (0, ∞), in Ω .

The system (1.1) was claimed to be nondissipative in [22] and a special inequality was introduced to prove the exponential stability of the system. The system (1.1) was also considered later in [23] where both the existence of strong (or weak) solution and the uniform decay rate of the solution were established. Motivated from these works, we consider, in this paper, an equation of the same form as (1.1) by replacing ∆ in (1.1) with ∆g for a Riemannian metric g = h·, ·ig . The system can be considered as a semilinear wave equation with variable coefficients, which covers the system (1.1) as a special case. The stability of a nondissipative system described by partial differential equations (PDEs) represents most often an extremely mathematical challenge. In recent years, the nondissipative systems have attracted much attention in PDEs. [24] developed the exponential stability for an abstract nondissipative linear system, and in [25], the Riesz basis property was developed for a beam equation with nondissipativity. A new inequality criterion was introduced in [22] to achieve the uniform stabilization for the system (1.1) which is regarded as a nondissipative system. However, in this paper, using a completely different approach and the Riemannian geometry method, we show that the system (1.1) is essentially a dissipative system by introducing an equivalent energy function of the system. The uniform stabilization of such a semilinear wave equation with variable coefficients is thus obtained. This approach not only generalizes the result from constant coefficients to variable coefficients but also simplifies significantly the proof for constant coefficients case considered in [22]. The main contributions of this paper are: (a) a detailed proof for the existence of the solution of the system is given by the Faedo–Galerkin method as in [23] and [5]; (b) the Riemannian geometry approach seems necessary to solve the problem, which is different from the classical analysis; (c) the generalization from the case of constant coefficients to variable ones is not direct, and it needs some additional geometry constraints on the Riemannian manifold formed by variable coefficients that is not needed for constant coefficients case. We proceed as follows. In Section 2, some necessary notations are introduced. The main results are presented in Section 3 and some preliminary results are proven. Section 4 is devoted to the proofs of the main results. 2. Some notation Let Ω ⊂ Rn (n ≥ 2) be an open bounded domain with C 2 -boundary ∂ Ω = Γ0 ∪ Γ1 , where Γ0 and Γ1 are nonempty ∂ closed and disjoint, and ν := ∂ν = νi ∂∂xi be the unit normal outer vector field on ∂ Ω . Here and in the rest of the paper, the symbol of Einstein summation is used. Let {aij (x), 1 ≤ i, j ≤ n} be smooth functions in Rn , satisfying aij (x) = aji (x),

λ|ξ |2 ≤ aij (x)ξi ξj ≤ Λ|ξ |2 ,

∀x ∈ Rn , ξ = (ξ1 , . . . , ξn ) ∈ Rn ,

(2.1)

where λ, Λ are positive constants. Let A(x) = (aij (x)) be an n × n matrix for x ∈ Rn and G(x) = (gij (x)) = A−1 (x) be its inverse matrix. For each x ∈ Rn define the inner product and norm on the tangent space Rnx = Rn by 1

g (X , Y ) = hX , Y ig = gij αi βj ,

|X |g = hX , X ig2 ,

∀X = αi

∂ ∂ , Y = βi ∈ Rnx . ∂ xi ∂ xi

(2.2)

Then (Rn , g ) is a Riemannian manifold with the Riemannian metric g [19]. Denote by D, ∇g , divg , and ∆g the Levi-Civita connection, the gradient operator, the divergence operator and the Beltrami–Laplace operator in terms of the Riemannian metric g, respectively. It can be easily shown that under the Euclidean coordinate,

 ∂u ∂ ∇g u(x) = aij (x) , ∂ xj ∂ xi 

|∇g u(x)|2g = aij (x)

∂u ∂u , ∂ xj ∂ xi

(2.3)

and

∂ ∆g u = √ G ∂ xi 1

 p

∂u Gaij ∂ xj 

where G = det(G), Au = ∂∂x i



aij ∂∂xu

j

= Au − h∇g ϕ, ∇g uig = Au − aij 

and ϕ =

1 2

log(det A).

∂ϕ ∂ u , ∂ xj ∂ xi

(2.4)

B.-Z. Guo, Z.-C. Shao / Nonlinear Analysis 71 (2009) 5961–5978

5963

Let νA = aij νj ∂∂x be the co-normal vector field on ∂ Ω . If we define i

akl νl ∂ νA ∂ = √ , (2.5) µ := = ∂µ |νA |g aij νi νj ∂ xk then µ is the unit normal vector field on ∂ Ω in terms of the metric g. Let H be a vector field on (Rn , g ). Then for each x ∈ Rn , the covariant differential DH of H determines a bilinear form on Rnx :

DH (X , Y ) = hDY H , X ig

∀ X , Y ∈ Rnx ,

(2.6)

where DY H stands for the covariant derivative of the vector field H with respect to Y . 3. The main results We consider the semilinear wave equation with variable coefficients under the nonlinear boundary feedback, which is of the same form of (1.1) by replacing ∆, ν with ∆g , µ produced by the Riemannian metric g introduced in Section 2:

 00 u − ∆g u + h(∇ u) + f (u) = 0 in Ω × (0, ∞),   u = 0 on Γ0 × (0, ∞),  ∂u (3.1) 0 + l(u ) = 0 on Γ1 × (0, ∞),   ∂µ   u(x, 0) = u0 (x), u0 (x, 0) = u1 (x) in Ω , where f , l : R → R and h : Rn → R are continuous nonlinear functions, and ∆g and µ are defined in (2.4) and (2.5), respectively. The operator ∇ stands for the gradient operator in the Euclidean space Rn . The following assumptions are needed. Assumption 3.1 (Assumption on f ). f : R → R is a C 1 -function deriving from a potential F : F (s) =

s

Z

f (τ )dτ ≥ 0 ∀ s ∈ R,

(3.2)

0

and satisfies

|f (s)| ≤ b1 |s|ρ + b2 ,

|f 0 (s)| ≤ b1 |s|ρ−1 + b2 ,

(3.3)

where b1 , b2 are positive constants, and the parameter ρ satisfies 2,

n ≤ 3,

( 1≤ρ≤

n

n−2

,

(3.4)

n ≥ 4.

Assumption 3.2 (Assumption on h). h : Rn → R is a C 1 -function, and there exist constants β > 0 and L > 0 such that

√ |h(ξ )| ≤ β λ|ξ | ∀ ξ ∈ Rn ,

(3.5)

|∇ h(ξ )| ≤ L ∀ ξ ∈ R .

(3.6)

n

Assumption 3.3 (Assumption on l). l : R → R is a C 1 -nondecreasing function, and there exist two positive constants c1 and c2 such that c1 |s|2 ≤ l(s)s ≤ c2 |s|2

∀ s ∈ R.

(3.7)

Remark 3.1. An example of the function f satisfying Assumption 3.1 and (3.11) in Theorem 3.2 later is given by 2,

( ρ−1

f (s) = γ |s|

s

for some constants γ > 0

and

1<ρ≤

n

n ≤ 3,

n−2

,

n ≥ 4.

Assumption 3.4. There exists a vector field on the Riemannian manifold (Rn , g ) such that DH (X , X ) = c (x)|X |2g

∀ x ∈ Ω , X ∈ Rnx ,

(3.8)

where b = minΩ c (x) > 0 and B = maxΩ c (x)



B < min b +

2b − 3ε0 n



, r b−

ε0  n



for some ε0 ∈ (0, b) and r > 1.

(3.9)

5964

B.-Z. Guo, Z.-C. Shao / Nonlinear Analysis 71 (2009) 5961–5978

Moreover, H ·ν ≤0

on Γ0

H · ν ≥ δ > 0 on Γ1

and

(3.10)

for some constant δ . Remark 3.2. For the constant coefficients case, aij = δij , the condition (3.8) is automatically satisfied by choosing H (x) = x − x0 for fixed x0 and c (x) ≡ 1. For the variable coefficients case, [19] presents some nontrivial examples for which the condition (3.8) is satisfied without constraint on B. Due to the continuity of the function c (x), there always exists a geodesic ball in (Rn , g ) with small radius centered at the minimum point of c (x). Hence the condition (3.8) is satisfied for any Ω located in the ball. An example in R2 will be given at the end of this section to meet Assumption 3.4. Let V = v ∈ H 1 (Ω )| v = 0 on Γ0





and W = H 2 (Ω ) ∩ V .

Definition 3.1. The function u is said to be a weak solution of system (3.1) if u ∈ L∞ (0, T ; V ), u0 ∈ L∞ (0, T ; L2 (Ω )) and satisfies T

Z

Z Ω

0

[uφ 00 + h∇g u, ∇g φig + h(∇ u)φ + f (u)φ]dxdt =

Z Ω

u1 φ(0)dx −

Z Ω

u0 φ 0 (0)dx −

T

Z 0

Z Γ1

l(u0 )φ dΓ dt

for any φ ∈ C 2 (0, T ; V ) with φ(T ) = φ 0 (T ) = 0. Now we state our main results. ∂u

Theorem 3.1. Under Assumptions 3.1–3.3, for any initial data (u0 , u1 ) ∈ W × W satisfying ∂µ0 + l(u1 ) = 0 on Γ1 , the system (3.1) admits a unique strong solution u : (0, ∞) → R such that u ∈ L∞ (0, ∞; V ),

u0 ∈ L∞ (0, ∞; V )

and

u00 ∈ L∞ (0, ∞; L2 (Ω )).

Moreover, if (u0 , u1 ) ∈ V × L2 (Ω ), then (3.1) possesses at least a weak solution in the space C ([0, ∞); V ) ∩ C 1 ([0, ∞); L2 (Ω )). Theorem 3.2. Let u be a (strong or weak) solution to the system (3.1). Suppose in addition that f satisfies 2rF (s) ≤ sf (s),

∀ s ∈ R for some constant r > 1,

(3.11)

and (3.2). Also suppose that h satisfies (3.5) and Assumptions 3.2–3.4 hold. If β in (3.5) is sufficiently small, then the energy of the system (3.1) defined by E (t ) =

Z Ω

 0 2  |u (t )| + |∇g u(t )|2g + 2F (u(t )) dx

(3.12)

decays exponentially to zero in the sense that E (t ) ≤ cE (0)e−ωt

∀t ≥ 0,

(3.13)

for some positive constants c and ω independent of initial values. In order to prove the main results, we need several lemmas. Lemma 3.1 ([26, pp. 128, 138]). Let u, v ∈ C 1 (Ω ) and H be a vector field on (Rn , g ). Then it has (a) divergence theorem:

Z

divg (uH ) = u divg (H ) + H (u),

Ω

divg (H ) dx =

Z ∂Ω

hH , µig dΓ ;

(b) the Green formula:

Z Ω

v ∆g u dx =

Z ∂Ω

v

∂u dΓ − ∂µ

Z Ω

h∇g u, ∇g vig dx,

where dx and dΓ stand for the volume elements of Ω and ∂ Ω , respectively. Lemma 3.2. Let H be a vector field on (Rn , g ). Let {ei }ni=1 be an orthonormal frame field on Ω and {w i }ni=1 be its dual frame field. Then we have divg H =

n n X X hDei H , ei ig = DH (ei , ei ). i =1

i=1

(3.14)

B.-Z. Guo, Z.-C. Shao / Nonlinear Analysis 71 (2009) 5961–5978

5965

Proof. Let D be the Levi-Civita connection in (Rn , g ) and the operator C11 be the contraction of tensor field. Denote by DH the (1, 1)-type tensor field. By the definition of operator divg , we have divg H = C11 (DH ) = C11 (DH (w i , ej )ei ⊗ w j )

=

n X

DH (w i , ei ) =

i=1

=

n X

n X

Dei H (w i ) =

w i (Dei H )

i =1

i=1

w i (Dei H )hei , ei ig =

n X

n n X X hDei H , ei ig . hw i (Dei H )ei , ei ig = i=1

i =1

i=1

(3.14) is then verified by virtue of the definition of DH (ei , ei ) in (2.6).



Remark 3.3. Combining (3.8) in Assumption 3.4 and Lemma 3.2, we have nb ≤ divg H ≤ nB in Ω .

(3.15)

This observation is important for the proof of the exponential stability claimed by Theorem 3.2. The following lemma is similar to (4) of Lemma 2.1 in [19]. Lemma 3.3. Let H be a vector field on (Rn , g ). For u ∈ C 1 (Ω ) and x ∈ Rn the following formula holds 1

1

2

2

h∇g u, ∇g (H (u))ig (x) = DH (∇g u, ∇g u)(x) + divg (|∇g u|2g H )(x) − |∇g u|2g (x)divg (H )(x).

(3.16)

Proof. Let x ∈ Rn be fixed. Let {Ei }ni=1 be a frame field normal at x on the Riemannian manifold (Rn , g ). This means that in some neighborhood of x, {Ei }ni=1 is a local basis satisfying

hEi , Ej ig = δij and DEi Ej (x) = 0 for 1 ≤ i, j ≤ n. Since H is a vector field, there exist functions h1 , h2 , . . . , hn such that H = H ( u) =

n X

Pn

i=1

hi Ei . Hence

hi Ei (u)

(3.17)

i=1

and

∇g u =

n X

Ei (u)Ei

and |∇g u|2g =

i=1

n X (Ei (u))2 ,

(3.18)

i =1

where Ei (u)(1 ≤ i ≤ n) is the covariant differential of f with regard to Ei in the Riemannian metric g. Notice that hEi , Ej ig = δij and DEi Ej (x) = 0 for 1 ≤ i, j ≤ n. We obtain DH (Ei , Ej )(x) = hDEi H , Ej ig (x) = hDEi (hk Ek ), Ej ig (x)

= hEi (hk )Ek + hk DEi Ek , Ej ig (x) = Ei (hj )(x).

(3.19)

Since divg (uH ) = udivg (H ) + H (u), it follows from (3.17), (3.18) and (3.19) that

h∇g u, ∇g (H (u))ig (x) = Ei (f )Ei (H (u))(x) = Ei (u)[Ei (hj )Ej (u) + hj Ei Ej (u)](x) = Ei (hj )Ei (u)Ej (u) + hj Ei (u)Ej Ei (u)(x) 1

= DH (∇g u, ∇g u)(x) + H (|∇g u|2g ) 2 1

1

2

2

= DH (∇g u, ∇g u)(x) + divg (|∇g u|2g H )(x) − |∇g u|2g (x)divg (H )(x),

(3.20)

where Ei Ej is the second covariant derivatives with regard to g. Notice that in the last step of (3.20), we used (a) of Lemma 3.1. The formula (3.16) then follows from the arbitrariness of x.  Proposition 3.1 ([27, pp. 390]). Suppose that there is a metric gˆ such that (Rn , g ) has zero curvature and there is a function u on Rn to meet the relation g = e2u gˆ

∀ x ∈ Rn .

5966

B.-Z. Guo, Z.-C. Shao / Nonlinear Analysis 71 (2009) 5961–5978

Given x0 ∈ Rn . Denote by ρ( ˆ x) the distance function from x0 to x in the metric gˆ . Set

ˆ ρ, H = ρˆ D ˆ

(3.21)

ˆ is the Levi-Civita connection on (Rn , gˆ ). Then where D DH (X , X ) = [1 + H (u)]|X |2g

∀ X ∈ Rnx , x ∈ Rn .

(3.22)

To end this section, we present an example which comes from [27] to meet Assumption 3.4. Example 3.1 ([27, pp. 391]). Let (R2 , g ) be a Riemannian manifold with the metric g =

1 1 + ax2 + by2

(dx2 + dy2 ),

where a and b are positive constants. For any (x, y) ∈ R2 , the Gaussian curvature is k(x, y) =

a + b + a(b − a)x2 + b(a − b)y2

(1 + ax2 + by2 )3

.

By (3.21) we have g = e2u gˆ with u = − 21 log(1 + x2 + y2 ) and gˆ = dx2 + dy2 . (R2 , gˆ ) is of zero curvature. Take ρˆ = Then H =

x ∂∂x

+

y ∂∂y

1 + H (u) =

p

x2 + y 2 .

and 1

1 + ax2 + by2

> 0 ∀ ( x, y ) ∈ R 2 .

It follows from Proposition 3.1 that for any given annuluses bounded by two concentric circles centered at the origin Ω ⊂ R2 , if the constants a and b are small enough, then Assumption 3.4 holds. 4. Proof of the main results Proof of Theorem 3.1. We use the Galerkin approximation to prove the existence of solution to (3.1). A variational formulation for problem (3.1) is given by

Z Ω

u00 w dx +

Z Ω

h∇g u, ∇g wig dx +

Z Ω

h(∇ u)w dx +

Z Ω

f (u)w dx +

Z Γ1

l(u0 )w dΓ = 0 ∀ w ∈ V .

The change of variable

v(x, t ) = u(x, t ) − φ(x, t )

(4.1)

where

φ(x, t ) = u0 (x) + tu1 (x),

(x, t ) ∈ Q , Ω × (0, T )

(4.2)

gives the equivalent problem to (3.1):

 00 v − ∆g v + h(∇v + ∇φ) + f (v + φ) = F   v = 0 on Γ × (0, T ),  0  ∂v  + l(v 0 + φ 0 ) = B on Γ1 × (0, T ),     ∂µ v(0) = v 0 (0) = 0,

in Q , (4.3)

where

F = ∆g φ and

B=−

∂φ . ∂µ

Let {wi }i∈N be a basis for W that is orthonormal in L2 (Ω ), and let Vm be the space spanned by w1 , . . . , wm . Let

vm (t ) =

m X j =1

γj (t )wj

(4.4)

B.-Z. Guo, Z.-C. Shao / Nonlinear Analysis 71 (2009) 5961–5978

5967

be the solution to the Cauchy problem:

Z

Z Z vm00 (t )w dx + h∇g vm (t ), ∇g wig dx + h(∇vm (t ) + ∇φ(t ))w dx Ω Ω Ω Z Z 0 l(vm (t ) + φ 0 (t ))w dΓ + f (vm (t ) + φ(t ))w dx + Ω

Z = Ω

Γ1

F (t )w dx +

Z Γ1

B (t )w dΓ

∀ w ∈ Vm , and vm (0) = vm0 (0) = 0.

(4.5)

The existence of solution on some interval [0, Tm ) for the system (4.5) can be proven by standard methods of ordinary differential equations. The existence of the solution in the whole interval [0, T ] is a consequence of the first estimate in Step 1 below. The proof will be accomplished by splitting into several steps. Step 1: The first-order estimate of vm . 0 Replace w by vm (t ) in (4.5) to get

1 d

Z

 0  |vm (x, t )|2 + |∇g vm (x, t )|2g + 2F (vm (x, t ) + φ(x, t )) dx 2 dt Ω Z Z Z 0 0 + h(∇vm + ∇φ)vm dx − f (vm + φ)φ 0 dx + l(vm + φ 0 )(vm0 + φ 0 )dΓ Ω

Z

Ω

F (t )vm (t )dx + 0

= Ω

Z

d dt

Γ1

Γ1

B (t )vm (t )dΓ −

Z

B (t )vm (t )dΓ + 0

Γ1

Z Γ1

0 l(vm + φ 0 )φ 0 dΓ .

(4.6)

Firstly, by Assumption 3.1, the Sobolev imbedding theorem, and the regularities of the initial data, we have

Z Ω

Z |vm + φ|ρ |u1 |dx + C |u1 |dx Ω Ω Z  Z ρ ρ ≤C |vm | |u1 |dx + |φ| |u1 |dx + C

f (vm + φ)φ 0 dx ≤ C

Z

Ω

Ω

Z

 21 Z

2ρ

≤C Ω

Z ≤C Ω

|vm | dx

2

Ω

 ρ2

|∇g vm (t )|

2 g dx

 21

|u1 | dx

Z +C Ω

(|u0 |ρ |u1 | + t ρ |u1 |ρ+1 )dx + C

+ Ct ρ + C .

(4.7)

Here and in what follows, we use constant C > 0 to denote some constants independent of functions involved although it may have different values in different contexts. Secondly, by Assumption 3.2, it has

Z Ω

0 h(∇vm (t ) + ∇φ(t ))vm (t )dx ≤

β2 2

Z Ω

|∇g vm (t ) + ∇g φ(t )|2g dx +

1 2

Z Ω

|vm0 (t )|2 dx.

(4.8)

Thirdly, by Assumption 3.3, one has

Z Γ1

0 l(vm + φ 0 )(vm0 + φ 0 )dΓ ≥ C

Z Γ1

|vm0 (t ) + u1 |2 dΓ

(4.9)

and

Z

l(vm + φ )φ dΓ ≤ C 0

Γ1

0

0

Z Γ1

≤η

Z

≤η

Z

Γ1

Γ1

|vm0 (t ) + u1 ||u1 |dΓ |vm0 (t ) + u1 |2 dΓ + C (η)

Z

|vm0 (t ) + u1 |2 dΓ + C (η),

where η > 0 is a constant that will be determined later.

Γ1

|u1 |2 dΓ (4.10)

5968

B.-Z. Guo, Z.-C. Shao / Nonlinear Analysis 71 (2009) 5961–5978

Combining (4.6)–(4.10), and the trace theorem that that 1 d

Z

2 dt

R

Γ1

|vm |2 dΓ ≤ C0

R


|vm0 (t )|2 + |∇g vm (t )|2g + 2F (vm (t ) + φ(t )) dx + (C − η)

Ω

Ω

|∇g vm |2g dx for some constant C0 > 0, it follows

Z Γ1

|vm0 (t ) + u1 |2 dΓ

 ρ2

Z Z β2 1 + Ct ρ + |∇g vm (t ) + ∇g φ(t )|2g dx + |vm0 (t )|2 dx 2 Ω 2 Ω Ω Z Z Z d 1 1 2 + B (t )vm (t )dΓ + |F (t )| dx + |vm0 (t )|2 dx Z

|∇g vm (t )|

≤C

dt

+

C02

2 g dx

2

Γ1

Z

2

Γ1

1

|B 0 (t )|2 dΓ +

2

2

Ω

Z Ω

Ω

|∇g vm (t )|2g dΓ + C (η) + C .

(4.11)

0 Integrate (4.11) over the interval (0, t ) and notice vm (0) = vm (0) = 0 and ρ2 ≤ 1, to obtain

Z


|vm0 (t )|2 + |∇g vm (t )|2g + 2F (vm (t ) + φ(t )) dx + 2(C − η)

Ω

Z tZ Γ1

0

Z t Z ≤C Ω

0

+ 2β 2

|∇g vm (s)|

ds + Ct ρ+1 + (1 + 2β 2 )

Z tZ Ω

0

Z tZ 0

 ρ2

2 g dx

Ω

|∇g φ(s)|2g dxds +

Z tZ Ω

0

≤ Ct + (C + 1 + 2β 2 )

Z tZ Ω

0

Z

|F (s)|2 dxds +

|∇g vm (s)|2g dxds + 2

Γ1

Z tZ Ω

0

|vm0 (t ) + u1 |2 dΓ ds

|∇g vm (s)|2g dxds + 2

Z tZ Ω

0

B (t )vm (t )dΓ + C02 t

|vm0 (s)|2 dxds + η

Z Ω

|vm0 (s)|2 dxds

Z ∂ u1 2 dΓ + Ct + C Γ1 ∂νA

|∇g vm (t )|2g dx + C (t ρ+1 + t 3 ) + C . (4.12)

Finally, choosing η sufficiently small, by Gronwall’s lemma and the fact F (s) ≥ 0 for any s ∈ R, we obtain the first-order estimate of vm

Z

|vm (t )| + |∇g vm (t )| + 2F (vm (t ) + φ(t )) dx + 0

Ω

2

2 g




Z tZ 0

Γ1

|vm0 (t ) + u1 |2 dΓ ds ≤ C1 ,

(4.13)

where C1 > 0 is a constant independent of m ∈ N and t ∈ [0, T ]. Step 2: The second-order estimate of vm . 00 We first estimate the term kvm (0)kL2 (Ω ) . Take t = 0 in (4.5) and notice the fact vm (0) = vm0 (0) = 0 in (4.4), to obtain

Z Ω

vm00 (0)w dx + Z

= Ω

Z Ω

h(∇ u0 )w dx +

∆g u0 w dx +

Z  Γ1

∂ u0 − ∂µ



Z Ω

f (u0 )w dx +

w dΓ

Z Γ1

l(u1 )w dΓ

∀ w ∈ Vm .

(4.14)

∂u

Since ∂µ0 + l(u1 ) = 0 on Γ1 , it has

Z Ω

vm00 (0)w dx +

Z Ω

h(∇ u0 )w dx +

Z Ω

f (u0 )w dx =

Z Ω

∆g u0 w dx ∀ w ∈ Vm .

(4.15)

This together with Assumptions 3.1 and 3.2 and the regularity of the initial value gives

kvm00 (0)kL2 (Ω ) ≤ C2 ,

(4.16)

where C2 is a positive constant independent of m ∈ N. 00 Next, differentiate (4.5) with respect to t and replace w by vm , to obtain 1 d

Z

2 dt

Ω

Z + Ω


|vm00 (t )|2 + |∇g vm0 (t )|2g dx + 0 f 0 (vm + φ)(vm + φ 0 )vm00 dx +

Z Ω

Z Γ1


∇ h(∇vm + ∇φ) · (∇vm0 + ∇φ 0 ) vm00 dx

0 l0 (vm + φ 0 )(vm00 )2 dΓ =

Z Ω

00 F 0 (t )vm (t )dx +

d dt

Z Γ1

0 B 0 (t )vm (t )dΓ .

(4.17)

B.-Z. Guo, Z.-C. Shao / Nonlinear Analysis 71 (2009) 5961–5978

5969

By Assumption 3.2, we have

Z Ω

  Z Z
∇ h(∇vm + ∇φ) · (∇vm0 + ∇φ 0 ) vm00 dx ≤ C 1 + |∇g vm0 (t )|2g dx + |vm00 (t )|2 dx . Ω

(4.18)

Ω

Applying Hölder’s inequality, (4.13), and the Sobolev imbedding theorem, we deduce, by noticing the assumption on f 0 , that

Z Ω

Z

0 f 0 (vm + φ)(vm + φ 0 )vm00 dx ≤

Z

2(ρ−1)

|vm |

≤C Ω

Z Ω

Z ≤C Ω

Z

0 2

|φ|

|vm | dx + C n

≤C

|vm |2(ρ−1) 2 dx

0 C (|vm |ρ−1 + |φ|ρ−1 + C )(|vm | + |φ 0 |)|vm00 |dx

Ω

Ω

 2n Z

n

Ω

|∇g vm0 (t )|2g dx +

Z Ω

2(ρ−1)

Z

0 2

|vm | dx + C

|vm0 |2 n−2 dx

 n−n 2

Ω

Z +C Ω

|vm00 |2 dx + C

|φ|2(ρ−1) |vm0 |2 dx + C

Z Ω

|vm00 |2 dx + C

 |vm00 (t )|2 dx + C ,

(4.19)

and

Z Ω

Z

00 F 0 (t )vm (t )dx ≤ C

Ω

B (t )vm (t )dΓ ≤ C 0

Γ1

Z

0

|vm00 (t )|2 dx + C ,

C02 4η

+η

Z Ω

(4.20)

|vm0 (t )|2g dx.

(4.21)

Finally, combining (4.18)–(4.21), integrating (4.17) over (0, t ) and choosing η sufficiently small, by Gronwall’s lemma and the fact l0 (s) ≥ 0 for any s ∈ R, we obtain the second-order estimate of vm

Z Ω

|vm00 (t )|2 dx +

Z Ω

|∇g vm0 (t )|2g dx +

Z tZ 0

Γ1

0 l0 (vm (s) + φ 0 (s))(vm00 (s))2 dΓ ds ≤ C3 ,

(4.22)

where C3 is a positive constant independent of m ∈ N and t ∈ [0, T ]. Step 3: Analysis of the nonlinear term. By (4.13) and Assumption 3.2, we know that

{h(∇vm + ∇φ)}

is bounded in L∞ (0, T ; L2 (Ω )).

(4.23)

By the growth condition of f in Assumption 3.1 and (4.13), we have

{f (vm + φ)} is bounded in L∞ (0, T ; L2 (Ω )).

(4.24)

Denoting still by {vm } its convergent subsequence obviously from the context without confusion, it follows from (4.23) and (4.24) that there exist χ, ζ ∈ L2 (Ω ) such that h(∇vm + ∇φ) → χ

weakly in L2 (0, T ; L2 (Ω )),

(4.25)

and f (vm + φ) → ζ

weakly in L2 (0, T ; L2 (Ω )).

(4.26)

Next, by (4.22) and Assumption 3.3, we have that

{l(vm0 + φ 0 )} is bounded in L∞ (0, T ; L2 (Γ1 )).

(4.27)

Hence there exists ψ ∈ L (0, T ; L (Γ1 )) such that 2

0 l(vm + φ0) → ψ

2

weakly in L2 (0, T ; L2 (Γ1 )).

(4.28)

1 2

On the other hand, since H (Γ1 ) ,→ L2 (Γ1 ) is continuous and compact and that

kvm (t )k

Z

2 1

H 2 (Γ1 )

≤C Ω

|∇g vm (t )|

2 g dx

Z and Γ1

|vm (t )| dx ≤ C 0

2

Z Ω

|∇g vm0 (t )|2g dx,

(4.29)

it follows from the Aubin–Lions theorem (see Theorem 5.1 on the pages 57–58 of [28]), (4.13), and (4.22) that

vm → v strongly in L2 (0, T ; L2 (Γ1 )).

(4.30)

5970

B.-Z. Guo, Z.-C. Shao / Nonlinear Analysis 71 (2009) 5961–5978

Furthermore, (4.13) and (4.22) also imply that

vm00 → v 00 weakly in L2 (0, T ; L2 (Ω )),

(4.31)

∇g vm → ∇g v and ∇vm → ∇v weakly in L (0, T ; (L (Ω )) ). 2

2

n

(4.32)

Passing to the limit as m → ∞ in (4.5), we obtain

v 00 − ∆g v + χ + ζ = F

in D 0 (Q ).

Since v 00 , χ , ζ ∈ L2 (Q ), the above equality shows that ∆g v ∈ L2 (Q ) and

v 00 − ∆g v + χ + ζ = F

in L2 (0, T ; L2 (Ω )).

(4.33)

Thirdly, by the generalized Green formula (see the Appendix) and (4.33), we have

∂v 1 + ψ = B in D 0 (0, T ; H − 2 (Γ1 )). ∂µ

(4.34)

This together with the fact ψ, B ∈ L2 (0, T ; L2 (Γ1 )) shows that

∂v + ψ = B in L2 (0, T ; L2 (Γ1 )). ∂µ

(4.35)

Now, we are in a position to show that (a) f (v + φ) = ζ ; (b) h(∇v + ∇φ) = χ ; (c) l(v 0 + φ 0 ) = ψ . To this end, replace w by vm in (4.5) and integrate over [0, T ] in t to yield

Z

vm00 (t )vm (t )dxdt +

Z

Q

h∇g vm (t ), ∇g vm (t )ig dxdt +

Z

Q

Z

f (vm (t ) + φ(t ))vm (t )dxdt +

+ Z

T

Z

Q

0

F (t )vm (t )dxdt +

=

h(∇vm (t ) + ∇φ(t ))vm (t )dxdt Q

T

Z

Q

Z Γ1

0

Z Γ1

0 l(vm (t ) + φ 0 (t ))vm (t )dΓ dt

B (t )vm (t )dΓ dt ,

vm (0) = vm0 (0) = 0.

(4.36)

This together with (4.13), (4.22), and the Aubin–Lions theorem (see Theorem 5.1 on the pages 57–58 of [28]) implies that

vm → v strongly in L2 (0, T ; L2 (Ω )),

(4.37)

vm → v

(4.38)

0

0

strongly in L (0, T ; L (Ω )). 2

2

Furthermore, by (4.37), vm → v a.e. in Ω × [0, T ], and hence f (vm + φ) → f (v + φ)

a.e. in Ω × [0, T ].

(4.39)

Combine (4.24), (4.39) and apply the Lions lemma (Lemma 1.3 of [28]), to obtain f (vm + φ) → ζ = f (v + φ) weakly in L2 (0, T ; L2 (Ω )).

(4.40)

The first claim (a) is thus proved. Similarly, by virtue of (4.25), (4.28), (4.30), (4.37), and passing to the limit as m → ∞ in (4.36) gives

Z

|∇g vm (t )|2g dxdt = −

lim

m→∞

Q

Z

v 00 (t )v(t )dxdt − Q

Z

− 0

χ (t )v(t )dxdt − Q

T

Z

Z

Γ1

ψ(t )v(t )dΓ dt +

Z

f (v(t ) + φ(t ))v(t )dxdt Q

Z

F (t )v(t )dxdt + Q

T

Z 0

Z Γ1

B (t )v(t )dΓ dt .

(4.41)

Applying the generalized Green formula (see (A.4) of the Appendix) again, we have, from (4.33), (4.35) and (4.41), that

Z

|∇g vm (t )|

2 g dxdt

lim

m→∞

Q

Z

|∇g v(t )|2g dxdt .

=

(4.42)

Q

By (4.32) and (4.42), it has

∇g vm → ∇g v and ∇vm → ∇v strongly in L2 (0, T ; (L2 (Ω ))n ),

(4.43)

B.-Z. Guo, Z.-C. Shao / Nonlinear Analysis 71 (2009) 5961–5978

5971

which leads to the convergence of the following

∇g vm + ∇g φ → ∇g v + ∇g φ a.e. in Ω , ∇vm + ∇φ → ∇v + ∇φ a.e. in Ω . Consequently h(∇vm + ∇φ) → h(∇v + ∇φ)

a.e. in Ω .

(4.44)

With the application of the Lions lemma (Lemma 1.3 of [28]), (4.44) and (4.23) it guarantees that h(∇vm + ∇φ) → χ = h(∇v + ∇φ)

weakly in L2 (0, T ; L2 (Ω )).

(4.45)

This is claim (b). 0 In order to prove claim (c), replace w by vm in (4.5) and integrate over [0, T ], to obtain

Z

vm00 (t )vm0 (t )dxdt +

Z

h∇g vm (t ), ∇g vm0 (t )ig dxdt +

Z

Z

0 f (vm (t ) + φ(t ))vm (t )dxdt +

+ Z

T

Z

Z Γ1

0

Q

0 F (t )vm (t )dxdt +

=

0 h(∇vm (t ) + ∇φ(t ))vm (t )dxdt Q

Q

Q

T

Z

Q

Z Γ1

0

0 l(vm (t ) + φ 0 (t ))vm0 (t )dΓ dt

0 B (t )vm (t )dΓ dt ,

vm (0) = vm0 (0) = 0.

(4.46)

Since

vm0 → v 0 weakly in L2 (0, T ; L2 (Γ1 )), ∇g vm → ∇g v 0

0

(4.47)

weakly in L (0, T ; (L (Ω )) ), 2

2

n

(4.48)

combine (4.46) with (4.30), (4.31), (4.38), (4.43), and (4.45) to deduce T

Z

Z

lim

m→∞

Γ1

0

Z

0 l(vm (t ) + φ 0 (t ))vm0 (t )dΓ dt

v 00 (t )v 0 (t )dxdt −

=− Q

Z

Z

h(∇v(t ) + ∇φ(t ))v 0 (t )dxdt − Q

f (v(t ) + φ(t ))v 0 (t )dxdt Q

h∇g v(t ), ∇g v (t )ig dxdt + 0

−

Z

Z

Q

F (t )v (t )dxdt + 0

T

Z

Q

0

Z Γ1

B (t )v 0 (t )dΓ dt .

(4.49)

Applying the generalized Green formula (see (A.4) of the Appendix) and noticing (4.33) and (4.35), it deduces from the above equality that T

Z

Z

lim

m→∞

Γ1

0

l(vm (t ) + φ (t ))vm (t )dΓ dt = 0

0

0

Z

T

Z

0

Γ1

ψ(t )v 0 (t )dxdt .

(4.50)

On the other hand, by the monotonicity of l, we have T

Z

Z

0 l(vm (t ) + φ 0 (t )) − l(ϕ)




Γ1

0


vm0 (t ) + φ 0 (t ) − ϕ dΓ dt ≥ 0,

∀ϕ ∈ L2 (Γ1 ),

or equivalently, T

Z 0

Z Γ1

0 l(vm (t ) + φ 0 (t ))ϕ dΓ dt +

T

Z

T

Z 0

Z

≤ Γ1

0

Z Γ1

0 l(ϕ)(vm (t ) + φ 0 (t ) − ϕ) dΓ dt

0 l(vm (t ) + φ 0 (t ))(vm0 (t ) + φ 0 (t )) dΓ dt .

(4.51)

Hence T

Z

Z

lim inf m→∞

0

Γ1 T

Z

0 l(vm (t ) + φ 0 (t ))ϕ dΓ dt + lim inf m→∞

Z

≤ lim inf m→∞

0

Γ1

T

Z 0

Z Γ1

0 l(ϕ)(vm (t ) + φ 0 (t ) − ϕ) dΓ dt

0 l(vm (t ) + φ 0 (t ))(vm0 (t ) + φ 0 (t )) dΓ dt .

(4.52)

5972

B.-Z. Guo, Z.-C. Shao / Nonlinear Analysis 71 (2009) 5961–5978

This together with (4.28), (4.47), and (4.50) gives T

Z

Z Γ1

0


(ψ(t ) − l(ϕ)) v 0 (t ) + φ 0 (t ) − ϕ dΓ dt ≥ 0 ∀ ϕ ∈ L2 (Γ1 ).

(4.53)

Replace ϕ by v 0 + φ 0 + εξ in (4.53) for arbitrary ξ ∈ L2 (Γ1 ) and ε > 0, to get T

Z

Z Γ1

0

(ψ(t ) − l((v 0 (t ) + φ 0 (t )) + εξ ))(−εξ ) dΓ dt ≥ 0,

and hence, T

Z

Z Γ1

0

(ψ(t ) − l((v 0 (t ) + φ 0 (t )) + εξ ))ξ dΓ dt ≤ 0 ∀ ξ ∈ L2 (Γ1 ).

(4.54)

Furthermore, since the operator l : L2 (Γ1 ) → (L2 (Γ1 ))0 = L2 (Γ1 ); v 7→ l(v) is hemicontinuous, (4.54) implies that T

Z

Z Γ1

0

(ψ(t ) − l(v 0 (t ) + φ 0 (t )))ξ dΓ dt ≤ 0 ∀ ξ ∈ L2 (Γ1 ).

Therefore, T

Z

Z Γ1

0

(ψ(t ) − l(v 0 (t ) + φ 0 (t )))ξ dΓ dt = 0 ∀ ξ ∈ L2 (Γ1 ),

which implies that ψ = l(v 0 + φ 0 ). The claim (c) follows. Step 4: Uniqueness. Suppose z1 and z2 are two smooth solutions to the system (3.1). Then z = z1 − z2 satisfies

Z Ω

z 00 (t )w dx +

Z = Ω

Z Ω

h∇g z (t ), ∇g wig dx + Z

(h(∇ z2 ) − h(∇ z1 ))wdx +

Ω

Z Γ1

(l(z10 ) − l(z20 ))w dΓ

(f (z2 ) − f (z1 ))w dx,

∀ w ∈ V.

(4.55)

Choose w = z 0 (t ) in (4.55) and notice the monotonicity of l, to get d

 Z

dt

2

1

Ω

 Z Z (|z 0 (t )|2 + |∇g z (t )|2g )dx ≤ (h(∇ z2 ) − h(∇ z1 ))z 0 (t )dx + (f (z2 ) − f (z1 ))z 0 (t )dx. Ω

(4.56)

Ω

First, by Assumption 3.3, the first term of the right-hand side of (4.56) satisfies

Z

(h(∇ z2 ) − h(∇ z1 ))z (t )dx ≤ C 0

Ω

Z Ω

|∇g z (t )|

2 g dx

Z + Ω



|z (t )| dx . 0

2

(4.57)

2 Next, notice Assumption 3.1 and the fact 0 ≤ ρ − 1 ≤ n− when n ≥ 4, and 0 ≤ ρ − 1 ≤ 1 when n ≤ 3. By a simple 2 application of Hölder’s inequality and the Sobolev imbedding theorem, we can get the estimate for the second term on the right-hand side of (4.56) that

Z

(f (z2 ) − f (z1 ))z (t )dx ≤ C 0

Ω

Z

ρ−1

Ω

(1 + |z1 |

Z ≤C Ω

Z ≤C Ω

ρ−1 2

+ | z2 |

) |z (t )| dx + C 2

Z Ω

|z 0 (t )|2 dx

 1q Z  1p Z (1 + |z1 |ρ−1 + |z2 |ρ−1 )2q dx |z (t )|2p dx +C |z 0 (t )|2 dx Ω

|∇g z (t )|2g dx +

Ω



Z Ω

|z 0 (t )|2 dx ,

(4.58)

n where q = 2n , p = n− . 2 Finally, substituting (4.57) R and (4.58) intoR (4.56), and then integrating over (0, t ), after applying Gronwall’s lemma, we can get the uniqueness that Ω |z 0 (t )|2 dx = Ω |∇g z (t )|2g dx = 0.

Step 5: Existence of weak solution. Now suppose that the initial data for the system (3.1) satisfy

{u0 , u1 } ∈ V × L2 (Ω ).

(4.59)

B.-Z. Guo, Z.-C. Shao / Nonlinear Analysis 71 (2009) 5961–5978

5973

∂u Since D(−∆g ) = {u ∈ V ∩ H 2 (Ω )| ∂µ = 0 on Γ1 } is dense in V and H01 (Ω ) ∩ H 2 (Ω ) is dense in L2 (Ω ), there exist two

sequences {u0m } ⊂ D(−∆g ) and {u1m } ⊂ H01 (Ω ) ∩ H 2 (Ω ) such that u0m → u0

strongly in V ,

(4.60)

u1m → u1

strongly in L (Ω ).

(4.61)

2

Under Assumption 3.3, it has

∂ u0m + l(u1m ) = 0 ∀ m ∈ N. ∂µ

(4.62)

By the first part of Theorem 3.1 that has just been proven, for each m ∈ N, there exists a smooth solution um : Ω ×(0, ∞) → R to the system:

 00  um − ∆g um + h(∇ um ) + f (um ) = 0   um = 0 ∂ um + l(u0m ) = 0    ∂µ  um (x, 0) = u0m (x), u0m (x, 0) = u1m (x)

in L2 (0, ∞; L2 (Ω )), on Γ0 × (0, ∞), (4.63)

in L2 (0, ∞; L2 (Γ1 )), in Ω .

By Assumptions 3.2 and 3.3, using the same arguments as in obtaining (4.13), we can get

Z Ω

|u0m (t )|2 dx +

Z tZ Ω

0

Z tZ Ω

0

Ω

|∇g um (t )|2g dx +

Z tZ Γ1

0

|u0m (s)|2 dΓ ds ≤ C ,

(4.64)

|h(∇ um (s))|2 dxds ≤ C ,

(4.65)

|f (um (s))|2 dxds ≤ C ,

(4.66)

|l(u0m (s))|2 dΓ ds ≤ C .

(4.67)

Z tZ Γ1

0

Z

Consider zmi = um − ui for all m, i ∈ N where um and ui are smooth solutions to (4.63). Repeating the process of the proof for the uniqueness of the strong solution to (3.1) and making use of (4.60) and (4.61), we conclude that there exists u : Ω × (0, ∞) → R such that um → u strongly in C 0 ([0, T ]; V ), 0

um → u

(4.68)

strongly in C ([0, T ]; L (Ω )).

0

0

2

(4.69)

On the other hand, by (4.64)–(4.67) we have weakly in L2 (0, T ; L2 (Γ1 )),

u0m → u0

h(∇ um ) → χ

(4.70)

weakly in L (0, T ; L (Ω )), 2

2

f ( um ) → ζ

weakly in L (0, T ; L (Ω )),

l(um ) → ψ

weakly in L (0, T ; L (Γ1 )).

0

2

(4.71)

2

2

(4.72)

2

(4.73)

Now, from (4.68) we can easily get that χ = h(∇ u) and ζ = f (u). Next, we show that ψ = l(u ). To this purpose, first, multiply both sides of the first equation in (4.63) by u0m and integrate over Ω in x, to obtain 0

Z

1 d 2 dt


|u0m (t )|2 + |∇g um (t )|2g dx +

Ω

Z + Ω

f (um (t ))u0m (t )dx +

Z Γ1

Z Ω

h(∇ um (t ))u0m (t )dx

l(u0m (t ))u0m (t )dΓ = 0.

(4.74)

Integrate (4.74) over (0, t ) in s to get 1 2

Z

(|um (t )| + |∇g um (t )| )dx + 0

Ω

2

2 g

Z tZ 0

Z tZ + 0

Ω

f (um (s))u0m (s)dxds +

Ω

Z tZ 0

Γ1

h(∇ um (s))u0m (s)dxds l(u0m (s))u0m (s)dΓ ds =

1 2

Z Ω

(|u1m |2 + |∇g u0m |2g )dx.

(4.75)

5974

B.-Z. Guo, Z.-C. Shao / Nonlinear Analysis 71 (2009) 5961–5978

This together with (4.60), (4.61), and (4.70)–(4.73) deduces that

Z tZ lim

m→∞

Γ1

0

l(u0m (s))u0m (s)dΓ ds = −

1

Z

2

Ω

(|u0 (t )|2 + |∇g u(t )|2g )dx −

Z tZ 0

Z tZ − Ω

0

1

f (u(s))u0 (s)dxds +

Z

2

Ω

Ω

h(∇ u(s))u0 (s)dxds

(|u1 |2 + |∇g u0 |2g )dx.

(4.76)

Next, suppose that y is a weak solution to the system

 00 y − ∆g y + h(∇ y) + f (y) = 0    y = 0 ∂y +ψ =0    ∂µ  y(x, 0) = u0 (x), y0 (x, 0) = u1 (x)

in Ω × (0, ∞), on Γ0 × (0, ∞), (4.77)

on Γ1 × (0, ∞), in Ω .

Then applying the same techniques as in the proofs of (4.76)– (4.77) gives

Z tZ 0

Γ1

ψ(s)y0 (s)dΓ ds = −

1

Z

2

Ω

(|y0 (t )|2 + |∇g y(t )|2g )dx −

0

Z tZ − 0

Z tZ

Ω

f (y(s))y0 (s)dxds +

1

Z

2

Ω

Ω

h(∇ y(s))y0 (s)dxds

(|u1 |2 + |∇g u0 |2g )dx.

(4.78)

Since u is a weak solution to (4.77), it follows from (4.76) and (4.78) that

Z tZ lim

m→∞

Γ1

0

l(u0m (s))u0m (s)dΓ ds =

Z tZ 0

Γ1

ψ(s)u0 (s)dΓ ds.

(4.79)

Now the same arguments after (4.50) as in Step 3 can be applied to conclude that ψ = l(u0 ). The proof is complete.



Proof of Theorem 3.2. We only prove the exponential stability for the strong solution to (3.1) since by a density argument, the same result holds true for the weak solution. By the first equation and the boundary conditions in (3.1), it has E˙ (t ) =

Z Ω

Z = Ω

(2u00 u0 + 2h∇g u, ∇g u0 ig + 2f (u)u0 )dx (2u0 ∆g u − 2u0 h(∇ u) + 2h∇g u, ∇g u0 ig ) dx

Z

2u0

= Γ1

∂u dΓ − ∂µ

Z Ω

2u0 h(∇ u)dx = −2

Z Γ1

u0 l(u0 )dΓ − 2

Z Ω

u0 h(∇ u)dx.

(4.80)

|u0 |2 dΓ ,

(4.81)

This together with Assumption 3.3 and (3.5) gives E˙ (t ) ≤ β

Z

(|u | + |∇g u| ) dx − 2c2 0 2

Ω

2 g

Z

|u | dΓ ≤ β E (t ) − 2c2 0 2

Γ1

Z Γ1

where E (t ) defined by (3.12) is the energy function of the system (3.1). From the above inequality, it seems that the system (3.1) is not dissipative, which was claimed by [22] for constant coefficients case. However, this is a wrong impression. Actually, by introducing an equivalent energy function, we will find that the system (3.1) is essentially dissipative. To this end, for any ε > 0, define a new energy function for the system (3.1) by Eε (t ) = E (t ) + ε p(t ),

(4.82)

where p(t ) = 2

Z Ω

u0 (x, t )H (u)(x, t )dx + (nb − ε0 )

Z Ω

u0 (x, t )u(x, t )dx

for some ε0 ∈ (0, b).

(4.83)

Observe that

|Eε (t ) − E (t )| = ε|p(t )| ≤ ε C1 E (t ) ∀ t ≥ 0

(4.84)

for some constant C1 > 0. So, Eε (t ) is an equivalent energy function of (3.1) for small ε . By this equivalence, the proof of exponential stability will be accomplished if we can show that there exists some constant C2 > 0 such that E˙ ε (t ) ≤ −C2 E (t )

∀ t ≥ 0.

(4.85)

B.-Z. Guo, Z.-C. Shao / Nonlinear Analysis 71 (2009) 5961–5978

5975

To this end, we estimate p˙ (t ). Differentiate (4.83) with respect to t, to get

Z Z Z ∆g uH (u)dx − 2 h(∇ u)H (u)dx − 2 f (u)H (u)dx + 2 u0 H (u0 )dx Ω Ω Ω Ω Z Z
+ (nb − ε0 ) ∆g u − h(∇ u) − f (u) u dx + (nb − ε0 ) |u0 |2 dx

p˙ (t ) = 2

Z

Ω

Ω

= I1 (t ) + I2 (t ) + I3 (t ) + I4 (t ),

(4.86)

where I1 (t ) =

Z Ω

I3 (t ) = −

u0 (2H (u0 ) + (nb − ε0 )u0 )dx,

Z

h(∇ u)(2H (u) + (nb − ε0 )u)dx,

Ω

Z

∆g u(2H (u) + (nb − ε0 )u)dx, Z I4 (t ) = − f (u)(2H (u) + (nb − ε0 )u)dx.

I2 ( t ) =

Ω

Ω

Now we estimate Ii (t ), i = 1, 2, 3, 4, respectively. I1 (t ) =

Z

H (|u0 |2 )dx +

Ω

Z

Z Ω

(nb − ε0 )|u0 |2 dx

|u | hH , µig dΓ − 0 2

= Γ1

Z

(divg H − nb)|u | dxdx − ε0 0 2

Ω

Z Ω

|u0 |2 dx,

(4.87)

where in the second step, we used the following identity H (|u0 |2 ) = divg (|u0 |2 H ) − |u0 |2 divg H and the divergence theorem in Lemma 3.1. Denoting by M = maxΩ |H |g and noticing the fact divg H ≥ nb from (3.15), we obtain

Z

I1 (t ) ≤ M

Γ1

| u0 | 2 d Γ − ε 0

Z Ω

|u0 |2 dx.

(4.88)

Next, we estimate I2 (t ). By the divergence theorem of Lemmas 3.1 and 3.3, we have

Z Z h∇g (H (u)), ∇g uig dx + (nb − ε0 ) divg (u∇g u)dx − (nb − ε0 ) |∇g u|2g dx Ω Ω Ω Ω Z  Z Z ∂u ∂u =2 H (u)dΓ − (nb − ε0 ) |∇g u|2g dx − dΓ u ∂ Ω ∂µ Ω Γ1 ∂µ  Z  1 1 2 2 −2 DH (∇g u, ∇g u) + divg (|∇g u|g H ) − |∇g u|g divg H dx 2 2 ZΩ Z = −2 DH (∇g u, ∇g u)dx + (divg H − nb + ε0 )|∇g u|2g dx

I2 (t ) = 2

Z

divg (∇g uH (u))dx − 2

Ω

Ω

Z  +

2 Γ1

Z

 Z ∂u ∂u H (u) − |∇g u|2g hH , µig + (nb − ε0 )u dΓ + |∇g u|2g hH , µig dΓ , ∂µ ∂µ Γ0

(4.89)

where the validity of the last step comes from the fact u = 0 on Γ0 and hence

∇g u = h∇g u, µig µ =

∂u µ; ∂µ

∂u ∂u H (u) = h∇g u, H ig = h∇g u, µi2g hH , µig = |∇g u|2g hH , µig on Γ0 . ∂µ ∂µ

Since

Z

∂u 2 H (u)dΓ ≤ Γ1 ∂µ

∂u |H |g |∇g u|g dΓ 2 ∂µ Γ1 ! Z δ Λ 2 ∂ u 2 2 ≤ |∇g u|g + M dΓ , Λ δ ∂µ Γ1 Z

(4.90)

5976

B.-Z. Guo, Z.-C. Shao / Nonlinear Analysis 71 (2009) 5961–5978

by (3.15) and the fact hH , µig = |ν 1 | H · ν > Λδ on Γ1 , we have, from (4.89), that A g

2 ! ∂u ∂ u δ Λ δ + |∇g u|2g + M 2 dΓ I2 ( t ) ≤ (divg H − (n + 2)b + ε0 )|∇g u|2g dx + − |∇g u|2g + (nb − ε0 )u Λ ∂µ Λ δ ∂µ Ω Γ1 ! 2 Z Z ∂u ∂u Λ (nb − ε0 )u |∇g u|2g dx + ≤ (nB − (n + 2)b + ε0 ) + M 2 dΓ . (4.91) ∂µ δ ∂µ Γ1 Ω R R Using the inequality Γ |v|2 dΓ ≤ c˜ Ω |∇g v|2g dx for any v ∈ V , the boundary condition, Assumption 3.3, and the inequality 1 ab ≤ ηa2 + η1 b2 , η > 0, we estimate the last term on the right-hand side of (4.91) as !  Z Z  ∂u Λ 2 ∂ u 2 Λ (nb − ε0 )u + M dΓ = − (nb − ε0 )ul(u0 ) − M 2 |l(u0 )|2 dΓ ∂µ δ ∂µ δ Γ1 Γ1  Z  Z c2 Λ 2 1 02 ≤ c2 (nb − ε0 ) η|u|2 + |u | dx + M |u0 |2 dΓ 4η δ Γ1 Γ1 Z  Z c2 ΛM 2 c2 (nb − ε0 ) 2 = c˜ c2 (nb − ε0 )η |∇g u|g dx + + | u0 | 2 d Γ 4η δ Ω Γ1 Z Z = ε0 |∇g u|2g dx + M1 | u0 | 2 d Γ , (4.92) Z

Z

Ω

ε

where η := c˜ c (nb0−ε ) and M1 := 2 0

c2 (nb−ε0 ) 4η

I2 (t ) ≤ (nB − (n + 2)b + 2ε0 )

Z Ω

+

c2 ΛM 2

δ

Γ1

were used in the last step. Substitute (4.92) into (4.91), to obtain

|∇g u|2g dx + M1

Z Γ1

|u0 |2 dΓ .

(4.93)

The estimation of I3 (t ) comes from (3.5) and the Cauchy inequality I3 ( t ) ≤ 2 β M

Z Ω

+ β(nb − ε0 )

2 g dx

|∇g u|

≤ β(2M + c¯ nb)

Z Ω

Z Ω

|∇g u|g |u|dx

|∇g u|2g dx,

(4.94)

where c¯ is the positive constant verifying

Z

|v|2 dx ≤ c¯ Ω

Z Ω

|∇g v|2g dx ∀ v ∈ V .

Finally, we estimate I4 (t ). By (3.11), the nonnegativity of F , F (0) = 0, and the divergence formula, we have I4 (t ) ≤ −(nb − ε0 )r

Z =− Ω

Z Ω

2F (u)dx −

Ω

2H (F (u))dx

((nb − ε0 )r − divg H )2F (u)dx −

≤ (nB − (nb − ε0 )r ) Let 0 < β <

Z

ε0 2M +¯c nb

Z Ω

Z Γ1

2F (u)hH , µig dΓ

2F (u)dx.

(4.95)

. Combine (4.87), (4.93), (4.94), (4.95) and (4.86) to obtain

p˙ (t ) ≤ −γ E (t ) + (M + M1 )

Z Γ1

|u0 |2 dΓ ,

(4.96)

where γ := min{(n + 2)b − nB − 3ε0 , (nb − ε0 )r − nB, ε0 } is positive by (3.9) in Assumption 3.4. By (4.81), (4.82) and (4.96), we finally obtain E˙ ε (t ) = E˙ (t ) + ε p˙ (t )

≤ −(εγ − β)E (t ) − (2c2 − ε(M + M1 ))

Z Γ1

| u0 | 2 d Γ .

B.-Z. Guo, Z.-C. Shao / Nonlinear Analysis 71 (2009) 5961–5978

5977

Taking ε > 0 sufficiently small such that 2c2 − ε(M + M1 ) > 0 and choosing β > 0 small enough such that εγ − β > 0, we obtain the desired inequality (4.85). The proof is complete.  To end this paper, we point out some facts for (1.1). Since (1.1) is a special case of system (3.1), the energy of system (1.1) is

Z

e E (t ) =

 Ω

 |u0 (t )|2 + |∇ u(t )|2 + 2F (u(t )) dx,

(4.97)

and from the proof of Theorem 3.2 we know that system (1.1) is actually dissipative under an equivalent energy function

Z

e E (t ) + 2ε

Z

u (x, t )(x − x0 ) · ∇ u(x, t )dx + ε(n − ε0 ) 0

Ω

Ω

u0 (x, t )u(x, t )dx

(4.98)

for some constants ε0 ∈ (0, 1) and sufficiently small ε > 0. We specially state the following Corollary 4.1 as a consequence of Theorem 3.2, which is the main result of [22] obtained by regarding (1.1) as a nondissipative system. Corollary 4.1. Let u ∈ C ([0, ∞); V ) ∩ C 1 ([0, ∞); L2 (Ω )) be a solution of (1.1). Under the conditions (3.2), (3.6), (3.7), (3.11), and (3.5) with λ = 1 and sufficiently small β , the energy e E (t ) of system (1.1), defined by (4.97), decays exponentially in the sense

e E (t ) ≤ c˜e E (0)e−ω˜ t ∀ t ≥ 0,

(4.99)

for some constants c˜ , ω ˜ independent of t. Appendix. Explanation of (4.34) and the generalized Green formula Similar to (4.5), for any w ∈ V , the following variation formula holds true T

Z

Z Ω

0

vm00 (t )wϕ(t )dxdt + Z

+ Ω

0 T

Z

Z Ω

0 T

Z

T

Z

Z

= Ω

0

h∇g vm (t ), ∇g wig ϕ(t )dxdt + T

Z 0

F (t )wϕ(t )dxdt +

Z Ω

0

f (vm (t ) + φ(t ))wϕ(t )dxdt + T

Z

T

Z

Z Γ1

0

Z Γ1

h(∇vm (t ) + ∇φ(t ))wϕ(t )dxdt

0 l(vm (t ) + φ 0 (t ))wϕ(t )dΓ dt

B (t )wϕ(t )dΓ dt

∀ ϕ ∈ D (0, T ) and vm (0) = vm0 (0) = 0.

(A.1)

Making use of (4.25), (4.26), (4.28), (4.31) and (4.32), and passing to the limit as m → ∞ in (A.1) yield T

Z

Z Ω

0

v 00 (t )wϕ(t )dxdt +

Z Ω

0 T Z

Z +

Ω

0 T

Z

T

Z

Z

= Ω

0

ζ wϕ(t )dxdt +

h∇g v(t ), ∇g wig ϕ(t )dxdt +

0

T Z

Z

Γ1

0

F (t )wϕ(t )dxdt +

T

Z

T

Z

Ω

χ wϕ(t )dxdt

ψwϕ(t )dΓ dt

Z Γ1

0

Z

B (t )wϕ(t )dΓ dt .

(A.2)

On the other hand, multiply both sides of (4.33) by wϕ(t ) and integrate over Ω × [0, T ] to give T

Z

Z Ω

0

v (t )wϕ(t )dxdt − 00

Z

+ 0

Ω

Ω

ζ wϕ(t )dxdt =

RT R

T

Z

∆g vwϕ(t )dxdt +

Z Ω

0

T

Z 0

Z Ω

χ wϕ(t )dxdt

F (t )wϕ(t )dxdt .

(A.3)

h∇g v(t ), ∇g wig ϕ(t )dxdt makes sense, one can define Z TZ Z TZ ∂v ∆g vwϕ(t )dxdt + h∇g v(t ), ∇g wig ϕ(t )dxdt , wϕ(t )dΓ dt 0 Ω 0 Ω 0 Γ1 ∂µ   ∂v ≡ (ϕ), w , 1 −1 ∂µ H 2 (Γ1 )×H 2 (Γ1 )

Since the term

Z

Z

0 T

Z

T

Z

T

0

Ω

Z

(A.4)

∂v which means that ∂µ ∈ D 0 (0, T ; H − 2 (Γ1 )) because w ∈ H 2 (Γ1 ) and ϕ ∈ D (0, T ). When v is smooth enough, the above equality is just the Green formula. We call (A.4) the generalized Green formula. (4.34) is a consequence of (A.2), (A.3), and (A.4). 1

1

5978

B.-Z. Guo, Z.-C. Shao / Nonlinear Analysis 71 (2009) 5961–5978

References [1] J. Lagnese, Note on boundary stabilization of wave equations, SIAM J. Control Optim. 26 (1988) 1250–1256. [2] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev. 20 (1978) 639–739. [3] R. Triggiani, Wave equation on a bounded domain with boundary dissipation: An operator approach, J. Math. Anal. Appl. 137 (1989) 438–461. [4] Y. You, Energy decay and exact controllability for the Petrovsky equation in a bounded domain, Adv. Appl. Math. 11 (1990) 372–388. [5] M. Aassila, M.M. Cavalcanti, V.N. Domingos Cavalcanti, Existence and uniform decay rate of the wave equation with nonlinear boundary damping and boundary memory source term, Calc. Var. Partial Differential Equations 15 (2002) 155–180. [6] M.M. Cavalcanti, V.N. Domingos Cavalcanti, P. Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations 203 (2004) 119–158. [7] F. Conrad, B. Rao, Decay of solutions of wave equation in a star-shaped domain with nonlinear boundary feedback, Asympt. Anal. 7 (1993) 159–177. [8] V. Komornik, On the nonlinear boundary stabilization of the wave equation, Chinese Ann. Math. Ser. B 14 (2) (1993) 153–164. [9] V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, John Wiley and Sons. Ltd., Chichester, 1994. [10] V. Komornik, E. Zuazua, A direct method for the boundary stabilization of wave equation, J. Math. Pure Appl. 69 (1990) 33–54. [11] I. Lasiecka, D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary condition, Differential Integral Equations 6 (1993) 507–533. [12] E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim. 28 (1990) 466–477. [13] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations 50 (1983) 163–182. [14] S.G. Chai, K.S. Liu, Boundary stabilization of the transmission of wave equations with variable coefficients, Chinese Ann. Math. Ser. A 26 (5) (2005) 605–612. [15] S.G. Chai, Y.X. Guo, Boundary stabilization of wave equations with variable coefficients and memory, Differential Integral Equations 17 (2004) 669–680. [16] S.J. Feng, D.X. Feng, Nonlinear boundary stabilization of wave equations with variable coefficients, Chinese Ann. Math. 24 (2) (2003) 239–248. [17] Y.X. Guo, P.F. Yao, Stabilization of Euler–Bernoulli plate equation with variable coefficients by nonlinear boundary feedback, J. Math. Anal. Appl. 317 (2006) 50–70. [18] M.M. Cavalcanti, A. Khemmoudj, M. Medjden, Uniform stabilization of the damped Cauchy–Ventcel problem with variable coefficients and dynamic boundary conditions, J. Math. Anal. Appl. 328 (2007) 900–930. [19] P.F. Yao, On the observability inequality for exact controllability of wave equations with variable coefficients, SIAM J. Control Optim. 37 (1999) 1568–1599. [20] I. Lasiecka, R. Triggiani, P.F. Yao, Exact controllability for second-order hyperbolic equations with variable coefficients-principal part and first-order term, Nonlinear Anal. TMA 30 (1997) 111–122. [21] I. Lasiecka, R. Triggiani, P.F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients, J. Math. Anal. Appl. 235 (1999) 13–57. [22] A. Guesmia, A new approach of stabilization of nondissipative distributed systems, SIAM J. Control Optim. 42 (2003) 24–52. [23] M. Aassila, M.M. Cavalcanti, On nonlinear hyperbolic problems with nonlinear boundary feedback, Bull. Belg. Math. Soc. Simon Stevin 7 (2000) 521–540. [24] K.S. Liu, Z.Y. Liu, B. Rao, Exponential stability of an abstract nondissipative linear system, SIAM J. Control Optim. 40 (2001) 149–165. [25] B.Z. Guo, J.M. Wang, S.P. Yung, On the C0 -semigroup generation and exponential stability resulting from a shear force feedback on a rotating beam, Systems Control Lett. 18 (2005) 1013–1038. [26] M.E. Taylor, Partial Differential Equations I: Basic Theory, Springer-Verlag, New York, 1996. [27] P.F. Yao, Observability inequalities for the Euler–Bernoulli plate with variable coefficients, Contemp. Math. 268 (2000) 383–406. [28] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linèaires, Dunod Gaulthier-Villars, Paris, 1969.