We study the stabilization of the wave equation with variable coefficients in a bounded domain and a time-varying delay term in the time-varying, weakly nonlinear boundary feedbacks. By the Riemannian geometry methods and a suitable assumption of nonlinearity, we obtain the uniform decay of the energy of the closed loop system.

1. Introduction

Let Ω be a bounded domain in ℝn with smooth boundary Γ. It is assumed that Γ consists of two parts Γ1 and Γ2(Γ=Γ1∪Γ2) with Γ2≠∅,Γ¯1∩Γ¯2=∅. Define
(1)𝒜u=-divA(x)∇uforu∈H1(Ω),
where div is the divergence operator of the standard metric of ℝn. A(x)=(aij(x)) is symmetric, positively definite matrices for each x∈ℝn and aij(x) are smooth functions on ℝn.

We consider the stabilization of the wave equations with variable coefficients and time-varying delay in the dissipative boundary feedback as follows:
(2)utt+𝒜u=0(x,t)∈Ω×(0,+∞),u(x,t)|Γ2=0t∈(0,+∞),∂u(x,t)∂ν𝒜+ϕ(t)(μg1(ut(x,t))+λg2(ut(x,t-τ(t))))=011111111111111111111111111(x,t)∈Γ1×∈(0,+∞),u(x,0)=u0(x),ut(x,0)=u1(x)x∈Ω,ut(x,t-τ(0))=f0(x,t-τ(0))(x,t)∈Γ1×(0,τ(0)),
where τ(t) satisfies
(3)0≤τ(t)≤τ0,d0≤τ′(t)≤d<1∀t≥0,
where τ0>0 and d0 and d are constants. g1∈C(ℝ) and there exist positive constants c1,p≥1 such that
(4)g1(0)=0,sg1(s)≥|s|2fors∈ℝ,(5)|g1(s)|≤c1|s|for|s|>1,s2+(g1(s))2≤c1(sg1(s))1/pfor|s|≤1.g2(s)∈C(ℝ) satisfies
(6)(g2(s))2≤sg1(s)fors∈ℝ,
and ϕ(t)∈C([-τ(0),+∞)) satisfies
(7)0<ϕ(t)≤ϕ0∀t≥-τ(0),(8)c2ϕ(t)≤ϕ(t-τ(t))≤c3ϕ(t)∀t≥0,limt→∞F(t)t=0,
where ϕ0, c2, and c3 are positive constants and F(t)=1/inf{ϕ(ρ)∣0≤ρ≤t}.

∂u/∂ν𝒜 is the conormal derivative
(9)∂u∂ν𝒜=〈A(x)u,ν〉,
where 〈·,·〉 denotes the standard metric of the Euclidean space ℝn and ν(x) is the outside unit normal vector for each x∈Γ. Moreover, μ>0, λ∈ℝ, λ≠0, and the initial data (u0,u1,f0,w0,w1,h0) belongs to a suitable space.

There is a specific example for ϕ(t). Let ϕ1>0 be a constant. If ϕ(t) satisfies
(10)ϕ1≤ϕ(t)≤ϕ0∀t≥0,
then
(11)ϕ1ϕ0ϕ(t)≤ϕ(t-τ(t))≤ϕ0ϕ1ϕ(t)∀t≥0,limt→∞F(t)t≤ϕ0t=0.
Conditions (7) and (8) hold.

In absence of delay (λ=0), the problem (2) was studied by [1–8] and many others. The decay rate of the energy (when t goes to infinity) depends on the function ϕ and the growth of g1.

The system (2) with constant coefficient (the case: A(x) is a constant matrix on Ω¯) was studied by [9–11] and many other authors. For the system (2) with variable coefficients, the main tools to cope with the system (2) are the differential geometrical methods which were introduced by [12] and have been applied in many papers. See [13–22] and references cited therein. For a survey on the differential geometric methods, see [23, 24].

The authors in [11] considered the system (2) with constant coefficients operator and dissipative boundary conditions of time dependent delay and proved the exponential decay of the energy by combining the multiplier method with the use of suitable integral inequalities. Different from this paper, g1 is assumed to be linearly bounded and ϕ is assumed to be a constant function in the paper [11].

Based on [11], the purpose of this paper is to solve the stability of the system (2) with variable coefficients and time-varying, weakly nonlinear terms. To obtain our stabilization result, we assume that
(12)(12c2+12)|λ|1-d<μ,
where c2 is defined in (8).

Define the energy of the system (2) by
(13)E(t)=12∫Ω(ut2+∑i,j=1naijuxiuxj)dx+ξ∫t-τ(t)t∫Γ1ϕ(ρ)uρ(x,ρ)g1(uρ(x,ρ))dΓdρ,
where ξ is a positive constant satisfying
(14)|λ|2c21-d<ξ<μ-|λ|21-d.

We define
(15)g=A-1(x)forx∈ℝn
as a Riemannian metric on ℝn and consider the couple (ℝn,g) as a Riemannian manifold with an inner product
(16)〈X,Y〉g=〈A-1(x)X,Y〉,|X|g2=〈X,X〉gX,Y∈ℝxn.

Let Dg denote the Levi-Civita connection of the metric g. For the variable coefficients, the main assumptions are as follows.

Assumption A.

There exists a vector field H on Ω¯ and a constant ρ0>0 such that
(17)DgH(X,X)≥ρ0|X|g2forX∈ℝxnx∈Ω¯.
Moreover, we assume that
(18)supx∈Ω¯divH<infx∈Ω¯divH+2ρ0,(19)H·ν≤0x∈Γ2,H·ν≥δx∈Γ1,
where δ is a positive constant.

Assumption (17) was introduced by [12] as a checkable assumption for the exact controllability of the wave equation with variable coefficients. Assumption A is also useful for the controllability and the stabilization of the quasilinear wave equation [15]. For the examples of the condition, see [12, 23].

Based on Assumption (17), Assumption A was given by [22] to study the stabilization of the wave equation with variable coefficients and boundary condition of memory type. The authors in [22] also constructed some examples of the condition based on the assumption that A(x)=a(x)I or A(x) is a perturbation of a symmetric positive definite matrix A.

Define
(20)HΓ21(Ω)={u∈H1(Ω)∣u|Γ2=0}.
To obtain the stabilization of the system (2), we assume that the system (2) is well posed such that
(21)u∈C1([0,+∞),L2(Ω))∩C([0,+∞),HΓ21(Ω))ut∈C([0,+∞),L2(Γ1×(t,t-τ(t)))).

The main result of this paper is the following.

Theorem 1.

Let Assumption A hold true. Then, there exists a constant C, such that
(22)E(t)≤C(E1/p(0)(t+τ0)1/p+F(t+τ0)t+τ0E(0))t>0.

Remark 2.

If p=1 and ϕ(t) satisfies
(23)ϕ1≤ϕ(t)≤ϕ0∀t≥0,
where ϕ0 and ϕ1 are positive constants, then it follows from (13) that there exist constants T0>0 and 0<C0<1 such that
(24)E(T0)≤C0E(0).
Then, the decay of the energy E(t) is exponential. Methods in [21, 22] are useful for Theorem 1.

2. Basic Inequality of the System

In this section, we work on Ω with two metrics at the same time: the standard dot metric 〈·,·〉 and the Riemannian metric g=〈·,·〉g given by (15).

If f∈C1(ℝn), we define the gradient ∇gf of f in the Riemannian metric g, via the Riesz representation theorem, by
(25)X(f)=〈∇gf,X〉g,
where X is any vector field on (ℝn,g). The following lemma provides further relations between the two metrics; see [12], Lemma 3.

Lemma 3.

Let x=(x1,…,xn) be the natural coordinate system in ℝn. Let f, h be functions and let ℋ, X be vector fields. Then,

(26)〈H(x),A(x)X(x)〉g=〈(x),X(x)〉,x∈ℝn;

(27)∇gf=∑i=1n(∑j=1naij(x)fxj)∂∂xi=A(x)∇f,x∈ℝn,

where ∇f is the gradient of f in the standard metric;

(28)∇gf(h)=〈∇gf,∇gh〉g=〈∇f,A(x)∇h〉,x∈ℝn,

where the matrix A(x) is given in formula (1).

To prove Theorem 1, we still further need several lemmas. Define
(29)E0(t)=12∫Ω(ut2+|∇gu|g2)dx.
Then, we have
(30)E(t)=E0(t)+ξ∫t-τ(t)t∫Γ1ut2(x,ρ)dΓdρ.

Lemma 4.

Suppose that condition (14) holds true. Let (u,w) be the solution of system (2). Then, there exist constants C1, C2>0 such that
(31)E(0)-E(T)≥C1∫0T∫Γ1ϕ(t)[(ut(x,t-τ(t)))ut(x,t)g1(ut(x,t))+ut(x,t-τ(t))1111111111111111×g1(ut(x,t-τ(t)))]dΓdt,
where T≥0. Assertion (31) implies that E(t) is decreasing.

Proof.

Differentiating (13), we obtain
(32)E′(t)=∫Ω(ututt+∇gu·∇ut)dx+∫Γ1ξϕ(t)ut(x,t)g1(ut(x,t))dΓ-∫Γ1ξϕ(t-τ(t))(1-τ′(t))ut×(x,t-τ(t))g1(ut(x,t-τ(t)))dΓ.

Applying Green’s formula and by integrating by parts with (3) and (8), we arrive at
(33)E′(t)=∫Γ1ϕ(t)[-μutg1(ut)-λutg2(ut(x,t-τ(t)))]dΓ+∫Γ1ξϕ(t)ut(x,t)g1(ut(x,t))dΓ-∫Γ1ξϕ(t-τ(t))(1-τ′(t))ut(x,t-τ(t))g1×(ut(x,t-τ(t)))dΓ≤∫Γ1ϕ(t)(-μutg1(ut)+|λ|21-dut2+ξutg1(ut))dΓ+∫Γ1ϕ(t)[1-d|λ|2-c2(1-d)ξut(x,t-τ(t))g1+∫Γ1ϕ11(t)×(ut(x,t-τ(t)))+∫Γ1ϕ11(t)+1-d|λ|2g22(ut(x,t-τ(t)))1-d|λ|2]dΓ.
It follows from (3), (4), (12), and (14) that
(34)E′(t)≤-Cϕ(t)∫Γ1[(ut(x,t-τ(t)))ut(x,t)g1(ut(x,t))11111111111+ut(x,t-τ(t))11111111111×g1(ut(x,t-τ(t)))]dΓ,
where C>0 satisfies
(35)C=min{μ-|λ|21-d-ξ,c2(1-d)ξ-1-d|λ|2}.
Then, inequality (31) follows directly from (34) integrating from 0 to T.

3. Proofs of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>

From Proposition 2.1 in [12], we have the following identities.

Lemma 5.

Suppose that u(x,t) solves equation utt+𝒜u=0,(x,t)∈Ω×(0,+∞) and that ℋ is a vector field defined on Ω¯. Then, for T≥0,
(36)∫0T∫Γ∂u∂ν𝒜ℋ(u)dΓdt+12∫0T∫Γ(ut2-|∇gu|g2)ℋ·νdΓdt=(ut,ℋ(u))|0T+∫0T∫ΩDgℋ(∇gu,∇gu)dxdt+12∫0T∫Ω(ut2-|∇gu|g2)divℋdxdt.

Moreover, assume that P∈C1(Ω¯). Then,
(37)∫0T∫Ω(ut2-|∇gu|g2)Pdxdt=(ut,uP)|0T+12∫0T∫Ω∇gP(u2)dxdt-∫0T∫ΓPu∂u∂ν𝒜dΓdt.

Lemma 6.

Suppose that all assumptions in Theorem 1 hold true. Let u be the solution of the system (2). Then, there exists a positive constant C for which
(38)∫0TE(t)dt≤CE0(T+τ0)+C∫0T+τ0∫Γ1(ut2+(∂u∂ν𝒜)2)dΓdt+C∫0T+τ0∫Γϕ(t)ut(x,t-τ(t))g1×(ut(x,t-τ(t)))dΓdt,
where T≥0.

Proof.

Let θ be a positive constant satisfying
(39)12supx∈Ω¯divH<θ<12infx∈Ω¯divH+ρ0.
Set
(40)ℋ=H,P=θ-ρ0.
Substituting identity (37) into identity (36), we have
(41)ΠΓ=(ut,H(u)+Pu)|0T+∫0T∫Ω(DgH(∇gu,∇gu)-ρ0|∇gu|g2)dxdt+∫0T∫Ω((12divH+ρ0-θ)ut211111111+(θ-12divH)|∇gu|g212)dxdt,
where
(42)ΠΓ=∫0T∫Γ∂u∂ν𝒜(H(u)+uP)dΓdt+12∫0T∫Γ(ut2-|∇gu|g2)H·νdΓdt.

We decompose ΠΓ as
(43)ΠΓ=ΠΓ1+ΠΓ2.
Since u|Γ2=0, we obtain ∇Γu|Γ2=0; that is,
(44)∇gu=∂u∂ν𝒜ν𝒜|ν𝒜|g2forx∈Γ2.
Similarly, we have
(45)H(u)=〈H,∇gu〉g=∂u∂ν𝒜H·ν|ν𝒜|g2forx∈Γ2.
Using formulas (44) and (45) in formula (42) on the portion Γ2, with (19), we obtain
(46)ΠΓ2=12∫0T∫Γ2(∂u∂ν𝒜)2H·ν|ν𝒜|g2dΓdt≤0.
From (19), we have
(47)ΠΓ1=∫0T∫Γ1∂u∂ν𝒜(H(u)+uP)dΓdt+12∫0T∫Γ1(ut2-|∇gu|g2)H·νdΓdt≤Cɛ∫0T∫Γ1(∂u∂ν𝒜)2dΓdt+ɛ∫0T∫Γ1(u2+|∇gu|g2)dΓdt+∫0T∫Γ1(Cut2-δ|∇gu|g2)dΓdt≤C∫0T∫Γ1(∂u∂ν𝒜)2dΓdt+ɛE0(t)+C∫0T∫Γ1ut2dΓdt.

Substituting formulas (46) and (47) into formula (41), with (39), we obtain
(48)∫0TE0(t)dt≤C(E0(0)+E0(T))+C∫0T∫Γ1(ut2+(∂u∂ν𝒜)2)dΓdt.
Let ϱ=ρ-τ(ρ), and from (3), (7), (8), and (30), we have
(49)∫0T+τ0E0(t)dt+∫0T+τ0∫Γϕ(t)ut(x,t-τ(t))g1×(ut(x,t-τ(t)))dΓdt≥∫0T+τ0E0(t)dt+1τ0∫0T∫tt+τ0∫Γ1ϕ(ρ)uρ×(x,ρ-τ(ρ))g1(ut(x,ρ-τ(ρ)))dΓdρdt=∫0T+τ0E0(t)dt+1τ0∫0T∫t-τ(t)t+T1-τ(t+τ0)∫Γ1ϕ(ρ)urho(x,ϱ)g1×(ut(x,ϱ))dΓdϱ1-τ′(ρ)dt≥∫0T+τ0E0(t)dt+c2(1-d0)τ0∫0T∫t-τ(t)t∫Γ1ϕ(ϱ)uϱ(x,ϱ)g1×(ut(x,ϱ))dΓdϱdt≥min{1,c2(1-d0)τ0}∫0TE(t)dt.
Since
(50)E0(0)=E0(T+τ0)-∫0T+τ0∫Γ1ut∂u∂ν𝒜dΓdt≤E0(T+τ0)+12∫0T+τ0∫Γ1(ut2+∂u∂ν𝒜)2dΓdt,
substituting formula (49) into formula (48), we obtain
(51)∫0TE(t)dt≤CE0(T+τ0)+C∫0T+τ0∫Γ1(ut2+(∂u∂ν𝒜)2)dΓdt+∫0T+τ0∫Γϕ(t)ut(x,t-τ(t))g1×(ut(x,t-τ(t)))dΓdt.
Inequality (38) holds.

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>.

Since E(t) is decreasing, from (38), for sufficiently large T, we have
(52)TE(T)≤C∫0T+τ0∫Γ1(ut2+(∂u∂ν𝒜)2)dΓdt+C∫0T+τ0∫Γϕ(t)ut(x,t-τ(t))g1(ut(x,t-τ(t)))dΓdt≤C∫0T+τ0∫Γ1(ϕ(t))2[(g1(ut))2+(g2(ut))2]dΓdt+C∫0T+τ0∫Γ1ut2(x,t)dΓdt+C∫0T+τ0∫Γϕ(t)ut(x,t-τ(t))g1(ut(x,t-τ(t)))dΓdt≤C∫0T+τ0∫Γ1(ϕ(t))2[(g1(ut))2+(g2(ut))2]dΓdt+CF(T+τ0)∫0T+τ0∫Γ1ϕ(t)ut2(x,t)dΓdt+C∫0T+τ0∫Γϕ(t)ut(x,t-τ(t))g1(ut(x,t-τ(t)))dΓdt,
where F(t) is defined in (8). With (4)–(8) and (31), we deduce that
(53)TE(T)≤C∫0T+τ0∫Γ1ϕ(t)(s2+(g1(s))2)dΓdt+CF(T+τ0)(E(0)-E(T+τ0))≤C∫0T+τ0∫{x∈Γ1,|ut(x,t)|≤1}ϕ(t)(utg1(ut))1/pdΓdt+CF(T+τ0)(E(0)-E(T+τ0))≤C∫0T+τ0∫Γ1ϕ(t)(utg1(ut))1/pdΓdt+CF(T+τ0)(E(0)-E(T+τ0))≤Cmeas(Γ1)∫0T+τ0ϕ(t)dt·(1meas(Γ1)∫0T+τ0ϕ(t)dt111111×∫0T+τ0∫Γ1ϕ(t)utg(ut)dΓdt1meas(Γ1)∫0T+τ0ϕ(t)dt)1/p+CF(T+τ0)(E(0)-E(T+τ0))≤C(T+τ0)1-1/p·(E(0)-E(T+τ0))1/p+CF(T+τ0)(E(0)-E(T+τ0)).
Therefore,
(54)E(t)≤C(E1/p(0)(T+τ0)1/p+F(T+τ0)T+τ0E(0)).
Note that E(t) is decreasing; estimate (22) holds.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the National Science Foundation of China (nos. 91328201 and 41130422) and the National Basic Research Program of China (no. 2011CB201103).

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