AMP Advances in Mathematical Physics 1687-9139 1687-9120 Hindawi Publishing Corporation 735341 10.1155/2014/735341 735341 Research Article Stabilization of the Wave Equation with Boundary Time-Varying Delay http://orcid.org/0000-0001-5414-4964 Li Hao 1 Lin Changsong 1 Wang Shupeng 2 Zhang Yanmei 3 Ma Wen-Xiu 1 School of Ocean Sciences China University of Geosciences Beijing 100083 China cug.edu.cn 2 Institute of Information Engineering Chinese Academy of Sciences Beijing 100093 China cas.cn 3 School of Information Engineering China University of Geosciences Beijing 100083 China cug.edu.cn 2014 2622014 2014 18 10 2013 10 01 2014 10 01 2014 26 2 2014 2014 Copyright © 2014 Hao Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 n with smooth boundary Γ. It is assumed that Γ consists of two parts Γ1 and Γ2(Γ=Γ1Γ2) with Γ2,Γ¯1Γ¯2=. Define (1)𝒜u=-divA(x)uforuH1(Ω), where div is the divergence operator of the standard metric of n. A(x)=(aij(x)) is symmetric, positively definite matrices for each xn and aij(x) are smooth functions on n.

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)xΩ,ut(x,t-τ(0))=f0(x,t-τ(0))(x,t)Γ1×(0,τ(0)), where τ(t) satisfies (3)0τ(t)τ0,d0τ(t)d<1t0, where τ0>0 and d0 and d are constants. g1C() and there exist positive constants c1,p1 such that (4)g1(0)=0,sg1(s)|s|2fors,(5)|g1(s)|c1|s|for|s|>1,s2+(g1(s))2c1(sg1(s))1/pfor|s|1.g2(s)C() satisfies (6)(g2(s))2sg1(s)fors, and ϕ(t)C([-τ(0),+)) satisfies (7)0<ϕ(t)ϕ0t-τ(0),(8)c2ϕ(t)ϕ(t-τ(t))c3ϕ(t)t0,limtF(t)t=0, where ϕ0, c2, and c3 are positive constants and F(t)=1/inf{ϕ(ρ)0ρt}.

u / ν 𝒜 is the conormal derivative (9)uν𝒜=A(x)u,ν, where ·,· denotes the standard metric of the Euclidean space n and ν(x) is the outside unit normal vector for each xΓ. Moreover, μ>0, λ, λ0, 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)ϕ1ϕ(t)ϕ0t0, then (11)ϕ1ϕ0ϕ(t)ϕ(t-τ(t))ϕ0ϕ1ϕ(t)t0,limtF(t)tϕ0t=0. Conditions (7) and (8) hold.

In absence of delay (λ=0), the problem (2) was studied by  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  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  and have been applied in many papers. See  and references cited therein. For a survey on the differential geometric methods, see [23, 24].

The authors in  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 .

Based on , 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)=12Ω(ut2+i,j=1naijuxiuxj)dx+ξt-τ(t)tΓ1ϕ(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)forxn as a Riemannian metric on n and consider the couple (n,g) as a Riemannian manifold with an inner product (16)X,Yg=A-1(x)X,Y,      |X|g2=X,XgX,Yxn.

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|g2forXxnxΩ¯. Moreover, we assume that (18)supxΩ¯divH<infxΩ¯divH+2ρ0,(19)H·ν0xΓ2,H·νδxΓ1, where δ is a positive constant.

Assumption (17) was introduced by  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 . For the examples of the condition, see [12, 23].

Based on Assumption (17), Assumption A was given by  to study the stabilization of the wave equation with variable coefficients and boundary condition of memory type. The authors in  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)={uH1(Ω)u|Γ2=0}. To obtain the stabilization of the system (2), we assume that the system (2) is well posed such that (21)uC1([0,+),L2(Ω))C([0,+),HΓ21(Ω))utC([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)ϕ1ϕ(t)ϕ0t0, 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 fC1(n), we define the gradient gf of f in the Riemannian metric g, via the Riesz representation theorem, by (25)X(f)=gf,Xg, where X is any vector field on (n,g). The following lemma provides further relations between the two metrics; see , Lemma 3.

Lemma 3.

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

(26)H(x),A(x)X(x)g=(x),X(x),xn;

(27)gf=i=1n(j=1naij(x)fxj)xi=A(x)f,xn,

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

(28)gf(h)=gf,ghg=f,A(x)h,xn,

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

To prove Theorem 1, we still further need several lemmas. Define (29)E0(t)=12Ω(ut2+|gu|g2)dx. Then, we have (30)E(t)=E0(t)+ξt-τ(t)tΓ1ut2(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)C10TΓ1ϕ(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 T0. 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)))dΓΓ1ϕ(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 , we have the following identities.

Lemma 5.

Suppose that u(x,t) solves equation utt+𝒜u=0,(x,t)Ω×(0,+) and that is a vector field defined on Ω¯. Then, for T0, (36)0TΓuν𝒜(u)dΓdt+120TΓ(ut2-|gu|g2)·νdΓdt=(ut,(u))|0T+0TΩDg(gu,gu)dxdt+120TΩ(ut2-|gu|g2)divdxdt.

Moreover, assume that PC1(Ω¯). Then, (37)0TΩ(ut2-|gu|g2)Pdxdt=(ut,uP)|0T+120TΩgP(u2)dxdt-0TΓPuuν𝒜dΓdt.

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)dtCE0(T+τ0)+C0T+τ0Γ1(ut2+(uν𝒜)2)dΓdt+C0T+τ0Γϕ(t)ut(x,t-τ(t))g1×(ut(x,t-τ(t)))dΓdt, where T0.

Proof.

Let θ be a positive constant satisfying (39)12supxΩ¯divH<θ<12infxΩ¯divH+ρ0. Set (40)=H,P=θ-ρ0. Substituting identity (37) into identity (36), we have (41)ΠΓ=(ut,H(u)+Pu)|0T+0TΩ(DgH(gu,gu)-ρ0|gu|g2)dxdt+0TΩ((12divH+ρ0-θ)ut211111111+(θ-12divH)|gu|g212)dxdt, where (42)ΠΓ=0TΓuν𝒜(H(u)+uP)dΓdt+120TΓ(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=uν𝒜ν𝒜|ν𝒜|g2forxΓ2. Similarly, we have (45)H(u)=H,gug=uν𝒜H·ν|ν𝒜|g2forxΓ2. Using formulas (44) and (45) in formula (42) on the portion Γ2, with (19), we obtain (46)ΠΓ2=120TΓ2(uν𝒜)2H·ν|ν𝒜|g2dΓdt0. From (19), we have (47)ΠΓ1=0TΓ1uν𝒜(H(u)+uP)dΓdt+120TΓ1(ut2-|gu|g2)H·νdΓdtCɛ0TΓ1(uν𝒜)2dΓdt+ɛ0TΓ1(u2+|gu|g2)dΓdt+0TΓ1(Cut2-δ|gu|g2)dΓdtC0TΓ1(uν𝒜)2dΓdt+ɛE0(t)+C0TΓ1ut2dΓdt.

Substituting formulas (46) and (47) into formula (41), with (39), we obtain (48)0TE0(t)dtC(E0(0)+E0(T))+C0TΓ1(ut2+(uν𝒜)2)dΓdt. Let ϱ=ρ-τ(ρ), and from (3), (7), (8), and (30), we have (49)0T+τ0E0(t)dt+0T+τ0Γϕ(t)ut(x,t-τ(t))g1×(ut(x,t-τ(t)))dΓdt0T+τ0E0(t)dt+1τ00Ttt+τ0Γ1ϕ(ρ)uρ×(x,ρ-τ(ρ))g1(ut(x,ρ-τ(ρ)))dΓdρdt=0T+τ0E0(t)dt+1τ00Tt-τ(t)t+T1-τ(t+τ0)Γ1ϕ(ρ)urho(x,ϱ)g1×(ut(x,ϱ))dΓdϱ1-τ(ρ)dt0T+τ0E0(t)dt+c2(1-d0)τ00Tt-τ(t)tΓ1ϕ(ϱ)uϱ(x,ϱ)g1×(ut(x,ϱ))dΓdϱdtmin{1,c2(1-d0)τ0}0TE(t)dt. Since (50)E0(0)=E0(T+τ0)-0T+τ0Γ1utuν𝒜dΓdtE0(T+τ0)+120T+τ0Γ1(ut2+uν𝒜)2dΓdt, substituting formula (49) into formula (48), we obtain (51)0TE(t)dtCE0(T+τ0)+C0T+τ0Γ1(ut2+(uν𝒜)2)dΓdt+0T+τ0Γϕ(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)C0T+τ0Γ1(ut2+(uν𝒜)2)dΓdt+C0T+τ0Γϕ(t)ut(x,t-τ(t))g1(ut(x,t-τ(t)))dΓdtC0T+τ0Γ1(ϕ(t))2[(g1(ut))2+(g2(ut))2]dΓdt+C0T+τ0Γ1ut2(x,t)dΓdt+C0T+τ0Γϕ(t)ut(x,t-τ(t))g1(ut(x,t-τ(t)))dΓdtC0T+τ0Γ1(ϕ(t))2[(g1(ut))2+(g2(ut))2]dΓdt+CF(T+τ0)0T+τ0Γ1ϕ(t)ut2(x,t)dΓdt+C0T+τ0Γϕ(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)C0T+τ0Γ1ϕ(t)(s2+(g1(s))2)dΓdt+CF(T+τ0)(E(0)-E(T+τ0))  C0T+τ0{xΓ1,|ut(x,t)|1}ϕ(t)(utg1(ut))1/pdΓdt+CF(T+τ0)(E(0)-E(T+τ0))C0T+τ0Γ1ϕ(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+τ0Γ1ϕ(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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