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

1. Introduction

There are many results concerning the boundary stabilization of classical wave equations. See [1–6] for linear cases and [7–12] for nonlinear ones. The stability of the wave equation with variable coefficients has attracted much attention. See [13–23], and many others. In [20], by the methods in [11, 24], the authors study the stability of the wave equation with nonlinear term and time-varying term. However, under the condition the nonlinear term has upper bound and the time-varying term has lower bound, the stability of the wave equation was not studied in [20]. In this paper, our purpose is to study the stability of the wave equation under the condition the nonlinear term has upper bound and the time-varying term has lower bound.

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:
(2)utt+Au=0(x,t)∈Ω×(0,+∞),u(x,t)|Γ2=0t∈(0,+∞),∂u(x,t)∂νA+ϕ(t)g1ut(x,t)=0(x,t)∈Γ1×∈(0,+∞),u(x,0)=u0(x),ut(x,0)=u1(x)x∈Ω,g1∈C(R) and there exists a positive constant c1 such that
(3)g1(0)=0,sg1(s)≤|s|2fors∈R,|g1(s)|≥c1|s|for|s|>1,
and ϕ(t)∈C([0,+∞)) satisfies
(4)ϕ(t)≥ϕ0∀t≥0,(5)limt->+∞F(t)t=0,
where ϕ0 is a positive constant and F(t)=max0≤ρ≤tϕ(ρ).

∂u/∂νA is the conormal derivative
(6)∂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, the initial data (u0,u1) belongs to a suitable space.

Define the energy of the system (2) by
(7)E(t)=12∫Ω(ut2+∑i,j=1naijuxiuxj)dx.

We define
(8)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:
(9)〈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
(10)DgH(X,X)≥ρ0|X|g2forX∈Rxn,x∈Ω¯.
Moreover we assume that
(11)supx∈Ω¯divH<infx∈Ω¯divH+2ρ0,(12)H·ν≤0x∈Γ2,H·ν≥δx∈Γ1,
where δ is a positive constant.

Assumption (10) was introduced by [13] as a checkable assumption for the exact controllability of the wave equation with variable coefficients. For examples on the condition, see [13, 14].

Based on Assumption (10), Assumption A was given by [19] to study the stabilization of the wave equation with variable coefficients and boundary condition of memory type.

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

The main result of this paper is stated as follows.

Theorem 1.

Let Assumption A holds true. Then there exist positive constants C,C2, such that
(15)E(t)≤C1h(C2E(0)t)+C1F(t)tE(0),t>0.

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 (8).

If f∈C1(Rn), we define the gradient ∇gf of f in the Riemannian metric g, via the Riesz representation theorem, by
(16)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 [13] in Lemma 2.1.

Lemma 2.

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

(17)〈H(x),A(x)X(x)〉g=〈H(x),X(x)〉,x∈Rn;

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

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

(19)∇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 need several lemmas further. Define
(20)E0(t)=12∫Ω(ut2+|∇gu|g2)dx.
Then, we have
(21)E(t)=E0(t)+ξ∫t-τ(t)t∫Γ1ut2(x,ρ)dΓdρ.

Lemma 3.

Let (u) be the solution of system (2). Then there exists a constant C1 such that
(22)E(0)-E(T)=C1∫0T∫Γ1ϕ(t)ut(x,t)g1(ut(x,t))dΓdt,
where T≥0. The assertion (22) implies that E(t) is decreasing.

Proof.

Differentiating (7), we obtain
(23)E′(t)=∫Ω(ututt+∇gu·∇ut)dx=∫Γ1ϕ(t)ut(x,t)g1(ut(x,t))dΓ.
Then the inequality (22) holds true.

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

From Proposition 2.1 in [13], we have the following identities.

Lemma 4.

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,
(24)∫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
(25)∫0T∫Ω(ut2-|∇gu|g2)Pdxdt=(ut,uP)|0T+12∫0T∫Ω∇gP(u2)dxdt-∫0T∫ΓPu∂u∂νAdΓdt.

Lemma 5.

Suppose that all assumptions in Theorem 1 hold true. Let u be the solution of the system (2). Then there exist positive constants C,T0 for which
(26)E(T)≤CT∫0T∫Γ1(ut2+(∂u∂νA)2)dΓdt,
where T≥T0.

Proof.

We let θ be a positive constant satisfying
(27)12supx∈Ω¯divH<θ<12infx∈Ω¯divH+ρ0.
Set
(28)H=H,P=θ-ρ0.
Substituting the identity (25) into the identity (24), we obtain
(29)ΠΓ=(ut,H(u)+Pu)|0T+∫0T∫Ω(DgH(∇gu,∇gu)-ρ0|∇gu|g2)dxdt+∫0T∫Ω((12divH+ρ0-θ)ut2+(θ-12divH)|∇gu|g2)dxdt,
where
(30)ΠΓ=∫0T∫Γ∂u∂νA(H(u)+uP)dΓdt+12∫0T∫Γ(ut2-|∇gu|g2)H·νdΓdt.

Decompose ΠΓ as
(31)ΠΓ=ΠΓ1+ΠΓ2.
Since u|Γ2=0, we obtain ∇Γu|Γ2=0; that is,
(32)∇gu=∂u∂νAνA|νA|g2forx∈Γ2.
Similarly, we have
(33)H(u)=〈H,∇gu〉g=∂u∂νAH·ν|νA|g2forx∈Γ2.
Using the formulas (32) and (33) in the formula (30) on the portion Γ2, with (12), we obtain
(34)ΠΓ2=12∫0T∫Γ2(∂u∂νA)2H·ν|νA|g2dΓdt≤0.
From (12), we have
(35)ΠΓ1=∫0T∫Γ1∂u∂νA(H(u)+uP)dΓdt+12∫0T∫Γ1(ut2-|∇gu|g2)H·νdΓdt≤Cε∫0T∫Γ1(∂u∂νA)2dΓdt+ε∫0T∫Γ1(u2+|∇gu|g2)dΓdt+∫0T∫Γ1(Cut2-δ|∇gu|g2)dΓdt≤C∫0T∫Γ1(∂u∂νA)2dΓdt+εE(t)+C∫0T∫Γ1ut2dΓdt.

Substituting the formulas (34) and (35) into the formula (29), with (27), we obtain
(36)∫0TE(t)dt≤C(E(0)+E(T))+C∫0T∫Γ1(ut2+(∂u∂νA)2)dΓdt.

It follows from (22) that
(37)∫0TE(t)dt≥TE(T).

Substituting the formulas (22) and (37) into the formula (36), the inequality (26) holds.

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

Since E(t) is decreasing, with (4) and (26), for sufficiently large T, we have
(38)E(T)≤CT∫0T∫Γ1(ϕ2(t)g2(ut)+ut2)dΓdt≤CT{∫0T∫Γ1(ut2+g2(ut))dΓdt+F(T)∫0T∫Γ1utϕ(t)g(ut)dΓdt}≤CT{∫0T∫{x∈Γ1,|ut|≤1}h(utg(ut))dΓdt+CF(T)E(0)}≤CT∫0T∫Γ1h(utg(ut))dΓdt+CF(T)TE(0)≤Cmeas(Γ1)h(∫0T∫Γ1utg(ut)dΓdtT·meas(Γ1))+CF(T)TE(0)≤C1h(C2E(0)T)+C1F(T)TE(0).
Note that E(t) is decreasing; the estimate (15) holds.

4. Application of the System (<xref ref-type="disp-formula" rid="EEq1.2">2</xref>)

Nonlinear feedback describes a property of a physical system; that is, the response by the physical system to an applied force is nonlinear in its effect. One of the applications of the system (2) is in sound waves, where the system (2) describes the reflection of sound in heterogeneous materials at surfaces of some materials with nonlinearity of interest in engineering practice. Theorem 1 indicates that the energy of the sound waves with the reflection of sound at surfaces in heterogeneous materials at surfaces of some materials with nonlinearity is uniform decay under a suitable assumption of the nonlinearity.

Conflict of Interests

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

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