1. Introduction
Many results concerning the boundary stabilization of classical wave equations are available in literatures. See [1–6] for linear cases and [7–14] for nonlinear ones. The stability of a nondissipative system described by partial differential equations (PDEs) has attracted much attention. Reference [15] developed the exponential stability for an abstract nondissipative linear system, and in [16], the Riesz basis property was developed for a beam equation with nondissipativity.

In [17], the following semilinear wave equation was considered:(1)utt-Δgu+h(∇u)+f(u)=0 (x,t)∈Ω×(0,+∞),u(x,t)|Γ2=0 t∈(0,+∞),∂u(x,t)∂μ+l(ut)=0 (x,t)∈Γ1×(0,+∞),u(x,0)=u0(x), ut(x,0)=u1(x) x∈Ω
and the well-posedness and uniform decay of the energy of the system (1) was also established with linearly bounded l(u) in [17].

Based on [17], we study the system (1) with time-varying and nonlinear feedback:
(2)∂u(x,t)∂μ+ϕ(t)l(u)=0 (x,t)∈Γ1×(0,+∞).
The decay rate of the energy (when t goes to infinity) of the wave equation with time-varying feedback was established under the assumption ϕ is decreasing [18–20] or ϕ has an upper bound [21].

In this paper, we consider the decay rate of the energy under suitable assumptions of the bound of the time-varying term ϕ(t) and the nonlinearity of the nonlinear term l(u).

2. Some Notation
Let Ω be a bounded domain in ℝn (n≥2) 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=∅.

Let A(x)=(aij(x)) be symmetric, positively definite matrices for each x∈ℝn, and aij(x) are smooth functions on ℝn. As in [22], we define
(3)g=A-1(x) for x∈ℝn
as a Riemannian metric on ℝn and consider the couple (ℝn,g) as a Riemannian manifold with an inner product:
(4)〈X,Y〉g=〈A-1(x)X,Y〉, |X|g2=〈X,X〉g X,Y∈ℝxn.

Denote by D, ∇g, divg, and Δg 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,
(5)∇gf=∑i=1n(∑j=1naij(x)∂∂xjf)∂∂xi=A(x)∇f,|∇gu|g2 =∑i,j=1naij(x)∂f∂xi∂f∂xj x∈ℝn,Δgf=1G∑i=1n∂∂xi(G∑j=1naij(x)∂∂xjf), x∈ℝn,
where ∇f is the gradient of f in the standard metric and G=det(g).

Let H be a vector field on (ℝxn,g). Then for each x∈ℝn, the covariant differential DH of H determines a bilinear form on ℝxn:
(6)DH(X,Y)=〈DYH,X〉g ∀X,Y∈ℝxn,
where DYH 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 time-varying and nonlinear boundary feedback:
(7)utt-Δgu+f(u)=0 (x,t)∈Ω×(0,+∞),u(x,t)|Γ2=0 t∈(0,+∞),∂u(x,t)∂μ+ϕ(t)l(ut)=0 (x,t)∈Γ1×(0,+∞),u(x,0)=u0(x), ut(x,0)=u1(x) x∈Ω,
where l, f are continuous nonlinear functions and μ(x) is the outside unit normal vector of the Riemannian manifold (Ω,g) for each x∈Γ. Different from [18–21], in this paper, we consider a general ϕ; that is, ϕ∈C1([0,+∞)) satisfies
(8)1Φ(t)≤ϕ≤Φ(t) ∀t≥0,
where Φ(t)∈C([0,+∞)) is a positive and nondecreasing function satisfying
(9)limt→+∞Φ(t)t=0.

Let Φ′(t)∈C([0,+∞)) be a positive and nondecreasing function with 0 as the limit. Then tΦ′(t) satisfies (9). There are many examples of Φ′(t) such as (1+t)α(α<0) and eβt(β<0).

The main assumptions are listed as follows.

Assumption A.
f
∈
C
1
(
ℝ
)
,
f
(
0
)
=
0
derives from a potential F:
(10) F(s)=∫0sf(τ)dτ≥0 ∀s∈ℝ,
and satisfies
(11)|f′(s)|≤b1|s|ρ+b2 ∀s∈ℝ,
where b1, b2 are positive constants, and the parameter ρ satisfies
(12)1≤ρ≤{2,n=2,nn-2,n≥3.

Being different from [17], we assume the nonlinear term l(u) has no growth restriction near zero as in [23, 24].

Assumption B.
l
∈
C
1
(
ℝ
)
is a nondecreasing function satisfying
(13)l(0)=0, c1|s|2≤sl(s)≤c2|s|2 ∀|s|≥1.

Assumption C.
There exists a vector field H on Ω¯ such that
(14)DH(X,X)=c(x)|X|g2 for X∈ℝxn x∈Ω¯,
where b=minΩ¯c(x) and B=maxΩ¯c(x)(15)B<min{b+2bn,rb},
where r>1 is a constant. Moreover we assume that
(16)〈H,μ〉g≤0 x∈Γ2, 〈H,μ〉g≥0 x∈Γ1.

Condition (14) as a checkable assumption is very useful to study the control and stabilization of the wave equation with variable coefficients and the quasilinear wave equation [22, 25]. For the examples of the condition, see [22, 26].

Based on condition (14), Assumption C was given by [17] to study the stabilization of the wave equation with variable coefficients and nonlinear boundary condition. Being different from [17], the lower bound of 〈H,μ〉g was relaxed on Γ1 from a positive constant to zero.

To facilitate the writing, we denote the volume element of (Ω,g) by dx and denote the volume element of (Γ,g) by dΓ. Define the energy of the system (7) by
(17)E(t)=∫Ω(ut2+|∇gu|g2+2F(u))dx.

As in [23, 24], we let h∈C([0,+∞)) be a concave increasing function such that
(18)h(0)=0, s2+(g(s))2≤h(sg(s)) for |s|≤1.
With (18), the stabilization of the wave equation with variable coefficients and time dependent delay was studied by [27].

The main result of this paper is as follows.

Theorem 1.
Let Assumptions A–C hold true. Assume that
(19)2rF(s)≤sf(s) ∀s∈ℝ,
where r is defined in (15).

(a) If the function l in (7) satisfies
(20)c1|s|2≤sl(s)≤c2|s|2 ∀|s|<1,
then there exist constants C>0 such that
(21)E(t)≤CΦ(t)tE(0) t>0.

(b) If the functions ϕ(t),l in (7) satisfy
(22)ϕ(t)≤ϕ0 ∀t≥0, sl(s)≥c1|s|2 ∀|s|<1,
where ϕ0 is a positive constant, then there exist constants C1,C2>0 such that
(23)C1h(C2Φ(T)TE(0))+C1Φ(T)TE(0) t>0.

(c) If the function Φ(t) in (8) is a constant function; that is,
(24)Φ(t)=Φ(0) ∀t≥0,
then there exist constants C1,C2>0 such that
(25)C1h(C2E(0)T)+C1TE(0) t>0.

4. Well Posedness of the System
Define
(26)HΓ21(Ω)={u∈H1∣(Ω)u|Γ2=0}.
By a similar proof as Lemma 7.1 in [17], we have the following result.

Theorem 2.
Let Assumptions A-B hold true. For any initial data (u0,u1)∈HΓ21(Ω)×L2(Ω), system (7) admits a unique weak solution u such that u∈C([0,+∞),HΓ21(Ω))∩C1([0,+∞),L2(Ω)).

To prove Theorem 1, we still need several lemmas further. Define
(27)E0(t)=∫Ω(ut2+|∇gu|g2)dx.
Then, we have
(28)E(t)=E0(t)+2∫ΩF(u)dx.

The following lemma shows the energy of the system (7) is decreasing.

Lemma 3.
Suppose that Assumptions A-B hold true. Let u be the solution of the system (7). Then
(29)E(0)-E(T)=2∫0T∫Γ1ϕ(t)utl(ut)dΓ dt.
The assertion (29) implies that E(t) is decreasing.

Proof.
Differentiating (17), we obtain
(30)E′(t)=∫Ω(2ututt+2〈∇gu,∇gut〉g+2f(u))dx=∫Γ12ϕ(t)utl(ut)dΓ.
Then the inequality (29) follows directly from (30) integrating from 0 to T.

5. Proofs of Theorem <xref ref-type="statement" rid="thm3.1">1</xref>
Lemma 4.
Let u(x,t) be the solution of the equation utt+Δgu+f(u)=0,(x,t)∈Ω×(0,+∞) and that ℋ is a vector field defined on Ω¯. Then for T≥0(31)∫0T∫Γ∂u∂μℋ(u)dΓ dt+12∫0T∫Γ(ut2-|∇gu|g2-2F(u))iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii×〈ℋ,μ〉gdΓ dt =(ut,ℋ(u))|0T+∫0T∫ΩDℋ(∇gu,∇gu)dx dt +12∫0T∫Ω(ut2-|∇gu|g2-2F(u))divgℋdx dt.

Moreover, assume that P∈C1(Ω¯). Then
(32)∫0T∫Ω(ut2-|∇gu|g2-uf(u))Pdx dt =(ut,uP)|0T+12∫0T∫Ω∇gP(u2)dx dt -∫0T∫ΓPu∂u∂μdΓ dt.

Proof.
Note that
(33)ℋ(u)f(u)=ℋ(F(u))= divg(F(u)ℋ)-F(u)divgℋ.
The equality (31) and the equality (32) follow from Proposition 2.1 in [22].

Lemma 5.
Suppose that all assumptions in Theorem 1 hold true. Let u solve the system (7). Then there exist positive constants T¯,C for which
(34)E(T)≤CT∫0T∫Γ1(ut2+(∂u∂μ)2)dΓ dt,
where T≥T¯.

Proof.
From (15), we choose a positive constant θ satisfying
(35)θ<nb2, b+θ-nB2>0, 2rθ>nB.
Set
(36)ℋ=H, P=θ.
We substitute the formula (32) into the formula (31), and we have
(37)ΠΓ=(ut,H(u)+Pu)|0T +∫0T∫Ω(DH(∇gu,∇gu)-b|∇gu|g2)dx dt +∫0T∫Ω((12divH-θ)ut2iiiiiiiiiiiiiiii+(b+θ-12divH)|∇gu|g2)dx dt +∫0T∫Ω[θ(uf(u)-2rF(u))iiiiiiiiiiiiiii+(2rθ-divH)F(u)(uf(u)-2rF(u))]dx dt,
where
(38)ΠΓ=∫0T∫Γ∂u∂μ(H(u)+uP)dΓ dtiiiiiiiiiiii+12∫0T∫Γ(ut2-|∇gu|g2-2F(u))〈H,μ〉gdΓ dt.

Decompose ΠΓ as
(39)ΠΓ=ΠΓ1+ΠΓ2,
where ΠΓ1(ΠΓ2) stands by the value of the terms on the right side of (38) integrating on Γ1(Γ2).

Similar to [5, 22], we deal with ΠΓ2 as follows.

Since u|Γ2=0, we have ∇Γu|Γ2=0; that is,
(40)∇gu=∂u∂μμ for x∈Γ2.
Similarly, we obtain
(41)H(u)=〈H,∇gu〉g=∂u∂μ〈H,μ〉g for x∈Γ2.
Using the equality (40) and (41) in the equality (38) on the portion Γ2, with (16) we obtain
(42)ΠΓ2=12∫0T∫Γ2(∂u∂μ)2〈H,μ〉gdΓ dt≤0.

Let H1 be a vector field on Ω¯ such that
(43)H1=μ x∈Γ1,H1=0 x∈Γ2.
Set ℋ=H1; it follows from (31) that
(44)∫0T∫Γ1(∂u∂μ)2dΓ dt+12∫0T∫Γ1(ut2-|∇gu|g2)dΓ dt =(ut,H1(u))|0T+∫0Tdt∫ΩDH1(∇gu,∇gu)dx +12∫0Tdt∫Ω(ut2-|∇gu|g2-2F(u))divgH1dx.
Then we obtain that
(45)∫0T∫Γ1|∇gu|g2dΓ dt ≤C∫0T∫Γ1(ut2+(∂u∂μ)2)dΓ dt+C(E0(0)+E0(T)) +C∫0T∫Ω(ut2+|∇gu|g2+2F(u))dx dt.
With (16) and (45), we have
(46)ΠΓ1=∫0T∫Γ1∂u∂μ(H(u)+uP)dΓ dt +12∫0T∫Γ1(ut2-|∇gu|g2-F(u))〈H,μ〉gdΓ dt≤Cɛ∫0T∫Γ1(∂u∂μ)2dΓ dt+ɛ∫0T∫Γ1(u2+|∇gu|g2)dΓ dt +C∫0T∫Γ1ut2dΓ dt≤C∫0T∫Γ1(∂u∂μ)2dΓ dt +ɛ(E0(0)+E0(T)+∫0TE(t)dt)+C∫0T∫Γ1ut2dΓ dt.

Note that
(47)nb≤ divgH≤nB ∀x∈Ω¯.
Substituting the formulas (42) and (46) into the formula (37), with (19) and (35), we obtain
(48)∫0TE(t)dt≤C(E0(0)+E0(T)) +C∫0T∫Γ1(ut2+(∂u∂μ)2) dΓ dt.

Since
(49)E0(0)=E0(T)-∫0T∫Γ1ut∂u∂μdΓ dt≤E0(T)+12∫0T∫Γ1(ut2+(∂u∂μ)2)dΓ dt,
from (48), we have
(50)∫0TE(t)dt≤CE(T)+C∫0T∫Γ1(ut2+(∂u∂μ)2)dΓ dt.

Since E(t) is decreasing, we deduce that
(51)∫0TE(t)dt≥TE(T).
Substituting the formulas (51) into the formula (50), for sufficiently large T, we have
(52)E(T)≤CT∫0T∫Γ1(ut2+(∂u∂μ)2)dΓ dt.
The inequality (34) holds.

Proof of Theorem <xref ref-type="statement" rid="thm3.1">1</xref>.
(a) From (8), (13), (20), (29), and (34), for T≥T¯ we deduce that
(53)E(T)≤CT∫0T∫Γ1(ϕ2(t)+1)ut2dΓ dt≤CT(1ϕ(t)sup{ϕ(t)∣0≤t≤T}iiiiiiiii+sup{1ϕ(t)∣0≤t≤T}) ×∫0T∫Γ1ϕ(t)ut2dΓ dt≤CΦ(T)TE(0).
Note that E(t) is decreasing, and the estimate (21) holds.

(b) From (8), (13), (22), (29), and (34), for T≥T¯ we deduce that
(54)E(T)≤CT∫0T∫Γ1(ϕ2(t)g2(ut)+ut2)dΓ dt≤CT{∫0T∫Γ1ϕ(t)g2(ut)dΓ dtiiiiiiiiiiiiiiiiii+Φ(T)∫0T∫Γ1ϕ(t)ut2dΓ dt}≤CT{∫0T∫{x∈Γ1,|ut|≤1}ϕ(t)g2(ut)dΓ dtiiiiiiiiiiiiiiii+Φ(T)∫0T∫Γ1ϕ(t)utg(ut)dΓ dt}≤CT∫0T∫{x∈Γ1,|ut|≤1}ϕ(t)h(utg(ut))dΓ dtiiiiiiiiiiiiiiiiii+CΦ(T)TE(0)≤CT∫0T∫Γ1ϕ(t)h(utg(ut))dΓ dt+CΦ(T)TE(0) ≤C∫0Tϕ(t)dt·meas(Γ1)Th ×(∫0T∫Γ1ϕ(t)utg(ut)dΓ dt∫0Tϕ(t)dt·meas(Γ1))+CΦ(T)TE(0)≤C1h(C2Φ(T)TE(0))+C1Φ(T)TE(0).
Note that E(t) is decreasing, and the estimate (23) holds.

(c) From (8), (13), (24), (29), and (34), for T≥T¯ we deduce that
(55)E(T)≤CT∫0T∫Γ1(ϕ2(t)g2(ut)+ut2)dΓ dt≤CT∫0T∫Γ1ϕ(t)(g2(ut)+ut2)dΓ dt≤CT∫0T∫{x∈Γ1,|ut|≤1}ϕ(t)h(utg(ut))dΓ dtiiiiiiiiiiiii+CT∫0T∫{x∈Γ1,|ut|>1}ϕ(t)ut2dΓ dt≤CT∫0T∫Γ1ϕ(t)h(utg(ut))dΓ dtiiiiiiiiiiiiii+CT∫0T∫Γ1ϕ(t)utg(ut)dΓ dt≤C∫0Tϕ(t)dt·meas(Γ1)Th ×(∫0T∫Γ1ϕ(t)utg(ut)dΓ dt∫0Tϕ(t)dt·meas(Γ1))+CTE(0)≤C1h(C2E(0)T)+C1TE(0).
Note that E(t) is decreasing, and the estimate (25) holds.