# Boundary Stabilization of a Semilinear Wave Equation with Variable Coefficients under the Time-Varying and Nonlinear Feedback

Authors: Bei Gong; Xiaopeng Zhao
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study the boundary stabilization of a semilinear wave equation with variable coefficients under the time-varying and nonlinear feedback. By the Riemannian geometry methods, we obtain the stability results of the system under suitable
assumptions of the bound of the time-varying term and the nonlinearity of the nonlinear term.


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## Body

## 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=0t∈(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 termϕ(t) and the nonlinearity of the nonlinear term l(u).


## 2. Some Notation

LetΩ 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=∅.LetA(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)forx∈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 byD, ∇g, div⁡g, 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∂xjx∈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).LetH 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=0t∈(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)lim⁡t→+∞Φ(t)t=0.LetΦ′(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 fieldH on Ω¯ such that
(14)DH(X,X)=c(x)|X|g2forX∈ℝxnx∈Ω¯,
where b=min⁡Ω¯c(x) and B=max⁡Ω¯c(x)(15)B<min⁡{b+2bn,rb},
where r>1 is a constant. Moreover we assume that
(16)〈H,μ〉g≤0x∈Γ2,〈H,μ〉g≥0x∈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 functionl 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 functionsϕ(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 functionΦ(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 solutionu such that u∈C([0,+∞),HΓ21(Ω))∩C1([0,+∞),L2(Ω)).To prove Theorem1, 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. Letu be the solution of the system (7). Then
(29)E(0)-E(T)=2∫0T‍∫Γ1‍ϕ(t)utl(ut)dΓdt.
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 Theorem1

Lemma 4.
Letu(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)dΓdt+12∫0T‍∫Γ‍(ut2-|∇gu|g2-2F(u))iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii×〈ℋ,μ〉gdΓdt=(ut,ℋ(u))|0T+∫0T‍∫Ω‍Dℋ(∇gu,∇gu)dxdt+12∫0T‍∫Ω‍(ut2-|∇gu|g2-2F(u))div⁡g⁡ℋdxdt.
Moreover, assume thatP∈C1(Ω¯). Then
(32)∫0T‍∫Ω‍(ut2-|∇gu|g2-uf(u))Pdxdt=(ut,uP)|0T+12∫0T‍∫Ω‍∇gP(u2)dxdt-∫0T‍∫Γ‍Pu∂u∂μdΓdt.Proof.
Note that(33)ℋ(u)f(u)=ℋ(F(u))=div⁡g(F(u)ℋ)-F(u)div⁡g⁡ℋ.
The equality (31) and the equality (32) follow from Proposition  2.1 in [22].Lemma 5.
Suppose that all assumptions in Theorem1 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)dΓdt,
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)dxdt+∫0T‍∫Ω‍((12div⁡H-θ)ut2iiiiiiiiiiiiiiii+(b+θ-12div⁡H)|∇gu|g2)dxdt+∫0T‍∫Ω‍[θ(uf(u)-2rF(u))iiiiiiiiiiiiiii+(2rθ-div⁡H)F(u)(uf(u)-2rF(u))]dxdt,
where
(38)ΠΓ=∫0T‍∫Γ‍∂u∂μ(H(u)+uP)dΓdtiiiiiiiiiiii+12∫0T‍∫Γ‍(ut2-|∇gu|g2-2F(u))〈H,μ〉gdΓdt.
DecomposeΠΓ 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.
Sinceu|Γ2=0, we have ∇Γu|Γ2=0; that is,
(40)∇gu=∂u∂μμforx∈Γ2.
Similarly, we obtain
(41)H(u)=〈H,∇gu〉g=∂u∂μ〈H,μ〉gforx∈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〉gdΓdt≤0.
LetH1 be a vector field on Ω¯ such that
(43)H1=μx∈Γ1,H1=0x∈Γ2.
Set ℋ=H1; it follows from (31) that
(44)∫0T‍∫Γ1‍(∂u∂μ)2dΓdt+12∫0T‍∫Γ1‍(ut2-|∇gu|g2)dΓdt=(ut,H1(u))|0T+∫0T‍dt∫Ω‍DH1(∇gu,∇gu)dx+12∫0T‍dt∫Ω‍(ut2-|∇gu|g2-2F(u))div⁡g⁡H1dx.
Then we obtain that
(45)∫0T‍∫Γ1‍|∇gu|g2dΓdt≤C∫0T‍∫Γ1‍(ut2+(∂u∂μ)2)dΓdt+C(E0(0)+E0(T))+C∫0T‍∫Ω‍(ut2+|∇gu|g2+2F(u))dxdt.
With (16) and (45), we have
(46)ΠΓ1=∫0T‍∫Γ1‍∂u∂μ(H(u)+uP)dΓdt+12∫0T‍∫Γ1‍(ut2-|∇gu|g2-F(u))〈H,μ〉gdΓdt≤Cɛ∫0T‍∫Γ1‍(∂u∂μ)2dΓdt+ɛ∫0T‍∫Γ1‍(u2+|∇gu|g2)dΓdt+C∫0T‍∫Γ1‍ut2dΓdt≤C∫0T‍∫Γ1‍(∂u∂μ)2dΓdt+ɛ(E0(0)+E0(T)+∫0T‍E(t)dt)+C∫0T‍∫Γ1‍ut2dΓdt.
Note that(47)nb≤div⁡g⁡H≤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)dΓdt.
Since(49)E0(0)=E0(T)-∫0T∫Γ1ut∂u∂μdΓdt‍‍≤E0(T)+12∫0T‍∫Γ1‍(ut2+(∂u∂μ)2)dΓdt,
from (48), we have
(50)∫0T‍E(t)dt≤CE(T)+C∫0T‍∫Γ1‍(ut2+(∂u∂μ)2)dΓdt.
SinceE(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)dΓdt.
The inequality (34) holds.Proof of Theorem1.
(a) From (8), (13), (20), (29), and (34), for T≥T¯ we deduce that
(53)E(T)≤CT∫0T‍∫Γ1‍(ϕ2(t)+1)ut2dΓdt≤CT(1ϕ(t)sup⁡{ϕ(t)∣0≤t≤T}iiiiiiiii+sup⁡{1ϕ(t)∣0≤t≤T})×∫0T‍∫Γ1‍ϕ(t)ut2dΓdt≤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)dΓdt≤CT{∫0T‍∫Γ1‍ϕ(t)g2(ut)dΓdtiiiiiiiiiiiiiiiiii+Φ(T)∫0T‍∫Γ1‍ϕ(t)ut2dΓdt}≤CT{∫0T‍∫{x∈Γ1,|ut|≤1}‍ϕ(t)g2(ut)dΓdtiiiiiiiiiiiiiiii+Φ(T)∫0T‍∫Γ1‍ϕ(t)utg(ut)dΓdt}≤CT∫0T‍∫{x∈Γ1,|ut|≤1}‍ϕ(t)h(utg(ut))dΓdtiiiiiiiiiiiiiiiiii+CΦ(T)TE(0)≤CT∫0T‍∫Γ1‍ϕ(t)h(utg(ut))dΓdt+CΦ(T)TE(0)≤C∫0T‍ϕ(t)dt·meas⁡(Γ1)Th×(∫0T‍∫Γ1‍ϕ(t)utg(ut)dΓdt∫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)dΓdt≤CT∫0T‍∫Γ1‍ϕ(t)(g2(ut)+ut2)dΓdt≤CT∫0T‍∫{x∈Γ1,|ut|≤1}‍ϕ(t)h(utg(ut))dΓdtiiiiiiiiiiiii+CT∫0T‍∫{x∈Γ1,|ut|>1}‍ϕ(t)ut2dΓdt≤CT∫0T‍∫Γ1‍ϕ(t)h(utg(ut))dΓdtiiiiiiiiiiiiii+CT∫0T‍∫Γ1‍ϕ(t)utg(ut)dΓdt≤C∫0T‍ϕ(t)dt·meas⁡(Γ1)Th×(∫0T‍∫Γ1‍ϕ(t)utg(ut)dΓdt∫0T‍ϕ(t)dt·meas⁡(Γ1))+CTE(0)≤C1h(C2E(0)T)+C1TE(0).
Note that E(t) is decreasing, and the estimate (25) holds.



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