Exponential Stability of Wave Equations with Potential and Indefinite Damping

Exponential Stability of Wave Equations with Potential and Indefinite Damping

Georg Menz

Konstanzer Schriften in Mathematik und Informatik Nr. 224, Februar 2007

ISSN 1430-3558

© Fachbereich Mathematik und Statistik

© Fachbereich Informatik und Informationswissenschaft Universität Konstanz

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URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-23350

Indefinite Damping

Georg Menz

Institute for Applied Mathematics University of Bonn

menz@iam.uni-bonn.de

Abstract

First, we consider the linear wave equationutt−uxx+a(x)ut+b(x)u= 0on a bounded interval (0, L)⊂R. The damping functionais allowed to change its sign. Ifa:=_L¹ RL

0 a(x)dxis positive and the spectrum of the operator(∂xx−b)is negative, exponential stability is proved for smallka−ak_L2. Explicit estimates of the decay rateωare given in terms ofaand the biggest eigenvalue of(∂xx−b).

Second, we show the existence of a global, small, smooth solution of the corresponding nonlinear wave equationutt−σ(ux)x+a(x)ut+b(x)u= 0, if, additionally, the negative part ofais small enough compared withω. This is an extension of the results of Racke and Muñoz Rivera [17](b=0) and Benaddi and Rao [1] (kak∞small).

1 Introduction

The linear wave equation

u_tt−u_xx+a(x)u_t+b(x)u= 0. (1) is considered on the domainΩ := (0, L), L > 0. We assume that the functionu= u(x, t),(x, t) ∈ Ω×(0,∞)satisfies the following initial and Dirichlet boundary conditions

u(x,0) =u0, ut(x,0) =u1forx∈Ω and u(0, t) =u(L, t) = 0fort∈(0,∞).

Convention. We will writeL^pandH₀¹instead ofL^p(Ω)andH₀¹(Ω).

We assume that the functions a ∈ L^∞ andb ∈ L^∞ are time independent. Our main interest is exponential stability of (1). Because there are a lot of results on decay rates if the damping is definite i.e. a≥ 0(see for example [5], [6], [8], [18], [19] and [25]), we will focus our attention on indefinite damping i.e.ais allowed to change its sign. Ifb= 0and the functionais positive definite (i.e.a(x)≥0 anda(x)>0on a subinterval ofΩ), it is a well known fact that the equation (1) is exponentially stable.

Thus the key problem is to discover a condition which describes the positiveness of the functionain the right manner. So Chen, Fulling, Narcovich and Sun formulated in [3] a conjecture concerning exponential stability for the caseb= 0.

Conjecture 1.1. Letb= 0. If there existsγ >0such that

∀n∈N Z L

0

a(x) sin²nπx L

dx ≥γ, (2)

is satisfied, then the energyE(t) = RL

0 u²_x+u²_tdxdecays exponentially in time; i.e there are constants C >0andω >0independent of the initial data, such thatE(t)≤ Ce^−ωtE(0)for allt∈[0,∞).

1

We will call the real numberω >0a (possible) decay rate. Behind condition (2) stands the intuitive idea that it should be more effective to damp the string at the locations with high amplitudes than with low amplitudes of vibrations. But Freitas [9] outlined that the conjecture above is, in general, false. He also showed that condition (2) is not sufficient to guarantee exponential stability ifkakL^∞ is large. By replacingabyεaFreitas and Zuazua were able to show the following result (see [10]).

Proposition 1.2 (Freitas and Zuazua 1996). Let˜a∈BV,b= 0and (2) be satisfied. Then there exists ε >0such that the equation (1) witha:=ε˜ais exponentially stable.

This result was extended to the caseb6= 0by Benaddi and Rao [1] (see Proposition 1.3) and to higher space dimensions by Liu, Rao and Zhang [16]. Whereas K. Liu, Z. Liu and Rao [15] gave an abstract treatment to these results.

Proposition 1.3 (Benaddi and Rao 2000). Letλ_n, n∈Nbe the eigenvalue of the operator(∂_xx−b) belonging to the eigenfunctionv_n, which is normalized inL². Let˜a∈BV andb∈L¹. If

(i) 0 > λ1≥λ2≥. . .≥λn → −∞ (n→ ∞) (ii) ∃γ >0.: ∀N3n≥1 RL

0 ˜a(x)v_n²(x)dx≥γ

holds, then there existsε >0such that the equation (1) witha:=ε˜ais exponentially stable.

We want to remark that in order to apply Proposition 1.2 and 1.3, the functionaneeds to be small in thek · kL^∞norm. Racke and Muñoz Rivera [17] were able to determine an easier condition by using the mean valuea:=_L¹ RL

0 a(x)dxand the deviationka−ak_L2to measure the positiveness of the functiona for the one dimensional case withb= 0. Their main result for the linear wave equation is stated as Proposition 1.4 (Racke and Muñoz Rivera 2004). Leta∈L^∞andb= 0. Ifa >0, then there exists ε >0such that ifka−ak_L2 < ε, then the equation (1) is exponentially stable. One can choseω = 2a0

as a decay rate, ifa0satisfies0< a0<−Re

−^a₂ + q a

2

²

−^π_L²2

.

As one sees, the functionaneeds not to be small in thek · kL^∞ norm. To illustrate this proposition in comparison to the result of Freitas and Zuazua (see Proposition 1.2), Racke and Muñoz Rivera [17] stated a simple example. Because Racke and Muñoz Rivera used an explicit determination of the spectrum and of the resolvent of the operator(∂xx−a∂t), the question remains open, if their condition of positiveness also makes sense in more general situations. For instance, ifb 6= 0, the spectrum and resolvent of the operator(∂_xx−a∂_t−b)cannot be determined easily in general. Our first main result approves that the result of Racke and Muñoz Rivera (see Proposition 1.4) can be generalized to the caseb 6= 0. More precisely, we prove the following proposition in section 2.

Proposition 1.5 (Linear Case). Let a ∈ L^∞,b ∈ L^∞ and let η˜₁ be the greatest eigenvalue of the operatorA˜0:L²⊃H₀¹∩H²=:D( ˜A0)→L², that is defined as

A˜0p:= (∂xx−b(x))p for p∈D( ˜A0). (3) Ifa >0and0>η˜1then there existsε >0such that if ka−ak_L2 < ε, then the equation (1) is exponen- tially stable. One can choseω= 2a0as a decay rate, ifa0satisfies0< a0<−Re

−^a₂+ q a

2

2

− |˜η1|

. Thus, we also improved the result of Benaddi and Rao (see Proposition 1.3) that had the stronger assumption of smallness ofkakL^∞. We also lifted the argument of Racke and Muñoz Rivera from a concrete level to an abstract one. Only knowledge about the distribution of the spectrum of the operator

A˜0 = (∂xx−b)is needed. Instead of an explicit determination of the resolvent we will use the Hilbert- Schmitt representation. Hence, the argument is now applicable to a wider class of models.

In the second part we will prove the existence of a global, smooth, small solution of the corresponding nonlinear wave equation

utt−σ(ux)x+a(x)ut+b(x)u= 0 (4) with initial and Dirichlet boundary conditions on the domainΩunder certain hypotheses (see Proposition 1.6). We assume that the functionsa∈C³andb∈C³are time independent. The non-linear functionσ is assumed to satisfy

σ∈C³(R), d0:=σ⁰(0)>0, andσ⁰⁰(0) = 0. (5) For example, condition (5) is satisfied for the vibrating string, whereσ(y) = √^y

1+y² holds. One can rewrite (4) as

utt−d0uxx+a(x)ut+b(x)u=c(ux)uxx, (6) wherec(ux)is defined as

c(ux) :=σ⁰(ux)−d0=σ⁰(ux)−σ⁰(0). (7) Thus the associated linear system of (4) is

u_tt−d₀u_xx+a(x)u_t+b(x)u= 0. (8) The exponential stability of the associated linear system (8) follows directly by a transformation of coor- dinates from Proposition 1.5 (see for instance [17]) under analog hypotheses. As a consequence we can apply a standard technique in nearly the same way as Racke and Muñoz Rivera to get the existence of a global, smooth, small solution of (4). Additionally, one only needs the negative part ofato be small enough compared with a decay rate of (8). In section 3 we will proof our second main result, which generalizes the statement of Racke and Muñoz Rivera adequately to the caseb6= 0.

Proposition 1.6 (Non-Linear Case). Leta∈C³,b∈C³and letσsatisfy the condition (5). Letη˜1be the greatest eigenvalue of the operatorA˜d₀:L²⊃H₀¹∩H²=:D( ˜Ad₀)→L², that is defined as

A˜d0p:= (d0∂xx−b(x))p for p∈D( ˜Ad0). (9) Leta > 0 and 0 > η˜₁. If the associated linear system (8) is exponentially stable with a decay rate ω = 2a₀ and if the negative part ofais sufficiently small, i.e. a⁻_∞ := |min_x∈Ω(0, a(x))|L^∞ < a₀, then there existsδ >0such that ifk(u₀, u₁)k_H4×H³ < δ, there exists a unique global solutionuof the non-linear system (4) satisfying

u∈

3

\

k=0

C^k [0,∞), H^4−k(Ω)∩H₀¹(Ω)

∩C⁴ [0,∞), L²(Ω) .

Moreover letV := (ux, ut)^T andV0 := (∂xu0, u1)^T, then there are constantsc0 = c0(V0) >0 and c₁>0such thatkV(t)k_H2≤c₀e^−a0tandkV(t)k_H3≤c₁kV₀k_H3e^a−∞t,t≥0.

2 Linear Case

To derive exponential stability of (1) we will use a standard result which was obtained by Gearhart [11]

and Huang [12] independently (see for example page 852 in [21]).

Theorem 2.1. TheC0-semigroupS(t) =e^Atis exponentially stable if and only if

(C1) sup{Reλ : λ∈σ(A)} < 0 and (C2) sup{kRλ(A)k : Reλ≥0} < ∞ holds. Moreover, if also for a fixedδ >0

(C3) sup{kRλ(A)k : Reλ≥ −δ+ε} < ∞, ∀ε >0

is satisfied, then one can choose anyωas a decay rate that satisfies0< ω <2δ.

Motivated by the article of Racke and Muñoz Rivera [17] we verify conditions (C1) and (C3) for an associated system, where the functionais exchanged in (1) by its mean valuea. In a second step we transfer the conditions (C1) and (C3) to the original system by using a fixed point argument. The first step represents the crucial part of the proof of Proposition 1.5.

For the rest of this section we assume the hypothesis of Proposition 1.5 to be satisfied. We now translate (1) into the language of operator theory. LetH:=H₀¹(Ω)×L²(Ω)be a Hilbert space endowed with the inner product

f₁ f₂

, g₁

g₂

H

:=h∇f1,∇g1i_L2+hg2, g₂i_L2 with f₁

f₂

, g₁

g₂

∈ H.

We define the operatorA:H ⊃D(A)→ Has A

p q

:=

O Id

∂xx−b(x) −a(x) p q

, where p

q

∈(H₀¹(Ω)∩H²(Ω)) =:D(A).

A straightforward calculation shows that the Dirichlet problem of (1) is equivalent to p

q

t

=A p

q

, where p

q

= u

ut

and

p q

(t= 0) = u0

u1

= u0

∂tu0

. (10) Since the operatorAgenerates aC0-semigroup we have existence and uniqueness of a strong solution of (10). One easily verifies the following statement (cp. for example [20]).

Lemma 2.2. There existsω >0such that the operator(A−ω)is dissipative. Therefore the following conditions are satisfied:

(i) k(λ+ω)x−Axk_H≥λkxk_H for allx∈D(A)andλ >0.

(ii) If for someλ0> ω,R(λ0Id−A) =H, thenR(λId−A) =Hfor allλ > ω.

2.1 The Associated Operator A and the Reduced Associated Operator A

The associated differential operatorAis defined as the operatorA, onlyahas to be exchanged bya. We want to verify (C1) and (C3) for the operatorA. For this purpose we consider the eigenvalue problem of the operatorA. Letλ ∈ Cbe arbitrary. For everyF = (f1, f2)^T ∈ H, we want to find a unique U = (p, q)^T ∈D(A)which solves the equation

λU−A U =F. (11)

The first component of (11) givesq=λp−f₁. Substitutingqin the second component leads to

(λ²+λ a+b(x))p−∂_xxu=f₂+ (λ+a)f₁. (12)

In (12) we add(bmin−bmin)p= 0, wherebmin:=−|min_x∈Ω(0, b(x))|L^∞. Thus we obtain (λ²+λ a+bmin)

| {z }

=:η

p−(∂xx+bmin−b(x))

| {z }

=:A₀

p=f2+ (λ+a)f1

| {z }

=:g

. (13)

By defining the reduced associated operatorA0:L²⊃H₀¹∩H²→L²as

A₀p:= (∂_xx+b_min−b(x))p for p∈D(A₀) :=H₀¹∩H²,

we derive a new eigenvalue problem from (13):ηp−A0p=g. In summary we have shown the following proposition, which contains a strong connection between the eigenvalue problems of the operatorAand the operatorA0.

Proposition 2.3. LetF := (f₁, f₂)^T ∈ H,λ∈C. Ifη =λ²+aλ+b_min ∈%(A₀), thenR_λ(A)F =:

(p, q)^T is determined asp=Rη(A0)(f2+ (λ+a)f1)andq=λp−f1.

Remark 2.4. Letη1andη˜1be the biggest eigenvalue of the operatorA0, respectivelyA˜0. Then (i) 0>η˜1 ⇐⇒ bmin>η˜1+bmin=η1

(ii) |η˜1|=|η1−bmin| (14)

holds. Therefore, we assume thatbmin> η1is satisfied.

In order to apply Proposition 2.3 it is necessary to know for whichλ∈Choldsη:=λ²+λ+b_min∈

%(A0). Proposition 2.6 provides an answer to this question.

Definition 2.5. Letν := Re

−^a₂+ q

a 2

²

− |η1−bmin|

and letε^∗ > 0be sufficiently small that ε^∗+ν <0. The area of parametersΓ⊂Cis defined asΓ :={z∈C : Rez≥ε^∗+ν}.

Proposition 2.6. Ifλ∈Γ, thenη:=λ²+λ+bmin ∈%(A0)holds.

Proof. Letλ=µ₁+i µ₂∈Γ. It follows thatR3µ₁≥ε^∗+ν ≥ε^∗−^a₂. One can determineη∈Cas η=λ²+aλ+bmin=µ²₁+aµ1+bmin−µ²₂+i(a+ 2µ1)µ2. (15) In the first case letImλ=µ26= 0. Because

(a+ 2µ1)≥(a+ 2ε^∗−2a

2) = 2ε^∗>0 (16)

we know thatImη = (a+ 2µ₁)µ₂ 6= 0. Because the spectrum of the self-adjoint operatorA₀is real, it follows thatη∈%(A₀).

In the second case letImλ=µ2= 0. Because2ν+a≥0, a short calculation gives

µ²₁+aµ1≥(ε^∗+ν)²+a(ε^∗+ν)≥(ε^∗)²+ν²+aν. (17) Let ^a₂²

− |η1−bmin|>0. It follows thatν²+aν=−|η1−bmin|. By using (17) we can estimateηas η=µ²₁+aµ1+bmin≥(ε^∗)²+ν²+aν+bmin= (ε^∗)²− |η1−bmin|+bmin> η1. (18) Becauseη₁is the biggest eigenvalue of the operatorA₀, we getη∈%(A₀).

Finally, let ^a₂²

− |η1−bmin|<0. Thenν =−^a₂ holds. So we can estimateηby using (17) as η=µ²₁+aµ1+bmin≥(ε^∗)²−a²

4 +bmin≥(ε^∗)²− |η1−bmin|+bmin> η1. (19) Again we getη∈%(A₀).

By using the concept of lower semi bounded operators (see for example [2]) we are able to compare the eigenvalues ofA0with the eigenvalues of the operator∂xx. Let us introduce the operatorB : L²⊃ H₀¹∩H²→L², which is defined asB = (∂xx− |bmin−b|L^∞). Applying now Theorem 4 on page 227 in [2] to the operators−A0and−Band to the operators−∂xxand−A0gives the following statement.

Lemma 2.7. Letηk,k∈Nbe the eigenvalues ofA0in decreasing order. Then

−π²

L²k²− |bmin−b|L^∞≤ηk≤ −π² L²k².

As a direct consequence we get a lot of information about the distribution of the spectrum ofA0. Corollary 2.8. There existsn∈N, such that for allN3k≥n,

−π²

L²(k+ 1)²≤ηk ≤ −π²

L²k². (20)

From the Hilbert-Schmidt theorem (see for example [23]) the following representation of the resolvent ofA0can be easily deduced.

Proposition 2.9. Letη∈%(A0)andg∈L². Then there is an orthonormal set{uk}^∞_k=1of eigenfunctions and corresponding eigenvalues06=ηk∈Rof the operatorA0, such that

Rη(A0)g=

∞

X

k=1

hg, u_ki_L2

η−ηk

uk and kRη(A0)k²_L2=

∞

X

k=1

1

|η−ηk|²| hg, uki_L2|².

The first sum converges in thek · kL²-norm.

2.2 Uniform Convergence of Sums

This subsection is devoted to the proof of Proposition 2.10. It contains the uniform convergence of certain sums, which will play a crucial role in the proof of Proposition 1.5 i.e. in the verification of (C1) and (C3) for the associated operatorAand in the fix point argument. Notice that Lemma 2.2 also holds for the associated operatorA.

Proposition 2.10. Letω > 0be as in Lemma 2.2 and letλ ∈Γ^I := Γ∩ {z∈C : Rez≤2ω}. Let η:=λ²+aλ+b_minand letη_kbe the ordered eigenvalues associated with the eigenfunctionsu_k, k∈N of the operatorA₀. Then there is a constant0< C <∞independent ofλ∈Γ^I, such that

∞

X

k=1

1

|η−ηk| ≤C

∞

X

k=1

|λ²|

|η−ηk|² ≤C

∞

X

k=1

|η|

|η−ηk|² ≤C.

and ∞

X

k=1

|λ⁴|

|ηk| 1

|η−ηk|² ≤C

∞

X

k=1

|λ²|

|ηk|

|η|

|η−ηk|² ≤C

∞

X

k=1

|λ²|

|ηk| 1

|η−ηk| ≤C.

Before proceeding to the proof of Proposition 2.10, some preparatory work is required. Letλ = µ₁+i µ₂ ∈ Γ^I. It follows that ε^∗ +ν ≤ µ₁ ≤ 2ω andµ₂ ∈ R is arbitrary. From (15) we get η=C₁−µ²₂+i C₂µ₂, where the variablesC₁:=µ²₁+aµ₁+b_minandC₂:= 2µ₁+aare independent ofµ₂. From (16) we getC₂>0. From (18) and (19) it follows directly that

m:= (2ω)²+ 2aω+bmin= max

λ∈Γ^IC1≥C1≥(ε^∗)²+η1. (21)

Without loss of generality, one only has to consider the case µ2 ≥ 0 for reasons of symmetry. So η :=η(µ2)can be handled as a function of the variableµ2. We will now define an auxiliary functionψ asψ: (−∞, C1)→Rwithψ(x) := Imλ=µ2, whereµ2is the unique non negative real number such thatRe(η(µ2)) =C1−µ²₂=x. A short calculation shows that the functionψis given by

ψ(x) =µ₂=p

C₁−x (22)

and is thus monotone decreasing. With the auxiliary functionψ, we get the following representation of the functionη

η(ψ(x)) =x+i C2

pC1−x. (23) With the help of (22) and (23),|η|can be estimated in the following way (See also Figure 1).

Lemma 2.11. Letn∈N. If−^π_L²2(n+ 1)²<Re(η)≤ −^π_L²2n², then rπ⁴

L⁴n⁴+C₂²(C1+π²

L²n²)≤ |η|<

rπ⁴

L⁴(n+ 1)⁴+C₂²(C1+π²

L²(n+ 1)²).

So it is also valid thatC2

q

C1+_L^π22n²≤ |Imη|< C2

q

C1+^π_L²2(n+ 1)².

Figure 1: Illustration of Lemma 2.11

In order to prove the uniform convergence, we will split the sums into two parts

∞

X

k=1

=

˜ n

X

k=1

+

∞

X

k=˜n+1

.

We determine now the index˜n∈N. Let˜n2∈Nbe the smallestn∈Nfor which n⁻²η1+π²

L² ≥ π²

2L² (24)

is satisfied. It follows directly with (21) that for allN3n≥n˜₂ C1+n²π²

L² ≥η1+n²π²

L² ≥n² π² 2L² ≥ π²

2L² (25)

holds. The indexn˜ ∈Nis defined as

˜

n= max(˜n1, n˜2), (26)

wheren˜₁is taken as in Corollary 2.8. Notice that˜nonly depends on the functionb.

Convention. In this section,Cis designated to be a generic constant0< C <∞, which is independent ofλ∈Γ^I.

We will proof now the uniform convergence of the first sum i.e.

Lemma 2.12. Letλ∈Γ^I. ThenP∞ k=1

1

|η−ηk|≤C <∞, where the constantCis independent ofλ.

Proof. As mentioned before, we divide the sum into two parts

∞

X

k=1

1

|η−ηk| =

˜ n

X

k=1

1

|η−ηk|+

∞

X

k=˜n+1

1

|η−ηk|. By an elementary estimation one can control the first termPn˜

k=1 1

|η−ηk|without any problems. Therefore we put our attention on the more interesting termP∞

k=˜n+1 1

|η−ηk|. Without loss of generality we assume that−^π_L²2 (n)² ≥Re(η)>−_L^π22 (n+ 1)²withN3n≥n˜+ 3. We divide again the second term in the following way

∞

X

k=˜n+1

1

|η−η_k| =

n−2

X

k=˜n+1

1

|η−η_k|

| {z }

=:I

+

n+1

X

k=n−1

1

|η−η_k|

| {z }

=:II

+

∞

X

k=n+2

1

|η−η_k|

| {z }

III

.

It is also helpful to compare the following calculations with Figure 2 to get an intuitive understanding of

Figure 2: Estimation of|η−ηn| the estimation. Let us now estimate the summandI.

I≤

n−2

X

k=˜n+1

1

|Re(η−ηk)| ≤

n−2

X

k=˜n+1

1

|^π_L²2(k+ 1)²−_L^π22n²| =L² π²

n−˜n−2

X

i=1

1

|(n−i)²−n²| ≤C.

We will now estimate the summandII. We usen >n, (16) and (25) to obtain˜ II≤

n+1

X

k=n−1

1

|Im(η)| ≤

n+1

X

k=n−1

1

|C2(C1+^π_L²2n²)¹²| ≤

n+1

X

k=n−1

1

C2|(_2L^π22)¹²| ≤C.

We will now estimate the summandIII.

III≤

∞

X

k=n+2

1

|Re(η−η_k)| ≤

∞

X

k=n+2

1

|_L^π22k²−_L^π22(n+ 1)²| ≤ L² π²

∞

X

i=1

1 i² ≤C.

Overall we have shown that

∞

X

k=˜n+1

1

|η−η_k| ≤C, where the constantCis independent ofλ∈Γ^I.

The proof of the uniform convergence of the second sumP∞ k=1

|λ²|

|η−ηk|² is roughly the same as the proof for the first sum. Hence, we will omit it. Because|η|=|λ²+aλ+bmin|, the uniform convergence of the third sumP∞

k=1

|η|

|η−ηk|² can be reduced to the uniform convergence of the second sumP∞ k=1

|λ|²

|η−ηk|². We will now show the uniform convergence of the 4th sum of Proposition 2.10. It is the hardest sum to control and the estimation is sophisticated.

Lemma 2.13. Letλ∈Γ^I. ThenP∞ k=1

|λ⁴|

|ηk||η−ηk|² ≤C <∞, where the constantCis independent ofλ.

Proof. Letλ=µ₁+iµ₂∈Γ^I. BecauseReλis bounded it is sufficient to estimateΣ :=P∞ k=1

µ⁴₂

|ηk||η−ηk|²

uniformly. We assume without loss of generality that µ²₂≥4

π²

L²(˜n+ 3)²+m

≥4 π²

L²(˜n+ 1)²+m

≥ π²

L²(˜n+ 1)²+m. (27) We split upΣintoPn˜

k=1+P∞

k=˜n+1=: Σ^I+ Σ^II, where the integern˜is defined as in (26). We put our attention on the sumΣ^I. From (27) it follows from straightforward calculation that

0<2µ⁻²₂ π²

L²(˜n+ 1)²+m

≤1

2. (28)

By using (27) and (20) a direct forward calculation also results in (see also Figure 3) Reη=C1−µ²₂≤m−µ²₂≤ −π²

L²(˜n+ 1)²≤Reη˜n. (29) So one can estimate for allk∈ {1,· · · ,˜n}

|η−ηk|²≥(Re(η−η˜n))²≥

C1−µ²₂+π²

L²(˜n+ 1)^{2 2}

≥µ⁴₂−2µ²₂ π²

L²(˜n+ 1)²+m

. (30) From (27) followsµ⁴₂−2µ²₂

π²

L²(˜n+ 1)²+m

>0. Finally by using (28) and (30) we obtain

Σ^I≤ C η₁

˜ n

X

k=1

µ⁴₂

µ⁴₂−2µ²₂ ^π_L²₂(˜n+ 1)²+m ≤C

˜ n

X

k=1

1

1−¹₂ ≤C.

We put now our attention onΣ^II. From (27) we also getReη=C1−µ²₂≤m−µ²₂≤ −_L^π22(˜n+ 3)². Thus we can assume, without loss of generality, that there is aN3n≥n˜+ 3, such that

−π²

L²(n+ 1)<Reη(µ₂)≤ −π² L²n.

Figure 3: Illustration of the situation in (29)

By using the monotony of the auxiliary functionψand Lemma 2.11 we get C₁+π²

L² n²≤µ²₂< C₁+π²

L²(n+ 1)²≤m+π²

L²(n+ 1)²≤Cn². Thus we can estimateΣ^IIin the following manner

Σ^II≤C













n−2

X

k=˜n+1

n⁴

|ηk||η−ηk|²

| {z }

=:I

+

n+1

X

k=n−1

n⁴

|ηk||η−ηk|²

| {z }

=:II

+

∞

X

k=n+1

n⁴

|ηk||η−ηk|²

| {z }

=:III











 .

We will estimate the termIIby the following calculation using (16) and (25) II≤C

n+1

X

k=n−1

n⁴

(n−1)²(Im(η))² ≤C

n+1

X

k=n−1

n² (n−1)²

n²

C₂²(C1+_L^π22n²) ≤C.

In the estimation of the termIone encounters difficulties. Ifn → ∞, a pole arises in the denominator (compare with the next calculation). To solve this problem we estimateIandIIItogether. There is an intuitive idea behind this proceeding. The sumIIIconverges so nice that one can use free capacities to compensate the difficulties of the termI. The termI + IIIis estimated by the following calculation

I + III≤C

n−2

X

k=˜n+1

n⁴

k²(n²−(k+ 1)²)²+C

∞

X

k=n+2

n⁴

k²(k²−(n+ 1)²)²

≤C

n−˜n−2

X

j=1

n⁴

(n−j−1)²(j²−2nj)²

| {z }

=:T_j¹

+C

∞

X

j=1

n⁴

(n+j+ 1)²(j²+ 2jn)²

| {z }

=:T_j²

≤C





n−˜n−2

X

j=1

T_j¹+T_j²− 1 j²



+C

n−˜n−2

X

j=1

1 j² +C

∞

X

j=n−˜n−1

T_j². (31)

As one easily sees,T_j² ≤ _j¹2 holds. Hence it only remains to estimate the termPn−˜n−2

j=1 T_j¹+T_j²−_j¹2. Before proceeding we need some auxiliary results. Let

A:= X

0≤r+s≤4

αr,sn^rj^s:= (n−j−1)²(2n−j)² and B:= X

0≤r+s≤4

βr,sn^rj^s:= (n+j+1)²(j+2n)².

By a straightforward calculation one gets

α_4,0= 4, α_3,1=−12, α_3,0=−8 and β_4,0= 4, β_3,1= 12, β_3,0= 8.

Thus we can estimate the amount of the free capacity as T_j²− 1

j² = 1 j²

n⁴−(n+j+ 1)²(j+ 2n)² (n+j+ 1)²(j+ 2n)²

≤ 1 j²

−3n⁴−12n³j−8n³ (n+j+ 1)²(j+ 2n)²

. The next calculation shows that terms of lower order can be neglected in the estimation. Let

D:=A B= (n−j−1)²(j−2n)²(n+j+ 1)²(j+ 2n)². ThenD >0forj ∈ {1, . . . , n−n˜−2}. Ifr≤2andr+s≤4, then

n−n−2˜

X

j=1

n⁴ j²

n^rj^s

D ≤

n−˜n−2

X

j=1

n⁶ D ≤

n−˜n−2

X

j=1

1

(n−j−1)² =

n−2

X

k=˜n+1

1 k² ≤C.

Now the preparatory work is completed and we can return to the estimation ofI + III.

n−˜n−2

X

j=1

T_j¹+T_j²− 1 j² ≤

n−n−2˜

X

j=1

1 j²

n⁴

A +−3n⁴−12n³j−8n³ B

≤

n−˜n−2

X

j=1

1 j²

n⁴(β_4,0n⁴+β_3,1n³j+β_3,0n³) + (−3n⁴−12n³j−8n³)(α_4,0n⁴+α_3,1n³j+α_3,0n³) D

+C X

0≤r+s≤4,r≤2

[|βr,s|+|αr,s|]

n−˜n−2

X

j=1

n⁶ D

≤

n−˜n−2

X

j=1

1 j²

n⁴(4n⁴+ 12n³j+ 8n³)−3n⁴(4n⁴−12n³j−8n³)−(12n³j+ 8n³)4n⁴ D

+

n−n−2˜

X

j=1

1 j²

(12n³j+ 8n³)(α3,1n³j+α3,0n³)

D +C

≤

n−˜n−2

X

j=1

1 j²

−2n⁴4n⁴+ 4n⁴(12n³j+ 8n³)−(12n³j+ 8n³)4n⁴ D

+C≤C.

Recalling (31) we getI + III≤C, which completes the proof.

The uniform convergence of the fifth sum is easily reduced to the uniform convergence of the fourth sum. Because

∞

X

k=1

|λ²|

|ηk| 1

|η−ηk| =

∞

X

k=1

|λ²|

|ηk|

|η−η_k|

|η−ηk|² ≤

∞

X

k=1

|λ²|

|ηk|

|η|

|η−ηk|²+

∞

X

k=1

|λ²|

|η−ηk|² ≤C,

we have reduced the uniform convergence of the sixth sum to the uniform convergence of the second and fifth sum. Therefore, the proof of Proposition 2.10 is finished.

2.3 Exponential Stability of the Associated System

This subsection is devoted to the proof of exponential stability of the associated linear wave equation i.e.

we will show the following statement

Proposition 2.14. The operatorAsatisfies the conditions of exponential stability

(C1) sup

Reλ : λ∈σ(A) < 0

(C3) sup

kRλ(A)k : Reλ≥ν+ε^∗ < ∞ ∀ε^∗>0, whereνis as in Definition 2.5.

First, we will proof thatRkλ(A)kis uniformly bounded inλ∈Γ^I.

Lemma 2.15. Letλ∈Γ^I, thenkRλ(A)k ≤C, where the constant is independent ofλ∈Γ^I.

Proof. Letλ∈Γ^IandF = (f1, f2)^T ∈ Hbe arbitrary. From Proposition 2.3 and 2.6 we know, that the equationλU−AU =Fhas a unique solutionRλ(A)F = (p, q)^T, wherepandqare given as

q=λp−f1 and p=Rη(A0)(f2+ (λ+a)f1).

We know thatkR_λ(A)Fk²_H =k∇pk²_L2+kqk²_L2 =k∇pk²_L2+kλp−f₁k²_L2. By applying Lemma 2.16 the statement is obtained.

Lemma 2.16. LetpandFbe defined as in the proof of Lemma 2.15. Then there is a constant0< C <∞ independent ofλ∈Γ^Isuch thatk∇pk²_L2 ≤CkFk²_Handkλpk²_L2 ≤CkFk²_H.

Proof. Letg:=f₂+ (λ+a)f₁andp=R_η(A₀)(g). Using Proposition 2.9 and Proposition 2.10 we can estimatek∇pkL²as

k∇pk²_L2=

p,−A0p

L²+hp,(bmin−b)pi_L2≤

p,−A0p

L²=hp, gi_L2+hp,−ηpi_L2

≤

∞

X

k=1

1

|η−ηk|+ |η|

|η−ηk|²

| {z }

=:χk

| hf2+ (λ+a)f1, uki_L2|²

≤Ckf2k²_L2+

∞

X

k=1

|λ²|χk| hf1, uki_L2|²

| {z }

=:Σ

+C a²kf1k²_L2 (32)

A proposition of Hilbert and Courant (see page 288 in [4]) states that there is a constantC, such that for allk∈Nholds|uk|L^∞ ≤C. Thus one can derive

|

f1,(bmin−b) u_k

√ηk

L²

|²≤Ckf1k²_L2 and k∇ u_k

√ηk

k²_L2 ≤C, (33) where0< Cis independent ofk∈N. Hence we can estimateΣby using Proposition 2.10 and (33) as

Σ≤

∞

X

k=1

|λ²|

|ηk|² χk| hf1, ∂xxuki_L2|²+

∞

X

k=1

|λ²|

|ηk| χk|

f1,(bmin−b) u_k

√ηk

L²

|²

≤

∞

X

k=1

|λ²|

|ηk| χkk∇f1k²_L2k∇ uk

√ηk

k²_L2+C

∞

X

k=1

|λ²|

|ηk| χkkf1k²_L2 ≤Ck∇f1k²_L2. (34)

By assembling the estimates (32) and (34) we obtain

k∇pk²_L2 ≤C(k∇f₁k²_L2+kf₂k²_L2) =CkFk²_H,

where the constantC > 0is independent ofλ ∈ Γ^I. The estimation ofkλpk²_L2 is very similar to the estimation ofk∇pk²_L2. Thus we get (compare with (32) and (34))

kλpk²_L2=

∞

X

k=1

|λ²|

|η−η_k|² | hf2+ (λ+a)f1, uki_L2|²

≤Ckf2k²_L2+

∞

X

k=1

|λ⁴|

|η−η_k|² | hf1, uki_L2|²+a²Ckf1k²_L2

≤Ckf2k²_L2+Ck∇f1k²_L2 ≤CkFk²_H, where the constantC >0is again independent ofλ∈Γ^I.

By analyzing the proof of Lemma 2.15 and regarding Lemma 2.2, that is also valid for the operator A, one sees that (C1) is satisfied. By using a standard argument for dissipative operators (see also Lemma 2.2) it is possible to expand the statement of Lemma 2.15 to the whole areaΓ. Therefore (C3) is also satisfied and the proof of Proposition 2.14 is complete.

2.4 The Fixed Point Argument

In this section we proof Proposition 1.5 by using a fixed point argument. Letλ∈ΓandF= (f1, f2)^T ∈ Hbe arbitrary. We will now define the mapΦ(λ, F) : H → H, which is supposed to have a fixed point. LetV = (v1, v2)^T ∈ H. ThenΦ(λ, F)(V) :=U ∈D(A), whereU is the unique solution of the equation

λU−AU =F− 0

(a−a)v2

.

The next lemma is verified by a straightforward calculation and shows why a fixed point of the map Φ(λ, F) would be very useful.

Lemma 2.17. LetU ∈D(A)be a unique fixed point of the mapΦ(λ, F), thenU is the unique solution of the equationλU−AU =F.

The next statement gives an answer to the question when the mapΦ(λ, F)has a unique fixed point.

Lemma 2.18. Letλ∈Γ^I, then there existsε >0independent ofλsuch that ifka−ak_L2 < ε, then the mapΦ(λ, F)is contracting. In this case, by using the Banach‘s fixed point theorem, we know that the mapΦ(λ, F)has a unique fixed point.

Proof. LetV¹= (v₁¹, v¹₂)^T ∈ HandV²= (v₁², v²₂)^T ∈ H. One has to show the property of contraction for the mapΦ(λ, F) :H → H. LetU¹ = (p¹₁, p¹₂)^T := Φ(λ, F)(V¹)∈D(A)andU²:= (p²₁, p²₂)^T :=

Φ(λ, F)(V²)∈D(A). Then(U¹−U²)is the unique solution of the equation λU˜ −AU˜ =

0 (a−a)(v₂¹−v₂²)

.

Thus, by using Proposition 2.3 and Proposition 2.6 it follows that U¹−U²=Rλ(A)

0 (a−a)(v¹₂−v₂²)

= p

λp

holds andpis determined asp=Rη(A0)[(a−a)(v₂¹−v²₂)]. Using Lemma 2.19 we obtain kU¹−U²k²_H=k∇pk²_L2+kλpk²_L2 ≤Cka−ak²_L2kV¹−V²k²_H,

where the constant0< C <∞is independent ofλ∈Γ^I. If we takeka−ak_L2 <_C¹ =:ε, then the map Φ(λ, F)is contracting.

Lemma 2.19. Letp,V¹ and V² be defined as in the proof of Lemma 2.18. Then there is a constant 0< C <∞independent ofλ∈Γ^Isuch that

k∇pk²_L2 ≤Cka−ak²_L2kV¹−V²k²_H and kλpk²_L2 ≤Cka−ak²_L2kV¹−V²k²_H.

Proof. The Estimation ofk∇pkL² andkλpkL² is performed in the same way as in the proof of Lemma 2.16. One only has to regard that the function g is now given by g := (a−a)(v₂¹−v₂²) and that

|

(a−a)(v₂¹−v₂²), u_k

L²|²≤Cka−ak²_L2kV¹−V²k²_His satisfied by using the proposition of Courant and Hilbert mentioned before.

We assume now thatka−ak_L2 < ε, whereε >0is taken as in Lemma 2.18. By regarding Lemma 2.2, Lemma 2.17 and Lemma 2.18 one sees that condition (C1) is satisfied for the operatorA. To verify condition (C3) we have to estimate the norm of the resolventRλ(A)forλ∈Γ.

Lemma 2.20. Letλ∈ Γ^Iand letka−ak_L2 < ε, whereε >0is taken as in Lemma 2.18. Then there exists a constant0< C <∞independent ofλ, such thatkRλ(A)k ≤C.

Proof. Letλ∈Γ^I andF ∈ Hbe arbitrary. By Lemma 2.18 the mapΦ(λ, F)has a unique fixed point U ∈ D(A). Let U˜ := Φ(λ, F)(0) = Rλ(A)F. Let 0 ≤ d < 1 be the constant of the property of contraction of the map Φ(λ, F). From Lemma 2.15 we know that there is a constant 0 < C < ∞ independent ofλ∈Γ^I, such thatkRλ(A)k ≤C. Thus we obtain

kUk_H− kUk˜ _H≤ kΦ(λ, F)(0)−Φ(λ, F)(U)k_H ≤dk0−Uk_H.

So it follows that

kUk_H≤ 1

1−dkU˜k_H= 1

1−dkRλ(A)Fk_H≤ C

1−dkFk_H, where the constant0< C <∞is independent ofλ.

As in section 2.3 we can extend the statement of the last lemma to the whole area of parametersΓwith the help of Lemma 2.2. Therefore (C3) is also satisfied for the OperatorA. As a consequence the linear wave equation (1) is exponentially stable. The statement about a possible decay rate follows directly from the definition ofΓ. Thus the proof of Proposition 1.5 is complete.

3 Non-Linear Case

This section is devoted to the proof of Proposition 1.6. The argumentation is very similar to the one that Racke and Muñoz Rivera used in [17] for their statement in the caseb= 0. After deducing a high energy estimate and a weighted a priori estimate the local solution of (4) is extended to a global solution by a continuation argument (compare for example [22] or [13]). The smallness of the global solution follows automatically from the weighted a priori estimate. We assume, for the whole section, that the hypotheses of Proposition 1.6 are satisfied.