Exponential stability for wave equations with non-dissipative damping

Exponential stability for wave equations with non-dissipative damping

Jaime E. Mu˜noz Rivera

, Reinhard Racke

aDepartment of Research and Development, National Laboratory for Scientific Computation, Rua Getulio Vargas 333, Quitandinha, CEP 25651-070 Petr´opolis, RJ, Brazil

bUFRJ, Rio de Janeiro, Brazil

cDepartment of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany

Abstract

We consider the nonlinear wave equationut t −σ (ux)x+a(x)ut = 0 in a bounded interval(0,L) ⊂ R¹. The functiona is allowed to change sign, but has to satisfya = ¹

L RL

0 a(x)dx > 0. For this non-dissipative situation we prove the exponential stability of the corresponding linearized system for: (I) possibly largekak_L∞ with smallka(·)−ak

L², and (II) a class of pairs (a,L)with possibly negative momentRL

0 a(x)sin²(πx/L)dx. Estimates for the decay rate are also given in terms ofa. Moreover, we show the global existence of smooth, small solutions to the corresponding nonlinear system if, additionally, the negative part of ais small enough.

MSC:35L70; 35B40

Keywords:Indefinite damping; Wave equation; Exponential stability

1. Introduction

We consider the following nonlinear wave equation:

ut t−σ (ux)x+a(x)ut =0 (1.1)

for a functionu=u(t,x),t ≥0,x ∈(0,L)⊂R¹,L>0 fixed, with initial conditions

u(t =0)=u₀,u_t(t=0)=u₁ (1.2)

and Dirichlet type boundary conditions

u(·,0)=u(·,L)=0. (1.3)

ISupported by a CNPq-DLR grant.

∗Corresponding author.

E-mail addresses:rivera@lncc.br(J.E. Mu˜noz Rivera),reinhard.racke@uni-konstanz.de(R. Racke).

Konstanzer Online-Publikations-System (KOPS) URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-96367

URL: http://kops.ub.uni-konstanz.de/volltexte/2009/9636/

We assume that a ∈ L^∞((0,L)) for the part on the exponential stability of the associated semigroup, anda ∈ C³([0,L])for the discussion of the nonlinear system, as well as

a:= 1 L

Z L 0

a(x)dx>0, (1.4)

and, in particular,a may change sign in[0,L]or be zero in open subsets. The nonlinear functionσ is assumed to satisfy

σ ∈C³(R), d₀:=σ⁰(0) >0, and σ⁰⁰(0)=0. (1.5)

Remark. This is, for instance, satisfied forσcorresponding to a vibrating string, σ (y)= y

p1+y². Rewriting(1.1)as

u_{t t} −d0u_{x x} +au_t =b(u_x)u_{x x} (1.6)

with

b(u_x):=σ⁰(u_x)−d₀=σ⁰(u_x)−σ⁰(0) (1.7)

the associated linearized system is

u_{t t} −d₀u_{x x} +au_t =0 (1.8)

together with the initial conditions(1.2)and the boundary conditions(1.3). Sinceamay change sign we have a non- dissipative system still regardingaut as a non-local but indefinite damping. There are many papers on solutions to (1.1)and on decay rates for(1.1)or(1.8)ifa ≥0, i.e. ifa does not change sign; see for example the papers of Cox and Overton [3], Cox and Zuazua [5], Kawashima et al. [9], Nakao [13,14], da Silva Ferreira [19] or Zuazua [20] and the references therein. Ifa(x) ≥ a0 >0 is strictly positive, the exponential decay of solutions to(1.8)and also to (1.1), for small data, easily follows.

The non-dissipative case with indefiniteaseems to have been posed first by Chen et al. [2] where it was conjectured that the energy

E0(t)= Z L

0 (u²_t +u²_x)(t,x)dx (1.9)

decays exponentially if

∃γ >0 ∀n =1,2, . . .: Z L

0

a(x)sin²(nπx/L)dx≥γ (1.10)

holds. Later Freitas [6] found that (1.10)is not sufficient to guarantee exponential stability whenkak_L∞ is large.

Replacingabyεa, Freitas and Zuazua [7] proved that whenais of bounded variation and(1.10)holds, then there is ε^∗ = ε^∗(a)such that for allε ∈ (0, ε^∗)the energy does indeed decay exponentially. This result was extended to a differential equation of the type

u_{t t} −u_{x x} +εa(x)u_t+b(x)u =0

by Benaddi and Rao [1]. Liu et al. [10] gave an abstract treatment of these results under certain conditions on the abstract damping operator. An extension to higher space dimensions was presented by Liu et al. [11].

Here we show that solutions to the linearized system(1.8),(1.2)and(1.3)decay exponentially for (I) possibly largekak_L∞with smallka(·)−ak_L2, and

(II) a class of pairs(a,L)with possibly negative momentRL

0 a(x)sin²(πx/L)dx.

In part (I) we improve the previous works that had the stronger assumption of smallness ofkak_L∞; now we may admit large values of|a|as long aska(·)−ak_L2is small enough; cf. the examples(a_µ)µin Section2withka_µk_L∞ =1 andka_µ(·)−a_µk_L2 →0 asµ→0.

Part (II) is not a contradiction to a result of Freitas in [6], saying that if(1.10)is not valid, then the solution is not exponentially decaying for sufficiently smallkak_L∞, because in our examples of admissible pairs(a,L), leading to exponential decay,a(resp.kak_L∞), andLare not independent.

We remark that the results are not “perturbation results for smalla”, since fora=0 we do not have any decay.

Estimates for the decay rate are also given in terms ofa. Moreover, we show the global existence of smooth, small solutions to the corresponding nonlinear system if, additionally, the negative part ofais small enough. More precisely:

Ifα0denotes the decay rate for the linear system, Z L

0 (u²_t +u²_x)(t,x)≤c1e^−2α0t, c1>0, thena⁻has to satisfy in particular

ka⁻k_L∞ < α0, (1.11)

see Section3.

The paper is organized as follows. In Section2we shall prove the exponential stability for the linearized system in either of the situations (I) or (II) above. This is the crucial part, and the method will be the spectral one characterizing exponentially stable semigroups in terms of the spectrum of the associated generator of the semigroup. It is possible to give an explicit lower bound on the decay rate which, in turn, is necessary to make(1.11)a reasonable condition in the nonlinear case.

In Section3the global existence of small solutions to the nonlinear system is investigated. Using the result from Section2and perturbation arguments, the condition(1.11)is shown to be sufficient to guarantee the global existence and also the exponential stability of the nonlinear system.

Summarizing the contributions of our paper, we present results on exponential stability for the wave equation when the functionamay change sign under conditions that extend the existing results to cases with indefinite dampinga with possibly largeL^∞-norm, and give examples of pairs(a,L)for which exponential stability holds but the moment RL

0 a(x)sin²(πx/L)dx is negative. We also present an explicit description of the decay rate and of the type of the associated semigroup, and also a discussion of a corresponding nonlinear problem with global existence and stability.

Finally, our approach can be applied to other one-dimensional models.

We use standard notation, e.g. for Sobolev spaces.

2. Exponential stability for the linearized system

We first consider the linearized system. Without loss of generality we assumed0=σ⁰(0)=1.

ut t−ux x+a(x)ut =0 in(0,∞)×(0,L) (2.1)

u(0,·)=u₀, u_t(0,·)=u₁ in(0,L) (2.2)

u(·,0)=u(·,L)=0 in(0,∞). (2.3)

We assume thata ∈ L^∞((0,L))and satisfies(1.4). The aim is to prove that the energy given in(1.9)decays to zero exponentially as timettends to infinity in either

• Case (I): ka(·)−ak

L² is sufficiently small – butkak_L∞may be large – or

• Case (II): kak_L∞ is small, (a,L)satisfy certain relations — but the momentRL

0 a(x)sin²(πx/L)dx may be negative.

We introduce the variables p:=u_t−u_x, q:=u_t+u_x such that

pt+px = −a(x)ut, qt−qx = −a(x)ut. (2.4)

Let U :=

p q

, K :=

1 0 0 −1

, B:=

1 1 1 1

, and letAdenote the operator given by

AU:= −K U_x−a 2BU with domain

D(A):=

p q

∈ H¹((0,L))×H¹((0,L))|p(0)+q(0)=p(L)+q(L)=0,Z L 0

p(s)−q(s)ds=0 in the Hilbert space

H:=

p q

∈L²((0,L))×L²((0,L))| Z L

0

p(s)−q(s)ds=0

with the L²((0,L)) inner product. D(A) is dense inH and A is the infinitesimal generator of a C₀-semigroup {e^At}_t≥0. We can rewrite(2.1)as

U_t =AU, U(0)=U₀, U ∈ D(A). (2.5)

In a standard way we obtain Lemma 2.1. A⁻¹is compact.

Lemma 2.1implies that the spectrumσ(A)ofAconsists of eigenvalues(λn)nonly, without any finite accumulation point.

First we consider Case (I), where finallyka(·)−ak_L2 will be chosen sufficiently small, and we will show below, by using fixed point arguments, that forε >0

Γ_ε^I :=







ε+α+iβ;α >R



−a 2 +

s a

2 2

−π² L²



 andβ ∈R







⊂%(A), and also that

sup

λ∈Γ_ε^I

k(λI −A)⁻¹k<∞.

This will imply, see e.g. the results by Pr¨uss [17] or [12, Thm 1.3.1], that the corresponding semigroup decays exponentially.

We shall regardAas a perturbation ofA₀^I defined by A₀^IU:= −K U_x−a

2BU on

D(A₀^I):=D(A).

Lemma 2.2. Letσ (A₀^I)denote the spectrum of A^I₀. Then we have that σ (A^I₀)=







−1 2a±

s 1

2a 2

−k²π² L²





 k∈N.

Proof. As forA, one can show thatA₀^Ihas a compact inverse. We now consider the equation

λU−A₀^IU =F (2.6)

whereF ∈H, which is equivalent to finding a functionUsuch that U_x +MU =K F

whereM :=K(λI+^a

2B). The solution to this equation is given by U(x)=e^{−M x}U₀+

Z x 0

e^−M(x−s)K F(s)ds. (2.7)

This is the solution of an initial value problem. Finding the set ofλbelonging to the resolvent set is equivalent to the problem of findingλsuch thatUsatisfies the boundary conditions of the problem and can be estimated appropriately byF. Defining

U = p

q

, U0= p0

q₀

we have to findU ∈ D(A^I₀). To get this we have to satisfy firstp0+q0=0 which implies p₀= −q₀.

Now the problem reduces to findingp0for which we have

p(L)+q(L)=0. (2.8)

Let us define

E(x,s):=e^−M(x−s)=:

e₁₁(x,s) e₁₂(x,s) e₂₁(x,s) e₂₂(x,s)

. Note that

p q

=E(x,0) p₀

−p₀

+ Z x

0

E(x,s) f(s)

−g(s)

ds. (2.9)

Using the expression above to verify relation(2.8)we conclude that p0should satisfy

− {e₁₁(L,0)−e₁₂(L,0)+e₂₁(L,0)−e₂₂(L,0)}p₀

= Z L

0

{e11(L,s)+e21(L,s)} f(s)− {e12(L,s)+e22(L,s)}g(s)ds. (2.10) It is not difficult to see that the above problem has a solution if and only if

e₁₁(L,0)−e₁₂(L,0)+e₂₁(L,0)−e₂₂(L,0)6=0 (2.11) and that thereforeλ ∈%(A₀^I)holds if and only if condition(2.11)holds. To characterize the spectrum precisely, we need to calculate the matrixE(x,s)explicitly. To do this we note that

M=

α β

−β −α

whereα:=λ+1

2a, β:= 1

2a. (2.12)

To get the explicit representation of the exponential matrixE we will use the eigenvector representation. Therefore our next step is to calculate the eigenvalues and eigenvectors. Since

det

µ−α −β +β µ+α

=µ²−(α²−β²) it follows that the eigenvalues are given byµ= ±p

α²−β². Let us defineµ0 :=p

α²−β²with non-negative real part. Then we have that the eigenvectors are given by

w1=

µ0+α

−β

, w2=

−µ0+α

−β

. Observing 0∈%(A₀^I)we may assumeλ6=0. Letting

D:=

µ0+α −µ0+α

−β −β

⇒D⁻¹= − 1 2βµ0

−β µ0−α β µ0+α

we have the representation E(x,0)=D

e^−µ0x 0 0 e^µ0x

D⁻¹ which implies

E(x,0)=







cosh(µ0x)− α µ0

sinh(µ0x) −β µ0

sinh(µ0x) β

µ0

sinh(µ0x) cosh(µ0x)+ α µ0

sinh(µ0x)





. (2.13)

The condition(2.11)now turns into λ

µ0

sinh(µ0L)=0⇒e^2µ0L =1.

Therefore we conclude thatµ0L =kπi, for integersk, or µ²₀L²= −k²π².

Recalling the definition ofµ0we get and using(2.12)we get λ²L²+L²λa+k²π²=0⇒λ= −1

2a± s

1 2a

2

−k²π² L² .

Finally, note that ifk=0 then we will haveµ0=0 and thereforeλ=0∈%(A^I₀). Lemma 2.3. Letλ=γ+iηwithγ >−^a

2, and let µ0=A(η)+iB(η)

define A and B. Then we have A²+B²=

qη⁴+ [2γ²+2γa+a²]η²+(γ²+γa)². (2.14)

ηlim→∞

α µ0

=1, lim

η→∞

β µ0

=0, lim

η→∞

λ µ0

=1, lim

η→∞A=γ+a

2. (2.15)

(γ,η)lim→(0,∞)

|sinh(µ0x)|

|µ0| =x, lim sup

η→∞

|sinh(µ0x)| ≤ s

1+sinh²

γ +a 2

L. (2.16)

Proof. Recalling that µ0=

qα²−β²=p λ²+λa we have

µ0=

qα²−β²=

qγ²+aγ+iη(a+2γ )−η²=A+iB. Squaring the above expression we get

γ²+aγ−η²=A²−B² η(a+2γ )=2A B.

Solving the equation forAwe conclude that A²=γ²+aγ

2 +

−η²+

qη⁴+ [2γ²+2γa+a²]η²+(γ²+γa)² 2

B²= −γ²+aγ

2 +η²+

qη⁴+ [2γ²+2γa+a²]η²+(γ²+γa)²

2 .

Summing up the above identities we get(2.14). Note that qη⁴+ [2γ²+2γa+a²]η²+(γ²+γa)²

2 ≤

q(η²+γ²+γa+^a2

2)²−a²(γ +^a

2)² 2

= η²+γ²+γa+^a2

2

2 .

On the other hand we have

qη⁴+ [2γ²+2γa+a²]η²+(γ²+γa)²

2 ≥

pη⁴+ [2γ²+2γa]η²+(γ²+γa)² 2

= η²+γ²+γa

2 .

Therefore we obtain γ²+aγ ≤ A(η)²≤

γ+a

2 2

. Similarly we have

η²≤ B(η)²≤η²+a² 4 . Thus we conclude

ηlim→∞

α µ0

2

= lim

η→∞

(γ +¹

2a)²+η² A(η)²+B(η)² =1 proving(2.15). Finally,

sinh(µ0x)= 1

2{e^Ax(cos(B x)+i sin(B x))−e^−Ax(cos(B x)−i sin(B x))}

=cos(B x)sinh(Ax)+i sin(B x)cosh(Ax).

This implies

|sinh(µ0x)| = q

cos²(B x)sinh²(Ax)+sin²(B x)cosh²(Ax)

= q

sinh²(Ax)+sin²(B x) and similarly we have

|cosh(µ0x)| = q

sinh²(Ax)+cos²(B x).

yielding(2.16).

Lemma 2.4. Let F =(f₁, f₂)andλ=γ+iηas above, and let I(λ,F):= 1

2 RL

0(a−a)(cosh((L−s)µ0)+^λ_µ^+a

0 sinh(µ0(L−s)))f1ds

−^λ

µ0sinh(µ0L)

−1 2

RL

0 (a−a)(cosh(µ0(L−s))+ ^λ

µ0 sinh(µ0(L−s)))f₂dx

−_µ^λ

0 sinh(µ0L) .

Then we have lim sup

(γ,η)→(0,∞)

|I(λ,F)| ≤ ka−ak

L²kFk

L²

1+a L a L

lim sup

η→∞

|I(λ,F)| ≤ ka−ak_L2kFk_L2 1+sinh((γ+^a

2)L) sinh((γ +^a

2)L)

!

lim sup

(γ,η)→(0,∞)

|E(x,s)| ≤1+a L 2 lim sup

η→∞

|E(x,s)| ≤2 s

1+sinh²

γ+a 2

L

.

Proof. We have

|I(λ,F)| ≤1 2

ka−ak

L²(2kcosh(µ0x)k_L∞+^|λ+a|+|λ|

|µ0| ksinh(µ0x)k_L∞)kFk

L²

|^λ

µ0||sinh(µ0L)|

wherek · k_∞denotes the sup-norm with respect tox. Our conclusion now follows fromLemma 2.3using

|sinh(µ0L)| ≥ |sinh(A(η)L)| →sinh

γ+a 2

L

asη→ ∞.

Corollary 2.5. There exists a positive constant C0, depending essentially only on|γ +^a

2|, such that for anyη ∈ R and any(x,s)we have

|I(λ,F)| ≤C₀ka−ak

L²kFk

L²,

|E(x,s)| ≤C0.

Remark. For the interval(0,1)and 0< µ < ¹₂, let the functiona_µ: [0,1] −→Rbe given by a_µ(x):=

−1 0≤x< µ 1 µ≤x <1. Then

a_µ=1−2µ and ka_µ−a_µk_L2 =2p

(1−µ)µ <1, and hence, asµ→0,

ka_µk_L∞ =1 and ka_µ−a_µk

L² →0.

Of course, from this one can easily derive examples inC^∞. Lemma 2.6. There existsτ >0such that whenka−ak

L² < τ, we have (i)Γ_ε^I ⊂%(A),

(ii)sup_λ∈ΓI

ε k(λ−A)⁻¹k<∞.

Proof. It suffices to show that for sufficiently smallτ > 0 and forλ ∈Γ_ε^I the equation(λ−A)U = F is solvable for any F ∈ H, andkUk_L2 ≤CkFk_L2 with a constantCat most depending onεandτ. We shall use a fixed point argument to prove this. Now letF =(f,g)⁰∈Hbe given as well asλ∈Γ_ε^I. Let

Φ: ˜H→ ˜H, V 7→U =ΦV, H˜ :=L²((0,L))×L²((0,L)), be defined as the solutionU=(U1,U2)⁰to

(λ−A₀^I)U =F−a−a 2 B V

which is well defined sinceλ∈%(A₀^I). Using the explicit representation ofUwe have U(x)=(ΦV)(x)=E0(x)U0+

Z x 0

E0(x−s)

K F(s)−a(s)−a

2 K B V(s)

ds (2.17)

whereU0=(p0,−p0)withp0=I(λ,V), cf.(2.10),

−p₀ = Z L

0

(e₁₁+e₂₁)(L−s)

f(s)−a(s)−a

2 (V₁(s)+V₂(s))

−(e₁₂+e₂₂)(L−s)

×

g(s)−a(s)−a

2 (V₁(s)+V₂(s))

ds

(e₁₁(L)−e₁₂(L)+e₂₁(L)−e₂₂(L)), (2.18) whereV =(V₁,V₂)⁰. Let

U^j :=ΦV^j, j =1,2. Then

U¹(x)−U²(x)=E₀(x)(U₀¹−U₀²)+ Z x

0

E₀(x−s)

a(s)−a

2 K B(V²(s)−V¹(s))

ds. This implies

|U¹(x)−U²(x)| ≤ |E₀(x,0)(U₀¹−U₀²)| +c₁ Z x

0

|(a−a)E₀(x,s)(V¹(s)−V²(s))|ds (2.19) andc1denotes here and in the sequel a positive constant at most depending onτ andε. We conclude fromCorollary 2.5 that

|U₀¹−U₀²| ≤c1ka−ak_L2kV¹−V²k_L2. (2.20)

The last two inequalities yield and usingCorollary 2.5once more we have kU¹−U²k_L2 ≤c₁ka−ak_L2kV¹−V²k_L2.

HenceΦis a contraction mapping if c₁ka−ak

L² <1 determiningτ asτ = ¹

c₁. LetU≡(p,q)⁰be the unique fixed point. It satisfies λU+K Ux+a

2BU =F. (2.21)

By definition we haveU ∈ D(A^I₀)=D(A). Thus we conclude that U ∈ D(A) and (λ−A)U =F.

Finally, we estimate the inverse(λ−A)⁻¹. LetUstill be the fixed point, and let U˜ :=Φ(0)

or, in other words, (λ−A^I₀)U˜ =F.

Then we get

kUk_L2− k ˜Uk_L2 ≤ kU− ˜Uk_L2 = kΦU−Φ(0)k_L2 ≤dkUk_L2

whered <1 describes the contraction mapping property. It follows that kUk_L2 ≤ 1

1−dk ˜Uk_L2.

On the other hand we obtain from(2.17)and (2.18)(cf.(2.19)and (2.20)) k ˜Uk

L² ≤c₁kFk

L². Hence we have proved

k(λ−A)⁻¹k ≤c₁ 1 1−d

which proves the lemma.

As a consequence we obtain the following theorem on the exponential decay.

Theorem 2.7. There existsτ >0such that whenka−ak

L² < τ, we have that the solution to the linearized system decays exponentially, that is

∃c₀>0∃α0>0∀t ≥0: E₀(t)≤c₀e^−2α0tE₀(0).

In particular we can take any α0>R



−a 2 +

s a

2 2

−π² L²



, e.g. α0= − ¹

4L

RL

0 a(x)dx whenRL

0 a(x)dx<2π.

Proof. The assertion follows from Lemma 2.6 by well known characterizations of the exponential stability of semigroups; see the results given by Pr¨uss [17], and cf. [12, Thm 1.3.1].

Now we consider Case (II), where finallykak_L∞will have to be sufficiently small, related toL, that is,(a,L)will satisfy certain relations. Contrasting these restrictions with respect to previous results for smalla, e.g. those of Freitas and Zuazua [7], we shall obtain examples where the momentRL

0 a(x)sin²(πx/L)dxis negative.

Similarly to in Case (I) above, we wish to prove that for smallε1>0 and anyε0> ε1we can choose(a,L)with asmall enough such that for anyε∈ [ε1, ε0]

Γ_ε^{I I} :=

−a

2 +ε+iη η∈R

⊂%(A) (2.22)

and that sup

ε∈[ε1,ε0], λ∈Γ_ε^{I I}

k(λ−A)⁻¹k<∞. (2.23)

We choose ε1:= a

4, ε0:=max{2ε1,3|a|_∞}, (2.24)

where|a|_∞:= kak_L∞, and we observe thatA−3|a|_∞is dissipative.

Now we shall regardAas a perturbation of the following operatorA₀^{I I}, given by A₀^{I I}U:= −K U_x−a

2U with domain

D(A₀^{I I}):=D(A)

in the same Hilbert spaceH. ForU ∈ D(A)we have AU =A₀^{I I}U−a

2W U with

W :=

0 1 1 0

. (A₀^{I I})⁻¹is compact.

Lemma 2.8. σ(A₀^{I I})=n

−^a

L +^kπi

L |k∈Z

o .

Proof. We investigate the solvability of(λ−A₀^{I I})U = Ffor any F =(f,g)⁰ ∈ H. Using ideas similar to those in the proof ofLemma 2.2we obtain

λ∈σ (A₀^{I I})⇔e^−2λL−

RL

0 a(y)dy =1⇔λ=λk = −a L +kπi

L , k∈Z.

For given 0< γ0< γ1we define the set of admissibleaandLas K(γ0, γ1):=

(a,L)

γ0≤

Z L 0

a(x)dx≤a∞L ≤γ1

.

We shall prove that, after fixingγ0andγ1, we can determine the maximal possibleL^∞-norm ofa, and the admissible values ofL, that imply exponential decay.

Lemma 2.9. There are pairs(a,L)∈K(γ0, γ1)such that (i)∀ε∈ [ε1, ε0] : Γ_ε = {−^a

2+ε+iη|η∈R} ⊂%(A).

(ii)sup_ε∈[ε

1,ε0], λ∈Γ_εk(λ−A)⁻¹k<∞.

Proof. We will show, for any admittedε, that forλ ∈Γ_ε^{I I} the equation(λ−A)U = Fis solvable for anyF ∈ H, and the postulated estimate on the inverse holds. We shall use again a fixed point argument to prove this. Now let F=(f,g)⁰∈Hbe given as well asλ∈Γ_ε. Let

Φ: H˜ −→ ˜H, V →U=ΦV be defined as solutionU =(U1,U2)⁰to

(λ−A^{I I}₀ )U=F−a 2W V

which is well defined sinceλ∈%(A₀^{I I}). Let us consider U(x)=(ΦV)(x)=E₀(x,0, λ)U₀+

Z x 0

E₀(x,s, λ)

K F(s)−a(s)

2 K W V(s)

ds (2.25)

where

E0(x,s, λ):= e^{−λ(x−s)−Rsx a(y)2 dy} 0 0 e^λ(x−s)+

Rx s

a(y) 2 dy

!

≡

e1(x,s, λ) 0 0 e₂(x,s, λ)

andU0=(p0,−p0)with p₀=

RL

0 e₁(L,s, λ)

f(s)−^a(s)

2 V₂(s)

−e₂(L,s, λ)

g(s)−^a(s)

2 V₁(s) ds

e2(L,0, λ)−e1(L,0, λ) . (2.26)

Let

U^j :=ΦV^j, j =1,2. Then

U¹(x)−U²(x)=E0(x,0, λ)(U₀¹−U₀²)+ Z x

0

E0(x,s, λ)a(s)

2 K W(V²(s)−V¹(s)) ds.

This implies

|U¹(x)−U²(x)|²≤2e^(a∞+2ε0)L|U₀¹−U₀²|²+e^(a∞+2ε0)La²_∞ 2

Z x 0

|V¹(s)−V²(s)|ds 2

. (2.27)

We conclude from(2.26)that

|U₀¹−U₀²|²≤ 4e^(a∞+2ε0)La²_∞ (sinh(ε1L))²

Z L 0

|V¹(s)−V²(s)|ds ²

. (2.28)

The last two inequalities yield

kU¹−U²k_L2 ≤a∞e^{(a∞2 +ε0)L} 2e^{(a∞2 +ε0L)L} sinh(ε1L) +1

!

kV¹−V²k_L2. Hence, observing(2.24),Φis a contraction mapping for(a,L)inKif

a∞< sinh(γ0/4)

(sinh(γ1/4)+e^7/2γ1)e^7/2γ1 =:c0(γ0, γ1). (2.29) (i) Now fixingγ0andγ1, the last inequality determines the maximal possibleL^∞-norm ofa.

(ii) Then the condition γ0

a∞

<L ≤ γ1

a∞

(2.30) determines the possible values ofL.

(iii) Now fixingL, sincea∞L > γ0, we can chooseasatisfying (i) as well as Z L

0

a(x)dx≥γ0. (2.31)

Altogether we have found in (i)–(iii) pairs(a,L)∈Kfor whichΦis a contraction.

LetU≡(p,q)⁰be the unique fixed point. It satisfies λU+K U_x+a

2BU=F. (2.32)

By definition we haveU∈ D(A₀^{I I})=D(A). Thus we conclude U ∈D(A) and (λ−A)U=F.

Finally, we obtain the uniform boundedness of the inverses as in Case (I); namely letUstill be the fixed point, and let U˜ :=Φ(0). Using the same arguments as in the proof ofLemma 2.6we get

kUk

L² ≤ckFk

L²

wherecdepends at most ona,L, γ0, γ1. This proves assertion (ii).

As in Case (I) we conclude the exponential stability. Let E(t):= 1

2kU(t,·)k²

L² =1

2ke^tAU0k²

L²

be the energy associated with problem(2.5). Then E(t)= 1

2

p q

(t,·)

2 L²

=1 2

ut −ux

ut +ux

(t,·)

2 L²

= Z L

0

(u²_t +u²_x)(t,x)dx.

Theorem 2.10. For sufficiently smallkak_∞and admissible(a,L)∈K(γ0, γ1), we have

∃c₀ >0∃α0>0∀t≥0:E(t)≤c₀e^−2α0tE(0).

UsingLemma 2.9which gives information on the spectral radiusω_σ(A) :=sup_λ∈σ(A)Rλ, a result of Neves et al. [16]

which says that the essential typeωe(A)is given by ωe(A)= − 1

2L Z L

0

a(y)dy (2.33)

as well as using the general characterization (see [15]) ω0(A)=max{ωe(A), ω_σ(A)},

whereω0(A)denotes the type of the semigroup, ω0(A)= lim

t→∞

lnke^Atk t . Using this we can establish

Theorem 2.11. Under the conditions of Theorem2.10we have

∃c0>0∃α0=α0

1 L

Z L 0

a(x)dx

>0∀t ≥0:E(t)≤c0e^−2α0tE(0).

α0can be chosen as any number−αwith 0> α >− 1

2L Z L

0

a(x)dx+ε1, e.g. α0:= 1 4L

Z L 0

a(x)dx.

We can now present an example of a functiona : [0,1] −→Rfor which exponential stability holds, but for which (1.10)is violated since we shall have

Z 1 0

a(x)sin²(πx)dx<0. (2.34)

This will not be a contradiction to a result of Freitas in [6] saying that if(1.10)is not valid, then the solution is not exponentially decaying if one replacesabyεafor sufficiently smallε >0, because in our example replacingabyεa is not allowed in general because of the admissibility criteria to be observed in the construction of(a,L)in Case (II).

Let 0< δ1< δ2, to be fixed later, and let a(x):=

δ1 forx∈ [0,1/4)∪(3/4,1]

−δ2 forx∈ [1/4,3/4]. Then 0<R1

0 a(x)dx=(δ1−δ2)/2 providedδ1> δ2. Moreover, we have Z 1

0

a(x)sin²(πx)dx= 1

2π((π−1)δ1−(π+1)δ2) <0 if and only if

δ1<π+1 π−1δ2.

Hence choosingδ1, δ2such that δ2< δ1< π+1

π−1δ2 (2.35)

we have a functionasatisfying 0<Z 1

0

a(x)dx and Z 1

0

a(x)sin²(πx)dx<0.

Chooseδ2:=αδ1such that(2.35)is satisfied, e.g.α:=1/1.92 for the rest of the exposition. Thenδ1is still free. Let us define the functionaˆas

aˆ(x):= a(x/L)

L forx∈(0,L)

whereLwill be chosen to satisfy(2.30). Sincea∞=δ1we haveaˆ∞=δ1/L. We are free to chooseγ0andγ1. Fix γ1. The condition(2.31)can be satisfied if

γ0≤(δ1−δ2)/2= 23 96δ1, i.e. we chooseδ1:= ⁹⁶

23γ0, and condition(2.29)can now be satisfied if δ1

L <c₀(γ0, γ1)=c_γ1sinh(γ0/4) where

c_γ1 := 1

(sinh(γ1/4)+exp(7γ1/2))exp(7γ1/2). Since, for smallγ0,

sinh(γ0/4)≥ γ0

5 it is hence sufficient to require

L > 480

23c_γ1. (2.36)

The last condition(2.30)can now be satisfied if ⁹⁶₂₃γ0≤γ1; e.g. takeγ0:=χ²³₉₆γ1with appropriateχ ∈ (0,1]. As a conclusion from the discussion of Case (II) above we get that with such anLand the givenaˆthe solution to

ut t −ux x + ˆa(x)ut =0 in(0,L)

plus initial and boundary conditions has exponentially decaying energy. In the way we have chosenaˆ anda, we have now foundusolving

u_{t t} −u_{x x} +a(x)u_t =0 in(0,1)

with exponentially decaying energy for a functionasatisfying 0<Z 1

0

a(x)dx and Z 1

0

a(x)sin²(πx)dx<0.

We remark that from this example one can of course also construct examples inC^∞.

We finish this section by giving some higher norm estimates valid in both Cases (I) and (II), i.e. whenever exponential stability is given. Differentiating the differential equation(2.1)with respect tot,

(∂t^ju)t t −(∂t^ju)x x+a(x)(∂t^ju)t =0, j ∈N,

and using the fact that derivatives with respect toxcan be computed from the differential equation successively, we get as a consequence ofa∈C⁰([0,L],R), anda∈C^s−2([0,L],R)ifs≥2 the following theorem.

Theorem 2.12. Under the conditions of theTheorems2.7and 2.10, respectively, we have

∀s∈N ∃C_s >0 ∀t ≥0:

ut

ux

(t,·)

_Hs((0,L))

≤C_se^−α0t

u1

u0,x

_Hs(0,L)

where α0 is given in Theorem2.11, and the data are assumed to be sufficiently smooth and to satisfy the usual compatibility conditions.

Remarks. 1. We only useda∈L^∞((0,L)). 2. Without loss of generality we studied the equation

ut t −d0ux x +a(x)ut =0, x∈(0,L),

ford0=1, because ifd0>0 is arbitrary we may define v(t,y):=u(t,p

d0y), y∈

0, L

√ d₀

. Thenvsatisfies

vt t−vyy+ ˜a(y)vt =0, y∈

0, L

√ d0

,

for which Theorem 2.11 can be applied directly replacing a by a˜ and L by L/√

d₀. The decay rate α˜0 = α˜0

^√

d0

L

RL/√ d0

0 a˜(y)dy

turns intoα˜0=α0

1 L

RL

0 a(y)dy

again since

√ d0

L

RL/√ d0

0 a˜(y)dy= ¹

L

RL 0 a(x)ds.

3. Global existence for the nonlinear system

We now return to the nonlinear system(1.1)–(1.3)assuming again the positivity of the mean value(1.4)and also the condition(1.5)on the nonlinearity, which, for example, is satisfied in the classical model for a nonlinear string, where

σ (u_x)= u_x p1+u²_x.

After recalling the local well-posedness it is our aim to prove a global existence result for data (u0,u1) being sufficiently small in H⁴((0,L)), and, quasi-simultaneously, to obtain the exponential stability. The method used imitates one which is well known for nonlinear evolution equations and systems; see [18] for a presentation of the general approach for Cauchy problems (x ∈ Rⁿ, no boundary). Here we shall have to prove so-called high energy estimates and a weighted a priori estimate – describing the expected exponential decay – for a boundary value problem and a non-dissipative problem reflected in the possible negativity of the functiona. To deal with the latter the condition (1.11)on the negative part ofa, i.e.

ka⁻k_L∞ < α0, will be used.

We assume that the conditions ona and on (a,L)are satisfied which assures the exponential stability of the linearized systems as given inTheorem 2.12.

Observing that the terma(x)u_t is of lower order, we can recall the following local existence theorem; see for instance [4] or [8, p. 97].

Theorem 3.1. There is T =T(k(u0,u1)k_H4×H³) >0such that(1.1)–(1.3)has a unique local solution u∈

3

\

k=0

C^k([0,T]),H^4−k((0,L))∩H₀¹((0,L))∩C⁴([0,T]),L²((0,L)).

Remark. Of courseu0,u1have to satisfy the usual compatibility conditions.

Now we turn to the high energy estimates. For this purpose it is useful to rewrite(1.1)–(1.3)as a first-order system for

V :=(u_t,u_x)⁰. ThenV satisfies

Vt+

a −d0∂x

−∂x 0

| {z }

=:−A

V =

b(ux)∂xux

0

=:F(V,Vx),

V(t =0)=(u1, ∂xu0)⁰=:V0.

The first formally defined operatorAgenerates aC0-semigroup as usual; forF =0 the solutionV is given by V(t)=e^{t A}V₀

and the (local) solution to(1.1)–(1.3)satisfies V(t)=e^{t A}V₀+

Z t 0

e^(t−r)AF(V,V_x)(r)dr. (3.1)

We conclude from Section2that V :=(ut,ux)

as a solution of the linear system(1.1)–(1.3)written in first-order form satisfies V(t)=e^{t A}V(t=0)

with aC₀-semigroup{e^{t A}}_t≥0satisfying

kV(t)k_Hs ≤c_se^−α0tkV(t =0)k_Hs (3.2)

fors =0,1,2 (cf. below). This follows fromTheorem 2.7fors =0 and is obtained fors =1,2 by differentiating Eq.(1.8)with respect tot once and then twice, as well as using the differential equation to obtain information for derivatives inx. Let

a⁻_∞:= ka⁻k_L∞. (3.3)

In the sequel we assume without loss of generality thatu_x is small enough a priori, i.e. such thatσ⁰(u_x)remains strictly positive (nearu_x =0; cf.(1.5)), or in other words we can assume that there isη >0 such that

d₀+b(u_x)≥ d0

2 >0 if|u_x|< η <1. (3.4)

Lemma 3.2. There are constants c₂,c₃ > 0, not depending on V₀ or T , such that the local solution given in Theorem3.1satisfies, for t ∈ [0,T],

kV(t)k²

H³ ≤c2kV0k²

H³e^2a∞−te^c3

Rt 0(kV(r)k

H2+kV(r)k²

H2+kV(r)k³

H2)dr

. Proof. Multiplying

ut t −d0ux x +aut =b(ux)ux x (3.5)

byu_t inL²we obtain R ≡RL

0

1

2 d dt

Z

u²_t +d0u²_xdx = − Z

au²_tdx+

Z b(ux)ux xutdx

≤ Z

a_∞⁻u²_t − Z

(∂xb(u_x))u_xu_tdx− Z

b(u_x)u_xu_{x t}

≡ I.1+I.2+I.3, (3.6)

|I.2| ≤ 1

2kb⁰(u_x)u_{x x}k_L∞

Z

u²_x+u²_tdx

≤ ckVk_H2

Z

u²_x+u²_tdx (3.7)

wherecwill denote a constant not depending onV⁰or onT. I.3= −1

2 d dt

Z

b(u_x)u²_xdx+1 2 Z

(∂tb(u_x))u²_x

≡ I.3.1+I.3.2. (3.8)

The termI.3.2 can be estimated in the same way asI.2 in(3.7):

|I.3.2| ≤ckVk_H2 Z

u²_x. (3.9)

The termI.3.1 can be incorporated into and be dominated by the left-hand side of inequality(3.6)after an integration with respect tot later on, since

Z t 0

I.3.1(r)dr= −1 2

Z b(ux)u²_xdx+1 2

Z b(ux(t=0))u²_x(t=0)dx. (3.10)

Summarizing(3.6)–(3.10)we have proved kV(t)k²

L² ≤ckV₀k²

L²+ Z t

0

(2a_∞⁻ +ckV(r)k_H2)kV(r)k²

L²dr. (3.11)

In order to get estimates for the higher order derivatives ofV anduwe differentiate Eq.(3.5)with respect totto get

u_{t t t}−d₀u_{t x x}+au_{t t} =b⁰(u_x)u_{x t}u_{x x} +b(u_x)u_{t x x}. (3.12)

Multiplying byu_{t t} inL²we obtain 1

2 d dt

Z

u²_{t t} +d0u²_{t x}dx ≤ Z

a⁻_∞u²_{t t}dx+ Z

b(ux)ut x xut tdx+ Z

b⁰(ux)ux tux xut tdx

≡ I.4+I.5+I.6. (3.13)

The termI.5 can be treated like the termI.2+I.3 from(3.6); see(3.7)–(3.11).

|I.6| ≤ckVk

H²

Z

u²_{t t} +u²_{t x}dx. (3.14)

Observe that the differential equation(3.5)yields the estimate

|ux x|²≤c(|ut t|²+ |ut|²). (3.15)

Thus we obtain from(3.11),(3.13)and(3.14) kV(t)k²

H¹ ≤ckV₀k²

H¹ + Z t

0

(2a_∞⁻ +ckV(r)k_H2)kV(r)k²

H¹dr. (3.16)

Differentiating the differential equation(3.12)once more with respect totwe get

u_{t t t t}−d₀u_{t t x x}+au_{t t t} =b⁰⁰(u_x)u²_{x t}u_{x x}+b⁰(u_x)u_{x t t}u_{x x}+2b⁰(u_x)u_{x t}u_{x x t}+b(u_x)u_{t t x x}. (3.17) Multiplying byu_{t t t}inL²we obtain

1 2

d dt

Z

u²_{t t t}+d₀u²_{t t x}dx ≤ Z

a_∞⁻u²_{t t t}dx+ Z

b⁰⁰(u_x)u²_{x t}u_{x x}u_{t t t}dx+ Z

b⁰(u_x)u_{x t t}u_{x x}u_{t t t}dx +2

Z

b⁰(ux)ux tux x tut t tdx+

Z b(ux)ut t x xut t tdx

≡ I.7+I.8+I.9+I.10+I.11. (3.18)

The termI.11 is again dealt with likeI.2+I.3 in(3.7)–(3.11).

|I.8| + |I.9| + |I.10| ≤c(kVk²

H²+ kVk_H2)Z

u²_{x t}+u²_{t t t}+u²_{t t t}+u²_{x t t}+u²_{x x t}dx. (3.19) Hence we obtain from(3.16)and (3.19)using(3.12)to estimateu_{t x x},

kV(t)k²

H² ≤ckV₀k²

H² + Z t

0

(2a_∞⁻ +c(kV(r)k

H² + kV(r)k²

H²))kV(r)k²

H²dr. (3.20)