# Initial Boundary Value Problem and Asymptotic Stabilization of the Two-Component Camassa-Holm Equation

Authors: Xiju Zong; Xingong Cheng; Zhonghua Wang; Zhenlai Han
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

The nonhomogeneous initial boundary value problem for the two-component Camassa-Holm equation, which describes a generalized formulation for the shallow water wave equation, on an interval is investigated. A local in time existence theorem and a uniqueness result are achieved. Next by using the fixed-point technique, a result on the global asymptotic stabilization problem by means of a boundary feedback law is considered.


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## Body

## 1. Introduction

In this paper, we are concerned with the initial boundary value problem and the asymptotic stabilization of the two-component Camassa-Holm equation on a compact interval by means of a stationary feedback law acting on the boundary. The two-component Camassa-Holm equation reads as follows:(1.1)ut-uxxt+3uux-2uxuxx-uuxxx+ρ¯ρ¯x=0,ρ¯t+ρ¯xu+ρ¯ux=0,
which was first derived as a bi-Hamiltonian models by Olver and Rosenau, see [1]. The system (1.1) shares many features with the Korteweg-De Vries equation Camassa-Holm equation and Degasperis-Procesi Equation; for instance, it has a Lax pair formulation, and it is integrable. In fact, the system (1.1) is related to the first negative flow of the AKNS hierarchy via a reciprocal transformation [2, 3]. In [4], Constantin and Ivanov deviated (1.1) in the context of shallow water waves theory. As well as they showed that it has global strong solutions and also finite time blow-up solutions. Well-posedness and blow-up results are obtained in [5, 6].Forρ¯≡0, the equation (1.1) becomes the Camassa-Holm equation, which is modeling the unidirectional propagation of shallow water waves over a flat bottom. Here u(t,x) stands for the fluid velocity at time t in the spatial x direction [7–11]. The Camassa-Holm equation is also a model for the propagation of axially symmetric waves in hyperelastic rods [12, 13]. It has a bi-Hamiltonian structure [3] and is completely integrable [7, 14]. Also there is a geometric interpretation of the equation (1.1) in terms of geodesic flow on the diffeomorphism group of the circle [15, 16]. Its solitary waves are peaked [17]. They are orbitally stable and interact like solitons [18, 19].The Cauchy problem and initial-boundary value problem for the Camassa-Holm equation have been studied extensively in [20–26] and references within. It has been shown that this equation is locally well posed [20–23, 26] for some initial data. The advantage of the Camassa-Holm equation in comparison with the KdV equation lies in the fact that the Camassa-Holm equation has peaked solitons and models wave breaking [27, 28] (by wave breaking we understand that the wave remains bounded while its slope becomes unbounded in finite time [29]).Forρ¯≠0, the Cauchy problems of (1.1) have been discussed in [5, 30], respectively. Recently, a new global existence result and several new blow-up results of strong solutions for the Cauchy problem of (1.1) were obtained in [6]. And a new local existence result and several new blow-up results and blow-up rate of strong solutions for the Cauchy problem of (1.1) defined in a torus were obtained in [31]. Guan and Yin proved the existence of global week solutions to (1.1) provided the initial data satisfying some certain conditions, see [32].As far as the initial boundary value problem and the asymptotic stabilization of the two-component Camassa-Holm equation on a compact interval are concerned, there are seldom results yet, to the authors’ knowledge. Our aim of this paper is to prove the existence of the initial boundary value problem and the asymptotic stabilization of the two-component Camassa-Holm equation on a compact interval by acting on the boundary feedback law, precisely,(1)
the exact controllability problem: given two states(u0,ρ¯0) and (u1,ρ¯1) and a time T>0, can one find a certain function v(t) such that the solution to (1.1) satisfies u(T)=u1, ρ¯(T)=ρ¯1? and(2)
the stabilizability problem: can one find a stationary feedback lawv(x), such that for any state (u0,ρ¯0) a solution pair (u(t),ρ¯(t)) to closed-loop system is global?To explain our boundary formulation of (1.1), let us first introduce some transformation, precisely, m=u-uxx and ρ=ρ¯-1, which lead the system (1.1) to be equivalent to the system:(1.2)mt+umx+2mux+ρρx+ρx=0,ρt+uρx+ρux+ux=0.
Let T be a positive number. In the following we take ΩT=[0,T]×[0,1]. Let vl and vr be in C0([0,T],R) and m0∈L∞(0,1), ρ0∈W1,∞(0,1). We set (1.3)Γl={t∈[0,T]∣vl(t)>0},Γr={t∈[0,T]∣vr(t)<0}.
In the following, we will always suppose that the sets (1.4)Pl={t∈[0,T]∣vl(t)=0},Pr={t∈[0,T]∣vr(t)=0}
have a finite number of connected components. Finally, let ml,ρl∈L∞(Γl)×W1,∞(Γl) and mr,ρr∈L∞(Γr)×W1,∞(Γr). The given functions vl, vr, ml, ρl, and mr, ρr will be the boundary values for the equation; m0, ρ0 are the initial data. Let now 𝔄(t,x) be the auxiliary function which lifts the boundary values vl and vr and is defined by(1.5)(1-∂xx)A(t,x)=0,∀(t,x)∈ΩT,A(t,0)=vl(t),A(t,1)=vr(t),∀t∈[0,T].
Setting u=θ+𝔄, we can further rewrite the system (1.1) as(1.6)m(t,x)=(1-∂xx)θ(t,x),θ(t,0)=θ(t,1)=0,(1.7)mt+(θ+A)mx=-2m∂x(θ+A)-ρρx-ρx,ρt+(θ+A)ρx=-(ρ+1)∂x(θ+A),m(0,⋅)=m0,m(⋅,0)|Γl=ml,m(⋅,1)|Γr=mr,ρ(0,⋅)=ρ0,ρ(⋅,0)|Γl=ρl,ρ(⋅,1)|Γr=ρr.
Let y=(mρ), y0=(m0ρ0), b(t,x)=(-2∂x(θ+𝔄)00-∂x(θ+𝔄)), f(t,x)=(-ρρx-ρx-∂x(θ+𝔄)), and the system (1.7) can be written as(1.8)∂ty+(θ+A)∂xy=b(t,x)y+f(t,x),y(0,⋅)=y0,y(⋅,0)|Γl=yl,y(⋅,1)|Γr=yr.
We first define what we mean by a weak solution to (1.8). Our test functions will be in the space:(1.9)adm(ΩT)={ψ(t,0)=0;∀t∈[0,T]Γr,ψ(t,1)=0}ψ∈C1(ΩT)×C1(ΩT)∣∀x∈[0,1],ψ(t,x)=0;∀t∈[0,T]/Γl,ψ(t,0)=0;∀t∈[0,T]/Γr,ψ(t,1)=0}.Definition 1.1.
Giveny0=(m0ρ0)∈L∞(ΩT)×W1,∞(ΩT), when θ∈L∞((0,T);Lip([0,1])), a function pair y=(mρ)∈L∞(ΩT)×W1,∞(ΩT) is a weak solution to (1.8) if y satisfies
(1.10)∬ΩT[y∂tψ+y(θ+A)∂xψ-yb(t,x)ψ-f(t,x)ψ]dtdx=-∫01y0ψ(0,x)dx+∫0T[ψ(t,1)vr(t)yr(t)-ψ(t,0)vl(t)yl(t)]dt.It is obvious thatC01(ΩT)×C01(ΩT)⊂adm(ΩT); therefore, a weak solution to (1.8) is also a solution to (1.8) in the distribution sense.Definition 1.2 (see [33]).
For(t,x)∈ΩT, let φ(·,t,x) be the C1 maximal solution to
(1.11)∂sφ(s,t,x)=a(s,φ(s,t,x)),φ(t,t,x)=x,
which is defined on a certain set [e(t,x),h(t,x)] (which is closed because [0,1] is compact) and with possibly e(t,x) and/or h(t,x)=t.We take into account the influence of the boundaries by introducing the sets:(1.12)P={(t,x)∈ΩT∣∃s∈[e(t,x),h(t,x)]suchthatφ(s,t,x)∈{0,1},a(s,φ(s,t,x))=0},∪{(s,φ(s,0,0))∣∀s∈[0,T]}∪{(s,φ(s,0,1))∣∀s∈[0,T]},I={(t,x)∈ΩT/P∣e(t,x)=0},L={(t,x)∈ΩT/P∣φ(e(t,x)t,x)=0},R={(t,x)∈ΩT/P∣φ(e(t,x)t,x)=1},Γl={t∈[0,T]∣a(t,0)>0},Γr={t∈[0,T]∣a(t,1)>0}.
The following lemma, see [33], will play an important role in proving the local time existence theorem and of a uniqueness result of the initial boundary value problem.Lemma 1.3.
Leta∈C0([0,T];C1([0,1])), b,f∈L∞(ΩT), y0∈L∞(0,1), yl∈L∞(Γl), and yr∈L∞(Γr). We will also suppose that the sets:
(1.13)Pl={t∈[0,T]∣vl(t)=0},Pr={t∈[0,T]∣vr(t)=0}
have at most a countable number of connected components. Then the function y, defined by the formula
(1.14)for(t,x)∈P,y(t,x)=0,for(t,x)∈I,y(t,x)=exp⁡(∫0tb(r,φ(r,t,x))dr)y0(φ(0,t,x))+∫0texp⁡(∫stb(r,φ(r,t,x))dr)f(s,φ(s,t,x))ds,for(t,x)∈L,y(t,x)=exp(∫e(t,x)tb(r,φ(r,t,x))dr)yl(e(t,x))+∫e(t,x)texp⁡(∫stb(r,φ(r,t,x))dr)f(s,φ(s,t,x))ds,for(t,x)∈R,y(t,x)=exp⁡(∫e(t,x)tb(r,φ(r,t,x))dr)yr(e(t,x))+∫e(t,x)texp⁡(∫stb(r,φ(r,t,x))dr)f(s,φ(s,t,x))ds,
is a weak solution of
(1.15)∂ty+(θ+A)∂xy=b(t,x)y+f(t,x)
and satisfies
(1.16)‖y‖L∞(ΩT)≤(max⁡{‖y0‖L∞(ΩT),‖yl‖L∞(Γl),‖yr‖L∞(Γr)}+T‖f‖L∞(ΩT))et‖b‖L∞(ΩT).However, if we letml,ρl∈L∞(Γl)×L∞(Γl) and mr,ρr∈L∞(Γr)×L∞(Γr), note that f depends on the unknown ρ which is not a data; therefore Lemma 1.3 does not hold, or rather Theorem 6 from the Appendix of [33] can not be applied directly to (1.8) (or (1.1)). Indeed if θ and 𝔄 are given, one can solve the equation on ρ (the equation on the second component in (1.8) (or (1.1))), but this result only guarantee that ρ is in L∞. Therefore the source term in the equation on m is not in L∞ anymore but in L∞((0,T);W-1,∞(0,1)), and then Lemma 1.3 cannot be used to solve the transport equation on m (the first component equation on (1.8)). One might try to get more regularity on ρ, but in this case more regularity is also needed on ρ0,ρl, ρr0 and even on θ and 𝔄 to get sufficient geometrical assumptions. Then, one might manage ρ0∈W1,∞(0,1), ρl∈W1,∞(Γl), ρr∈W1,∞(Γr) to obtain at least Lipschitz solution of the scalar transport equation on ρ and then get a weak solution on m.The rest of this paper is organized as follows. In Section2, the main results of the present paper are stated. Section 3 will be devoted to the proofs of a local time existence theorem and of a uniqueness result of the initial boundary value problem for the system (1.8) (or (1.1)). The problem of asymptotic stabilization for the system is analyzed, and a feedback control law will be investigated in Section 4.


## 2. Main Results

Theorem 2.1.
ForT>0, we consider vl∈C0(Γl), vr∈C0(Γl) such that the sets Pl and Pr have only a finite number of connected components. Let y0∈L∞(0,1)×W1,∞(0,1), yl∈L∞(Γl)×W1,∞(Γl), and yr∈L∞(Γr)×W1,∞(Γr). There exist T>0 and (θ,y) a weak solution of the system (1.8) (or (1.1)) with θ∈L∞((0,T);C1,1([0,1])∩Lip((0,T);H01(0,1). Moreover any such solution θ is in fact in C0([0,T];W2,p(0,1))∩C1([0,T];W01,p(0,1)), for all p<+∞. Furthermore the existence time of a maximal solution is larger than min(T̃,T*), with
(2.1)T*=max⁡β>0(ln⁡⁡(1+β/C)2[C1+(2+sinh⁡(1))(β+C)]),C=max⁡{‖y0‖L∞×W1,∞,‖yl‖L∞×W1,∞,‖yr‖L∞×W1,∞},C1=1tanh(1)(‖vl‖L∞(0,T),‖vr‖L∞(0,T)).In a second step, we will show a weak-strong uniqueness property.Theorem 2.2.
Letθ∈L∞((0,T);C1,1([0,1])∩Lip((0,T);H01(0,1), and let y∈Lip([0,1])×Lip([0,1]) be a weak solution of (1.8) (or (1.1)); then it is unique in the function space [L∞((0,T);C1,1([0,1])×L∞(ΩT))]2.LetAl>2sinh(1), Ar>Alcosh(1)+sinh(2), T>0 and M a symmetric matrix, and assume that ρ0,ρl, and ρr have compact supports in (0,1)/Γl/Γr, respectively. Our feedback law for (1.8) (or (1.1)) reads(2.2)y∈C0([0,1])×C0([0,1])⟼{vl(y)=Al‖y‖C0([0,1])×C0([0,1]),vr(y)=Ar‖y‖C0([0,1])×C0([0,1]),ẏl(t)=Myl(t).Theorem 2.3.
For anyy0∈[C0([0,1])]2 there exists (y,v)∈[C0(ΩT)]2×C2([0,1]) a weak solution of (1.1) and (2.2) satisfying
(2.3)∀x∈[0,1],y(0,x)=y0(x).
Furthermore any maximal solution of (1.1), (2.2), and (2.3) is global, and if we let
(2.4)c=min⁡{Al-2sinh⁡(1),Ar-Alcosh⁡(1)+sinh⁡(2)sinh⁡(1)},τ=1‖M‖2ln⁡⁡(c‖y‖C0([0,1])‖M‖2),
then we have
(2.5)‖y‖C0([0,1])≤‖M‖2c[1+‖M‖2(t-τ)].Remark 2.4.
Forρ≡0, the system (1.1) becomes the classical Camassa-Holm equation, and the above theorems degenerate those of  [33] with k=0.


## 3. Proofs of the Main Theorems

### 3.1. Local Existence Theorem

This strategy is borrowed from [33]. We want to solve (1.6) (1.8) (or (1.1)). Equation (1.6) is a linear elliptic equation, and with θ fixed (1.8) is a linear transport equation in y, with boundary data.Givenθ∈L∞((0,T);C1,1([0,1]))∩Lip((0,T);H01(0,1)), we will define y=(mρ) to be the solution to (1.8), and once we have m in L∞(ΩT), we introduce θ̃ solution of(3.1)(1-∂xx)θ̃=m.
Then ℱ is defined as the operator θ̃=ℱ(θ)Lemma 3.1.
The function𝔄 defined by (1.5) satisfies
(3.2)∀(t,x)∈ΩT,A(t,x)∈C0([0,T];C∞([0,1])),A(t,x)=1sinh⁡(1)(sinh⁡(x)vr(t)+sinh⁡(1-x)vl(t)),‖A(t,x)‖L∞((0,T);C1,1([0,1]))≤cosh⁡(1)sinh⁡(1)(‖vl‖L∞(0,T)+‖vr‖L∞(0,T)).Then for a functionθ∈L∞((0,T);C1,1([0,1]))∩Lip((0,T);H01(0,1)), we consider φ the flow of θ+𝔄.Lemma 3.2.
The flowφ satisfies the following properties.(1)
φ is C1 with the following partial derivatives:(3.3)∂1φ(s,t,x)=(θ+A)(s,φ(s,t,x)),∂2φ(s,t,x)=-(θ+A)exp⁡(∫ts∂x(θ+A)(r,φ(r,t,x))dr),∂3φ(s,t,x)=exp⁡(∫ts∂x(θ+A)(r,φ(r,t,x))dr).(2)
For allj=1,2,3,(3.4)‖∂jφ‖C0≤(1+‖θ+A‖C0(ΩT))eT‖θ+A‖C0(ΩT).
For (t,x)∈ΩT, φ(·,t,x) is defined on a set [e(t,x),h(t,x)], here e(t,x) is basically the entrance time in ΩT of the characteristic curve going through (t,x).(3)
Ife(t,x)>0, then φ(e(t,x),t,x)∈{0,1}.(4)
Ifh(t,x)<T, then φ(h(t,x),t,x)∈{0,1}.Forθ∈L∞((0,T);C1,1([0,1]))∩Lip((0,T);H01(0,1)), we define the solution to (1.8) y∈L∞(ΩT)×W1,∞(ΩT) by(3.5)for(t,x)∈I,y(t,x)=(exp⁡∫0tE1dr00exp⁡∫0tE2dr)y0(φ(0,t,x))+∫0t(exp⁡∫stE1dr00exp⁡∫stE2dr)(-ρρx-ρx-∂x(θ+A))ds,for(t,x)∈L,y(t,x)=(exp⁡∫e(t,x)tE1dr00exp⁡∫e(t,x)tE2dr)yl(e(t,x))+∫e(t,x)t(exp⁡∫stE1dr00exp⁡∫stE2dr)(-ρρx-ρx-∂x(θ+A))ds,for(t,x)∈R,y(t,x)=(exp⁡∫e(t,x)tE1dr00exp⁡∫e(t,x)tE2dr)yr(e(t,x))+∫e(t,x)t(exp⁡∫stE1dr00exp⁡∫stE2dr)(-ρρx-ρx-∂x(θ+A))ds
with (3.6)E1=-2∂x(θ+A)(r,φ(r,t,x)),E2=-∂x(θ+A)(r,φ(r,t,x)).
And we have(1)
the functiony is the unique weak solution of (1.8) in the sense of Definition 4.1; thanks to Lemma 1.3. y is in L∞∩W1,∞.Remark 3.3.
y is the only weak solution of (1.8), and also y is in L∞∩W1,∞ which is crucial for the stabilization problem because of the coupling between the two components of y. However, rather thanks to the regularity on the boundary data ρ is indeed Lipschitz inside the zones L, R, and I; it ensures that the transition between those zones should be continuous under the kind of compatibility conditions between ρ0, ρl, and ρr; for example, all three have a compact support in (0,1)/Γl/Γr.(2)
Sincey∈L∞(ΩT)×W1,∞(ΩT), we immediately get y∈W1,∞(0,T;H-1(0,1))×W2,∞(0,T;H-1(0,1)) and satisfies (1.8). Also we can get the estimates:(3.7)‖y(t,x)‖L∞(ΩT)×L∞(ΩT)≤C0exp⁡[2T(‖∂xθ‖L∞(ΩT)+‖∂xA‖L∞(ΩT))],‖∂ty(t,x)‖L∞(0,T;H-1)×L∞(0,T;H-1)≤3C0exp⁡[2T(‖∂xθ‖L∞(ΩT)+‖∂xA‖L∞(ΩT))],×(‖θ‖L∞((0,T);Lip([0,1]))+‖A‖L∞((0,T);Lip([0,1]))),where(3.8)C0=max⁡{‖y0‖L∞×W1,∞,‖yl‖L∞×W1,∞,‖yr‖L∞×W1,∞}+T[(‖ρ‖L∞+1)‖ρx‖L∞((0,T);Lip([0,1]))+‖∂xθ‖L∞(ΩT)+‖∂xA‖L∞(ΩT)].(3)
If(t,x)∈I∪L∪R and if (s,s′)∈[e(t,x),h(t,x)]2, one has the following property:(3.9)y(s,φ(s,t,x))=(exp⁡∫0tE1dr00exp⁡∫0tE2dr)y(s′,φ(s′,t,x)).From the elliptic equation we can getLemma 3.4.
There exists a uniqueθ̃∈L∞((0,T);H01(0,1)) such that
(3.10)(1-∂xx)θ̃=m
holds in D′(0,1) for all (t,x)∈ΩT. Furthermore θ̃∈L∞((0,T);C1,1([0,1]))∩Lip([0,T];H01(0,1)), since m∈L∞(ΩT)∩Lip([0,T];H-1(0,1)). Moreover we have the bounds:
(3.11)‖θ̃‖L∞((0,T);C1,1([0,1]))≤[1+sinh⁡(1)]‖m‖L∞(ΩT)≤[1+sinh⁡(1)]‖y‖L∞(ΩT)×W1,∞(ΩT),‖∂tθ̃‖L∞((0,T);H01(0,1))≤‖∂tm‖L∞((0,T);H-1)≤‖∂ty‖L∞((0,T);H-1)×W1,∞((0,T);H-1).The proof can be found in [33] and omitted.Thus, forθ∈L∞((0,T);C1,1(0,1))∩Lip([0,T];H01(0,1)), the operator ℱ can be defined as θ̃=ℱ(θ)∈L∞((0,T);C1,1(0,1))∩Lip([0,T];H01(0,1)).LetB0 and B1 be positive numbers, then we set(3.12)CB0,B1,T={θ∈L∞((0,T);C1,1(0,1))∩Lip([0,T];H01(0,1))∣both‖θ‖L∞((0,T);C1,1(0,1))≤B0,‖θ‖Lip([0,T];H-1(0,1))≤B1}.
Obviously 𝒞B0,B1,T is convex. We will endow 𝒞B0,B1,T with the norm ∥θ∥L∞([0,T];Lip(0,1)).Lemma 3.5.
There exist positive numbersB0, B1 and T, such that ℱ maps 𝒞B0,B1,T into itself.Proof.
The proceeding of proof is similar to that of [33, Lemma 3], but the constant C0 differs slightly from that of [33, Lemma 3]. Let us first introduce the two following constants:
(3.13)C0=max⁡{‖y0‖L∞×W1,∞,‖yl‖L∞×W1,∞,‖yr‖L∞×W1,∞}+T[(‖ρ‖L∞+1)‖ρx‖L∞((0,T);Lip([0,1]))+‖∂xθ‖L∞(ΩT)+‖∂xA‖L∞(ΩT)],C1=1tanh(1)(‖vl‖L∞(0,T),‖vr‖L∞(0,T)).
Estimates (3.7), and (3.11) on y and θ now read,
(3.14)‖y(t,x)‖L∞(ΩT)×W1,∞(ΩT)≤C0exp⁡[2T(‖∂xθ‖L∞(ΩT)+C1)],‖∂ty(t,x)‖L∞((0,T);H-1)×L∞((0,T);H-1)≤3C0exp⁡[2T(‖∂xθ‖L∞(ΩT)+C1)]×(‖θ‖L∞((0,T);Lip([0,1]))+C1),‖θ̃‖L∞((0,T);C1,1([0,1]))≤[1+sinh⁡(1)]‖y(t,x)‖L∞(ΩT)×W1,∞(ΩT),‖∂tθ̃‖L∞((0,T);H01(0,1))≤‖∂ty(t,x)‖L∞((0,T);H-1)×L∞((0,T);H-1).
Combining those estimates we get for all θ∈𝒞B0,B1,T:
(3.15)‖θ̃‖L∞((0,T);C1,1([0,1]))≤C0[1+sinh⁡(1)]exp⁡[2T(‖∂xθ‖L∞(ΩT)+C1)]≤C0[1+sinh⁡(1)]exp⁡[2T(B0+C1)],‖∂tθ̃‖L∞((0,T);H01(0,1))≤3C0exp⁡[2T(‖∂xθ‖L∞(ΩT)+C1)](‖θ‖L∞((0,T);Lip([0,1]))+C1)≤3C0exp⁡[2T(B0+C1)](B0+C1).
To obtain θ̃∈𝒞B0,B1,T, it is sufficient that
(3.16)C0[1+sinh⁡(1)]exp⁡[2T(B0+C1)]≤B0,B0+3C0exp⁡[2T(B0+C1)](B0+C1)≤B1.
Once we have chosenT and B0, it is easy to choose B1 to satisfy the second inequality. For the first one we just choose B0 sufficiently large and then T close to 0. More precisely,
(3.17)C=max⁡{‖y0‖L∞×W1,∞,‖yl‖L∞×W1,∞,‖yr‖L∞×W1,∞},B0>C[1+sinh⁡(1)],T≤ln⁡B0/(C[1+sinh⁡(1)])2(B0+C1),
we complete the proof by taking B0/(C[1+sinh(1)])=C+β,β>0.Lemma 3.6.
𝒞B0,B1,T is compact with respect to the norm ∥·∥L∞([0,T];Lip(0,1)).The proof is very similar to that appeared in [33] and omitted.Lemma 3.7.
Fory∈Hs×Hs-1, s>2, f(y) is bounded on bounded sets in Hs×Hs-1. Therefore, f(y) is bounded on bounded sets in L∞×L∞ by the embedding theorem.The proof is very similar to that appeared in [33] and omitted.Lemma 3.8.
The operatorℱ:𝒞B0,B1,T→𝒞B0,B1,T is continuous with respect to the norm ∥θ∥L∞([0,T];Lip(0,1)).Proof.
Take a sequence{θn} which tends to θ with respect to ∥·∥L∞([0,T];Lip(0,1)), set θ̃n=ℱθn and θ̃=ℱθ, denote by φn the flow of θn+𝔄 and φ the flow of θ+𝔄, and we have that φn→φ locally in C1 as n→∞, thanks to Proposition A.4 in [33]. What we will need to do is to show that mn→m in L1(0,1) as n→∞ and ρn→ρ in L1(0,1) as n→∞.
Lett∈[0,T], having supposed that Pl and Pr have only a finite number of connected components, we can assume, reducing t if necessary that vl and vr do not change sign on [0,t]. Since the characteristics of φn and φ may or may not cross before time t, we only consider the case that φ(t,0,0)≤φn(t,0,0)≤φ(t,0,1)≤φn(t,0,1), without loss of generality. The other cases are proved in the same way. We first point out that since θn∈𝒞B0,B1,T we have a bound for {yn} in L∞(ΩT). Now
(3.18)∫01|y(t,x)-yn(t,x)|dx=(∫0φ(t,0,0)+∫φ(t,0,0)φn(t,0,0)+∫φn(t,0,0)φ(t,0,1)+∫φ(t,0,1)φn(t,0,1)+∫φn(t,0,1)1)|y(t,x)-yn(t,x)|dx=I1+I2+I3+I4+I5.
Since φn(t,0,0)→φ(t,0,0) as n→∞ and φn(t,0,1)→φ(t,0,1) as n→∞ and thanks to the uniform bound on ∥yn∥L∞, we see that both I2 and I4 tend to 0 when n goes to infinity.
ForI1 we have
(3.19)I1=∫0φ(t,0,0)|y(t,x)-yn(t,x)|dx=∫0φ(t,0,0)|(exp⁡∫e(t,x)tE1dr00exp⁡∫e(t,x)tE2dr)yl(e(t,x))+∫e(t,x)t(exp⁡∫stE1dr00exp⁡∫stE2dr)(-ρρx-ρx-∂x(θ+A))ds-(exp⁡∫en(t,x)tE1ndr00exp⁡∫en(t,x)tE2ndr)yl(en(t,x))-∫en(t,x)t(exp⁡∫stE1ndr00exp⁡∫stE2ndr)(-ρρx-ρx-∂x(θn+A))ds|dx,
where
(3.20)E1n=-2∂x(θn+A)(r,φn(r,t,x)),E2=-∂x(θn+A)(r,φn(r,t,x)).
Thanks to the boundedness on the ∥f∥L∞×L∞ (Lemma 3.7) and Proposition A.2 of [33], if (t,x)∈¯P we have φn(t,0,0)→φ(t,0,0) as n→∞. This implies that if yl was continuous, since we have a uniform bound on ∥θn∥L∞((0,T);C1,1(0,1)) the dominated convergence theorem would provide: I1→0.I3→0 and I5→0 which can be obtained by using the same method. Therefore, for y0, yl and yr continuous we have ∥y(t,·)-yn(t,·)∥L1→0.
From the inequality (56) in [33], we obtain
(3.21)‖y(t,⋅)‖L1(0,1)×L1(0,1)≤[‖y0‖L1(0,1)×L1(0,1)+‖yl‖L1(0,t)∩Γl×L1(0,t)∩Γl+‖yr‖L1(0,t)∩Γr×L1(0,t)∩Γr+(‖ρ‖L1(0,1)×L1(0,1)+1)‖ρx‖L1(0,1)×L1(0,1)+‖∂x(θ+A)‖L1(0,1)]‖∂x(θ+A)‖L∞(ΩT)exp⁡(3t‖∂x(θ+A)‖L∞(ΩT)),‖yn(t,⋅)‖L1(0,1)×L1(0,1)≤[‖y0‖L1(0,1)×L1(0,1)+‖yl‖L1(0,t)∩Γl×L1(0,t)∩Γl+‖yr‖L1(0,t)∩Γr×L1(0,t)∩Γr+(‖ρn‖L1(0,1)×L1(0,1)+1)‖ρn,x‖L1(0,1)×L1(0,1)+‖∂x(θn+A)‖L1(0,1)]‖∂x(θn+A)‖L∞(ΩT)exp⁡(3t‖∂x(θn+A)‖L∞(ΩT)).
So the general case of convergence ∥y(t,·)-yn(t,·)∥L1→0 follows from the density of C0 in L1 and the uniform bound on ∥θn∥L∞((0,T);Lip(0,1)).
Now only the restriction ont remains; we recall that until now we supposed that vl and vr did not change sign on [0,t]. If vl and vr do not change sign on [0,t1] and then on [t1,t], we have
(3.22)‖y(t1,⋅)-yn(t1,⋅)‖L1⟶0.
Let ỹn the solution of
(3.23)∂tỹn+(θn+A)∂xỹn=bn(t,x)ỹn+fn(t,x),ỹn(t1,⋅)=yt1,ỹn(⋅,0)|Γl=yl,ỹn(⋅,1)|Γr=yr.
We can conclude that as n→∞,
(3.24)‖y(t,⋅)-yn(t,⋅)‖L1(0,1)×L1(0,1)≤‖y(t,⋅)-ỹn(t,⋅)‖L1(0,1)×L1(0,1)+‖ỹn(t,⋅)-yn(t,⋅)‖L1(0,1)×L1(0,1)≤‖y(t1,⋅)-ỹn(t1,⋅)‖L1(0,1)×L1(0,1)exp⁡(3(t-t1)‖∂x(θn+A)‖L∞(ΩT))+‖y(t,⋅)-ỹn(t,⋅)‖L1(0,1)×L1(0,1)≤‖y(t1,⋅)-yn(t1,⋅)‖L1(0,1)×L1(0,1)exp⁡(3(t-t1)‖∂x(θn+A)‖L∞(ΩT))+‖y(t,⋅)-ỹn(t,⋅)‖L1(0,1)×L1(0,1)⟶0.
Thus the convergence in L1(0,1) propagates on each interval where vl and vr do not change sign; thanks to the hypothesis on Pr and Pl we have that for all t∈[0,T], ∥y(t,·)-yn(t,·)∥L1(0,1)→0, as n→∞. Combining this first convergence result with the uniform bound and using the dominated convergence theorem in the time variable, we obtain ∥y-yn∥L1(ΩT)→0 which implies that
(3.25)‖θ̃-θ̃n‖L1(0,T;W2,1(0,1))⟶0.
From the compactness of𝒞B0,B1,T, we get that θ̃n→θ̃ holds in 𝒞B0,B1,T.All the above lemmas result in the application of Schauders fixed point theorem toℱ and we get a solution (3.26)θ∈L∞((0,T);C1,1(0,1))∩Lip([0,T];H01(0,1)).From the construction of F and from Proposition A.8 in [33] the additional regularity properties of any solution θ,(3.27)θ∈C0([0,T];W2,p(0,1))∩C1([0,T];W01,p(0,1)),∀p<+∞.


### 3.2. Uniqueness

In this subsection, we will show that the solution to the system (1.6) and (1.8) is unique; that to say, given (y,θ) and (ỹ,θ̃) be two solutions of (1.6) and (1.8) for the same initial and boundary data, we will get y=ỹ and θ=θ̃.LetY=y-ỹ, Θ=θ-θ̃, B(t,x)=b(t,x)-b̃(t,x), F(t,x)=f(t,x)-f̃(t,x), then Θ∈Lip([0,T];H01(0,1)). And we have M=m-m̃∈L∞(ΩT) and Y∈L∞(ΩT)×L∞(ΩT) is the solution to(3.28)∂tY+(θ+A)∂xY=bY+B(t,x)ỹ-Θ∂xỹ+F(t,x),Y(0,⋅)=0,Y(⋅,0)|Γl=0,Y(⋅,1)|Γr=0.
Using the lemma again with b(t,x)=(-2∂x(θ+𝔄)00-∂x(θ+𝔄)) and f¯=B(t,x)ỹ-U∂xỹ+F(t,x), we get(3.29)for(t,x)∈P,Y(t,x)=0,for(t,x)∈I,Y(t,x)=∫0texp⁡(∫stb(r,φ(r,t,x))dr)f¯(s,φ(s,t,x))ds,for(t,x)∈L,Y(t,x)=∫e(t,x)texp⁡(∫stb(r,φ(r,t,x))dr)f¯(s,φ(s,t,x))ds,for(t,x)∈R,Y(t,x)=∫e(t,x)texp⁡(∫stb(r,φ(r,t,x))dr)f¯(s,φ(s,t,x))ds.Since∥U(t,·)∥L∞(0,1)≤C′∥Y(t,·)∥L∞×L∞ for some positive constant C′, and y,∂xy bounded, we see that for some C′′>0,(3.30)‖f¯(t,⋅)‖L∞×L∞≤C′′‖Y(t,⋅)‖L∞×L∞.
And since b(t,x) is bounded, we get that(3.31)‖Y(t,⋅)‖L∞×L∞≤C′′∫0s‖Y(s,⋅)‖L∞×L∞ds.Then we complete the proof of the uniqueness by using Gronwall’s lemma.


## 3.1. Local Existence Theorem

This strategy is borrowed from [33]. We want to solve (1.6) (1.8) (or (1.1)). Equation (1.6) is a linear elliptic equation, and with θ fixed (1.8) is a linear transport equation in y, with boundary data.Givenθ∈L∞((0,T);C1,1([0,1]))∩Lip((0,T);H01(0,1)), we will define y=(mρ) to be the solution to (1.8), and once we have m in L∞(ΩT), we introduce θ̃ solution of(3.1)(1-∂xx)θ̃=m.
Then ℱ is defined as the operator θ̃=ℱ(θ)Lemma 3.1.
The function𝔄 defined by (1.5) satisfies
(3.2)∀(t,x)∈ΩT,A(t,x)∈C0([0,T];C∞([0,1])),A(t,x)=1sinh⁡(1)(sinh⁡(x)vr(t)+sinh⁡(1-x)vl(t)),‖A(t,x)‖L∞((0,T);C1,1([0,1]))≤cosh⁡(1)sinh⁡(1)(‖vl‖L∞(0,T)+‖vr‖L∞(0,T)).Then for a functionθ∈L∞((0,T);C1,1([0,1]))∩Lip((0,T);H01(0,1)), we consider φ the flow of θ+𝔄.Lemma 3.2.
The flowφ satisfies the following properties.(1)
φ is C1 with the following partial derivatives:(3.3)∂1φ(s,t,x)=(θ+A)(s,φ(s,t,x)),∂2φ(s,t,x)=-(θ+A)exp⁡(∫ts∂x(θ+A)(r,φ(r,t,x))dr),∂3φ(s,t,x)=exp⁡(∫ts∂x(θ+A)(r,φ(r,t,x))dr).(2)
For allj=1,2,3,(3.4)‖∂jφ‖C0≤(1+‖θ+A‖C0(ΩT))eT‖θ+A‖C0(ΩT).
For (t,x)∈ΩT, φ(·,t,x) is defined on a set [e(t,x),h(t,x)], here e(t,x) is basically the entrance time in ΩT of the characteristic curve going through (t,x).(3)
Ife(t,x)>0, then φ(e(t,x),t,x)∈{0,1}.(4)
Ifh(t,x)<T, then φ(h(t,x),t,x)∈{0,1}.Forθ∈L∞((0,T);C1,1([0,1]))∩Lip((0,T);H01(0,1)), we define the solution to (1.8) y∈L∞(ΩT)×W1,∞(ΩT) by(3.5)for(t,x)∈I,y(t,x)=(exp⁡∫0tE1dr00exp⁡∫0tE2dr)y0(φ(0,t,x))+∫0t(exp⁡∫stE1dr00exp⁡∫stE2dr)(-ρρx-ρx-∂x(θ+A))ds,for(t,x)∈L,y(t,x)=(exp⁡∫e(t,x)tE1dr00exp⁡∫e(t,x)tE2dr)yl(e(t,x))+∫e(t,x)t(exp⁡∫stE1dr00exp⁡∫stE2dr)(-ρρx-ρx-∂x(θ+A))ds,for(t,x)∈R,y(t,x)=(exp⁡∫e(t,x)tE1dr00exp⁡∫e(t,x)tE2dr)yr(e(t,x))+∫e(t,x)t(exp⁡∫stE1dr00exp⁡∫stE2dr)(-ρρx-ρx-∂x(θ+A))ds
with (3.6)E1=-2∂x(θ+A)(r,φ(r,t,x)),E2=-∂x(θ+A)(r,φ(r,t,x)).
And we have(1)
the functiony is the unique weak solution of (1.8) in the sense of Definition 4.1; thanks to Lemma 1.3. y is in L∞∩W1,∞.Remark 3.3.
y is the only weak solution of (1.8), and also y is in L∞∩W1,∞ which is crucial for the stabilization problem because of the coupling between the two components of y. However, rather thanks to the regularity on the boundary data ρ is indeed Lipschitz inside the zones L, R, and I; it ensures that the transition between those zones should be continuous under the kind of compatibility conditions between ρ0, ρl, and ρr; for example, all three have a compact support in (0,1)/Γl/Γr.(2)
Sincey∈L∞(ΩT)×W1,∞(ΩT), we immediately get y∈W1,∞(0,T;H-1(0,1))×W2,∞(0,T;H-1(0,1)) and satisfies (1.8). Also we can get the estimates:(3.7)‖y(t,x)‖L∞(ΩT)×L∞(ΩT)≤C0exp⁡[2T(‖∂xθ‖L∞(ΩT)+‖∂xA‖L∞(ΩT))],‖∂ty(t,x)‖L∞(0,T;H-1)×L∞(0,T;H-1)≤3C0exp⁡[2T(‖∂xθ‖L∞(ΩT)+‖∂xA‖L∞(ΩT))],×(‖θ‖L∞((0,T);Lip([0,1]))+‖A‖L∞((0,T);Lip([0,1]))),where(3.8)C0=max⁡{‖y0‖L∞×W1,∞,‖yl‖L∞×W1,∞,‖yr‖L∞×W1,∞}+T[(‖ρ‖L∞+1)‖ρx‖L∞((0,T);Lip([0,1]))+‖∂xθ‖L∞(ΩT)+‖∂xA‖L∞(ΩT)].(3)
If(t,x)∈I∪L∪R and if (s,s′)∈[e(t,x),h(t,x)]2, one has the following property:(3.9)y(s,φ(s,t,x))=(exp⁡∫0tE1dr00exp⁡∫0tE2dr)y(s′,φ(s′,t,x)).From the elliptic equation we can getLemma 3.4.
There exists a uniqueθ̃∈L∞((0,T);H01(0,1)) such that
(3.10)(1-∂xx)θ̃=m
holds in D′(0,1) for all (t,x)∈ΩT. Furthermore θ̃∈L∞((0,T);C1,1([0,1]))∩Lip([0,T];H01(0,1)), since m∈L∞(ΩT)∩Lip([0,T];H-1(0,1)). Moreover we have the bounds:
(3.11)‖θ̃‖L∞((0,T);C1,1([0,1]))≤[1+sinh⁡(1)]‖m‖L∞(ΩT)≤[1+sinh⁡(1)]‖y‖L∞(ΩT)×W1,∞(ΩT),‖∂tθ̃‖L∞((0,T);H01(0,1))≤‖∂tm‖L∞((0,T);H-1)≤‖∂ty‖L∞((0,T);H-1)×W1,∞((0,T);H-1).The proof can be found in [33] and omitted.Thus, forθ∈L∞((0,T);C1,1(0,1))∩Lip([0,T];H01(0,1)), the operator ℱ can be defined as θ̃=ℱ(θ)∈L∞((0,T);C1,1(0,1))∩Lip([0,T];H01(0,1)).LetB0 and B1 be positive numbers, then we set(3.12)CB0,B1,T={θ∈L∞((0,T);C1,1(0,1))∩Lip([0,T];H01(0,1))∣both‖θ‖L∞((0,T);C1,1(0,1))≤B0,‖θ‖Lip([0,T];H-1(0,1))≤B1}.
Obviously 𝒞B0,B1,T is convex. We will endow 𝒞B0,B1,T with the norm ∥θ∥L∞([0,T];Lip(0,1)).Lemma 3.5.
There exist positive numbersB0, B1 and T, such that ℱ maps 𝒞B0,B1,T into itself.Proof.
The proceeding of proof is similar to that of [33, Lemma 3], but the constant C0 differs slightly from that of [33, Lemma 3]. Let us first introduce the two following constants:
(3.13)C0=max⁡{‖y0‖L∞×W1,∞,‖yl‖L∞×W1,∞,‖yr‖L∞×W1,∞}+T[(‖ρ‖L∞+1)‖ρx‖L∞((0,T);Lip([0,1]))+‖∂xθ‖L∞(ΩT)+‖∂xA‖L∞(ΩT)],C1=1tanh(1)(‖vl‖L∞(0,T),‖vr‖L∞(0,T)).
Estimates (3.7), and (3.11) on y and θ now read,
(3.14)‖y(t,x)‖L∞(ΩT)×W1,∞(ΩT)≤C0exp⁡[2T(‖∂xθ‖L∞(ΩT)+C1)],‖∂ty(t,x)‖L∞((0,T);H-1)×L∞((0,T);H-1)≤3C0exp⁡[2T(‖∂xθ‖L∞(ΩT)+C1)]×(‖θ‖L∞((0,T);Lip([0,1]))+C1),‖θ̃‖L∞((0,T);C1,1([0,1]))≤[1+sinh⁡(1)]‖y(t,x)‖L∞(ΩT)×W1,∞(ΩT),‖∂tθ̃‖L∞((0,T);H01(0,1))≤‖∂ty(t,x)‖L∞((0,T);H-1)×L∞((0,T);H-1).
Combining those estimates we get for all θ∈𝒞B0,B1,T:
(3.15)‖θ̃‖L∞((0,T);C1,1([0,1]))≤C0[1+sinh⁡(1)]exp⁡[2T(‖∂xθ‖L∞(ΩT)+C1)]≤C0[1+sinh⁡(1)]exp⁡[2T(B0+C1)],‖∂tθ̃‖L∞((0,T);H01(0,1))≤3C0exp⁡[2T(‖∂xθ‖L∞(ΩT)+C1)](‖θ‖L∞((0,T);Lip([0,1]))+C1)≤3C0exp⁡[2T(B0+C1)](B0+C1).
To obtain θ̃∈𝒞B0,B1,T, it is sufficient that
(3.16)C0[1+sinh⁡(1)]exp⁡[2T(B0+C1)]≤B0,B0+3C0exp⁡[2T(B0+C1)](B0+C1)≤B1.
Once we have chosenT and B0, it is easy to choose B1 to satisfy the second inequality. For the first one we just choose B0 sufficiently large and then T close to 0. More precisely,
(3.17)C=max⁡{‖y0‖L∞×W1,∞,‖yl‖L∞×W1,∞,‖yr‖L∞×W1,∞},B0>C[1+sinh⁡(1)],T≤ln⁡B0/(C[1+sinh⁡(1)])2(B0+C1),
we complete the proof by taking B0/(C[1+sinh(1)])=C+β,β>0.Lemma 3.6.
𝒞B0,B1,T is compact with respect to the norm ∥·∥L∞([0,T];Lip(0,1)).The proof is very similar to that appeared in [33] and omitted.Lemma 3.7.
Fory∈Hs×Hs-1, s>2, f(y) is bounded on bounded sets in Hs×Hs-1. Therefore, f(y) is bounded on bounded sets in L∞×L∞ by the embedding theorem.The proof is very similar to that appeared in [33] and omitted.Lemma 3.8.
The operatorℱ:𝒞B0,B1,T→𝒞B0,B1,T is continuous with respect to the norm ∥θ∥L∞([0,T];Lip(0,1)).Proof.
Take a sequence{θn} which tends to θ with respect to ∥·∥L∞([0,T];Lip(0,1)), set θ̃n=ℱθn and θ̃=ℱθ, denote by φn the flow of θn+𝔄 and φ the flow of θ+𝔄, and we have that φn→φ locally in C1 as n→∞, thanks to Proposition A.4 in [33]. What we will need to do is to show that mn→m in L1(0,1) as n→∞ and ρn→ρ in L1(0,1) as n→∞.
Lett∈[0,T], having supposed that Pl and Pr have only a finite number of connected components, we can assume, reducing t if necessary that vl and vr do not change sign on [0,t]. Since the characteristics of φn and φ may or may not cross before time t, we only consider the case that φ(t,0,0)≤φn(t,0,0)≤φ(t,0,1)≤φn(t,0,1), without loss of generality. The other cases are proved in the same way. We first point out that since θn∈𝒞B0,B1,T we have a bound for {yn} in L∞(ΩT). Now
(3.18)∫01|y(t,x)-yn(t,x)|dx=(∫0φ(t,0,0)+∫φ(t,0,0)φn(t,0,0)+∫φn(t,0,0)φ(t,0,1)+∫φ(t,0,1)φn(t,0,1)+∫φn(t,0,1)1)|y(t,x)-yn(t,x)|dx=I1+I2+I3+I4+I5.
Since φn(t,0,0)→φ(t,0,0) as n→∞ and φn(t,0,1)→φ(t,0,1) as n→∞ and thanks to the uniform bound on ∥yn∥L∞, we see that both I2 and I4 tend to 0 when n goes to infinity.
ForI1 we have
(3.19)I1=∫0φ(t,0,0)|y(t,x)-yn(t,x)|dx=∫0φ(t,0,0)|(exp⁡∫e(t,x)tE1dr00exp⁡∫e(t,x)tE2dr)yl(e(t,x))+∫e(t,x)t(exp⁡∫stE1dr00exp⁡∫stE2dr)(-ρρx-ρx-∂x(θ+A))ds-(exp⁡∫en(t,x)tE1ndr00exp⁡∫en(t,x)tE2ndr)yl(en(t,x))-∫en(t,x)t(exp⁡∫stE1ndr00exp⁡∫stE2ndr)(-ρρx-ρx-∂x(θn+A))ds|dx,
where
(3.20)E1n=-2∂x(θn+A)(r,φn(r,t,x)),E2=-∂x(θn+A)(r,φn(r,t,x)).
Thanks to the boundedness on the ∥f∥L∞×L∞ (Lemma 3.7) and Proposition A.2 of [33], if (t,x)∈¯P we have φn(t,0,0)→φ(t,0,0) as n→∞. This implies that if yl was continuous, since we have a uniform bound on ∥θn∥L∞((0,T);C1,1(0,1)) the dominated convergence theorem would provide: I1→0.I3→0 and I5→0 which can be obtained by using the same method. Therefore, for y0, yl and yr continuous we have ∥y(t,·)-yn(t,·)∥L1→0.
From the inequality (56) in [33], we obtain
(3.21)‖y(t,⋅)‖L1(0,1)×L1(0,1)≤[‖y0‖L1(0,1)×L1(0,1)+‖yl‖L1(0,t)∩Γl×L1(0,t)∩Γl+‖yr‖L1(0,t)∩Γr×L1(0,t)∩Γr+(‖ρ‖L1(0,1)×L1(0,1)+1)‖ρx‖L1(0,1)×L1(0,1)+‖∂x(θ+A)‖L1(0,1)]‖∂x(θ+A)‖L∞(ΩT)exp⁡(3t‖∂x(θ+A)‖L∞(ΩT)),‖yn(t,⋅)‖L1(0,1)×L1(0,1)≤[‖y0‖L1(0,1)×L1(0,1)+‖yl‖L1(0,t)∩Γl×L1(0,t)∩Γl+‖yr‖L1(0,t)∩Γr×L1(0,t)∩Γr+(‖ρn‖L1(0,1)×L1(0,1)+1)‖ρn,x‖L1(0,1)×L1(0,1)+‖∂x(θn+A)‖L1(0,1)]‖∂x(θn+A)‖L∞(ΩT)exp⁡(3t‖∂x(θn+A)‖L∞(ΩT)).
So the general case of convergence ∥y(t,·)-yn(t,·)∥L1→0 follows from the density of C0 in L1 and the uniform bound on ∥θn∥L∞((0,T);Lip(0,1)).
Now only the restriction ont remains; we recall that until now we supposed that vl and vr did not change sign on [0,t]. If vl and vr do not change sign on [0,t1] and then on [t1,t], we have
(3.22)‖y(t1,⋅)-yn(t1,⋅)‖L1⟶0.
Let ỹn the solution of
(3.23)∂tỹn+(θn+A)∂xỹn=bn(t,x)ỹn+fn(t,x),ỹn(t1,⋅)=yt1,ỹn(⋅,0)|Γl=yl,ỹn(⋅,1)|Γr=yr.
We can conclude that as n→∞,
(3.24)‖y(t,⋅)-yn(t,⋅)‖L1(0,1)×L1(0,1)≤‖y(t,⋅)-ỹn(t,⋅)‖L1(0,1)×L1(0,1)+‖ỹn(t,⋅)-yn(t,⋅)‖L1(0,1)×L1(0,1)≤‖y(t1,⋅)-ỹn(t1,⋅)‖L1(0,1)×L1(0,1)exp⁡(3(t-t1)‖∂x(θn+A)‖L∞(ΩT))+‖y(t,⋅)-ỹn(t,⋅)‖L1(0,1)×L1(0,1)≤‖y(t1,⋅)-yn(t1,⋅)‖L1(0,1)×L1(0,1)exp⁡(3(t-t1)‖∂x(θn+A)‖L∞(ΩT))+‖y(t,⋅)-ỹn(t,⋅)‖L1(0,1)×L1(0,1)⟶0.
Thus the convergence in L1(0,1) propagates on each interval where vl and vr do not change sign; thanks to the hypothesis on Pr and Pl we have that for all t∈[0,T], ∥y(t,·)-yn(t,·)∥L1(0,1)→0, as n→∞. Combining this first convergence result with the uniform bound and using the dominated convergence theorem in the time variable, we obtain ∥y-yn∥L1(ΩT)→0 which implies that
(3.25)‖θ̃-θ̃n‖L1(0,T;W2,1(0,1))⟶0.
From the compactness of𝒞B0,B1,T, we get that θ̃n→θ̃ holds in 𝒞B0,B1,T.All the above lemmas result in the application of Schauders fixed point theorem toℱ and we get a solution (3.26)θ∈L∞((0,T);C1,1(0,1))∩Lip([0,T];H01(0,1)).From the construction of F and from Proposition A.8 in [33] the additional regularity properties of any solution θ,(3.27)θ∈C0([0,T];W2,p(0,1))∩C1([0,T];W01,p(0,1)),∀p<+∞.


## 3.2. Uniqueness

In this subsection, we will show that the solution to the system (1.6) and (1.8) is unique; that to say, given (y,θ) and (ỹ,θ̃) be two solutions of (1.6) and (1.8) for the same initial and boundary data, we will get y=ỹ and θ=θ̃.LetY=y-ỹ, Θ=θ-θ̃, B(t,x)=b(t,x)-b̃(t,x), F(t,x)=f(t,x)-f̃(t,x), then Θ∈Lip([0,T];H01(0,1)). And we have M=m-m̃∈L∞(ΩT) and Y∈L∞(ΩT)×L∞(ΩT) is the solution to(3.28)∂tY+(θ+A)∂xY=bY+B(t,x)ỹ-Θ∂xỹ+F(t,x),Y(0,⋅)=0,Y(⋅,0)|Γl=0,Y(⋅,1)|Γr=0.
Using the lemma again with b(t,x)=(-2∂x(θ+𝔄)00-∂x(θ+𝔄)) and f¯=B(t,x)ỹ-U∂xỹ+F(t,x), we get(3.29)for(t,x)∈P,Y(t,x)=0,for(t,x)∈I,Y(t,x)=∫0texp⁡(∫stb(r,φ(r,t,x))dr)f¯(s,φ(s,t,x))ds,for(t,x)∈L,Y(t,x)=∫e(t,x)texp⁡(∫stb(r,φ(r,t,x))dr)f¯(s,φ(s,t,x))ds,for(t,x)∈R,Y(t,x)=∫e(t,x)texp⁡(∫stb(r,φ(r,t,x))dr)f¯(s,φ(s,t,x))ds.Since∥U(t,·)∥L∞(0,1)≤C′∥Y(t,·)∥L∞×L∞ for some positive constant C′, and y,∂xy bounded, we see that for some C′′>0,(3.30)‖f¯(t,⋅)‖L∞×L∞≤C′′‖Y(t,⋅)‖L∞×L∞.
And since b(t,x) is bounded, we get that(3.31)‖Y(t,⋅)‖L∞×L∞≤C′′∫0s‖Y(s,⋅)‖L∞×L∞ds.Then we complete the proof of the uniqueness by using Gronwall’s lemma.


## 4. Stabilization

### 4.1. Preliminary Results

The equilibrium state that we want to stabilize isy=0, θ=𝔄=0. A natural idea is using Lyapunov indirection method to investigate whether the linearized system around the equilibrium state is stabilizable or not. Its stabilization would provide a local stabilization result on the nonlinear system. Unfortunately, the linearized system is not stabilizable, for the state of the linearized system around the equilibrium state is constant. We see that the sign of θ+𝔄 controls the geometry of the characteristics, and the sign of ∂x(θ+𝔄) controls the ingredient information of y along the characteristics. We will use the return method that Coron introduced in [34]. We would like our feedback law, vl(y)=Al∥y∥C0([0,1])×C0([0,1]), vr(y)=Ar∥y∥C0([0,1])×C0([0,1]), to provide θ+𝔄≥0 and ∂x(θ+𝔄)≥0. However, there is a difficulty in the stabilization problem. It needs not to be true that the transition between those zones is continuous rather thanks to the regularity on the boundary data ρ is indeed Lipschitz inside the zones L, R, and I. To achieve this target, we have to prescribe yl, and we just need to make a continuous transition at (t,x)=(0,0) and let yl asymptotically converge in time; we assume that compatibility conditions hold; precisely, ρ0, ρl and ρr have compact supports in (0,1)/Γl/Γr, respectively. To achieve this target, we have to prescribe yl, and we just need to make a continuous transition at (t,x)=(0,0) and let yl asymptotically converge in time. This is guaranteed by(4.1)∂tyl=Myl,
where M, symmetric matrix, is the unique matrix solution to the matrix function:(4.2)PM+MTP=-Q,
for some symmetric positive-definite matrices, P and Q. Indeed, let V(t,yl)=ylTPyl be the Lyapunov candidate, and that yl asymptotically converges in time is equivalent to that the time derivative of the V, V̇=ylT(PM+MTP)yl is strictly negative. A fixed-point strategy will be used again to prove the existence of a solution to the closed-loop system. We begin by defining the domain of the operator.Definition 4.1.
LetX be the space of (g,N)∈C0([0,T];[0,1])2×C0([0,T]) satisfying(1)
g(0,x)=y0(x), g(t,0)=y0(0)eMt,(2)
∥g(t,·)∥C0([0,1])≤N(t),(3)
N(t) is nonincreasing and N(0)≤∥y0∥C0([0,1])×C0([0,1]).Lemma 4.2.
The domainX is nonempty, convex, bounded, and closed with respect to the uniform topology.TakingM=(λ,00,λ), λ<0, satisfying (4.1) and (4.2), and (y0(x)eMt,∥y0∥C0([0,1])2eMt)∈X, so X is nonempty.Now for(y,N(t))∈X we define θ̆ and 𝔄̆ as the solutions of(4.3)m(t,x)=(1-∂xx)θ̆(t,x),ŭ(t,0)=θ̆(t,1)=0,∀(t,x)∈ΩT,(1-∂xx)Ă(t,x)=0,Ă(t,0)=AlN(t),Ă(t,1)=ArN(t),∀(t,x)∈ΩT.
That is,(4.4)∀(t,x)∈ΩT,ŭ(t,x)=-∫0xsinh⁡(x-ξ)m(t,ξ)dξ,Ă(t,x)=N(t)sinh⁡(1)[sinh⁡(x)Ar+sinh⁡(1-x)Al].Thus, we have the estimates:(4.5)|θ̆(t,x)|≤2sinh⁡(1)‖y‖C0([0,1])×C0([0,1]),|∂xθ̆(t,x)|≤2cosh⁡(1)‖y‖C0([0,1])×C0([0,1]),|∂xxθ̆(t,x)|≤[1+2sinh⁡(1)]‖y‖C0([0,1])×C0([0,1]),|∂xĂ(t,x)|≥Ar-2cosh⁡(1)sinh⁡(1)N(t),|Ă(t,x)|≥AlN(t).
And in turn,(4.6)(θ̆+Ă)(t,x)≥[Al-2sinh⁡(1)]‖y‖C0([0,1])×C0([0,1]),(4.7)∂x(θ̆+Ă)(t,x)≥Ar-2cosh⁡(1)Al-sinh⁡(2)sinh⁡(1)‖y‖C0([0,1])×C0([0,1]).
Now, if φ is the flow of θ̆+𝔄̆, e is C1 and since ŭ+𝔄̆≥0, φ(·,t,x) is nondecreasing. Thus we can define the entrance time and then the operator 𝒮 as follows.Lete(t,x)=min{s∈[0,T]∣φ(s,t,x)=0}, for (t,x)∈[0,T]×[0,1],𝒮(y,N)=(ỹ,Ñ) with(1)
ifx≥φ(t,0,0),(4.8)ỹ(t,x)=(exp⁡∫0tE1dr00exp⁡∫0tE2dr)y0(φ(0,t,x))+∫0t(exp⁡∫stE1dr00exp⁡∫stE2dr)(-ρρx-ρx-∂x(θ+A))ds,(2)
ifx≤φ(t,0,0),(4.9)ỹ(t,x)=(exp⁡∫0tE1dr00exp⁡∫0tE2dr)eMe(t,x)y0(0)+∫e(t,x)t(exp⁡∫stE1dr00exp⁡∫stE2dr)(-ρρx-ρx-∂x(θ+A))ds,(3)
N(t)=∥y(t,·)∥C0([0,1])×C0([0,1]).From Lemma1.3 we know that ỹ is the weak solution of(4.10)∂tỹ+(θ+A)∂xỹ=b(t,x)ỹ+f(t,x),ỹ(0,⋅)=0,ỹ(t,0)=eMty0(0).Lemma 4.3.
(1) The operator𝒮 maps X to X.
(2) The family𝒮(X) is uniformly bounded and equicontinuous.
(3)S is continuous with respect to the uniform topology.The proof is very similar to [33] except for here the state y is a two-component vector and the proof is omitted.We can apply Schauder’s fixed-point theorem toS and get (y,N) fixed point of S.


### 4.2. Stabilization and Global Existence

From (4.6) and (4.7), for all (t,x)∈ΩT, (4.11)(θ̆+Ă)(t,x)≥c‖y(r,⋅)‖C0([0,1])×C0([0,1]),∂x(θ̆+Ă)(t,x)≥c‖y(r,⋅)‖C0([0,1])×C0([0,1]),y is the solution of the transport equation (1.8) and it satisfies(4.12)y(t,x)=(exp⁡∫stE1dr′00exp⁡∫stE2dr)y(s,φ(s,t,x))+∫st(exp⁡∫rtE1dr00exp⁡∫rtE2dr′)f(r,x)dr.
We get for t≥s(4.13)|y(t,x)|≤exp⁡(-∫stc‖y(r,⋅)‖C0([0,1])×C0([0,1])dr)y|s,φ(s,t,x)|.This implies that|y(t,x)|=|m(t,x)|2+|ρ(t,x)|2 decreases along the characteristics (strictly for the times where y(t,·)≠0). But we have also imposed y(t,0)=y(s,0)eM(t-s); therefore |y(t,x)| also decreases along x=0. This already shows, thanks to the existence theorem, that a maximal solution of the closed loop system is global. To get a more precise statement, we consider all the characteristics between time t and s, and we obtain(4.14)‖y(t,⋅)‖C0([0,1])×C0([0,1])≤max⁡r∈[s,t]{e‖M‖2(s-r)exp⁡(-∫stc‖y(r,⋅)‖C0([0,1])×C0([0,1])dr)}×‖y(t,⋅)‖C0([0,1])×C0([0,1]).Now we define(4.15)g(r)=e‖M‖2(s-r)exp⁡(-∫stc‖y(r,⋅)‖C0([0,1])×C0([0,1])dr).Then(4.16)g′(r)=[c‖y(t,⋅)‖C0([0,1])×C0([0,1])-‖M‖2]g(r),
and we know that as long as the quantity ∥y(t,·)∥C0([0,1])×C0([0,1]) is not equal to zero, it strictly decreases. So if ∥y0∥C0([0,1])×C0([0,1])>∥M∥2/c, for t small enough ∥y(t,·)∥C0([0,1])×C0([0,1])≥∥M∥2/c, and we have(4.17)‖y(t,⋅)‖C0([0,1])×C0([0,1])≤e-‖M‖2t‖y0‖C0([0,1])×C0([0,1]),
which implies ∥y(τ,·)∥C0([0,1])×C0([0,1])≤∥M∥2/c. This provides for τ≤s≤t, the inequality(4.18)‖y(t,⋅)‖C0([0,1])×C0([0,1])≤e-∫stc‖y(r,⋅)‖C0([0,1])×C0([0,1])dr‖y(s,⋅)‖C0([0,1])×C0([0,1]).
And we conclude with a classical comparison principle for ODES.


## 4.1. Preliminary Results

The equilibrium state that we want to stabilize isy=0, θ=𝔄=0. A natural idea is using Lyapunov indirection method to investigate whether the linearized system around the equilibrium state is stabilizable or not. Its stabilization would provide a local stabilization result on the nonlinear system. Unfortunately, the linearized system is not stabilizable, for the state of the linearized system around the equilibrium state is constant. We see that the sign of θ+𝔄 controls the geometry of the characteristics, and the sign of ∂x(θ+𝔄) controls the ingredient information of y along the characteristics. We will use the return method that Coron introduced in [34]. We would like our feedback law, vl(y)=Al∥y∥C0([0,1])×C0([0,1]), vr(y)=Ar∥y∥C0([0,1])×C0([0,1]), to provide θ+𝔄≥0 and ∂x(θ+𝔄)≥0. However, there is a difficulty in the stabilization problem. It needs not to be true that the transition between those zones is continuous rather thanks to the regularity on the boundary data ρ is indeed Lipschitz inside the zones L, R, and I. To achieve this target, we have to prescribe yl, and we just need to make a continuous transition at (t,x)=(0,0) and let yl asymptotically converge in time; we assume that compatibility conditions hold; precisely, ρ0, ρl and ρr have compact supports in (0,1)/Γl/Γr, respectively. To achieve this target, we have to prescribe yl, and we just need to make a continuous transition at (t,x)=(0,0) and let yl asymptotically converge in time. This is guaranteed by(4.1)∂tyl=Myl,
where M, symmetric matrix, is the unique matrix solution to the matrix function:(4.2)PM+MTP=-Q,
for some symmetric positive-definite matrices, P and Q. Indeed, let V(t,yl)=ylTPyl be the Lyapunov candidate, and that yl asymptotically converges in time is equivalent to that the time derivative of the V, V̇=ylT(PM+MTP)yl is strictly negative. A fixed-point strategy will be used again to prove the existence of a solution to the closed-loop system. We begin by defining the domain of the operator.Definition 4.1.
LetX be the space of (g,N)∈C0([0,T];[0,1])2×C0([0,T]) satisfying(1)
g(0,x)=y0(x), g(t,0)=y0(0)eMt,(2)
∥g(t,·)∥C0([0,1])≤N(t),(3)
N(t) is nonincreasing and N(0)≤∥y0∥C0([0,1])×C0([0,1]).Lemma 4.2.
The domainX is nonempty, convex, bounded, and closed with respect to the uniform topology.TakingM=(λ,00,λ), λ<0, satisfying (4.1) and (4.2), and (y0(x)eMt,∥y0∥C0([0,1])2eMt)∈X, so X is nonempty.Now for(y,N(t))∈X we define θ̆ and 𝔄̆ as the solutions of(4.3)m(t,x)=(1-∂xx)θ̆(t,x),ŭ(t,0)=θ̆(t,1)=0,∀(t,x)∈ΩT,(1-∂xx)Ă(t,x)=0,Ă(t,0)=AlN(t),Ă(t,1)=ArN(t),∀(t,x)∈ΩT.
That is,(4.4)∀(t,x)∈ΩT,ŭ(t,x)=-∫0xsinh⁡(x-ξ)m(t,ξ)dξ,Ă(t,x)=N(t)sinh⁡(1)[sinh⁡(x)Ar+sinh⁡(1-x)Al].Thus, we have the estimates:(4.5)|θ̆(t,x)|≤2sinh⁡(1)‖y‖C0([0,1])×C0([0,1]),|∂xθ̆(t,x)|≤2cosh⁡(1)‖y‖C0([0,1])×C0([0,1]),|∂xxθ̆(t,x)|≤[1+2sinh⁡(1)]‖y‖C0([0,1])×C0([0,1]),|∂xĂ(t,x)|≥Ar-2cosh⁡(1)sinh⁡(1)N(t),|Ă(t,x)|≥AlN(t).
And in turn,(4.6)(θ̆+Ă)(t,x)≥[Al-2sinh⁡(1)]‖y‖C0([0,1])×C0([0,1]),(4.7)∂x(θ̆+Ă)(t,x)≥Ar-2cosh⁡(1)Al-sinh⁡(2)sinh⁡(1)‖y‖C0([0,1])×C0([0,1]).
Now, if φ is the flow of θ̆+𝔄̆, e is C1 and since ŭ+𝔄̆≥0, φ(·,t,x) is nondecreasing. Thus we can define the entrance time and then the operator 𝒮 as follows.Lete(t,x)=min{s∈[0,T]∣φ(s,t,x)=0}, for (t,x)∈[0,T]×[0,1],𝒮(y,N)=(ỹ,Ñ) with(1)
ifx≥φ(t,0,0),(4.8)ỹ(t,x)=(exp⁡∫0tE1dr00exp⁡∫0tE2dr)y0(φ(0,t,x))+∫0t(exp⁡∫stE1dr00exp⁡∫stE2dr)(-ρρx-ρx-∂x(θ+A))ds,(2)
ifx≤φ(t,0,0),(4.9)ỹ(t,x)=(exp⁡∫0tE1dr00exp⁡∫0tE2dr)eMe(t,x)y0(0)+∫e(t,x)t(exp⁡∫stE1dr00exp⁡∫stE2dr)(-ρρx-ρx-∂x(θ+A))ds,(3)
N(t)=∥y(t,·)∥C0([0,1])×C0([0,1]).From Lemma1.3 we know that ỹ is the weak solution of(4.10)∂tỹ+(θ+A)∂xỹ=b(t,x)ỹ+f(t,x),ỹ(0,⋅)=0,ỹ(t,0)=eMty0(0).Lemma 4.3.
(1) The operator𝒮 maps X to X.
(2) The family𝒮(X) is uniformly bounded and equicontinuous.
(3)S is continuous with respect to the uniform topology.The proof is very similar to [33] except for here the state y is a two-component vector and the proof is omitted.We can apply Schauder’s fixed-point theorem toS and get (y,N) fixed point of S.


## 4.2. Stabilization and Global Existence

From (4.6) and (4.7), for all (t,x)∈ΩT, (4.11)(θ̆+Ă)(t,x)≥c‖y(r,⋅)‖C0([0,1])×C0([0,1]),∂x(θ̆+Ă)(t,x)≥c‖y(r,⋅)‖C0([0,1])×C0([0,1]),y is the solution of the transport equation (1.8) and it satisfies(4.12)y(t,x)=(exp⁡∫stE1dr′00exp⁡∫stE2dr)y(s,φ(s,t,x))+∫st(exp⁡∫rtE1dr00exp⁡∫rtE2dr′)f(r,x)dr.
We get for t≥s(4.13)|y(t,x)|≤exp⁡(-∫stc‖y(r,⋅)‖C0([0,1])×C0([0,1])dr)y|s,φ(s,t,x)|.This implies that|y(t,x)|=|m(t,x)|2+|ρ(t,x)|2 decreases along the characteristics (strictly for the times where y(t,·)≠0). But we have also imposed y(t,0)=y(s,0)eM(t-s); therefore |y(t,x)| also decreases along x=0. This already shows, thanks to the existence theorem, that a maximal solution of the closed loop system is global. To get a more precise statement, we consider all the characteristics between time t and s, and we obtain(4.14)‖y(t,⋅)‖C0([0,1])×C0([0,1])≤max⁡r∈[s,t]{e‖M‖2(s-r)exp⁡(-∫stc‖y(r,⋅)‖C0([0,1])×C0([0,1])dr)}×‖y(t,⋅)‖C0([0,1])×C0([0,1]).Now we define(4.15)g(r)=e‖M‖2(s-r)exp⁡(-∫stc‖y(r,⋅)‖C0([0,1])×C0([0,1])dr).Then(4.16)g′(r)=[c‖y(t,⋅)‖C0([0,1])×C0([0,1])-‖M‖2]g(r),
and we know that as long as the quantity ∥y(t,·)∥C0([0,1])×C0([0,1]) is not equal to zero, it strictly decreases. So if ∥y0∥C0([0,1])×C0([0,1])>∥M∥2/c, for t small enough ∥y(t,·)∥C0([0,1])×C0([0,1])≥∥M∥2/c, and we have(4.17)‖y(t,⋅)‖C0([0,1])×C0([0,1])≤e-‖M‖2t‖y0‖C0([0,1])×C0([0,1]),
which implies ∥y(τ,·)∥C0([0,1])×C0([0,1])≤∥M∥2/c. This provides for τ≤s≤t, the inequality(4.18)‖y(t,⋅)‖C0([0,1])×C0([0,1])≤e-∫stc‖y(r,⋅)‖C0([0,1])×C0([0,1])dr‖y(s,⋅)‖C0([0,1])×C0([0,1]).
And we conclude with a classical comparison principle for ODES.



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*Source: 635851-2011-09-29.xml*