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Abstract and Applied Analysis
Volume 2011, Article ID 635851, 20 pages
http://dx.doi.org/10.1155/2011/635851
Research Article

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

1School of Control Science & Engineering, University of Jinan, Jinan 250022, Shandong, China
2School of Science, University of Jinan, Jinan 250022, China

Received 9 May 2011; Revised 23 June 2011; Accepted 20 July 2011

Academic Editor: Dirk Aeyels

Copyright © 2011 Xiju Zong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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:𝑢𝑡𝑢𝑥𝑥𝑡+3𝑢𝑢𝑥2𝑢𝑥𝑢𝑥𝑥𝑢𝑢𝑥𝑥𝑥+𝜌𝜌𝑥=0,𝜌𝑡+𝜌𝑥𝑢+𝜌𝑢𝑥=0,(1.1) 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].

For 𝜌0, 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 [711]. 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 [2026] and references within. It has been shown that this equation is locally well posed [2023, 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]).

For 𝜌0, 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,𝜌0) and (𝑢1,𝜌1) 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)=𝜌1? and(2)the stabilizability problem: can one find a stationary feedback law 𝑣(𝑥), such that for any state (𝑢0,𝜌0) 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=𝑢𝑢𝑥𝑥 and 𝜌=𝜌1, which lead the system (1.1) to be equivalent to the system:𝑚𝑡+𝑢𝑚𝑥+2𝑚𝑢𝑥+𝜌𝜌𝑥+𝜌𝑥𝜌=0,𝑡+𝑢𝜌𝑥+𝜌𝑢𝑥+𝑢𝑥=0.(1.2) 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 𝑚0𝐿(0,1), 𝜌0𝑊1,(0,1). We set Γ𝑙=[]𝑡0,𝑇𝑣𝑙(𝑡)>0,Γ𝑟=[]𝑡0,𝑇𝑣𝑟(𝑡)<0.(1.3) In the following, we will always suppose that the sets 𝑃𝑙=[]𝑡0,𝑇𝑣𝑙(𝑡)=0,𝑃𝑟=[]𝑡0,𝑇𝑣𝑟(𝑡)=0(1.4) have a finite number of connected components. Finally, let 𝑚𝑙,𝜌𝑙𝐿(Γ𝑙)×𝑊1,(Γ𝑙) and 𝑚𝑟,𝜌𝑟𝐿(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 by1𝜕𝑥𝑥𝔄(𝑡,𝑥)=0,(𝑡,𝑥)Ω𝑇,𝔄(𝑡,0)=𝑣𝑙(𝑡),𝔄(𝑡,1)=𝑣𝑟[].(𝑡),𝑡0,𝑇(1.5) Setting 𝑢=𝜃+𝔄, we can further rewrite the system (1.1) as𝑚(𝑡,𝑥)=1𝜕𝑥𝑥𝜃𝑚(𝑡,𝑥),𝜃(𝑡,0)=𝜃(𝑡,1)=0,(1.6)𝑡+(𝜃+𝔄)𝑚𝑥=2𝑚𝜕𝑥(𝜃+𝔄)𝜌𝜌𝑥𝜌𝑥,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.(1.7) Let 𝑦=𝑚𝜌, 𝑦0=𝑚0𝜌0, 𝑏(𝑡,𝑥)=2𝜕𝑥(𝜃+𝔄)00𝜕𝑥(𝜃+𝔄), 𝑓(𝑡,𝑥)=𝜌𝜌𝑥𝜌𝑥𝜕𝑥(𝜃+𝔄), and the system (1.7) can be written as𝜕𝑡𝑦+(𝜃+𝔄)𝜕𝑥𝑦𝑦=𝑏(𝑡,𝑥)𝑦+𝑓(𝑡,𝑥),(0,)=𝑦0,𝑦(,0)|Γ𝑙=𝑦𝑙,𝑦(,1)|Γ𝑟=𝑦𝑟.(1.8) We first define what we mean by a weak solution to (1.8). Our test functions will be in the space:Ωadm𝑇=𝜓𝐶1Ω𝑇×𝐶1Ω𝑇[][]𝑥0,1,𝜓(𝑡,𝑥)=0;𝑡0,𝑇/Γ𝑙,𝜓[](𝑡,0)=0;𝑡0,𝑇/Γ𝑟.,𝜓(𝑡,1)=0(1.9)

Definition 1.1. Given 𝑦0=𝑚0𝜌0𝐿(Ω𝑇)×𝑊1,(Ω𝑇), when 𝜃𝐿((0,𝑇);Lip([0,1])), a function pair 𝑦=𝑚𝜌𝐿(Ω𝑇)×𝑊1,(ΩT) is a weak solution to (1.8) if 𝑦 satisfies Ω𝑇𝑦𝜕𝑡𝜓+𝑦(𝜃+𝔄)𝜕𝑥𝜓𝑦𝑏(𝑡,𝑥)𝜓𝑓(𝑡,𝑥)𝜓d𝑡d𝑥=10𝑦0𝜓(0,𝑥)d𝑥+𝑇0𝜓(𝑡,1)𝑣𝑟(𝑡)𝑦𝑟(𝑡)𝜓(𝑡,0)𝑣𝑙(𝑡)𝑦𝑙(𝑡)d𝑡.(1.10)

It is obvious that 𝐶10(Ω𝑇)×𝐶10(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 𝜕𝑠𝜑𝜑(𝑠,𝑡,𝑥)=𝑎(𝑠,𝜑(𝑠,𝑡,𝑥)),(𝑡,𝑡,𝑥)=𝑥,(1.11) which is defined on a certain set [𝑒(𝑡,𝑥),(𝑡,𝑥)] (which is closed because [0,1] is compact) and with possibly 𝑒(𝑡,𝑥) and/or (𝑡,𝑥)=𝑡.

We take into account the influence of the boundaries by introducing the sets: 𝑃=(𝑡,𝑥)Ω𝑇[𝑒],[][]𝑠(𝑡,𝑥),(𝑡,𝑥)suchthat𝜑(𝑠,𝑡,𝑥){0,1},𝑎(𝑠,𝜑(𝑠,𝑡,𝑥))=0{(𝑠,𝜑(𝑠,0,0))𝑠0,𝑇}{(𝑠,𝜑(𝑠,0,1))𝑠0,𝑇},𝐼=(𝑡,𝑥)Ω𝑇,/𝑃𝑒(𝑡,𝑥)=0𝐿=(𝑡,𝑥)Ω𝑇,(/𝑃𝜑(𝑒(𝑡,𝑥)𝑡,𝑥)=0𝑅=𝑡,𝑥)Ω𝑇,Γ/𝑃𝜑(𝑒(𝑡,𝑥)𝑡,𝑥)=1𝑙[]Γ={𝑡0,𝑇𝑎(𝑡,0)>0},𝑟=[]{𝑡0,𝑇𝑎(𝑡,1)>0}.(1.12) 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. Let 𝑎𝐶0([0,𝑇];𝐶1([0,1])), 𝑏,𝑓𝐿(Ω𝑇), 𝑦0𝐿(0,1), 𝑦𝑙𝐿(Γ𝑙), and 𝑦𝑟𝐿(Γ𝑟). We will also suppose that the sets: 𝑃𝑙=[]𝑡0,𝑇𝑣𝑙(𝑡)=0,𝑃𝑟=[]𝑡0,𝑇𝑣𝑟(𝑡)=0(1.13) have at most a countable number of connected components. Then the function 𝑦, defined by the formula for(𝑡,𝑥)𝑃,𝑦(𝑡,𝑥)=0,for(𝑡,𝑥)𝐼,𝑦(𝑡,𝑥)=exp𝑡0𝑦𝑏(𝑟,𝜑(𝑟,𝑡,𝑥))d𝑟0(+𝜑(0,𝑡,𝑥))𝑡0exp𝑡𝑠𝑏(𝑟,𝜑(𝑟,𝑡,𝑥))d𝑟𝑓(𝑠,𝜑(𝑠,𝑡,𝑥))d𝑠,for(𝑡,𝑥)𝐿,𝑦(𝑡,𝑥)=exp𝑡𝑒(𝑡,𝑥)𝑦𝑏(𝑟,𝜑(𝑟,𝑡,𝑥))d𝑟𝑙(+𝑒(𝑡,𝑥))𝑡𝑒(𝑡,𝑥)exp𝑡𝑠𝑏(𝑟,𝜑(𝑟,𝑡,𝑥))d𝑟𝑓(𝑠,𝜑(𝑠,𝑡,𝑥))d𝑠,for(𝑡,𝑥)𝑅,𝑦(𝑡,𝑥)=exp𝑡𝑒(𝑡,𝑥)𝑦𝑏(𝑟,𝜑(𝑟,𝑡,𝑥))d𝑟𝑟+(𝑒(𝑡,𝑥))𝑡𝑒(𝑡,𝑥)exp𝑡𝑠𝑏(𝑟,𝜑(𝑟,𝑡,𝑥))d𝑟𝑓(𝑠,𝜑(𝑠,𝑡,𝑥))d𝑠,(1.14) is a weak solution of 𝜕𝑡𝑦+(𝜃+𝔄)𝜕𝑥𝑦=𝑏(𝑡,𝑥)𝑦+𝑓(𝑡,𝑥)(1.15) and satisfies 𝑦𝐿(Ω𝑇)𝑦max0𝐿(Ω𝑇),𝑦𝑙𝐿(Γ𝑙),𝑦𝑟𝐿(Γ𝑟)+𝑇𝑓𝐿(Ω𝑇)𝑒𝑡𝑏𝐿𝑇)(Ω.(1.16)

However, if we let 𝑚𝑙,𝜌𝑙𝐿(Γ𝑙)×𝐿(Γ𝑙) and 𝑚𝑟,𝜌𝑟𝐿(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);𝑊1,(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 𝜌0𝑊1,(0,1), 𝜌𝑙𝑊1,(Γ𝑙), 𝜌𝑟𝑊1,(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 Section 2, 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. For 𝑇>0, we consider 𝑣𝑙𝐶0(Γ𝑙), 𝑣𝑟𝐶0(Γ𝑙) such that the sets 𝑃𝑙 and 𝑃𝑟 have only a finite number of connected components. Let 𝑦0𝐿(0,1)×𝑊1,(0,1), 𝑦𝑙𝐿(Γ𝑙)×𝑊1,(Γ𝑙), and 𝑦𝑟𝐿(Γ𝑟)×𝑊1,(Γ𝑟). There exist 𝑇>0 and (𝜃,𝑦) a weak solution of the system (1.8) (or (1.1)) with 𝜃𝐿((0,𝑇);𝐶1,1([0,1])Lip((0,𝑇);𝐻10(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(𝑇,𝑇), with 𝑇=max𝛽>0ln(1+𝛽/𝐶)2𝐶1,𝑦+(2+sinh(1))(𝛽+𝐶)𝐶=max0𝐿×𝑊1,,𝑦𝑙𝐿×𝑊1,,𝑦𝑟𝐿×𝑊1,,𝐶1=1𝑣tanh(1)𝑙𝐿(0,𝑇),𝑣𝑟𝐿(0,𝑇).(2.1)

In a second step, we will show a weak-strong uniqueness property.

Theorem 2.2. Let 𝜃𝐿((0,𝑇);𝐶1,1([0,1])Lip((0,𝑇);𝐻10(0,1), and let 𝑦Lip([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.

Let 𝐴𝑙>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)) reads𝑦𝐶0([]0,1)×𝐶0([]𝑣0,1)𝑙(𝑦)=𝐴𝑙𝑦𝐶0([0,1])×𝐶0([0,1]),𝑣𝑟(𝑦)=𝐴𝑟𝑦𝐶0([0,1])×𝐶0([0,1]),̇𝑦𝑙(𝑡)=𝑀𝑦𝑙(𝑡).(2.2)

Theorem 2.3. For any 𝑦0[𝐶0([0,1])]2 there exists (𝑦,𝑣)[𝐶0(Ω𝑇)]2×𝐶2([0,1]) a weak solution of (1.1) and (2.2) satisfying []𝑥0,1,𝑦(0,𝑥)=𝑦0(𝑥).(2.3) Furthermore any maximal solution of (1.1), (2.2), and (2.3) is global, and if we let 𝐴𝑐=min𝑙𝐴2sinh(1),𝑟𝐴𝑙cosh(1)+sinh(2)1sinh(1),𝜏=𝑀2ln𝑐𝑦𝐶0([0,1])𝑀2,(2.4) then we have 𝑦𝐶0([0,1])𝑀2𝑐1+𝑀2.(𝑡𝜏)(2.5)

Remark 2.4. For 𝜌0, 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.

Given 𝜃𝐿((0,𝑇);𝐶1,1([0,1]))Lip((0,𝑇);𝐻10(0,1)), we will define 𝑦=𝑚𝜌 to be the solution to (1.8), and once we have 𝑚 in 𝐿(Ω𝑇), we introduce ̃𝜃 solution of1𝜕𝑥𝑥̃𝜃=𝑚.(3.1) Then is defined as the operator ̃𝜃=(𝜃)

Lemma 3.1. The function 𝔄 defined by (1.5) satisfies (𝑡,𝑥)Ω𝑇,𝔄(𝑡,𝑥)𝐶0([]0,𝑇;𝐶([]10,1)),𝔄(𝑡,𝑥)=sinh(1)sinh(𝑥)𝑣𝑟(𝑡)+sinh(1𝑥)𝑣𝑙,(𝑡)𝔄(𝑡,𝑥)𝐿((0,𝑇);𝐶1,1([0,1]))cosh(1)𝑣sinh(1)𝑙𝐿(0,𝑇)+𝑣𝑟𝐿(0,𝑇).(3.2)

Then for a function 𝜃𝐿((0,𝑇);𝐶1,1([0,1]))Lip((0,𝑇);𝐻10(0,1)), we consider 𝜑 the flow of 𝜃+𝔄.

Lemma 3.2. The flow 𝜑 satisfies the following properties. (1)𝜑 is 𝐶1 with the following partial derivatives:𝜕1𝜕𝜑(𝑠,𝑡,𝑥)=(𝜃+𝔄)(𝑠,𝜑(𝑠,𝑡,𝑥)),2𝜑(𝑠,𝑡,𝑥)=(𝜃+𝔄)exp𝑠𝑡𝜕𝑥,𝜕(𝜃+𝔄)(𝑟,𝜑(𝑟,𝑡,𝑥))d𝑟3𝜑(𝑠,𝑡,𝑥)=exp𝑠𝑡𝜕𝑥.(𝜃+𝔄)(𝑟,𝜑(𝑟,𝑡,𝑥))d𝑟(3.3)(2)For all 𝑗=1,2,3,𝜕𝑗𝜑𝐶01+𝜃+𝔄𝐶0(Ω𝑇)𝑒𝑇𝜃+𝔄𝐶0𝑇)(Ω.(3.4) For (𝑡,𝑥)Ω𝑇, 𝜑(,𝑡,𝑥) is defined on a set [𝑒(𝑡,𝑥),(𝑡,𝑥)], here 𝑒(𝑡,𝑥) is basically the entrance time in Ω𝑇 of the characteristic curve going through (𝑡,𝑥). (3) If 𝑒(𝑡,𝑥)>0, then 𝜑(𝑒(𝑡,𝑥),𝑡,𝑥){0,1}.(4) If (𝑡,𝑥)<𝑇, then 𝜑((𝑡,𝑥),𝑡,𝑥){0,1}.

For 𝜃𝐿((0,𝑇);𝐶1,1([0,1]))Lip((0,𝑇);𝐻10(0,1)), we define the solution to (1.8) 𝑦𝐿(Ω𝑇)×𝑊1,(Ω𝑇) byfor(𝑡,𝑥)𝐼,𝑦(𝑡,𝑥)=exp𝑡0𝐸1d𝑟00exp𝑡0𝐸2𝑦d𝑟0+(𝜑(0,𝑡,𝑥))𝑡0exp𝑡𝑠𝐸1d𝑟00exp𝑡𝑠𝐸2d𝑟𝜌𝜌𝑥𝜌𝑥𝜕𝑥(𝜃+𝔄)d𝑠,for(𝑡,𝑥)𝐿,𝑦(𝑡,𝑥)=exp𝑡𝑒(𝑡,𝑥)𝐸1d𝑟00exp𝑡𝑒(𝑡,𝑥)𝐸2𝑦d𝑟𝑙+(𝑒(𝑡,𝑥))𝑡𝑒(𝑡,𝑥)exp𝑡𝑠𝐸1d𝑟00exp𝑡𝑠𝐸2d𝑟𝜌𝜌𝑥𝜌𝑥𝜕𝑥(𝜃+𝔄)d𝑠,for(𝑡,𝑥)𝑅,𝑦(𝑡,𝑥)=exp𝑡𝑒(𝑡,𝑥)𝐸1d𝑟00exp𝑡𝑒(𝑡,𝑥)𝐸2𝑦d𝑟𝑟+(𝑒(𝑡,𝑥))𝑡𝑒(𝑡,𝑥)exp𝑡𝑠𝐸1d𝑟00exp𝑡𝑠𝐸2d𝑟𝜌𝜌𝑥𝜌𝑥𝜕𝑥(𝜃+𝔄)d𝑠(3.5) with 𝐸1=2𝜕𝑥(𝜃+𝔄)(𝑟,𝜑(𝑟,𝑡,𝑥)),𝐸2=𝜕𝑥(𝜃+𝔄)(𝑟,𝜑(𝑟,𝑡,𝑥)).(3.6) And we have(1) the function 𝑦 is the unique weak solution of (1.8) in the sense of Definition 4.1; thanks to Lemma 1.3. 𝑦 is in 𝐿𝑊1,.

Remark 3.3. 𝑦 is the only weak solution of (1.8), and also 𝑦 is in 𝐿𝑊1, 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)Since 𝑦𝐿(Ω𝑇)×𝑊1,(Ω𝑇), we immediately get 𝑦𝑊1,(0,𝑇;𝐻1(0,1))×𝑊2,(0,𝑇;𝐻1(0,1)) and satisfies (1.8). Also we can get the estimates:𝑦(𝑡,𝑥)𝐿(Ω𝑇)×𝐿(Ω𝑇)𝐶0𝜕exp2𝑇𝑥𝜃𝐿(Ω𝑇)+𝜕𝑥𝔄𝐿(Ω𝑇),𝜕𝑡𝑦(𝑡,𝑥)𝐿(0,𝑇;𝐻1)×𝐿(0,𝑇;𝐻1)3𝐶0𝜕exp2𝑇𝑥𝜃𝐿(Ω𝑇)+𝜕𝑥𝔄𝐿(Ω𝑇),×𝜃𝐿((0,𝑇);Lip([0,1]))+𝔄𝐿((0,𝑇);Lip([0,1])),(3.7)

where𝐶0𝑦=max0𝐿×𝑊1,,𝑦𝑙𝐿×𝑊1,,𝑦𝑟𝐿×𝑊1,+𝑇𝜌𝐿𝜌+1𝑥𝐿((0,𝑇);Lip([0,1]))+𝜕𝑥𝜃𝐿(Ω𝑇)+𝜕𝑥𝔄𝐿(Ω𝑇).(3.8)(3) If (𝑡,𝑥)𝐼𝐿𝑅 and if (𝑠,𝑠)[𝑒(𝑡,𝑥),(𝑡,𝑥)]2, one has the following property:𝑦(𝑠,𝜑(𝑠,𝑡,𝑥))=exp𝑡0𝐸1d𝑟00exp𝑡0𝐸2𝑦𝑠d𝑟𝑠,𝜑,𝑡,𝑥.(3.9)

From the elliptic equation we can get

Lemma 3.4. There exists a unique ̃𝜃𝐿((0,𝑇);𝐻10(0,1)) such that 1𝜕𝑥𝑥̃𝜃=𝑚(3.10) holds in 𝐷(0,1) for all (𝑡,𝑥)Ω𝑇. Furthermore ̃𝜃𝐿((0,𝑇);𝐶1,1([0,1]))Lip([0,𝑇];𝐻10(0,1)), since 𝑚𝐿(Ω𝑇)Lip([0,𝑇];𝐻1(0,1)). Moreover we have the bounds: ̃𝜃𝐿((0,𝑇);𝐶1,1([0,1]))[]1+sinh(1)𝑚𝐿(Ω𝑇)[]1+sinh(1)𝑦𝐿(Ω𝑇)×𝑊1,(Ω𝑇),𝜕𝑡̃𝜃𝐿((0,𝑇);𝐻10(0,1))𝜕𝑡𝑚𝐿((0,𝑇);𝐻1)𝜕𝑡𝑦𝐿((0,𝑇);𝐻1)×𝑊1,((0,𝑇);𝐻1).(3.11)

The proof can be found in [33] and omitted.

Thus, for 𝜃𝐿((0,𝑇);𝐶1,1(0,1))Lip([0,𝑇];𝐻10(0,1)), the operator can be defined as ̃𝜃=(𝜃)𝐿((0,𝑇);𝐶1,1(0,1))Lip([0,𝑇];𝐻10(0,1)).

Let 𝐵0 and 𝐵1 be positive numbers, then we set𝒞𝐵0,𝐵1,𝑇=𝜃𝐿(0,𝑇);𝐶1,1[](0,1)Lip0,𝑇;𝐻10(0,1)both𝜃𝐿((0,𝑇);𝐶1,1(0,1))𝐵0,𝜃Lip([0,𝑇];𝐻1(0,1))𝐵1.(3.12) Obviously 𝒞𝐵0,𝐵1,𝑇 is convex. We will endow 𝒞𝐵0,𝐵1,𝑇 with the norm 𝜃𝐿([0,𝑇];Lip(0,1)).

Lemma 3.5. There exist positive numbers 𝐵0, 𝐵1 and 𝑇, such that 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: 𝐶0𝑦=max0𝐿×𝑊1,,𝑦𝑙𝐿×𝑊1,,𝑦𝑟𝐿×𝑊1,+𝑇𝜌𝐿𝜌+1𝑥𝐿((0,𝑇);Lip([0,1]))+𝜕𝑥𝜃𝐿(Ω𝑇)+𝜕𝑥𝔄𝐿(Ω𝑇),𝐶1=1𝑣tanh(1)𝑙𝐿(0,𝑇),𝑣𝑟𝐿(0,𝑇).(3.13)
Estimates (3.7), and (3.11) on 𝑦 and 𝜃 now read, 𝑦(𝑡,𝑥)𝐿(Ω𝑇)×𝑊1,(Ω𝑇)𝐶0𝜕exp2𝑇𝑥𝜃𝐿(Ω𝑇)+𝐶1,𝜕𝑡𝑦(𝑡,𝑥)𝐿((0,𝑇);𝐻1)×𝐿((0,𝑇);𝐻1)3𝐶0𝜕exp2𝑇𝑥𝜃𝐿(Ω𝑇)+𝐶1×𝜃𝐿((0,𝑇);Lip([0,1]))+𝐶1,̃𝜃𝐿((0,𝑇);𝐶1,1([0,1]))[]1+sinh(1)𝑦(𝑡,𝑥)𝐿(Ω𝑇)×𝑊1,(Ω𝑇),𝜕𝑡̃𝜃𝐿((0,𝑇);𝐻10(0,1))𝜕𝑡𝑦(𝑡,𝑥)𝐿((0,𝑇);𝐻1)×𝐿((0,𝑇);𝐻1).(3.14) Combining those estimates we get for all 𝜃𝒞𝐵0,𝐵1,𝑇: ̃𝜃𝐿((0,𝑇);𝐶1,1([0,1]))𝐶0[]𝜕1+sinh(1)exp2𝑇𝑥𝜃𝐿(Ω𝑇)+𝐶1𝐶0[]𝐵1+sinh(1)exp2𝑇0+𝐶1,𝜕𝑡̃𝜃𝐿((0,𝑇);𝐻10(0,1))3𝐶0𝜕exp2𝑇𝑥𝜃𝐿(Ω𝑇)+𝐶1𝜃𝐿((0,𝑇);Lip([0,1]))+𝐶13𝐶0𝐵exp2𝑇0+𝐶1𝐵0+𝐶1.(3.15) To obtain ̃𝜃𝒞𝐵0,𝐵1,𝑇, it is sufficient that 𝐶0[]𝐵1+sinh(1)exp2𝑇0+𝐶1𝐵0,𝐵0+3𝐶0𝐵exp2𝑇0+𝐶1𝐵0+𝐶1𝐵1.(3.16)
Once we have chosen 𝑇 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, 𝑦𝐶=max0𝐿×𝑊1,,𝑦𝑙𝐿×𝑊1,,𝑦𝑟𝐿×𝑊1,,𝐵0[]>𝐶1+sinh(1),𝑇ln𝐵0[])/(𝐶1+sinh(1)2𝐵0+𝐶1,(3.17) 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. For 𝑦𝐻𝑠×𝐻𝑠1, 𝑠>2, 𝑓(𝑦) is bounded on bounded sets in 𝐻𝑠×𝐻𝑠1. 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 operator 𝒞𝐵0,𝐵1,𝑇𝒞𝐵0,𝐵1,𝑇 is continuous with respect to the norm 𝜃𝐿([0,𝑇];Lip(0,1)).

Proof. Take a sequence {𝜃𝑛} which tends to 𝜃 with respect to 𝐿([0,𝑇];Lip(0,1)), set ̃𝜃𝑛=𝜃𝑛 and ̃𝜃=𝜃, denote by 𝜑𝑛 the flow of 𝜃𝑛+𝔄 and 𝜑 the flow of 𝜃+𝔄, and we have that 𝜑𝑛𝜑 locally in 𝐶1 as 𝑛, thanks to Proposition A.4 in [33]. What we will need to do is to show that 𝑚𝑛𝑚 in 𝐿1(0,1) as 𝑛 and 𝜌𝑛𝜌 in 𝐿1(0,1) as 𝑛.
Let 𝑡[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 𝜃𝑛𝒞𝐵0,𝐵1,𝑇 we have a bound for {𝑦𝑛} in 𝐿(Ω𝑇). Now 10||𝑦(𝑡,𝑥)𝑦𝑛||=(𝑡,𝑥)d𝑥0𝜑(𝑡,0,0)+𝜑𝑛(𝑡,0,0)𝜑(𝑡,0,0)+𝜑𝜑(𝑡,0,1)𝑛(𝑡,0,0)+𝜑𝑛(𝑡,0,1)𝜑(𝑡,0,1)+1𝜑𝑛(𝑡,0,1)||𝑦(𝑡,𝑥)𝑦𝑛||(𝑡,𝑥)d𝑥=𝐼1+𝐼2+𝐼3+𝐼4+𝐼5.(3.18) 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 𝑦𝑛𝐿, we see that both 𝐼2 and 𝐼4 tend to 0 when 𝑛 goes to infinity.
For 𝐼1 we have 𝐼1=0𝜑(𝑡,0,0)||𝑦(𝑡,𝑥)𝑦𝑛||=(𝑡,𝑥)d𝑥0𝜑(𝑡,0,0)|||||||exp𝑡𝑒(𝑡,𝑥)𝐸1d𝑟00exp𝑡𝑒(𝑡,𝑥)𝐸2𝑦d𝑟𝑙+(𝑒(𝑡,𝑥))𝑡𝑒(𝑡,𝑥)exp𝑡𝑠𝐸1d𝑟00exp𝑡𝑠𝐸2d𝑟𝜌𝜌𝑥𝜌𝑥𝜕𝑥(𝜃+𝔄)d𝑠exp𝑡𝑒𝑛(𝑡,𝑥)𝐸1𝑛d𝑟00exp𝑡𝑒𝑛(𝑡,𝑥)𝐸2𝑛𝑦d𝑟𝑙𝑒𝑛(𝑡,𝑥)𝑡𝑒𝑛(𝑡,𝑥)exp𝑡𝑠𝐸1𝑛d𝑟00exp𝑡𝑠𝐸2𝑛d𝑟𝜌𝜌𝑥𝜌𝑥𝜕𝑥𝜃𝑛|||||||+𝔄d𝑠d𝑥,(3.19) where 𝐸1𝑛=2𝜕𝑥𝜃𝑛+𝔄𝑟,𝜑𝑛(𝑟,𝑡,𝑥),𝐸2=𝜕𝑥𝜃𝑛+𝔄𝑟,𝜑𝑛(𝑟,𝑡,𝑥).(3.20) Thanks to the boundedness on the 𝑓𝐿×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 𝜃𝑛𝐿((0,𝑇);𝐶1,1(0,1)) the dominated convergence theorem would provide: 𝐼10.𝐼30 and 𝐼50 which can be obtained by using the same method. Therefore, for 𝑦0, 𝑦𝑙 and 𝑦𝑟 continuous we have 𝑦(𝑡,)𝑦𝑛(𝑡,)𝐿10.
From the inequality (56) in [33], we obtain 𝑦(𝑡,)𝐿1(0,1)×𝐿1(0,1)𝑦0𝐿1(0,1)×𝐿1(0,1)+𝑦𝑙𝐿1(0,𝑡)Γ𝑙×𝐿1(0,𝑡)Γ𝑙+𝑦𝑟𝐿1(0,𝑡)Γ𝑟×𝐿1(0,𝑡)Γ𝑟+𝜌𝐿1(0,1)×𝐿1(0,1)𝜌+1𝑥𝐿1(0,1)×𝐿1(0,1)+𝜕𝑥(𝜃+𝔄)𝐿1(0,1)𝜕𝑥(𝜃+𝔄)𝐿(Ω𝑇)𝜕exp3𝑡𝑥(𝜃+𝔄)𝐿(Ω𝑇),𝑦𝑛(𝑡,)𝐿1(0,1)×𝐿1(0,1)𝑦0𝐿1(0,1)×𝐿1(0,1)+𝑦𝑙𝐿1(0,𝑡)Γ𝑙×𝐿1(0,𝑡)Γ𝑙+𝑦𝑟𝐿1(0,𝑡)Γ𝑟×𝐿1(0,𝑡)Γ𝑟+𝜌𝑛𝐿1(0,1)×𝐿1(0,1)𝜌+1𝑛,𝑥𝐿1(0,1)×𝐿1(0,1)+𝜕𝑥𝜃𝑛+𝔄𝐿1(0,1)𝜕𝑥𝜃𝑛+𝔄𝐿(Ω𝑇)𝜕exp3𝑡𝑥𝜃𝑛+𝔄𝐿(Ω𝑇).(3.21) So the general case of convergence 𝑦(𝑡,)𝑦𝑛(𝑡,)𝐿10 follows from the density of 𝐶0 in 𝐿1 and the uniform bound on 𝜃𝑛𝐿((0,𝑇);Lip(0,1)).
Now only the restriction on 𝑡 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 𝑦𝑡1,𝑦𝑛𝑡1,𝐿10.(3.22) Let ̃𝑦𝑛 the solution of 𝜕𝑡̃𝑦𝑛+𝜃𝑛𝜕+𝔄𝑥̃𝑦𝑛=𝑏𝑛(𝑡,𝑥)̃𝑦𝑛+𝑓𝑛(𝑡,𝑥),̃𝑦𝑛𝑡1,=𝑦𝑡1,̃𝑦𝑛(,0)|Γ𝑙=𝑦𝑙,̃𝑦𝑛(,1)|Γ𝑟=𝑦𝑟.(3.23) We can conclude that as 𝑛, 𝑦(𝑡,)𝑦𝑛(𝑡,)𝐿1(0,1)×𝐿1(0,1)𝑦(𝑡,)̃𝑦𝑛(𝑡,)𝐿1(0,1)×𝐿1(0,1)+̃𝑦𝑛(𝑡,)𝑦𝑛(𝑡,)𝐿1(0,1)×𝐿1(0,1)𝑦𝑡1,̃𝑦𝑛𝑡1,𝐿1(0,1)×𝐿1(0,1)3exp𝑡𝑡1𝜕𝑥(𝜃𝑛+𝔄)𝐿(Ω𝑇)+𝑦(𝑡,)̃𝑦𝑛(𝑡,)𝐿1(0,1)×𝐿1(0,1)𝑦𝑡1,𝑦𝑛𝑡1,𝐿1(0,1)×𝐿1(0,1)3exp𝑡𝑡1𝜕𝑥𝜃𝑛+𝔄𝐿(Ω𝑇)+𝑦(𝑡,)̃𝑦𝑛(𝑡,)𝐿1(0,1)×𝐿1(0,1)0.(3.24) 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 𝑦𝑦𝑛𝐿1(Ω𝑇)0 which implies that ̃̃𝜃𝜃𝑛𝐿1(0,𝑇;𝑊2,1(0,1))0.(3.25)
From the compactness of 𝒞𝐵0,𝐵1,𝑇, we get that ̃𝜃𝑛̃𝜃 holds in 𝒞𝐵0,𝐵1,𝑇.

All the above lemmas result in the application of Schauders fixed point theorem to and we get a solution 𝜃𝐿(0,𝑇);𝐶1,1[](0,1)Lip0,𝑇;𝐻10.(0,1)(3.26)

From the construction of F and from Proposition A.8 in [33] the additional regularity properties of any solution 𝜃,𝜃𝐶0[]0,𝑇;𝑊2,𝑝(0,1)𝐶1[]0,𝑇;𝑊01,𝑝(0,1),𝑝<+.(3.27)

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 ̃(̃𝑦,𝜃) be two solutions of (1.6) and (1.8) for the same initial and boundary data, we will get 𝑦=̃𝑦 and ̃𝜃𝜃=.

Let 𝑌=𝑦̃𝑦, ̃𝜃Θ=𝜃, ̃𝐵(𝑡,𝑥)=𝑏(𝑡,𝑥)𝑏(𝑡,𝑥), 𝐹(𝑡,𝑥)=𝑓(𝑡,𝑥)𝑓(𝑡,𝑥), then ΘLip([0,𝑇];𝐻10(0,1)). And we have 𝑀=𝑚𝑚𝐿(Ω𝑇) and 𝑌𝐿(Ω𝑇)×𝐿(Ω𝑇) is the solution to𝜕𝑡𝑌+(𝜃+𝔄)𝜕𝑥𝑌=𝑏𝑌+𝐵(𝑡,𝑥)̃𝑦Θ𝜕𝑥𝑌̃𝑦+𝐹(𝑡,𝑥),(0,)=0,𝑌(,0)|Γ𝑙=0,𝑌(,1)|Γ𝑟=0.(3.28) Using the lemma again with 𝑏(𝑡,𝑥)=2𝜕𝑥(𝜃+𝔄)00𝜕𝑥(𝜃+𝔄) and 𝑓=𝐵(𝑡,𝑥)̃𝑦𝑈𝜕𝑥̃𝑦+𝐹(𝑡,𝑥), we getfor(𝑡,𝑥)𝑃,𝑌(𝑡,𝑥)=0,for(𝑡,𝑥)𝐼,𝑌(𝑡,𝑥)=𝑡0exp𝑡𝑠𝑏(𝑟,𝜑(𝑟,𝑡,𝑥))d𝑟𝑓(𝑠,𝜑(𝑠,𝑡,𝑥))d𝑠,for(𝑡,𝑥)𝐿,𝑌(𝑡,𝑥)=𝑡𝑒(𝑡,𝑥)exp𝑡𝑠𝑏(𝑟,𝜑(𝑟,𝑡,𝑥))d𝑟𝑓(𝑠,𝜑(𝑠,𝑡,𝑥))d𝑠,for(𝑡,𝑥)𝑅,𝑌(𝑡,𝑥)=𝑡𝑒(𝑡,𝑥)exp𝑡𝑠𝑏(𝑟,𝜑(𝑟,𝑡,𝑥))d𝑟𝑓(𝑠,𝜑(𝑠,𝑡,𝑥))d𝑠.(3.29)

Since 𝑈(𝑡,)𝐿(0,1)𝐶𝑌(𝑡,)𝐿×𝐿 for some positive constant 𝐶, and 𝑦,𝜕𝑥𝑦 bounded, we see that for some 𝐶>0,𝑓(𝑡,)𝐿×𝐿𝐶𝑌(𝑡,)𝐿×𝐿.(3.30) And since 𝑏(𝑡,𝑥) is bounded, we get that𝑌(𝑡,)𝐿×𝐿𝐶𝑠0𝑌(𝑠,)𝐿×𝐿d𝑠.(3.31)

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 is 𝑦=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)=𝐴𝑙𝑦𝐶0([0,1])×𝐶0([0,1]), 𝑣𝑟(𝑦)=𝐴𝑟𝑦𝐶0([0,1])×𝐶0([0,1]), to provide 𝜃+𝔄0 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𝜕𝑡𝑦𝑙=𝑀𝑦𝑙,(4.1) where 𝑀, symmetric matrix, is the unique matrix solution to the matrix function:𝑃𝑀+𝑀𝑇𝑃=𝑄,(4.2) for some symmetric positive-definite matrices, 𝑃 and 𝑄. Indeed, let 𝑉(𝑡,𝑦𝑙)=𝑦𝑇𝑙𝑃𝑦𝑙 be the Lyapunov candidate, and that 𝑦𝑙 asymptotically converges in time is equivalent to that the time derivative of the 𝑉, ̇𝑉=𝑦𝑇𝑙(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. Let 𝑋 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)𝑦0𝐶0([0,1])×𝐶0([0,1]).

Lemma 4.2. The domain X is nonempty, convex, bounded, and closed with respect to the uniform topology.

Taking 𝑀=𝜆,00,𝜆, 𝜆<0, satisfying (4.1) and (4.2), and (𝑦0(𝑥)𝑒𝑀𝑡,𝑦0𝐶0([0,1])2𝑒𝑀𝑡)𝑋, so 𝑋 is nonempty.

Now for (𝑦,𝑁(𝑡))𝑋 we define ̆𝜃 and ̆𝔄 as the solutions of𝑚(𝑡,𝑥)=1𝜕𝑥𝑥̆̆𝜃𝜃(𝑡,𝑥),̆𝑢(𝑡,0)=(𝑡,1)=0,(𝑡,𝑥)Ω𝑇,1𝜕𝑥𝑥̆̆𝔄(𝑡,𝑥)=0,𝔄(𝑡,0)=𝐴𝑙̆𝑁(𝑡),𝔄(𝑡,1)=𝐴𝑟𝑁(𝑡),(𝑡,𝑥)Ω𝑇.(4.3) That is,(𝑡,𝑥)Ω𝑇,̆𝑢(𝑡,𝑥)=𝑥0̆sinh(𝑥𝜉)𝑚(𝑡,𝜉)d𝜉,𝔄(𝑡,𝑥)=𝑁(𝑡)sinh(1)sinh(𝑥)𝐴𝑟+sinh(1𝑥)𝐴𝑙.(4.4)

Thus, we have the estimates:||̆||𝜃(𝑡,𝑥)2sinh(1)𝑦𝐶0([0,1])×𝐶0([0,1]),||𝜕𝑥̆||𝜃(𝑡,𝑥)2cosh(1)𝑦𝐶0([0,1])×𝐶0([0,1]),||𝜕𝑥𝑥̆||[]𝜃(𝑡,𝑥)1+2sinh(1)𝑦𝐶0([0,1])×𝐶0([0,1]),||𝜕𝑥̆||𝐴𝔄(𝑡,𝑥)𝑟2cosh(1)||̆||sinh(1)𝑁(𝑡),𝔄(𝑡,𝑥)𝐴𝑙𝑁(𝑡).(4.5) And in turn,̆̆𝔄𝐴𝜃+(𝑡,𝑥)𝑙2sinh(1)𝑦𝐶0([0,1])×𝐶0([0,1]),𝜕(4.6)𝑥̆̆𝔄𝐴𝜃+(𝑡,𝑥)𝑟2cosh(1)𝐴𝑙sinh(2)sinh(1)𝑦𝐶0([0,1])×𝐶0([0,1]).(4.7) Now, if 𝜑 is the flow of ̆̆𝔄𝜃+, 𝑒 is 𝐶1 and since ̆̆𝑢+𝔄0, 𝜑(,𝑡,𝑥) is nondecreasing. Thus we can define the entrance time and then the operator 𝒮 as follows.

Let 𝑒(𝑡,𝑥)=min{𝑠[0,𝑇]𝜑(𝑠,𝑡,𝑥)=0}, for (𝑡,𝑥)[0,𝑇]×[0,1],𝒮(𝑦,𝑁)=(̃𝑦,𝑁) with(1) if 𝑥𝜑(𝑡,0,0),̃𝑦(𝑡,𝑥)=exp𝑡0𝐸1d𝑟00exp𝑡0𝐸2𝑦d𝑟0+(𝜑(0,𝑡,𝑥))𝑡0exp𝑡𝑠𝐸1d𝑟00exp𝑡𝑠𝐸2d𝑟𝜌𝜌𝑥𝜌𝑥𝜕𝑥(𝜃+𝔄)d𝑠,(4.8)(2)if 𝑥𝜑(𝑡,0,0),̃𝑦(𝑡,𝑥)=exp𝑡0𝐸1d𝑟00exp𝑡0𝐸2𝑒d𝑟𝑀𝑒(𝑡,𝑥)𝑦0+(0)𝑡𝑒(𝑡,𝑥)exp𝑡𝑠𝐸1d𝑟00exp𝑡𝑠𝐸2d𝑟𝜌𝜌𝑥𝜌𝑥𝜕𝑥(𝜃+𝔄)d𝑠,(4.9)(3)𝑁(𝑡)=𝑦(𝑡,)𝐶0([0,1])×𝐶0([0,1]).

From Lemma 1.3 we know that ̃𝑦 is the weak solution of𝜕𝑡̃𝑦+(𝜃+𝔄)𝜕𝑥̃y=𝑏(𝑡,𝑥)̃𝑦+𝑓(𝑡,𝑥),̃𝑦(0,)=0,̃𝑦(𝑡,0)=𝑒𝑀𝑡𝑦0(0).(4.10)

Lemma 4.3. (1) The operator 𝒮 maps 𝑋 to 𝑋.
(2) The family 𝒮(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 to 𝑆 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, ̆̆𝔄𝜃+(𝑡,𝑥)𝑐𝑦(𝑟,)𝐶0([0,1])×𝐶0([0,1]),𝜕𝑥̆̆𝔄𝜃+(𝑡,𝑥)𝑐𝑦(𝑟,)𝐶0([0,1])×𝐶0([0,1]),(4.11)𝑦 is the solution of the transport equation (1.8) and it satisfies𝑦(𝑡,𝑥)=exp𝑡𝑠𝐸1d𝑟00exp𝑡𝑠𝐸2+d𝑟𝑦(𝑠,𝜑(𝑠,𝑡,𝑥))𝑡𝑠exp𝑡𝑟𝐸1d𝑟00exp𝑡𝑟𝐸2d𝑟𝑓(𝑟,𝑥)d𝑟.(4.12) We get for 𝑡𝑠||||𝑦(𝑡,𝑥)exp𝑡𝑠𝑐𝑦(𝑟,)𝐶0([0,1])×𝐶0([0,1])𝑦||||d𝑟𝑠,𝜑(𝑠,𝑡,𝑥).(4.13)

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(𝑡𝑠); 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(𝑦𝑡,)𝐶0([0,1])×𝐶0([0,1])max[]𝑟𝑠,𝑡𝑒𝑀2(𝑠𝑟)exp𝑡𝑠𝑐𝑦(𝑟,)𝐶0([0,1])×𝐶0([0,1])d𝑟×𝑦(𝑡,)𝐶0([0,1])×𝐶0([0,1]).(4.14)

Now we define𝑔(𝑟)=𝑒𝑀2(𝑠𝑟)exp𝑡𝑠𝑐𝑦(𝑟,)𝐶0([0,1])×𝐶0([0,1]).d𝑟(4.15)

Then 𝑔(𝑟)=𝑐𝑦(𝑡,)𝐶0([0,1])×𝐶0([0,1])𝑀2𝑔(𝑟),(4.16) 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 𝑦0𝐶0([0,1])×𝐶0([0,1])>𝑀2/𝑐, for 𝑡 small enough 𝑦(𝑡,)𝐶0([0,1])×𝐶0([0,1])𝑀2/𝑐, and we have(𝑦𝑡,)𝐶0([0,1])×𝐶0([0,1])𝑒𝑀2t𝑦0𝐶0([0,1])×𝐶0([0,1]),(4.17) which implies 𝑦(𝜏,)𝐶0([0,1])×𝐶0([0,1])𝑀2/𝑐. This provides for 𝜏𝑠𝑡, the inequality𝑦(𝑡,)𝐶0([0,1])×𝐶0([0,1])𝑒𝑡𝑠𝑐𝑦(𝑟,)𝐶00([0,1])×𝐶([0,1])d𝑟𝑦(𝑠,)𝐶0([0,1])×𝐶0