Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 635851 | https://doi.org/10.1155/2011/635851

Xiju Zong, Xingong Cheng, Zhonghua Wang, Zhenlai Han, "Initial Boundary Value Problem and Asymptotic Stabilization of the Two-Component Camassa-Holm Equation", Abstract and Applied Analysis, vol. 2011, Article ID 635851, 20 pages, 2011. https://doi.org/10.1155/2011/635851

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

Academic Editor: Dirk Aeyels
Received09 May 2011
Revised23 Jun 2011
Accepted20 Jul 2011
Published29 Sep 2011

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:๐‘ข๐‘กโˆ’๐‘ข๐‘ฅ๐‘ฅ๐‘ก+3๐‘ข๐‘ข๐‘ฅโˆ’2๐‘ข๐‘ฅ๐‘ข๐‘ฅ๐‘ฅโˆ’๐‘ข๐‘ข๐‘ฅ๐‘ฅ๐‘ฅ+๐œŒ๐œŒ๐‘ฅ=0,๐œŒ๐‘ก+๐œŒ๐‘ฅ๐‘ข+๐œŒ๐‘ข๐‘ฅ=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 ๐œŒโ‰ก0, the equation (1.1) becomes the Camassa-Holm equation, which is modeling the unidirectional propagation of shallow water waves over a flat bottom. Here ๐‘ข(๐‘ก,๐‘ฅ) stands for the fluid velocity at time ๐‘ก in the spatial ๐‘ฅ direction [7โ€“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 [20โ€“26] and references within. It has been shown that this equation is locally well posed [20โ€“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]).

For ๐œŒโ‰ 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 authorsโ€™ 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 (๐‘ข0,๐œŒ0) and (๐‘ข1,๐œŒ1) and a time ๐‘‡>0, can one find a certain function ๐‘ฃ(๐‘ก) such that the solution to (1.1) satisfies ๐‘ข(๐‘‡)=๐‘ข1, ๐œŒ(๐‘‡)=๐œŒ1? and(2)the stabilizability problem: can one find a stationary feedback law ๐‘ฃ(๐‘ฅ), such that for any state (๐‘ข0,๐œŒ0) a solution pair (๐‘ข(๐‘ก),๐œŒ(๐‘ก)) to closed-loop system is global?

To explain our boundary formulation of (1.1), let us first introduce some transformation, precisely, ๐‘š=๐‘ขโˆ’๐‘ข๐‘ฅ๐‘ฅ and ๐œŒ=๐œŒโˆ’1, which lead the system (1.1) to be equivalent to the system:๐‘š๐‘ก+๐‘ข๐‘š๐‘ฅ+2๐‘š๐‘ข๐‘ฅ+๐œŒ๐œŒ๐‘ฅ+๐œŒ๐‘ฅ๐œŒ=0,๐‘ก+๐‘ข๐œŒ๐‘ฅ+๐œŒ๐‘ข๐‘ฅ+๐‘ข๐‘ฅ=0.(1.2) Let ๐‘‡ be a positive number. In the following we take ฮฉ๐‘‡=[0,๐‘‡]ร—[0,1]. Let ๐‘ฃ๐‘™ and ๐‘ฃ๐‘Ÿ be in ๐ถ0([0,๐‘‡],๐‘…) and ๐‘š0โˆˆ๐ฟโˆž(0,1), ๐œŒ0โˆˆ๐‘Š1,โˆž(0,1). We set ฮ“๐‘™=๎€ฝ[]๐‘กโˆˆ0,๐‘‡โˆฃ๐‘ฃ๐‘™๎€พ(๐‘ก)>0,ฮ“๐‘Ÿ=๎€ฝ[]๐‘กโˆˆ0,๐‘‡โˆฃ๐‘ฃ๐‘Ÿ๎€พ(๐‘ก)<0.(1.3) In the following, we will always suppose that the sets ๐‘ƒ๐‘™=๎€ฝ[]๐‘กโˆˆ0,๐‘‡โˆฃ๐‘ฃ๐‘™๎€พ(๐‘ก)=0,๐‘ƒ๐‘Ÿ=๎€ฝ[]๐‘กโˆˆ0,๐‘‡โˆฃ๐‘ฃ๐‘Ÿ๎€พ(๐‘ก)=0(1.4) have a finite number of connected components. Finally, let ๐‘š๐‘™,๐œŒ๐‘™โˆˆ๐ฟโˆž(ฮ“๐‘™)ร—๐‘Š1,โˆž(ฮ“๐‘™) and ๐‘š๐‘Ÿ,๐œŒ๐‘Ÿโˆˆ๐ฟโˆž(ฮ“๐‘Ÿ)ร—๐‘Š1,โˆž(ฮ“๐‘Ÿ). The given functions ๐‘ฃ๐‘™, ๐‘ฃ๐‘Ÿ, ๐‘š๐‘™, ๐œŒ๐‘™, and ๐‘š๐‘Ÿ, ๐œŒ๐‘Ÿ will be the boundary values for the equation; ๐‘š0, ๐œŒ0 are the initial data. Let now ๐”„(๐‘ก,๐‘ฅ) be the auxiliary function which lifts the boundary values ๐‘ฃ๐‘™ and ๐‘ฃ๐‘Ÿ and is defined by๎€ท1โˆ’๐œ•๐‘ฅ๐‘ฅ๎€ธ๐”„(๐‘ก,๐‘ฅ)=0,โˆ€(๐‘ก,๐‘ฅ)โˆˆฮฉ๐‘‡,๐”„(๐‘ก,0)=๐‘ฃ๐‘™(๐‘ก),๐”„(๐‘ก,1)=๐‘ฃ๐‘Ÿ[].(๐‘ก),โˆ€๐‘กโˆˆ0,๐‘‡(1.5) Setting ๐‘ข=๐œƒ+๐”„, we can further rewrite the system (1.1) as๐‘š๎€ท(๐‘ก,๐‘ฅ)=1โˆ’๐œ•๐‘ฅ๐‘ฅ๎€ธ๐œƒ๐‘š(๐‘ก,๐‘ฅ),๐œƒ(๐‘ก,0)=๐œƒ(๐‘ก,1)=0,(1.6)๐‘ก+(๐œƒ+๐”„)๐‘š๐‘ฅ=โˆ’2๐‘š๐œ•๐‘ฅ(๐œƒ+๐”„)โˆ’๐œŒ๐œŒ๐‘ฅโˆ’๐œŒ๐‘ฅ,๐œŒ๐‘ก+(๐œƒ+๐”„)๐œŒ๐‘ฅ=โˆ’(๐œŒ+1)๐œ•๐‘ฅ(๐œƒ+๐”„),๐‘š(0,โ‹…)=๐‘š0,๐‘š(โ‹…,0)|ฮ“๐‘™=๐‘š๐‘™,๐‘š(โ‹…,1)|ฮ“๐‘Ÿ=๐‘š๐‘Ÿ,๐œŒ(0,โ‹…)=๐œŒ0,๐œŒ(โ‹…,0)|ฮ“๐‘™=๐œŒ๐‘™,๐œŒ(โ‹…,1)|ฮ“๐‘Ÿ=๐œŒ๐‘Ÿ.(1.7) Let ๎€ท๐‘ฆ=๐‘š๐œŒ๎€ธ, ๐‘ฆ0=๎€ท๐‘š0๐œŒ0๎€ธ, ๎‚€๐‘(๐‘ก,๐‘ฅ)=โˆ’2๐œ•๐‘ฅ(๐œƒ+๐”„)00โˆ’๐œ•๐‘ฅ(๐œƒ+๐”„)๎‚, ๎€ท๐‘“(๐‘ก,๐‘ฅ)=โˆ’๐œŒ๐œŒ๐‘ฅโˆ’๐œŒ๐‘ฅโˆ’๐œ•๐‘ฅ(๐œƒ+๐”„)๎€ธ, and the system (1.7) can be written as๐œ•๐‘ก๐‘ฆ+(๐œƒ+๐”„)๐œ•๐‘ฅ๐‘ฆ๐‘ฆ=๐‘(๐‘ก,๐‘ฅ)๐‘ฆ+๐‘“(๐‘ก,๐‘ฅ),(0,โ‹…)=๐‘ฆ0,๐‘ฆ(โ‹…,0)|ฮ“๐‘™=๐‘ฆ๐‘™,๐‘ฆ(โ‹…,1)|ฮ“๐‘Ÿ=๐‘ฆ๐‘Ÿ.(1.8) We first define what we mean by a weak solution to (1.8). Our test functions will be in the space:๎€ทฮฉadm๐‘‡๎€ธ=๎‚ป๐œ“โˆˆ๐ถ1๎€ทฮฉ๐‘‡๎€ธร—๐ถ1๎€ทฮฉ๐‘‡๎€ธ[][]โˆฃโˆ€๐‘ฅโˆˆ0,1,๐œ“(๐‘ก,๐‘ฅ)=0;โˆ€๐‘กโˆˆ0,๐‘‡/ฮ“๐‘™,๐œ“[](๐‘ก,0)=0;โˆ€๐‘กโˆˆ0,๐‘‡/ฮ“๐‘Ÿ๎€พ.,๐œ“(๐‘ก,1)=0(1.9)

Definition 1.1. Given ๐‘ฆ0=๎€ท๐‘š0๐œŒ0๎€ธโˆˆ๐ฟโˆž(ฮฉ๐‘‡)ร—๐‘Š1,โˆž(ฮฉ๐‘‡), when ๐œƒโˆˆ๐ฟโˆž((0,๐‘‡);Lip([0,1])), a function pair ๎€ท๐‘ฆ=๐‘š๐œŒ๎€ธโˆˆ๐ฟโˆž(ฮฉ๐‘‡)ร—๐‘Š1,โˆž(ฮฉT) is a weak solution to (1.8) if ๐‘ฆ satisfies ๎€ฮฉ๐‘‡๎€บ๐‘ฆ๐œ•๐‘ก๐œ“+๐‘ฆ(๐œƒ+๐”„)๐œ•๐‘ฅ๎€ป๎€œ๐œ“โˆ’๐‘ฆ๐‘(๐‘ก,๐‘ฅ)๐œ“โˆ’๐‘“(๐‘ก,๐‘ฅ)๐œ“d๐‘กd๐‘ฅ=โˆ’10๐‘ฆ0๎€œ๐œ“(0,๐‘ฅ)d๐‘ฅ+๐‘‡0๎€บ๐œ“(๐‘ก,1)๐‘ฃ๐‘Ÿ(๐‘ก)๐‘ฆ๐‘Ÿ(๐‘ก)โˆ’๐œ“(๐‘ก,0)๐‘ฃ๐‘™(๐‘ก)๐‘ฆ๐‘™๎€ป(๐‘ก)d๐‘ก.(1.10)

It is obvious that ๐ถ10(ฮฉ๐‘‡)ร—๐ถ10(ฮฉ๐‘‡)โŠ‚adm(ฮฉ๐‘‡); therefore, a weak solution to (1.8) is also a solution to (1.8) in the distribution sense.

Definition 1.2 (see [33]). For (๐‘ก,๐‘ฅ)โˆˆฮฉ๐‘‡, let ๐œ‘(โ‹…,๐‘ก,๐‘ฅ) be the ๐ถ1 maximal solution to ๐œ•๐‘ ๐œ‘๐œ‘(๐‘ ,๐‘ก,๐‘ฅ)=๐‘Ž(๐‘ ,๐œ‘(๐‘ ,๐‘ก,๐‘ฅ)),(๐‘ก,๐‘ก,๐‘ฅ)=๐‘ฅ,(1.11) which is defined on a certain set [๐‘’(๐‘ก,๐‘ฅ),โ„Ž(๐‘ก,๐‘ฅ)] (which is closed because [0,1] is compact) and with possibly ๐‘’(๐‘ก,๐‘ฅ) and/or โ„Ž(๐‘ก,๐‘ฅ)=๐‘ก.

We take into account the influence of the boundaries by introducing the sets: ๎€ฝ๐‘ƒ=(๐‘ก,๐‘ฅ)โˆˆฮฉ๐‘‡[๐‘’]๎€พ,[][]๎€ฝโˆฃโˆƒ๐‘ โˆˆ(๐‘ก,๐‘ฅ),โ„Ž(๐‘ก,๐‘ฅ)suchthat๐œ‘(๐‘ ,๐‘ก,๐‘ฅ)โˆˆ{0,1},๐‘Ž(๐‘ ,๐œ‘(๐‘ ,๐‘ก,๐‘ฅ))=0โˆช{(๐‘ ,๐œ‘(๐‘ ,0,0))โˆฃโˆ€๐‘ โˆˆ0,๐‘‡}โˆช{(๐‘ ,๐œ‘(๐‘ ,0,1))โˆฃโˆ€๐‘ โˆˆ0,๐‘‡},๐ผ=(๐‘ก,๐‘ฅ)โˆˆฮฉ๐‘‡๎€พ,๎€ฝ/๐‘ƒโˆฃ๐‘’(๐‘ก,๐‘ฅ)=0๐ฟ=(๐‘ก,๐‘ฅ)โˆˆฮฉ๐‘‡๎€พ,๎€ฝ(/๐‘ƒโˆฃ๐œ‘(๐‘’(๐‘ก,๐‘ฅ)๐‘ก,๐‘ฅ)=0๐‘…=๐‘ก,๐‘ฅ)โˆˆฮฉ๐‘‡๎€พ,ฮ“/๐‘ƒโˆฃ๐œ‘(๐‘’(๐‘ก,๐‘ฅ)๐‘ก,๐‘ฅ)=1๐‘™[]ฮ“={๐‘กโˆˆ0,๐‘‡โˆฃ๐‘Ž(๐‘ก,0)>0},๐‘Ÿ=[]{๐‘กโˆˆ0,๐‘‡โˆฃ๐‘Ž(๐‘ก,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 ๐‘Žโˆˆ๐ถ0([0,๐‘‡];๐ถ1([0,1])), ๐‘,๐‘“โˆˆ๐ฟโˆž(ฮฉ๐‘‡), ๐‘ฆ0โˆˆ๐ฟโˆž(0,1), ๐‘ฆ๐‘™โˆˆ๐ฟโˆž(ฮ“๐‘™), and ๐‘ฆ๐‘Ÿโˆˆ๐ฟโˆž(ฮ“๐‘Ÿ). We will also suppose that the sets: ๐‘ƒ๐‘™=๎€ฝ[]๐‘กโˆˆ0,๐‘‡โˆฃ๐‘ฃ๐‘™๎€พ(๐‘ก)=0,๐‘ƒ๐‘Ÿ=๎€ฝ[]๐‘กโˆˆ0,๐‘‡โˆฃ๐‘ฃ๐‘Ÿ๎€พ(๐‘ก)=0(1.13) have at most a countable number of connected components. Then the function ๐‘ฆ, defined by the formula ๎‚ต๎€œfor(๐‘ก,๐‘ฅ)โˆˆ๐‘ƒ,๐‘ฆ(๐‘ก,๐‘ฅ)=0,for(๐‘ก,๐‘ฅ)โˆˆ๐ผ,๐‘ฆ(๐‘ก,๐‘ฅ)=exp๐‘ก0๎‚ถ๐‘ฆ๐‘(๐‘Ÿ,๐œ‘(๐‘Ÿ,๐‘ก,๐‘ฅ))d๐‘Ÿ0(+๎€œ๐œ‘(0,๐‘ก,๐‘ฅ))๐‘ก0๎‚ต๎€œexp๐‘ก๐‘ ๎‚ถ๎‚ต๎€œ๐‘(๐‘Ÿ,๐œ‘(๐‘Ÿ,๐‘ก,๐‘ฅ))d๐‘Ÿ๐‘“(๐‘ ,๐œ‘(๐‘ ,๐‘ก,๐‘ฅ))d๐‘ ,for(๐‘ก,๐‘ฅ)โˆˆ๐ฟ,๐‘ฆ(๐‘ก,๐‘ฅ)=exp๐‘ก๐‘’(๐‘ก,๐‘ฅ)๎‚ถ๐‘ฆ๐‘(๐‘Ÿ,๐œ‘(๐‘Ÿ,๐‘ก,๐‘ฅ))d๐‘Ÿ๐‘™(+๎€œ๐‘’(๐‘ก,๐‘ฅ))๐‘ก๐‘’(๐‘ก,๐‘ฅ)๎‚ต๎€œexp๐‘ก๐‘ ๎‚ถ๎‚ต๎€œ๐‘(๐‘Ÿ,๐œ‘(๐‘Ÿ,๐‘ก,๐‘ฅ))d๐‘Ÿ๐‘“(๐‘ ,๐œ‘(๐‘ ,๐‘ก,๐‘ฅ))d๐‘ ,for(๐‘ก,๐‘ฅ)โˆˆ๐‘…,๐‘ฆ(๐‘ก,๐‘ฅ)=exp๐‘ก๐‘’(๐‘ก,๐‘ฅ)๎‚ถ๐‘ฆ๐‘(๐‘Ÿ,๐œ‘(๐‘Ÿ,๐‘ก,๐‘ฅ))d๐‘Ÿ๐‘Ÿ+๎€œ(๐‘’(๐‘ก,๐‘ฅ))๐‘ก๐‘’(๐‘ก,๐‘ฅ)๎‚ต๎€œexp๐‘ก๐‘ ๎‚ถ๐‘(๐‘Ÿ,๐œ‘(๐‘Ÿ,๐‘ก,๐‘ฅ))d๐‘Ÿ๐‘“(๐‘ ,๐œ‘(๐‘ ,๐‘ก,๐‘ฅ))d๐‘ ,(1.14) is a weak solution of ๐œ•๐‘ก๐‘ฆ+(๐œƒ+๐”„)๐œ•๐‘ฅ๐‘ฆ=๐‘(๐‘ก,๐‘ฅ)๐‘ฆ+๐‘“(๐‘ก,๐‘ฅ)(1.15) and satisfies โ€–๐‘ฆโ€–๐ฟโˆž(ฮฉ๐‘‡)โ‰ค๎‚€๎‚†โ€–โ€–๐‘ฆmax0โ€–โ€–๐ฟโˆž(ฮฉ๐‘‡),โ€–โ€–๐‘ฆ๐‘™โ€–โ€–๐ฟโˆž(ฮ“๐‘™),โ€–โ€–๐‘ฆ๐‘Ÿโ€–โ€–๐ฟโˆž(ฮ“๐‘Ÿ)๎‚‡+๐‘‡โ€–๐‘“โ€–๐ฟโˆž(ฮฉ๐‘‡)๎‚๐‘’๐‘กโ€–๐‘โ€–๐ฟโˆž๐‘‡)(ฮฉ.(1.16)

However, if we let ๐‘š๐‘™,๐œŒ๐‘™โˆˆ๐ฟโˆž(ฮ“๐‘™)ร—๐ฟโˆž(ฮ“๐‘™) and ๐‘š๐‘Ÿ,๐œŒ๐‘Ÿโˆˆ๐ฟโˆž(ฮ“๐‘Ÿ)ร—๐ฟโˆž(ฮ“๐‘Ÿ), note that ๐‘“ depends on the unknown ๐œŒ 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 ๐œƒ and ๐”„ are given, one can solve the equation on ๐œŒ (the equation on the second component in (1.8) (or (1.1))), but this result only guarantee that ๐œŒ is in ๐ฟโˆž. Therefore the source term in the equation on ๐‘š is not in ๐ฟโˆž anymore but in ๐ฟโˆž((0,๐‘‡);๐‘Šโˆ’1,โˆž(0,1)), and then Lemma 1.3 cannot be used to solve the transport equation on ๐‘š (the first component equation on (1.8)). One might try to get more regularity on ๐œŒ, but in this case more regularity is also needed on ๐œŒ0,๐œŒ๐‘™, ๐œŒ๐‘Ÿ0 and even on ๐œƒ and ๐”„ to get sufficient geometrical assumptions. Then, one might manage ๐œŒ0โˆˆ๐‘Š1,โˆž(0,1), ๐œŒ๐‘™โˆˆ๐‘Š1,โˆž(ฮ“๐‘™), ๐œŒ๐‘Ÿโˆˆ๐‘Š1,โˆž(ฮ“๐‘Ÿ) to obtain at least Lipschitz solution of the scalar transport equation on ๐œŒ and then get a weak solution on ๐‘š.

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 ๐‘‡>0, we consider ๐‘ฃ๐‘™โˆˆ๐ถ0(ฮ“๐‘™), ๐‘ฃ๐‘Ÿโˆˆ๐ถ0(ฮ“๐‘™) such that the sets ๐‘ƒ๐‘™ and ๐‘ƒ๐‘Ÿ have only a finite number of connected components. Let ๐‘ฆ0โˆˆ๐ฟโˆž(0,1)ร—๐‘Š1,โˆž(0,1), ๐‘ฆ๐‘™โˆˆ๐ฟโˆž(ฮ“๐‘™)ร—๐‘Š1,โˆž(ฮ“๐‘™), and ๐‘ฆ๐‘Ÿโˆˆ๐ฟโˆž(ฮ“๐‘Ÿ)ร—๐‘Š1,โˆž(ฮ“๐‘Ÿ). There exist ๐‘‡>0 and (๐œƒ,๐‘ฆ) a weak solution of the system (1.8) (or (1.1)) with ๐œƒโˆˆ๐ฟโˆž((0,๐‘‡);๐ถ1,1([0,1])โˆฉLip((0,๐‘‡);๐ป10(0,1). Moreover any such solution ๐œƒ is in fact in ๐ถ0([0,๐‘‡];๐‘Š2,๐‘(0,1))โˆฉ๐ถ1([0,๐‘‡];๐‘Š01,๐‘(0,1)), for all ๐‘<+โˆž. Furthermore the existence time of a maximal solution is larger than ๎‚min(๐‘‡,๐‘‡โˆ—), with ๐‘‡โˆ—=max๐›ฝ>0๎ƒฉln(1+๐›ฝ/๐ถ)2๎€บ๐ถ1๎€ป๎ƒช,๎€ฝโ€–โ€–๐‘ฆ+(2+sinh(1))(๐›ฝ+๐ถ)๐ถ=max0โ€–โ€–๐ฟโˆžร—๐‘Š1,โˆž,โ€–โ€–๐‘ฆ๐‘™โ€–โ€–๐ฟโˆžร—๐‘Š1,โˆž,โ€–โ€–๐‘ฆ๐‘Ÿโ€–โ€–๐ฟโˆžร—๐‘Š1,โˆž๎€พ,๐ถ1=1๎€ทโ€–โ€–๐‘ฃtanh(1)๐‘™โ€–โ€–๐ฟโˆž(0,๐‘‡),โ€–โ€–๐‘ฃ๐‘Ÿโ€–โ€–๐ฟโˆž(0,๐‘‡)๎€ธ.(2.1)

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

Theorem 2.2. Let ๐œƒโˆˆ๐ฟโˆž((0,๐‘‡);๐ถ1,1([0,1])โˆฉLip((0,๐‘‡);๐ป10(0,1), and let ๐‘ฆโˆˆLip([0,1])ร—Lip([0,1]) be a weak solution of (1.8) (or (1.1)); then it is unique in the function space [๐ฟโˆž((0,๐‘‡);๐ถ1,1([0,1])ร—๐ฟโˆž(ฮฉ๐‘‡))]2.

Let ๐ด๐‘™>2sinh(1), ๐ด๐‘Ÿ>๐ด๐‘™cosh(1)+sinh(2), ๐‘‡>0 and ๐‘€ a symmetric matrix, and assume that ๐œŒ0,๐œŒ๐‘™, and ๐œŒ๐‘Ÿ have compact supports in (0,1)/ฮ“๐‘™/ฮ“๐‘Ÿ, respectively. Our feedback law for (1.8) (or (1.1)) reads๐‘ฆโˆˆ๐ถ0([]0,1)ร—๐ถ0([]โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๐‘ฃ0,1)โŸผ๐‘™(๐‘ฆ)=๐ด๐‘™โ€–๐‘ฆโ€–๐ถ0([0,1])ร—๐ถ0([0,1]),๐‘ฃ๐‘Ÿ(๐‘ฆ)=๐ด๐‘Ÿโ€–๐‘ฆโ€–๐ถ0([0,1])ร—๐ถ0([0,1]),ฬ‡๐‘ฆ๐‘™(๐‘ก)=๐‘€๐‘ฆ๐‘™(๐‘ก).(2.2)

Theorem 2.3. For any ๐‘ฆ0โˆˆ[๐ถ0([0,1])]2 there exists (๐‘ฆ,๐‘ฃ)โˆˆ[๐ถ0(ฮฉ๐‘‡)]2ร—๐ถ2([0,1]) a weak solution of (1.1) and (2.2) satisfying []โˆ€๐‘ฅโˆˆ0,1,๐‘ฆ(0,๐‘ฅ)=๐‘ฆ0(๐‘ฅ).(2.3) Furthermore any maximal solution of (1.1), (2.2), and (2.3) is global, and if we let ๎‚ป๐ด๐‘=min๐‘™๐ดโˆ’2sinh(1),๐‘Ÿโˆ’๐ด๐‘™cosh(1)+sinh(2)๎‚ผ1sinh(1),๐œ=โ€–๐‘€โ€–2๎‚ตln๐‘โ€–๐‘ฆโ€–๐ถ0([0,1])โ€–๐‘€โ€–2๎‚ถ,(2.4) then we have โ€–๐‘ฆโ€–๐ถ0([0,1])โ‰คโ€–๐‘€โ€–2๐‘๎€บ1+โ€–๐‘€โ€–2๎€ป.(๐‘กโˆ’๐œ)(2.5)

Remark 2.4. For ๐œŒโ‰ก0, the system (1.1) becomes the classical Camassa-Holm equation, and the above theorems degenerate those of โ€‰[33] with ๐‘˜=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 ๐œƒ fixed (1.8) is a linear transport equation in ๐‘ฆ, with boundary data.

Given ๐œƒโˆˆ๐ฟโˆž((0,๐‘‡);๐ถ1,1([0,1]))โˆฉLip((0,๐‘‡);๐ป10(0,1)), we will define ๎€ท๐‘ฆ=๐‘š๐œŒ๎€ธ to be the solution to (1.8), and once we have ๐‘š in ๐ฟโˆž(ฮฉ๐‘‡), we introduce ฬƒ๐œƒ solution of๎€ท1โˆ’๐œ•๐‘ฅ๐‘ฅ๎€ธฬƒ๐œƒ=๐‘š.(3.1) Then โ„ฑ is defined as the operator ฬƒ๐œƒ=โ„ฑ(๐œƒ)

Lemma 3.1. The function ๐”„ defined by (1.5) satisfies โˆ€(๐‘ก,๐‘ฅ)โˆˆฮฉ๐‘‡,๐”„(๐‘ก,๐‘ฅ)โˆˆ๐ถ0([]0,๐‘‡;๐ถโˆž([]10,1)),๐”„(๐‘ก,๐‘ฅ)=๎€ทsinh(1)sinh(๐‘ฅ)๐‘ฃ๐‘Ÿ(๐‘ก)+sinh(1โˆ’๐‘ฅ)๐‘ฃ๐‘™๎€ธ,โ€–(๐‘ก)โ€–๐”„(๐‘ก,๐‘ฅ)๐ฟโˆž((0,๐‘‡);๐ถ1,1([0,1]))โ‰คcosh(1)๎€ทโ€–โ€–๐‘ฃsinh(1)๐‘™โ€–โ€–๐ฟโˆž(0,๐‘‡)+โ€–โ€–๐‘ฃ๐‘Ÿโ€–โ€–๐ฟโˆž(0,๐‘‡)๎€ธ.(3.2)

Then for a function ๐œƒโˆˆ๐ฟโˆž((0,๐‘‡);๐ถ1,1([0,1]))โˆฉLip((0,๐‘‡);๐ป10(0,1)), we consider ๐œ‘ the flow of ๐œƒ+๐”„.

Lemma 3.2. The flow ๐œ‘ satisfies the following properties. (1)๐œ‘ is ๐ถ1 with the following partial derivatives:๐œ•1๐œ•๐œ‘(๐‘ ,๐‘ก,๐‘ฅ)=(๐œƒ+๐”„)(๐‘ ,๐œ‘(๐‘ ,๐‘ก,๐‘ฅ)),2๎‚ต๎€œ๐œ‘(๐‘ ,๐‘ก,๐‘ฅ)=โˆ’(๐œƒ+๐”„)exp๐‘ ๐‘ก๐œ•๐‘ฅ๎‚ถ,๐œ•(๐œƒ+๐”„)(๐‘Ÿ,๐œ‘(๐‘Ÿ,๐‘ก,๐‘ฅ))d๐‘Ÿ3๎‚ต๎€œ๐œ‘(๐‘ ,๐‘ก,๐‘ฅ)=exp๐‘ ๐‘ก๐œ•๐‘ฅ๎‚ถ.(๐œƒ+๐”„)(๐‘Ÿ,๐œ‘(๐‘Ÿ,๐‘ก,๐‘ฅ))d๐‘Ÿ(3.3)(2)For all ๐‘—=1,2,3,โ€–โ€–๐œ•๐‘—๐œ‘โ€–โ€–๐ถ0โ‰ค๎€ท1+โ€–๐œƒ+๐”„โ€–๐ถ0(ฮฉ๐‘‡)๎€ธ๐‘’๐‘‡โ€–๐œƒ+๐”„โ€–๐ถ0๐‘‡)(ฮฉ.(3.4) For (๐‘ก,๐‘ฅ)โˆˆฮฉ๐‘‡, ๐œ‘(โ‹…,๐‘ก,๐‘ฅ) is defined on a set [๐‘’(๐‘ก,๐‘ฅ),โ„Ž(๐‘ก,๐‘ฅ)], here ๐‘’(๐‘ก,๐‘ฅ) is basically the entrance time in ฮฉ๐‘‡ of the characteristic curve going through (๐‘ก,๐‘ฅ). (3) If ๐‘’(๐‘ก,๐‘ฅ)>0, then ๐œ‘(๐‘’(๐‘ก,๐‘ฅ),๐‘ก,๐‘ฅ)โˆˆ{0,1}.(4) If โ„Ž(๐‘ก,๐‘ฅ)<๐‘‡, then ๐œ‘(โ„Ž(๐‘ก,๐‘ฅ),๐‘ก,๐‘ฅ)โˆˆ{0,1}.

For ๐œƒโˆˆ๐ฟโˆž((0,๐‘‡);๐ถ1,1([0,1]))โˆฉLip((0,๐‘‡);๐ป10(0,1)), we define the solution to (1.8) ๐‘ฆโˆˆ๐ฟโˆž(ฮฉ๐‘‡)ร—๐‘Š1,โˆž(ฮฉ๐‘‡) byโŽ›โŽœโŽœโŽœโŽ๎€œfor(๐‘ก,๐‘ฅ)โˆˆ๐ผ,๐‘ฆ(๐‘ก,๐‘ฅ)=exp๐‘ก0๐ธ1๎€œd๐‘Ÿ00exp๐‘ก0๐ธ2โŽžโŽŸโŽŸโŽŸโŽ ๐‘ฆd๐‘Ÿ0+๎€œ(๐œ‘(0,๐‘ก,๐‘ฅ))๐‘ก0โŽ›โŽœโŽœโŽœโŽ๎€œexp๐‘ก๐‘ ๐ธ1๎€œd๐‘Ÿ00exp๐‘ก๐‘ ๐ธ2โŽžโŽŸโŽŸโŽŸโŽ ๎ƒฉd๐‘Ÿโˆ’๐œŒ๐œŒ๐‘ฅโˆ’๐œŒ๐‘ฅโˆ’๐œ•๐‘ฅ๎ƒชโŽ›โŽœโŽœโŽœโŽ๎€œ(๐œƒ+๐”„)d๐‘ ,for(๐‘ก,๐‘ฅ)โˆˆ๐ฟ,๐‘ฆ(๐‘ก,๐‘ฅ)=exp๐‘ก๐‘’(๐‘ก,๐‘ฅ)๐ธ1๎€œd๐‘Ÿ00exp๐‘ก๐‘’(๐‘ก,๐‘ฅ)๐ธ2โŽžโŽŸโŽŸโŽŸโŽ ๐‘ฆd๐‘Ÿ๐‘™+๎€œ(๐‘’(๐‘ก,๐‘ฅ))๐‘ก๐‘’(๐‘ก,๐‘ฅ)โŽ›โŽœโŽœโŽœโŽ๎€œexp๐‘ก๐‘ ๐ธ1๎€œd๐‘Ÿ00exp๐‘ก๐‘ ๐ธ2โŽžโŽŸโŽŸโŽŸโŽ ๎ƒฉd๐‘Ÿโˆ’๐œŒ๐œŒ๐‘ฅโˆ’๐œŒ๐‘ฅโˆ’๐œ•๐‘ฅ๎ƒชโŽ›โŽœโŽœโŽœโŽ๎€œ(๐œƒ+๐”„)d๐‘ ,for(๐‘ก,๐‘ฅ)โˆˆ๐‘…,๐‘ฆ(๐‘ก,๐‘ฅ)=exp๐‘ก๐‘’(๐‘ก,๐‘ฅ)๐ธ1๎€œd๐‘Ÿ00exp๐‘ก๐‘’(๐‘ก,๐‘ฅ)๐ธ2โŽžโŽŸโŽŸโŽŸโŽ ๐‘ฆd๐‘Ÿ๐‘Ÿ+๎€œ(๐‘’(๐‘ก,๐‘ฅ))๐‘ก๐‘’(๐‘ก,๐‘ฅ)โŽ›โŽœโŽœโŽœโŽ๎€œexp๐‘ก๐‘ ๐ธ1๎€œd๐‘Ÿ00exp๐‘ก๐‘ ๐ธ2โŽžโŽŸโŽŸโŽŸโŽ ๎ƒฉd๐‘Ÿโˆ’๐œŒ๐œŒ๐‘ฅโˆ’๐œŒ๐‘ฅโˆ’๐œ•๐‘ฅ๎ƒช(๐œƒ+๐”„)d๐‘ (3.5) with ๐ธ1=โˆ’2๐œ•๐‘ฅ(๐œƒ+๐”„)(๐‘Ÿ,๐œ‘(๐‘Ÿ,๐‘ก,๐‘ฅ)),๐ธ2=โˆ’๐œ•๐‘ฅ(๐œƒ+๐”„)(๐‘Ÿ,๐œ‘(๐‘Ÿ,๐‘ก,๐‘ฅ)).(3.6) And we have(1) the function ๐‘ฆ is the unique weak solution of (1.8) in the sense of Definition 4.1; thanks to Lemma 1.3. ๐‘ฆ is in ๐ฟโˆžโˆฉ๐‘Š1,โˆž.

Remark 3.3. ๐‘ฆ is the only weak solution of (1.8), and also ๐‘ฆ is in ๐ฟโˆžโˆฉ๐‘Š1,โˆž which is crucial for the stabilization problem because of the coupling between the two components of ๐‘ฆ. However, rather thanks to the regularity on the boundary data ๐œŒ is indeed Lipschitz inside the zones ๐ฟ, ๐‘…, and ๐ผ; it ensures that the transition between those zones should be continuous under the kind of compatibility conditions between ๐œŒ0, ๐œŒ๐‘™, and ๐œŒ๐‘Ÿ; for example, all three have a compact support in (0,1)/ฮ“๐‘™/ฮ“๐‘Ÿ.

(2)Since ๐‘ฆโˆˆ๐ฟโˆž(ฮฉ๐‘‡)ร—๐‘Š1,โˆž(ฮฉ๐‘‡), we immediately get ๐‘ฆโˆˆ๐‘Š1,โˆž(0,๐‘‡;๐ปโˆ’1(0,1))ร—๐‘Š2,โˆž(0,๐‘‡;๐ปโˆ’1(0,1)) and satisfies (1.8). Also we can get the estimates:โ€–๐‘ฆ(๐‘ก,๐‘ฅ)โ€–๐ฟโˆž(ฮฉ๐‘‡)ร—๐ฟโˆž(ฮฉ๐‘‡)โ‰ค๐ถ0๎‚ƒ๎‚€โ€–โ€–๐œ•exp2๐‘‡๐‘ฅ๐œƒโ€–โ€–๐ฟโˆž(ฮฉ๐‘‡)+โ€–โ€–๐œ•๐‘ฅ๐”„โ€–โ€–๐ฟโˆž(ฮฉ๐‘‡),โ€–โ€–๐œ•๎‚๎‚„๐‘กโ€–โ€–๐‘ฆ(๐‘ก,๐‘ฅ)๐ฟโˆž(0,๐‘‡;๐ปโˆ’1)ร—๐ฟโˆž(0,๐‘‡;๐ปโˆ’1)โ‰ค3๐ถ0๎‚ƒ๎‚€โ€–โ€–๐œ•exp2๐‘‡๐‘ฅ๐œƒโ€–โ€–๐ฟโˆž(ฮฉ๐‘‡)+โ€–โ€–๐œ•๐‘ฅ๐”„โ€–โ€–๐ฟโˆž(ฮฉ๐‘‡),ร—๎€ท๎‚๎‚„โ€–๐œƒโ€–๐ฟโˆž((0,๐‘‡);Lip([0,1]))+โ€–๐”„โ€–๐ฟโˆž((0,๐‘‡);Lip([0,1]))๎€ธ,(3.7)

where๐ถ0๎€ฝโ€–โ€–๐‘ฆ=max0โ€–โ€–๐ฟโˆžร—๐‘Š1,โˆž,โ€–โ€–๐‘ฆ๐‘™โ€–โ€–๐ฟโˆžร—๐‘Š1,โˆž,โ€–โ€–๐‘ฆ๐‘Ÿโ€–โ€–๐ฟโˆžร—๐‘Š1,โˆž๎€พ๎‚ƒ๎€ท+๐‘‡โ€–๐œŒโ€–๐ฟโˆž๎€ธโ€–โ€–๐œŒ+1๐‘ฅโ€–โ€–๐ฟโˆž((0,๐‘‡);Lip([0,1]))+โ€–โ€–๐œ•๐‘ฅ๐œƒโ€–โ€–๐ฟโˆž(ฮฉ๐‘‡)+โ€–โ€–๐œ•๐‘ฅ๐”„โ€–โ€–๐ฟโˆž(ฮฉ๐‘‡)๎‚„.(3.8)(3) If (๐‘ก,๐‘ฅ)โˆˆ๐ผโˆช๐ฟโˆช๐‘… and if (๐‘ ,๐‘ ๎…ž)โˆˆ[๐‘’(๐‘ก,๐‘ฅ),โ„Ž(๐‘ก,๐‘ฅ)]2, one has the following property:โŽ›โŽœโŽœโŽœโŽ๎€œ๐‘ฆ(๐‘ ,๐œ‘(๐‘ ,๐‘ก,๐‘ฅ))=exp๐‘ก0๐ธ1๎€œd๐‘Ÿ00exp๐‘ก0๐ธ2โŽžโŽŸโŽŸโŽŸโŽ ๐‘ฆ๎€ท๐‘ d๐‘Ÿ๎…ž๎€ท๐‘ ,๐œ‘๎…ž,๐‘ก,๐‘ฅ๎€ธ๎€ธ.(3.9)

From the elliptic equation we can get

Lemma 3.4. There exists a unique ฬƒ๐œƒโˆˆ๐ฟโˆž((0,๐‘‡);๐ป10(0,1)) such that ๎€ท1โˆ’๐œ•๐‘ฅ๐‘ฅ๎€ธฬƒ๐œƒ=๐‘š(3.10) holds in ๐ท๎…ž(0,1) for all (๐‘ก,๐‘ฅ)โˆˆฮฉ๐‘‡. Furthermore ฬƒ๐œƒโˆˆ๐ฟโˆž((0,๐‘‡);๐ถ1,1([0,1]))โˆฉLip([0,๐‘‡];๐ป10(0,1)), since ๐‘šโˆˆ๐ฟโˆž(ฮฉ๐‘‡)โˆฉLip([0,๐‘‡];๐ปโˆ’1(0,1)). Moreover we have the bounds: โ€–โ€–ฬƒ๐œƒโ€–โ€–๐ฟโˆž((0,๐‘‡);๐ถ1,1([0,1]))โ‰ค[]1+sinh(1)โ€–๐‘šโ€–๐ฟโˆž(ฮฉ๐‘‡)โ‰ค[]1+sinh(1)โ€–๐‘ฆโ€–๐ฟโˆž(ฮฉ๐‘‡)ร—๐‘Š1,โˆž(ฮฉ๐‘‡),โ€–โ€–๐œ•๐‘กฬƒ๐œƒโ€–โ€–๐ฟโˆž((0,๐‘‡);๐ป10(0,1))โ‰คโ€–โ€–๐œ•๐‘ก๐‘šโ€–โ€–๐ฟโˆž((0,๐‘‡);๐ปโˆ’1)โ‰คโ€–โ€–๐œ•๐‘ก๐‘ฆโ€–โ€–๐ฟโˆž((0,๐‘‡);๐ปโˆ’1)ร—๐‘Š1,โˆž((0,๐‘‡);๐ปโˆ’1).(3.11)

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

Thus, for ๐œƒโˆˆ๐ฟโˆž((0,๐‘‡);๐ถ1,1(0,1))โˆฉLip([0,๐‘‡];๐ป10(0,1)), the operator โ„ฑ can be defined as ฬƒ๐œƒ=โ„ฑ(๐œƒ)โˆˆ๐ฟโˆž((0,๐‘‡);๐ถ1,1(0,1))โˆฉLip([0,๐‘‡];๐ป10(0,1)).

Let ๐ต0 and ๐ต1 be positive numbers, then we set๐’ž๐ต0,๐ต1,๐‘‡=๎€ฝ๐œƒโˆˆ๐ฟโˆž๎€ท(0,๐‘‡);๐ถ1,1๎€ธ๎€ท[](0,1)โˆฉLip0,๐‘‡;๐ป10๎€ธ(0,1)โˆฃbothโ€–๐œƒโ€–๐ฟโˆž((0,๐‘‡);๐ถ1,1(0,1))โ‰ค๐ต0,โ€–๐œƒโ€–Lip([0,๐‘‡];๐ปโˆ’1(0,1))โ‰ค๐ต1๎€พ.(3.12) Obviously ๐’ž๐ต0,๐ต1,๐‘‡ is convex. We will endow ๐’ž๐ต0,๐ต1,๐‘‡ with the norm โ€–๐œƒโ€–๐ฟโˆž([0,๐‘‡];Lip(0,1)).

Lemma 3.5. There exist positive numbers ๐ต0, ๐ต1 and ๐‘‡, such that โ„ฑ maps ๐’ž๐ต0,๐ต1,๐‘‡ into itself.

Proof. The proceeding of proof is similar to that of [33, Lemma 3], but the constant ๐ถ0 differs slightly from that of [33, Lemma 3]. Let us first introduce the two following constants: ๐ถ0๎€ฝโ€–โ€–๐‘ฆ=max0โ€–โ€–๐ฟโˆžร—๐‘Š1,โˆž,โ€–โ€–๐‘ฆ๐‘™โ€–โ€–๐ฟโˆžร—๐‘Š1,โˆž,โ€–โ€–๐‘ฆ๐‘Ÿโ€–โ€–๐ฟโˆžร—๐‘Š1,โˆž๎€พ๎‚ƒ๎€ท+๐‘‡โ€–๐œŒโ€–๐ฟโˆž๎€ธโ€–โ€–๐œŒ+1๐‘ฅโ€–โ€–๐ฟโˆž((0,๐‘‡);Lip([0,1]))+โ€–โ€–๐œ•๐‘ฅ๐œƒโ€–โ€–๐ฟโˆž(ฮฉ๐‘‡)+โ€–โ€–๐œ•๐‘ฅ๐”„โ€–โ€–๐ฟโˆž(ฮฉ๐‘‡)๎‚„,๐ถ1=1๎€ทโ€–โ€–๐‘ฃtanh(1)๐‘™โ€–โ€–๐ฟโˆž(0,๐‘‡),โ€–โ€–๐‘ฃ๐‘Ÿโ€–โ€–๐ฟโˆž(0,๐‘‡)๎€ธ.(3.13)
Estimates (3.7), and (3.11) on ๐‘ฆ and ๐œƒ now read, โ€–๐‘ฆ(๐‘ก,๐‘ฅ)โ€–๐ฟโˆž(ฮฉ๐‘‡)ร—๐‘Š1,โˆž(ฮฉ๐‘‡)โ‰ค๐ถ0๎‚ƒ๎‚€โ€–โ€–๐œ•exp2๐‘‡๐‘ฅ๐œƒโ€–โ€–๐ฟโˆž(ฮฉ๐‘‡)+๐ถ1,โ€–โ€–๐œ•๎‚๎‚„๐‘กโ€–โ€–๐‘ฆ(๐‘ก,๐‘ฅ)๐ฟโˆž((0,๐‘‡);๐ปโˆ’1)ร—๐ฟโˆž((0,๐‘‡);๐ปโˆ’1)โ‰ค3๐ถ0๎‚ƒ๎‚€โ€–โ€–๐œ•exp2๐‘‡๐‘ฅ๐œƒโ€–โ€–๐ฟโˆž(ฮฉ๐‘‡)+๐ถ1ร—๎€ท๎‚๎‚„โ€–๐œƒโ€–๐ฟโˆž((0,๐‘‡);Lip([0,1]))+๐ถ1๎€ธ,โ€–โ€–ฬƒ๐œƒโ€–โ€–๐ฟโˆž((0,๐‘‡);๐ถ1,1([0,1]))โ‰ค[]1+sinh(1)โ€–๐‘ฆ(๐‘ก,๐‘ฅ)โ€–๐ฟโˆž(ฮฉ๐‘‡)ร—๐‘Š1,โˆž(ฮฉ๐‘‡),โ€–โ€–๐œ•๐‘กฬƒ๐œƒโ€–โ€–๐ฟโˆž((0,๐‘‡);๐ป10(0,1))โ‰คโ€–โ€–๐œ•๐‘กโ€–โ€–๐‘ฆ(๐‘ก,๐‘ฅ)๐ฟโˆž((0,๐‘‡);๐ปโˆ’1)ร—๐ฟโˆž((0,๐‘‡);๐ปโˆ’1).(3.14) Combining those estimates we get for all ๐œƒโˆˆ๐’ž๐ต0,๐ต1,๐‘‡: โ€–โ€–ฬƒ๐œƒโ€–โ€–๐ฟโˆž((0,๐‘‡);๐ถ1,1([0,1]))โ‰ค๐ถ0[]๎‚ƒ๎‚€โ€–โ€–๐œ•1+sinh(1)exp2๐‘‡๐‘ฅ๐œƒโ€–โ€–๐ฟโˆž(ฮฉ๐‘‡)+๐ถ1๎‚๎‚„โ‰ค๐ถ0[]๎€บ๎€ท๐ต1+sinh(1)exp2๐‘‡0+๐ถ1,โ€–โ€–๐œ•๎€ธ๎€ป๐‘กฬƒ๐œƒโ€–โ€–๐ฟโˆž((0,๐‘‡);๐ป10(0,1))โ‰ค3๐ถ0๎‚ƒ๎‚€โ€–โ€–๐œ•exp2๐‘‡๐‘ฅ๐œƒโ€–โ€–๐ฟโˆž(ฮฉ๐‘‡)+๐ถ1๎€ทโ€–๎‚๎‚„๐œƒโ€–๐ฟโˆž((0,๐‘‡);Lip([0,1]))+๐ถ1๎€ธโ‰ค3๐ถ0๎€บ๎€ท๐ตexp2๐‘‡0+๐ถ1๐ต๎€ธ๎€ป๎€ท0+๐ถ1๎€ธ.(3.15) To obtain ฬƒ๐œƒโˆˆ๐’ž๐ต0,๐ต1,๐‘‡, it is sufficient that ๐ถ0[]๎€บ๎€ท๐ต1+sinh(1)exp2๐‘‡0+๐ถ1๎€ธ๎€ปโ‰ค๐ต0,๐ต0+3๐ถ0๎€บ๎€ท๐ตexp2๐‘‡0+๐ถ1๐ต๎€ธ๎€ป๎€ท0+๐ถ1๎€ธโ‰ค๐ต1.(3.16)
Once we have chosen ๐‘‡ and ๐ต0, it is easy to choose ๐ต1 to satisfy the second inequality. For the first one we just choose ๐ต0 sufficiently large and then ๐‘‡ close to 0. More precisely, ๎€ฝโ€–โ€–๐‘ฆ๐ถ=max0โ€–โ€–๐ฟโˆžร—๐‘Š1,โˆž,โ€–โ€–๐‘ฆ๐‘™โ€–โ€–๐ฟโˆžร—๐‘Š1,โˆž,โ€–โ€–๐‘ฆ๐‘Ÿโ€–โ€–๐ฟโˆžร—๐‘Š1,โˆž๎€พ,๐ต0[]>๐ถ1+sinh(1),๐‘‡โ‰คln๐ต0[])/(๐ถ1+sinh(1)2๎€ท๐ต0+๐ถ1๎€ธ,(3.17) we complete the proof by taking ๐ต0/(๐ถ[1+sinh(1)])=๐ถ+๐›ฝ,๐›ฝ>0.

Lemma 3.6. ๐’ž๐ต0,๐ต1,๐‘‡ is compact with respect to the norm โ€–โ‹…โ€–๐ฟโˆž([0,๐‘‡];Lip(0,1)).

The proof is very similar to that appeared in [33] and omitted.

Lemma 3.7. For ๐‘ฆโˆˆ๐ป๐‘ ร—๐ป๐‘ โˆ’1, ๐‘ >2, ๐‘“(๐‘ฆ) is bounded on bounded sets in ๐ป๐‘ ร—๐ป๐‘ โˆ’1. Therefore, ๐‘“(๐‘ฆ) is bounded on bounded sets in ๐ฟโˆžร—๐ฟโˆž by the embedding theorem.

The proof is very similar to that appeared in [33] and omitted.

Lemma 3.8. The operator โ„ฑโˆถ๐’ž๐ต0,๐ต1,๐‘‡โ†’๐’ž๐ต0,๐ต1,๐‘‡ is continuous with respect to the norm โ€–๐œƒโ€–๐ฟโˆž([0,๐‘‡];Lip(0,1)).

Proof. Take a sequence {๐œƒ๐‘›} which tends to ๐œƒ with respect to โ€–โ‹…โ€–๐ฟโˆž([0,๐‘‡];Lip(0,1)), set ฬƒ๐œƒ๐‘›=โ„ฑ๐œƒ๐‘› and ฬƒ๐œƒ=โ„ฑ๐œƒ, denote by ๐œ‘๐‘› the flow of ๐œƒ๐‘›+๐”„ and ๐œ‘ the flow of ๐œƒ+๐”„, and we have that ๐œ‘๐‘›โ†’๐œ‘ locally in ๐ถ1 as ๐‘›โ†’โˆž, thanks to Proposition A.4 in [33]. What we will need to do is to show that ๐‘š๐‘›โ†’๐‘š in ๐ฟ1(0,1) as ๐‘›โ†’โˆž and ๐œŒ๐‘›โ†’๐œŒ in ๐ฟ1(0,1) as ๐‘›โ†’โˆž.
Let ๐‘กโˆˆ[0,๐‘‡], having supposed that ๐‘ƒ๐‘™ and ๐‘ƒ๐‘Ÿ have only a finite number of connected components, we can assume, reducing ๐‘ก if necessary that ๐‘ฃ๐‘™ and ๐‘ฃ๐‘Ÿ do not change sign on [0,๐‘ก]. Since the characteristics of ๐œ‘๐‘› and ๐œ‘ may or may not cross before time ๐‘ก, we only consider the case that ๐œ‘(๐‘ก,0,0)โ‰ค๐œ‘๐‘›(๐‘ก,0,0)โ‰ค๐œ‘(๐‘ก,0,1)โ‰ค๐œ‘๐‘›(๐‘ก,0,1), without loss of generality. The other cases are proved in the same way. We first point out that since ๐œƒ๐‘›โˆˆ๐’ž๐ต0,๐ต1,๐‘‡ we have a bound for {๐‘ฆ๐‘›} in ๐ฟโˆž(ฮฉ๐‘‡). Now ๎€œ10||๐‘ฆ(๐‘ก,๐‘ฅ)โˆ’๐‘ฆ๐‘›||=๎‚ต๎€œ(๐‘ก,๐‘ฅ)d๐‘ฅ0๐œ‘(๐‘ก,0,0)+๎€œ๐œ‘๐‘›(๐‘ก,0,0)๐œ‘(๐‘ก,0,0)+๎€œ๐œ‘๐œ‘(๐‘ก,0,1)๐‘›(๐‘ก,0,0)+๎€œ๐œ‘๐‘›(๐‘ก,0,1)๐œ‘(๐‘ก,0,1)+๎€œ1๐œ‘๐‘›(๐‘ก,0,1)๎‚ถ||๐‘ฆ(๐‘ก,๐‘ฅ)โˆ’๐‘ฆ๐‘›||(๐‘ก,๐‘ฅ)d๐‘ฅ=๐ผ1+๐ผ2+๐ผ3+๐ผ4+๐ผ5.(3.18) Since ๐œ‘๐‘›(๐‘ก,0,0)โ†’๐œ‘(๐‘ก,0,0) as ๐‘›โ†’โˆž and ๐œ‘๐‘›(๐‘ก,0,1)โ†’๐œ‘(๐‘ก,0,1) as ๐‘›โ†’โˆž and thanks to the uniform bound on โ€–๐‘ฆ๐‘›โ€–๐ฟโˆž, we see that both ๐ผ2 and ๐ผ4 tend to 0 when ๐‘› goes to infinity.
For ๐ผ1 we have ๐ผ1=๎€œ0๐œ‘(๐‘ก,0,0)||๐‘ฆ(๐‘ก,๐‘ฅ)โˆ’๐‘ฆ๐‘›||=๎€œ(๐‘ก,๐‘ฅ)d๐‘ฅ0๐œ‘(๐‘ก,0,0)|||||||โŽ›โŽœโŽœโŽœโŽ๎€œexp๐‘ก๐‘’(๐‘ก,๐‘ฅ)๐ธ1๎€œd๐‘Ÿ00exp๐‘ก๐‘’(๐‘ก,๐‘ฅ)๐ธ2โŽžโŽŸโŽŸโŽŸโŽ ๐‘ฆd๐‘Ÿ๐‘™+๎€œ(๐‘’(๐‘ก,๐‘ฅ))๐‘ก๐‘’(๐‘ก,๐‘ฅ)โŽ›โŽœโŽœโŽœโŽ๎€œexp๐‘ก๐‘ ๐ธ1๎€œd๐‘Ÿ00exp๐‘ก๐‘ ๐ธ2โŽžโŽŸโŽŸโŽŸโŽ ๎ƒฉd๐‘Ÿโˆ’๐œŒ๐œŒ๐‘ฅโˆ’๐œŒ๐‘ฅโˆ’๐œ•๐‘ฅ๎ƒชโˆ’โŽ›โŽœโŽœโŽœโŽ๎€œ(๐œƒ+๐”„)d๐‘ exp๐‘ก๐‘’๐‘›(๐‘ก,๐‘ฅ)๐ธ1๐‘›๎€œd๐‘Ÿ00exp๐‘ก๐‘’๐‘›(๐‘ก,๐‘ฅ)๐ธ2๐‘›โŽžโŽŸโŽŸโŽŸโŽ ๐‘ฆd๐‘Ÿ๐‘™๎€ท๐‘’๐‘›๎€ธโˆ’๎€œ(๐‘ก,๐‘ฅ)๐‘ก๐‘’๐‘›(๐‘ก,๐‘ฅ)โŽ›โŽœโŽœโŽœโŽ๎€œexp๐‘ก๐‘ ๐ธ1๐‘›๎€œd๐‘Ÿ00exp๐‘ก๐‘ ๐ธ2๐‘›โŽžโŽŸโŽŸโŽŸโŽ ๎ƒฉd๐‘Ÿโˆ’๐œŒ๐œŒ๐‘ฅโˆ’๐œŒ๐‘ฅโˆ’๐œ•๐‘ฅ๎€ท๐œƒ๐‘›๎€ธ๎ƒช|||||||+๐”„d๐‘ d๐‘ฅ,(3.19) where ๐ธ1๐‘›=โˆ’2๐œ•๐‘ฅ๎€ท๐œƒ๐‘›+๐”„๎€ธ๎€ท๐‘Ÿ,๐œ‘๐‘›๎€ธ(๐‘Ÿ,๐‘ก,๐‘ฅ),๐ธ2=โˆ’๐œ•๐‘ฅ๎€ท๐œƒ๐‘›+๐”„๎€ธ๎€ท๐‘Ÿ,๐œ‘๐‘›๎€ธ(๐‘Ÿ,๐‘ก,๐‘ฅ).(3.20) Thanks to the boundedness on the โ€–๐‘“โ€–๐ฟโˆžร—๐ฟโˆž (Lemma 3.7) and Proposition A.2 of [33], if (๐‘ก,๐‘ฅ)โˆˆ๐‘ƒ we have ๐œ‘๐‘›(๐‘ก,0,0)โ†’๐œ‘(๐‘ก,0,0) as ๐‘›โ†’โˆž. This implies that if ๐‘ฆ๐‘™ was continuous, since we have a uniform bound on โ€–๐œƒ๐‘›โ€–๐ฟโˆž((0,๐‘‡);๐ถ1,1(0,1)) the dominated convergence theorem would provide: ๐ผ1โ†’0.๐ผ3โ†’0 and ๐ผ5โ†’0 which can be obtained by using the same method. Therefore, for ๐‘ฆ0, ๐‘ฆ๐‘™ and ๐‘ฆ๐‘Ÿ continuous we have โ€–๐‘ฆ(๐‘ก,โ‹…)โˆ’๐‘ฆ๐‘›(๐‘ก,โ‹…)โ€–๐ฟ1โ†’0.
From the inequality (56) in [33], we obtain โ€–๐‘ฆ(๐‘ก,โ‹…)โ€–๐ฟ1(0,1)ร—๐ฟ1(0,1)โ‰ค๎‚ƒโ€–โ€–๐‘ฆ0โ€–โ€–๐ฟ1(0,1)ร—๐ฟ1(0,1)+โ€–โ€–๐‘ฆ๐‘™โ€–โ€–๐ฟ1(0,๐‘ก)โˆฉฮ“๐‘™ร—๐ฟ1(0,๐‘ก)โˆฉฮ“๐‘™+โ€–โ€–๐‘ฆ๐‘Ÿโ€–โ€–๐ฟ1(0,๐‘ก)โˆฉฮ“๐‘Ÿร—๐ฟ1(0,๐‘ก)โˆฉฮ“๐‘Ÿ+๎€ทโ€–๐œŒโ€–๐ฟ1(0,1)ร—๐ฟ1(0,1)๎€ธโ€–โ€–๐œŒ+1๐‘ฅโ€–โ€–๐ฟ1(0,1)ร—๐ฟ1(0,1)+โ€–โ€–๐œ•๐‘ฅโ€–โ€–(๐œƒ+๐”„)๐ฟ1(0,1)๎€ปโ€–โ€–๐œ•๐‘ฅโ€–โ€–(๐œƒ+๐”„)๐ฟโˆž(ฮฉ๐‘‡)๎‚€โ€–โ€–๐œ•exp3๐‘ก๐‘ฅโ€–โ€–(๐œƒ+๐”„)๐ฟโˆž(ฮฉ๐‘‡)๎‚,โ€–โ€–๐‘ฆ๐‘›โ€–โ€–(๐‘ก,โ‹…)๐ฟ1(0,1)ร—๐ฟ1(0,1)โ‰ค๎‚ƒโ€–โ€–๐‘ฆ0โ€–โ€–๐ฟ1(0,1)ร—๐ฟ1(0,1)+โ€–โ€–๐‘ฆ๐‘™โ€–โ€–๐ฟ1(0,๐‘ก)โˆฉฮ“๐‘™ร—๐ฟ1(0,๐‘ก)โˆฉฮ“๐‘™+โ€–โ€–๐‘ฆ๐‘Ÿโ€–โ€–๐ฟ1(0,๐‘ก)โˆฉฮ“๐‘Ÿร—๐ฟ1(0,๐‘ก)โˆฉฮ“๐‘Ÿ+๎€ทโ€–โ€–๐œŒ๐‘›โ€–โ€–๐ฟ1(0,1)ร—๐ฟ1(0,1)๎€ธโ€–โ€–๐œŒ+1๐‘›,๐‘ฅโ€–โ€–๐ฟ1(0,1)ร—๐ฟ1(0,1)+โ€–โ€–๐œ•๐‘ฅ๎€ท๐œƒ๐‘›๎€ธโ€–โ€–+๐”„๐ฟ1(0,1)๎‚„โ€–โ€–๐œ•๐‘ฅ๎€ท๐œƒ๐‘›๎€ธโ€–โ€–+๐”„๐ฟโˆž(ฮฉ๐‘‡)๎‚€โ€–โ€–๐œ•exp3๐‘ก๐‘ฅ๎€ท๐œƒ๐‘›๎€ธโ€–โ€–+๐”„๐ฟโˆž(ฮฉ๐‘‡)๎‚.(3.21) So the general case of convergence โ€–๐‘ฆ(๐‘ก,โ‹…)โˆ’๐‘ฆ๐‘›(๐‘ก,โ‹…)โ€–๐ฟ1โ†’0 follows from the density of ๐ถ0 in ๐ฟ1 and the uniform bound on โ€–๐œƒ๐‘›โ€–๐ฟโˆž((0,๐‘‡);Lip(0,1)).
Now only the restriction on ๐‘ก remains; we recall that until now we supposed that ๐‘ฃ๐‘™ and ๐‘ฃ๐‘Ÿ did not change sign on [0,๐‘ก]. If ๐‘ฃ๐‘™ and ๐‘ฃ๐‘Ÿ do not change sign on [0,๐‘ก1] and then on [๐‘ก1,๐‘ก], we have โ€–โ€–๐‘ฆ๎€ท๐‘ก1๎€ธ,โ‹…โˆ’๐‘ฆ๐‘›๎€ท๐‘ก1๎€ธโ€–โ€–,โ‹…๐ฟ1โŸถ0.(3.22) Let ฬƒ๐‘ฆ๐‘› the solution of ๐œ•๐‘กฬƒ๐‘ฆ๐‘›+๎€ท๐œƒ๐‘›๎€ธ๐œ•+๐”„๐‘ฅฬƒ๐‘ฆ๐‘›=๐‘๐‘›(๐‘ก,๐‘ฅ)ฬƒ๐‘ฆ๐‘›+๐‘“๐‘›(๐‘ก,๐‘ฅ),ฬƒ๐‘ฆ๐‘›๎€ท๐‘ก1๎€ธ,โ‹…=๐‘ฆ๐‘ก1,ฬƒ๐‘ฆ๐‘›(โ‹…,0)|ฮ“๐‘™=๐‘ฆ๐‘™,ฬƒ๐‘ฆ๐‘›(โ‹…,1)|ฮ“๐‘Ÿ=๐‘ฆ๐‘Ÿ.(3.23) We can conclude that as ๐‘›โ†’โˆž, โ€–โ€–๐‘ฆ(๐‘ก,โ‹…)โˆ’๐‘ฆ๐‘›โ€–โ€–(๐‘ก,โ‹…)๐ฟ1(0,1)ร—๐ฟ1(0,1)โ‰คโ€–โ€–๐‘ฆ(๐‘ก,โ‹…)โˆ’ฬƒ๐‘ฆ๐‘›โ€–โ€–(๐‘ก,โ‹…)๐ฟ1(0,1)ร—๐ฟ1(0,1)+โ€–โ€–ฬƒ๐‘ฆ๐‘›(๐‘ก,โ‹…)โˆ’๐‘ฆ๐‘›โ€–โ€–(๐‘ก,โ‹…)๐ฟ1(0,1)ร—๐ฟ1(0,1)โ‰คโ€–โ€–๐‘ฆ๎€ท๐‘ก1๎€ธ,โ‹…โˆ’ฬƒ๐‘ฆ๐‘›๎€ท๐‘ก1๎€ธโ€–โ€–,โ‹…๐ฟ1(0,1)ร—๐ฟ1(0,1)๎‚€3๎€ทexp๐‘กโˆ’๐‘ก1๎€ธโ€–โ€–๐œ•๐‘ฅ(๐œƒ๐‘›โ€–โ€–+๐”„)๐ฟโˆž(ฮฉ๐‘‡)๎‚+โ€–โ€–๐‘ฆ(๐‘ก,โ‹…)โˆ’ฬƒ๐‘ฆ๐‘›โ€–โ€–(๐‘ก,โ‹…)๐ฟ1(0,1)ร—๐ฟ1(0,1)โ‰คโ€–โ€–๐‘ฆ๎€ท๐‘ก1๎€ธ,โ‹…โˆ’๐‘ฆ๐‘›๎€ท๐‘ก1๎€ธโ€–โ€–,โ‹…๐ฟ1(0,1)ร—๐ฟ1(0,1)๎‚€3๎€ทexp๐‘กโˆ’๐‘ก1๎€ธโ€–โ€–๐œ•๐‘ฅ๎€ท๐œƒ๐‘›๎€ธโ€–โ€–+๐”„๐ฟโˆž(ฮฉ๐‘‡)๎‚+โ€–โ€–๐‘ฆ(๐‘ก,โ‹…)โˆ’ฬƒ๐‘ฆ๐‘›โ€–โ€–(๐‘ก,โ‹…)๐ฟ1(0,1)ร—๐ฟ1(0,1)โŸถ0.(3.24) Thus the convergence in ๐ฟ1(0,1) propagates on each interval where ๐‘ฃ๐‘™ and ๐‘ฃ๐‘Ÿ do not change sign; thanks to the hypothesis on ๐‘ƒ๐‘Ÿ and ๐‘ƒ๐‘™ we have that for all ๐‘กโˆˆ[0,๐‘‡], โ€–๐‘ฆ(๐‘ก,โ‹…)โˆ’๐‘ฆ๐‘›(๐‘ก,โ‹…)โ€–๐ฟ1(0,1)โ†’0, as ๐‘›โ†’โˆž. Combining this first convergence result with the uniform bound and using the dominated convergence theorem in the time variable, we obtain โ€–๐‘ฆโˆ’๐‘ฆ๐‘›โ€–๐ฟ1(ฮฉ๐‘‡)โ†’0 which implies that โ€–โ€–ฬƒฬƒ๐œƒ๐œƒโˆ’๐‘›โ€–โ€–๐ฟ1(0,๐‘‡;๐‘Š2,1(0,1))โŸถ0.(3.25)
From the compactness of ๐’ž๐ต0,๐ต1,๐‘‡, we get that ฬƒ๐œƒ๐‘›โ†’ฬƒ๐œƒ holds in ๐’ž๐ต0,๐ต1,๐‘‡.

All the above lemmas result in the application of Schauders fixed point theorem to โ„ฑ and we get a solution ๐œƒโˆˆ๐ฟโˆž๎€ท(0,๐‘‡);๐ถ1,1๎€ธ๎€ท[](0,1)โˆฉLip0,๐‘‡;๐ป10๎€ธ.(0,1)(3.26)

From the construction of F and from Proposition A.8 in [33] the additional regularity properties of any solution ๐œƒ,๐œƒโˆˆ๐ถ0๎€ท[]0,๐‘‡;๐‘Š2,๐‘๎€ธ(0,1)โˆฉ๐ถ1๎‚€[]0,๐‘‡;๐‘Š01,๐‘๎‚(0,1),โˆ€๐‘<+โˆž.(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 (๐‘ฆ,๐œƒ) and ฬƒ(ฬƒ๐‘ฆ,๐œƒ) be two solutions of (1.6) and (1.8) for the same initial and boundary data, we will get ๐‘ฆ=ฬƒ๐‘ฆ and ฬƒ๐œƒ๐œƒ=.

Let ๐‘Œ=๐‘ฆโˆ’ฬƒ๐‘ฆ, ฬƒ๐œƒฮ˜=๐œƒโˆ’, ฬƒ๐ต(๐‘ก,๐‘ฅ)=๐‘(๐‘ก,๐‘ฅ)โˆ’๐‘(๐‘ก,๐‘ฅ), ๎‚๐น(๐‘ก,๐‘ฅ)=๐‘“(๐‘ก,๐‘ฅ)โˆ’๐‘“(๐‘ก,๐‘ฅ), then ฮ˜โˆˆLip([0,๐‘‡];๐ป10(0,1)). And we have ๐‘€=๐‘šโˆ’๎‚๐‘šโˆˆ๐ฟโˆž(ฮฉ๐‘‡) and ๐‘Œโˆˆ๐ฟโˆž(ฮฉ๐‘‡)ร—๐ฟโˆž(ฮฉ๐‘‡) is the solution to๐œ•๐‘ก๐‘Œ+(๐œƒ+๐”„)๐œ•๐‘ฅ๐‘Œ=๐‘๐‘Œ+๐ต(๐‘ก,๐‘ฅ)ฬƒ๐‘ฆโˆ’ฮ˜๐œ•๐‘ฅ๐‘Œฬƒ๐‘ฆ+๐น(๐‘ก,๐‘ฅ),(0,โ‹…)=0,๐‘Œ(โ‹…,0)|ฮ“๐‘™=0,๐‘Œ(โ‹…,1)|ฮ“๐‘Ÿ=0.(3.28) Using the lemma again with ๎‚€๐‘(๐‘ก,๐‘ฅ)=โˆ’2๐œ•๐‘ฅ(๐œƒ+๐”„)00โˆ’๐œ•๐‘ฅ(๐œƒ+๐”„)๎‚ and ๐‘“=๐ต(๐‘ก,๐‘ฅ)ฬƒ๐‘ฆโˆ’๐‘ˆ๐œ•๐‘ฅฬƒ๐‘ฆ+๐น(๐‘ก,๐‘ฅ), we get๎€œfor(๐‘ก,๐‘ฅ)โˆˆ๐‘ƒ,๐‘Œ(๐‘ก,๐‘ฅ)=0,for(๐‘ก,๐‘ฅ)โˆˆ๐ผ,๐‘Œ(๐‘ก,๐‘ฅ)=๐‘ก0๎‚ต๎€œexp๐‘ก๐‘ ๎‚ถ๐‘(๐‘Ÿ,๐œ‘(๐‘Ÿ,๐‘ก,๐‘ฅ))d๐‘Ÿ๎€œ๐‘“(๐‘ ,๐œ‘(๐‘ ,๐‘ก,๐‘ฅ))d๐‘ ,for(๐‘ก,๐‘ฅ)โˆˆ๐ฟ,๐‘Œ(๐‘ก,๐‘ฅ)=๐‘ก๐‘’(๐‘ก,๐‘ฅ)๎‚ต๎€œexp๐‘ก๐‘ ๎‚ถ๐‘(๐‘Ÿ,๐œ‘(๐‘Ÿ,๐‘ก,๐‘ฅ))d๐‘Ÿ๎€œ๐‘“(๐‘ ,๐œ‘(๐‘ ,๐‘ก,๐‘ฅ))d๐‘ ,for(๐‘ก,๐‘ฅ)โˆˆ๐‘…,๐‘Œ(๐‘ก,๐‘ฅ)=๐‘ก๐‘’(๐‘ก,๐‘ฅ)๎‚ต๎€œexp๐‘ก๐‘ ๎‚ถ๐‘(๐‘Ÿ,๐œ‘(๐‘Ÿ,๐‘ก,๐‘ฅ))d๐‘Ÿ๐‘“(๐‘ ,๐œ‘(๐‘ ,๐‘ก,๐‘ฅ))d๐‘ .(3.29)

Since โ€–๐‘ˆ(๐‘ก,โ‹…)โ€–๐ฟโˆž(0,1)โ‰ค๐ถ๎…žโ€–๐‘Œ(๐‘ก,โ‹…)โ€–๐ฟโˆžร—๐ฟโˆž for some positive constant ๐ถ๎…ž, and ๐‘ฆ,๐œ•๐‘ฅ๐‘ฆ bounded, we see that for some ๐ถ๎…ž๎…ž>0,โ€–โ€–โ€–โ€–๐‘“(๐‘ก,โ‹…)๐ฟโˆžร—๐ฟโˆžโ‰ค๐ถ๎…ž๎…žโ€–๐‘Œ(๐‘ก,โ‹…)โ€–๐ฟโˆžร—๐ฟโˆž.(3.30) And since ๐‘(๐‘ก,๐‘ฅ) is bounded, we get thatโ€–๐‘Œ(๐‘ก,โ‹…)โ€–๐ฟโˆžร—๐ฟโˆžโ‰ค๐ถ๎…ž๎…ž๎€œ๐‘ 0โ€–๐‘Œ(๐‘ ,โ‹…)โ€–๐ฟโˆžร—๐ฟโˆžd๐‘ .(3.31)

Then we complete the proof of the uniqueness by using Gronwallโ€™s lemma.

4. Stabilization

4.1. Preliminary Results

The equilibrium state that we want to stabilize is ๐‘ฆ=0, ๐œƒ=๐”„=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 ๐œƒ+๐”„ controls the geometry of the characteristics, and the sign of ๐œ•๐‘ฅ(๐œƒ+๐”„) controls the ingredient information of ๐‘ฆ along the characteristics. We will use the return method that Coron introduced in [34]. We would like our feedback law, ๐‘ฃ๐‘™(๐‘ฆ)=๐ด๐‘™โ€–๐‘ฆโ€–๐ถ0([0,1])ร—๐ถ0([0,1]), ๐‘ฃ๐‘Ÿ(๐‘ฆ)=๐ด๐‘Ÿโ€–๐‘ฆโ€–๐ถ0([0,1])ร—๐ถ0([0,1]), to provide ๐œƒ+๐”„โ‰ฅ0 and ๐œ•๐‘ฅ(๐œƒ+๐”„)โ‰ฅ0. 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 ๐œŒ is indeed Lipschitz inside the zones ๐ฟ, ๐‘…, and ๐ผ. To achieve this target, we have to prescribe ๐‘ฆ๐‘™, and we just need to make a continuous transition at (๐‘ก,๐‘ฅ)=(0,0) and let ๐‘ฆ๐‘™ asymptotically converge in time; we assume that compatibility conditions hold; precisely, ๐œŒ0, ๐œŒ๐‘™ and ๐œŒ๐‘Ÿ have compact supports in (0,1)/ฮ“๐‘™/ฮ“๐‘Ÿ, respectively. To achieve this target, we have to prescribe ๐‘ฆ๐‘™, and we just need to make a continuous transition at (๐‘ก,๐‘ฅ)=(0,0) and let ๐‘ฆ๐‘™ asymptotically converge in time. This is guaranteed by๐œ•๐‘ก๐‘ฆ๐‘™=๐‘€๐‘ฆ๐‘™,(4.1) where ๐‘€, symmetric matrix, is the unique matrix solution to the matrix function:๐‘ƒ๐‘€+๐‘€๐‘‡๐‘ƒ=โˆ’๐‘„,(4.2) for some symmetric positive-definite matrices, ๐‘ƒ and ๐‘„. Indeed, let ๐‘‰(๐‘ก,๐‘ฆ๐‘™)=๐‘ฆ๐‘‡๐‘™๐‘ƒ๐‘ฆ๐‘™ be the Lyapunov candidate, and that ๐‘ฆ๐‘™ asymptotically converges in time is equivalent to that the time derivative of the ๐‘‰, ฬ‡๐‘‰=๐‘ฆ๐‘‡๐‘™(๐‘ƒ๐‘€+๐‘€๐‘‡๐‘ƒ)๐‘ฆ๐‘™ 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 ๐‘‹ be the space of (๐‘”,๐‘)โˆˆ๐ถ0([0,๐‘‡];[0,1])2ร—๐ถ0([0,๐‘‡]) satisfying(1)๐‘”(0,๐‘ฅ)=๐‘ฆ0(๐‘ฅ), ๐‘”(๐‘ก,0)=๐‘ฆ0(0)๐‘’๐‘€๐‘ก,(2)โ€–๐‘”(๐‘ก,โ‹…)โ€–๐ถ0([0,1])โ‰ค๐‘(๐‘ก),(3)๐‘(๐‘ก) is nonincreasing and ๐‘(0)โ‰คโ€–๐‘ฆ0โ€–๐ถ0([0,1])ร—๐ถ0([0,1]).

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

Taking ๎€ท๐‘€=๐œ†,00,๐œ†๎€ธ, ๐œ†<0, satisfying (4.1) and (4.2), and (๐‘ฆ0(๐‘ฅ)๐‘’๐‘€๐‘ก,โ€–๐‘ฆ0โ€–๐ถ0([0,1])2๐‘’๐‘€๐‘ก)โˆˆ๐‘‹, so ๐‘‹ is nonempty.

Now for (๐‘ฆ,๐‘(๐‘ก))โˆˆ๐‘‹ we define ฬ†๐œƒ and ฬ†๐”„ as the solutions of๎€ท๐‘š(๐‘ก,๐‘ฅ)=1โˆ’๐œ•๐‘ฅ๐‘ฅ๎€ธฬ†ฬ†๐œƒ๐œƒ(๐‘ก,๐‘ฅ),ฬ†๐‘ข(๐‘ก,0)=(๐‘ก,1)=0,โˆ€(๐‘ก,๐‘ฅ)โˆˆฮฉ๐‘‡,๎€ท1โˆ’๐œ•๐‘ฅ๐‘ฅ๎€ธฬ†ฬ†๐”„(๐‘ก,๐‘ฅ)=0,๐”„(๐‘ก,0)=๐ด๐‘™ฬ†๐‘(๐‘ก),๐”„(๐‘ก,1)=๐ด๐‘Ÿ๐‘(๐‘ก),โˆ€(๐‘ก,๐‘ฅ)โˆˆฮฉ๐‘‡.(4.3) That is,โˆ€(๐‘ก,๐‘ฅ)โˆˆฮฉ๐‘‡๎€œ,ฬ†๐‘ข(๐‘ก,๐‘ฅ)=โˆ’๐‘ฅ0ฬ†sinh(๐‘ฅโˆ’๐œ‰)๐‘š(๐‘ก,๐œ‰)d๐œ‰,๐”„(๐‘ก,๐‘ฅ)=๐‘(๐‘ก)๎€บsinh(1)sinh(๐‘ฅ)๐ด๐‘Ÿ+sinh(1โˆ’๐‘ฅ)๐ด๐‘™๎€ป.(4.4)

Thus, we have the estimates:||ฬ†||๐œƒ(๐‘ก,๐‘ฅ)โ‰ค2sinh(1)โ€–๐‘ฆโ€–๐ถ0([0,1])ร—๐ถ0([0,1]),||๐œ•๐‘ฅฬ†||๐œƒ(๐‘ก,๐‘ฅ)โ‰ค2cosh(1)โ€–๐‘ฆโ€–๐ถ0([0,1])ร—๐ถ0([0,1]),||๐œ•๐‘ฅ๐‘ฅฬ†||โ‰ค[]๐œƒ(๐‘ก,๐‘ฅ)1+2sinh(1)โ€–๐‘ฆโ€–๐ถ0([0,1])ร—๐ถ0([0,1]),||๐œ•๐‘ฅฬ†||โ‰ฅ๐ด๐”„(๐‘ก,๐‘ฅ)๐‘Ÿโˆ’2cosh(1)||ฬ†||sinh(1)๐‘(๐‘ก),๐”„(๐‘ก,๐‘ฅ)โ‰ฅ๐ด๐‘™๐‘(๐‘ก).(4.5) And in turn,๎€ทฬ†ฬ†๐”„๎€ธ๎€บ๐ด๐œƒ+(๐‘ก,๐‘ฅ)โ‰ฅ๐‘™๎€ปโˆ’2sinh(1)โ€–๐‘ฆโ€–๐ถ0([0,1])ร—๐ถ0([0,1]),๐œ•(4.6)๐‘ฅ๎€ทฬ†ฬ†๐”„๎€ธ๐ด๐œƒ+(๐‘ก,๐‘ฅ)โ‰ฅ๐‘Ÿโˆ’2cosh(1)๐ด๐‘™โˆ’sinh(2)sinh(1)โ€–๐‘ฆโ€–๐ถ0([0,1])ร—๐ถ0([0,1]).(4.7) Now, if ๐œ‘ is the flow of ฬ†ฬ†๐”„๐œƒ+, ๐‘’ is ๐ถ1 and since ฬ†ฬ†๐‘ข+๐”„โ‰ฅ0, ๐œ‘(โ‹…,๐‘ก,๐‘ฅ) is nondecreasing. Thus we can define the entrance time and then the operator ๐’ฎ as follows.

Let ๐‘’(๐‘ก,๐‘ฅ)=min{๐‘ โˆˆ[0,๐‘‡]โˆฃ๐œ‘(๐‘ ,๐‘ก,๐‘ฅ)=0}, for (๐‘ก,๐‘ฅ)โˆˆ[0,๐‘‡]ร—[0,1],๎‚๐’ฎ(๐‘ฆ,๐‘)=(ฬƒ๐‘ฆ,๐‘) with(1) if ๐‘ฅโ‰ฅ๐œ‘(๐‘ก,0,0),โŽ›โŽœโŽœโŽœโŽ๎€œฬƒ๐‘ฆ(๐‘ก,๐‘ฅ)=exp๐‘ก0๐ธ1๎€œd๐‘Ÿ00exp๐‘ก0๐ธ2โŽžโŽŸโŽŸโŽŸโŽ ๐‘ฆd๐‘Ÿ0+๎€œ(๐œ‘(0,๐‘ก,๐‘ฅ))๐‘ก0โŽ›โŽœโŽœโŽœโŽ๎€œexp๐‘ก๐‘ ๐ธ1๎€œd๐‘Ÿ00exp๐‘ก๐‘ ๐ธ2โŽžโŽŸโŽŸโŽŸโŽ ๎ƒฉd๐‘Ÿโˆ’๐œŒ๐œŒ๐‘ฅโˆ’๐œŒ๐‘ฅโˆ’๐œ•๐‘ฅ๎ƒช(๐œƒ+๐”„)d๐‘ ,(4.8)(2)if ๐‘ฅโ‰ค๐œ‘(๐‘ก,0,0),โŽ›โŽœโŽœโŽœโŽ๎€œฬƒ๐‘ฆ(๐‘ก,๐‘ฅ)=exp๐‘ก0๐ธ1๎€œd๐‘Ÿ00exp๐‘ก0๐ธ2โŽžโŽŸโŽŸโŽŸโŽ ๐‘’d๐‘Ÿ๐‘€๐‘’(๐‘ก,๐‘ฅ)๐‘ฆ0+๎€œ(0)๐‘ก๐‘’(๐‘ก,๐‘ฅ)โŽ›โŽœโŽœโŽœโŽ๎€œexp๐‘ก๐‘ ๐ธ1๎€œd๐‘Ÿ00exp๐‘ก๐‘ ๐ธ2โŽžโŽŸโŽŸโŽŸโŽ ๎ƒฉd๐‘Ÿโˆ’๐œŒ๐œŒ๐‘ฅโˆ’๐œŒ๐‘ฅโˆ’๐œ•๐‘ฅ๎ƒช(๐œƒ+๐”„)d๐‘ ,(4.9)(3)๐‘(๐‘ก)=โ€–๐‘ฆ(๐‘ก,โ‹…)โ€–๐ถ0([0,1])ร—๐ถ0([0,1]).

From Lemma 1.3 we know that