Initial Boundary Value Problem of the General Three-Component Camassa-Holm Shallow Water System on an Interval

Abstract

We study the initial boundary value problem of the general three-component Camassa-Holm shallow water system on an interval subject to inhomogeneous boundary conditions. First we prove a local in time existence theorem and present a weak-strong uniqueness result. Then, we establish a asymptotic stabilization of this system by a boundary feedback. Finally, we obtain a result of blow-up solution with certain initial data and boundary profiles.

1. Introduction

It is well known that the Camassa-Holm equation has attracted much attention in the past decade. It is a nonlinear dispersive wave equation that models the propagation of unidirectional irrotational shallow water waves over a flat bed, as well as water waves moving over an underlying shear flow. It was first introduced by Fo kas and Fuchssteiner as a bi-Hamiltonian model. Cauchy problem and initial boundary value problem for Camassa-Holm equation have been studied extensively in a number of papers (see [1–15] and the references within).

Fu and Qu in [16] proposed a coupled Camassa-Holm equation, 𝑚_𝑡=2𝑚𝑢_𝑥+𝑚_𝑥𝑢+(𝑚𝑣)_𝑥+𝑛𝑣_𝑥,𝑛_𝑡=2𝑛𝑣_𝑥+𝑛_𝑥𝑣+(𝑛𝑢)_𝑥+𝑚𝑢_𝑥; (1) with 𝑚=𝑢−𝑢_𝑥𝑥,𝑛=𝑣−𝑣_𝑥𝑥, which has peakon solitons in the form of a superposition of multipeakons and may as well be integrable. They investigated the local well-posedness and blow-up solutions of system (1) by means of Kato’s semigroup approach to nonlinear hyperbolic evolution equation and obtained a criterion and condition on the initial data guaranteeing the development of singularities in finite time for strong solutions of system (1) by energy estimates. Recently the initial boundary value problem for the system (1) has been established in [17]; moreover, the local well-posedness and blow-up phenomena for the coupled Camassa-Holm equation were also established in [16, 18–32]. In [33], Tian and Xu obtained the compact and bounded absorbing set and the existence of the global attractor for viscous system (1) with the periodic boundary condition in by uniform prior estimate.

Recently, Fu and Qu in [34] introduced a general three-component Camassa-Holm equation as follows: 𝑚_𝑡=2𝑚𝑢_𝑥+𝑚_𝑥𝑢+(𝑚𝑣+𝑚𝑤)_𝑥+𝑛𝑣_𝑥+𝑙𝑤_𝑥,𝑛_𝑡=2𝑛𝑣_𝑥+𝑛_𝑥𝑣+(𝑛𝑢+𝑛𝑤)_𝑥+𝑚𝑢_𝑥+𝑙𝑤_𝑥,𝑙_𝑡=2𝑙𝑤_𝑥+𝑙_𝑥𝑤+(𝑙𝑢+𝑙𝑣)_𝑥+𝑚𝑢_𝑥+𝑛𝑣_𝑥, (2) where 𝑚=𝑢−𝑢_𝑥𝑥, 𝑛=𝑣−𝑣_𝑥𝑥, and 𝑙=𝑤−𝑤_𝑥𝑥. Equation (2) also has peakon solitons in the form of a superposition of multipeakons. Such system also conserves the 𝐻¹-norm conservation law. Moreover, the well-posedness and blow-up phenomena for system (2) with peakons have been established in [35]. To our knowledge, the initial boundary value problem of (2) has not been studied yet. The first aim of this paper is to consider an initial boundary value problem of the following 𝑚_𝑡=2𝑚𝑢_𝑥+𝑚_𝑥𝑢+(𝑚𝑣+𝑚𝑤)_𝑥+𝑛𝑣_𝑥+𝑙𝑤_𝑥,𝑛_𝑡=2𝑛𝑣_𝑥+𝑛_𝑥𝑣+(𝑛𝑢+𝑛𝑤)_𝑥+𝑚𝑢_𝑥+𝑙𝑤_𝑥,𝑙_𝑡=2𝑙𝑤_𝑥+𝑙_𝑥𝑤+(𝑙𝑢+𝑙𝑣)_𝑥+𝑚𝑢_𝑥+𝑛𝑣_𝑥,𝑚(0,⋅)=𝑚₀,𝑛(0,⋅)=𝑛₀,𝑙(0,⋅)=𝑙₀,𝑚(⋅,0)_|Γ𝑙=𝑚_𝑙,𝑛(⋅,0)_|Γ𝑙=𝑛_𝑙,𝑙(⋅,0)_|Γ𝑙=𝑙_𝑙,𝑚(⋅,1)_|Γ𝑟=𝑚_𝑟,𝑛(⋅,1)_|Γ𝑟=𝑛_𝑟,𝑙(⋅,1)_|Γ𝑟=𝑙_𝑟, (3) where Γ_𝑙={𝑡∈[0,𝑇]∣(𝑢+𝑣+𝑤)(𝑡,𝑥)<0},Γ_𝑟={𝑡∈[0,𝑇]∣(𝑢+𝑣+𝑤)(𝑡,𝑥)>0}.

Then, we will consider the asymptotic stabilization of (3) by means of a stationary feedback law acting on the inhomogeneous boundary condition. Following the step in [11], we convert the initial boundary value problem of (3) on the interval into an ODE system and two PDE systems. Then, we can consider the system (3) easily. Consequently, we obtain a local in time existence theorem, a weak-strong uniqueness result, asymptotic stabilization result on the interval, and a result of blow-up solution, respectively.

Our paper is organized as follows. In Section 2, we will consider an initial boundary value problem and the uniqueness of the solution to (3). By using the feedback law enjoyed by (3), the asymptotic stabilization on an interval is considered in Section 3. Finally, in Section 4, a result of blow-up solution with certain initial data and boundary profiles will be established.

First, we begin with a general remark that will be used many times later.

Remark 1. Let 𝑇 be a positive number and Ω_𝑇=[0,𝑇]×[0,1]. Changing 𝑢(𝑡,𝑥) in−𝑢(𝑡,1−𝑥), 𝑣(𝑡,𝑥) in−𝑣(𝑡,1−𝑥),  𝑤(𝑡,𝑥) in−𝑤(𝑡,1−𝑥), and 𝑡 in 𝑇−𝑡, and it will be more convenient for us to analysis the system, if we define the following sets 𝑃_𝑙={[]∣()},𝑃𝑡∈0,𝑇𝑢+𝑣+𝑤)(𝑡,0=0_𝑟={𝑡∈[0,𝑇]∣(𝑢+𝑣+𝑤)(𝑡,1)=0}. (4)
Let Λ=(1−𝜕²_𝑥)^1/2, then the operator Λ⁻² can be expressed as Λ⁻²𝑓(𝑥)(𝑥)=1=𝐺∗𝑓2¹₀𝑒^{−|𝑥−𝑦|}𝑓𝑦𝑑𝑦, (5) where 𝐺=(1/2)𝑒^−|𝑥|. Now, let 𝐴_𝑖=Λ⁻²𝐵_𝑖=𝐺∗𝐵_𝑖,  𝑖=1,2,3, where 𝐵_𝑖 is an auxiliary function which lifts the boundary values 𝑚_𝑙, 𝑚_𝑟, 𝑛_𝑙, 𝑛_𝑟, and 𝑙_𝑙 defined by 𝐵_𝑖(𝑡,𝑥)=0,(𝑡,𝑥)∈[0,𝑇]×[0,1],𝐺∗𝐵_𝑖(𝑡,0)=𝑣_𝑙𝑖(𝑡),𝐺∗𝐵_𝑖(𝑡,1)=𝑣_𝑟𝑖(𝑡),∀𝑡∈[0,𝑇], (6) where 𝑖=1,2,3.
Setting 𝑢=𝑝+𝐺∗𝐵₁, 𝑣=𝑞+𝐺∗𝐵₂, and 𝑤=𝑟+𝐺∗𝐵₃, we can further rewrite the system (3) as 𝑝()()()()𝑟()()𝑚𝑝=𝐺∗𝑚,𝑞=𝐺∗𝑛,𝑟=𝐺∗𝑙,𝑡,0=𝑝𝑡,1=0,𝑞𝑡,0=𝑞𝑡,1=0,𝑡,0=𝑟𝑡,1=0,_𝑡=𝑝+𝐺∗𝐵₁+𝑣+𝑤𝑚_𝑥+2𝑝+𝐺∗𝐵_1𝑥+𝑣_𝑥+𝑤_𝑥𝑚+𝑛𝑣_𝑥+𝑙𝑤_𝑥,𝑛_𝑡=𝑞+𝐺∗𝐵₂+𝑢+𝑤𝑛_𝑥+2𝑞+𝐺∗𝐵_2𝑥+𝑢_𝑥+𝑤_𝑥𝑛+𝑚𝑢_𝑥+𝑙𝑤_𝑥,𝑙_𝑡=𝑟+𝐺∗𝐵₃+𝑢+𝑣𝑙_𝑥+2𝑟+𝐺∗𝐵_3𝑥+𝑢_𝑥+𝑣_𝑥𝑙+𝑚𝑢_𝑥+𝑛𝑣_𝑥,𝑚(0,⋅)=𝑚₀,𝑚(⋅,0)_|Γ𝑙=𝑚_𝑙,𝑚(⋅,1)_|Γ𝑟=𝑚_𝑟,𝑛(0,⋅)=𝑛₀,𝑛(⋅,0)_|Γ𝑙=𝑛_𝑙,𝑛(⋅,1)_|Γ𝑟=𝑛_𝑟,𝑙(0,⋅)=𝑙₀,𝑙(⋅,0)_|Γ𝑙=𝑙_𝑙,𝑙(⋅,1)_|Γ𝑟=𝑙_𝑟, (7)(8) where functions 𝑚_𝑙,𝑚_𝑟,𝑛_𝑙,𝑛_𝑟,𝑙_𝑙, and 𝑙_𝑟 in 𝐶⁰([0,1],𝑅) are the boundary values and 𝑚₀,𝑛₀, and 𝑙₀ in 𝐿^∞(0,1) are the initial datum.

Lemma 2. We have 𝐴_𝑖=𝐺∗𝐵_𝑖∈𝐶⁰([0,𝑇];𝐶^∞[0,1])∈𝐶⁰([0,𝑇];𝐶^∞[0,1]) and 𝑝,𝑞,𝑟̃∈𝐿^∞((0,𝑇),𝐶^1,1([0,1]))∩Lip((0,𝑇),𝐻¹₀(0,1)), 𝑖=1,2,3,𝑚,𝑛,𝑙∈𝐿^∞(Ω_𝑇)∩Lip((0,𝑇),𝐻⁻¹(0,1)). Moreover, we also have the bounds 𝐴_{𝑖𝐿∞((0,𝑇);𝐶1,1[0,1])}≤cosh(1)(1)𝑣sinh^{𝑟𝑖𝐿∞(0,𝑇)}+𝑣_{𝑙𝑖𝐿∞(0,𝑇)},(),𝑝𝑖=1,2,3_{𝐿∞((0,𝑇);𝐶1,1[0,1])}≤2(1+cosh(1))‖𝑚‖_𝐿^∞_(Ω𝑇),𝜕_𝑡𝑝_{𝐿∞((0,𝑇);𝐻10[0,1])}≤𝜕_𝑡𝑚_{𝐿∞((0,𝑇);𝐻−1[0,1])},𝑞_{𝐿∞((0,𝑇);𝐶1,1[0,1])}≤2(1+cosh(1))‖𝑛‖_𝐿^∞_(Ω𝑇),𝜕_𝑡𝑞_{𝐿∞((0,𝑇);𝐻10[0,1])}≤𝜕_𝑡𝑛_{𝐿∞((0,𝑇);𝐻−1[0,1])},‖𝑟‖̃_𝐿^∞_{((0,𝑇);𝐶}^1,1_[0,1])≤2(1+cosh(1))‖𝑙‖_𝐿^∞_(Ω𝑇),𝜕_𝑡𝑟̃_{𝐿∞((0,𝑇);𝐻10[0,1])}≤𝜕_𝑡𝑙_{𝐿∞((0,𝑇);𝐻−1[0,1])}. (9)(10)

Proof. 𝐴_𝑖,𝑝,𝑞,𝑟,̃(𝑖=1,2,3) can be expressed, respectively, as 𝐴_𝑖(𝑡,𝑥)=𝐺∗𝐵_𝑖()=1𝑡,𝑥(1)(𝑥)𝑣sinhsinh_𝑟𝑖(𝑡)+sinh(1−𝑥)𝑣_𝑙𝑖(𝑡),(),𝑝()𝑖=1,2,3𝑡,𝑥=−^𝑥₀(𝑥)𝑚(𝑥)𝑑𝑥+(𝑥)sinh𝑥−𝑡,sinhsinh(1)¹₀(𝑥)𝑚(𝑥)𝑑𝑞()sinh1−𝑡,𝑥,𝑡,𝑥=−^𝑥₀(𝑥)𝑛(𝑥)𝑑𝑥+(𝑥)sinh𝑥−𝑡,sinhsinh(1)¹₀(𝑥)𝑛(𝑥)𝑑𝑟̃()sinh1−𝑡,𝑥,𝑡,𝑥=−^𝑥₀(𝑥)𝑙(𝑥)𝑑𝑥+(𝑥)sinh𝑥−𝑡,sinhsinh(1)¹₀sinh(1−𝑥)𝑙(𝑡,𝑥)𝑑𝑥. (11) Estimates (9) and (10) can be easily obtained from the above expressions.

2. Initial Boundary Value Problem

First, we define what we mean by a weak solution to (8). Our test functions will be in the space: AdmΩ_𝑇=𝜑∈𝐶¹Ω_𝑇∣𝜑(𝑡,𝑥)=0on[0,𝑇]⧵Γ_𝑙×{0}∪[0,𝑇]⧵Γ_𝑟×{0}∪{𝑇}×[0,1]. (12)

Definition 3. When (𝑝,𝑞,𝑟)∈𝐿^∞((0,𝑇);Lip[0,1])×𝐿^∞((0,𝑇);Lip[0,1])×𝐿^∞((0,𝑇);Lip[0,1]), the function (𝑚,𝑛,𝑙)∈𝐿^∞(Ω_𝑇)×𝐿^∞(Ω_𝑇)×𝐿^∞(Ω_𝑇) is the weak solution to (8) if for all 𝜑∈Adm(Ω_𝑇): _Ω𝑇𝑚𝜑_𝑡−(𝑢+𝑣+𝑤)𝜑_𝑥+𝑝_𝑥+𝜕_𝑥𝐺∗𝐵₁𝜑𝑑𝑡𝑑𝑥=−_Ω𝑇𝑛𝑣_𝑥+𝑙𝑤_𝑥𝜑(𝑡,𝑥)𝑑𝑡𝑑𝑥−¹₀𝑚₀(𝑥)𝜑()+0,𝑥𝑑𝑡_𝑇0()𝜑()𝑚()−()𝜑()𝑚()𝑢+𝑣+𝑤)(𝑡,0𝑡,0𝑡,0𝑢+𝑣+𝑤)(𝑡,1𝑡,1𝑡,1𝑑𝑡,_Ω𝑇𝑛𝜑_𝑡−(𝑢+𝑣+𝑤)𝜑_𝑥+𝑞_𝑥+𝜕_𝑥𝐺∗𝐵₂𝜑𝑑𝑡𝑑𝑥=−_Ω𝑇𝑚𝑢_𝑥+𝑙𝑤_𝑥𝜑(𝑡,𝑥)𝑑𝑡𝑑𝑥−¹₀𝑛₀(𝑥)𝜑(0,𝑥)𝑑𝑥+^𝑇₀()𝜑()𝑛()−()𝜑()𝑛()𝑢+𝑣+𝑤)(𝑡,0𝑡,0𝑡,0𝑢+𝑣+𝑤)(𝑡,1𝑡,1𝑡,1𝑑𝑡,_Ω𝑇𝑙𝜑_𝑡−(𝑢+𝑣+𝑤)𝜑_𝑥+𝑟_𝑥+𝜕_𝑥𝐺∗𝐵₃𝜑𝑑𝑡𝑑𝑥=−_Ω𝑇𝑚𝑢_𝑥+𝑛𝑣_𝑥𝜑(𝑡,𝑥)𝑑𝑡𝑑𝑥−¹₀𝑙₀(𝑥)𝜑()+0,𝑥𝑑𝑥_𝑇0()𝜑()𝑙()−()𝜑()𝑙(𝑢+𝑣+𝑤)(𝑡,0𝑡,0𝑡,0𝑢+𝑣+𝑤)(𝑡,1𝑡,1𝑡,1)𝑑𝑡. (13)
It is obvious that 𝐶¹₀(Ω_𝑇)⊂Adm(Ω_𝑇); therefore, a weak solution to system (8) is also a solution to (8) in the distribution sense. And it is clear that a regular weak solution is a classical solution.

Definition 4. For (𝑡,𝑥)∈Ω_𝑇, we consider 𝜔(⋅,𝑡,𝑥) the maximal solution satisfying 𝜔_𝑡((,𝜔()=−𝑢+𝑣+𝑤)(𝑡,𝜔𝑡,𝑥))0,𝑥=𝑥. (14)
We consider that 𝜔 is the flow of (𝑢(𝑡,𝑥),𝑣(𝑡,𝑥),𝑤(𝑡,𝑥)). For (𝑡,𝑥)∈Ω_𝑇, 𝜔(⋅,𝑡,𝑥) is defined on a set [𝑒(𝑡,𝑥),ℎ(𝑡,𝑥)]. Here 𝑒(𝑡,𝑥) is basically the entrance time in Ω_𝑇 of the characteristic curve going through (𝑡,𝑥).

Remark 5. Obviously 𝑒(𝑡,𝑥)>0 implies that 𝜔(𝑒(𝑡,𝑥),𝑡,𝑥)∈{0,1}.

In the following, we consider a partition of Ω_𝑇, which allows us to distinguish the different influence zones in Ω_𝑇.

Definition 6. Let 𝑃={(𝑡,𝑥)∈Ω_𝑇|∃𝑠∈[𝑒(𝑡,𝑥),ℎ(𝑡,𝑥)] such that 𝜔∈{0,1} and (𝑢+𝑣+𝑤)(𝑠,𝜔(𝑠,𝑡,𝑥))=0}∪{(𝑠,𝜔(𝑠,0,0))∣forall𝑠∈[0,𝑇]}∪{(𝑠,𝜔(𝑠,0,1))∣forall𝑠∈[0,𝑇]}, 𝐼=(𝑡,𝑥)∈Ω_𝑇(),)⧵𝑝∣𝑒𝑡,𝑥=0𝐿=(𝑡,𝑥∈Ω_𝑇(𝑒()),)⧵𝑝∣𝜔𝑡,𝑥,𝑡,𝑥=0𝑅=(𝑡,𝑥∈Ω_𝑇⧵𝑝∣𝜔(𝑒(𝑡,𝑥),𝑡,𝑥)=1. (15)

Those points of the set 𝑃 are tangent to the boundary, which are precisely the singular points of 𝑒 and ℎ. It's obviously that the sets 𝑃,𝐼,𝐿, and 𝑅 constitute a partition of Ω_𝑇. Furthermore, if (𝑡,𝑥)∈𝐿, then 𝑒(𝑡,𝑥)∈Γ_𝑙, and if (𝑡,𝑥)∈𝑅, then 𝑒(𝑡,𝑥)∈Γ_𝑟.

Definition 7. Here, we consider the case of data (𝑢,𝑣,𝑤)∈𝐿^∞([0,𝑇];𝐶¹([0,1]))×𝐿^∞([0,𝑇];𝐶¹([0,1]))×𝐿^∞([0,𝑇];𝐶¹([0,1])),(𝑚_𝑙,𝑛_𝑙,𝑙_𝑙)∈𝐶¹_𝑐(Γ_𝑙𝐶)×_1𝑐(Γ_𝑙)×𝐶¹_𝑐(Γ_𝑙); (𝑚_𝑟,𝑛_𝑟,𝑙_𝑟)∈𝐶¹_𝑐(Γ_𝑟)×𝐶¹_𝑐(Γ_𝑟)×𝐶¹_𝑐(Γ_𝑟),(𝑚₀,𝑛₀,𝑙₀)∈𝐶¹_𝑐(0,1)×𝐶¹_𝑐(0,1)×𝐶¹_𝑐(0,1). We define the functions 𝑚,𝑛, and 𝑙 in the following way.
When (𝑡,𝑥)∈𝑃,  𝑚(𝑡,𝑥)=0,  𝑛(𝑡,𝑥)=0, and  𝑙(𝑡,𝑥)=0, when (𝑡,𝑥)∈𝐼, 𝑚(𝑡,𝑥)=𝑚₀(𝜔(0,𝑡,𝑥))×exp^𝑡₀2𝑝_𝑥+𝜕_𝑥𝐺∗𝐵₁+𝑣_𝑥+𝑤_𝑥×((+𝑠,𝜔𝑠,𝑡,𝑥))𝑑𝑠_𝑡0𝑛𝑣_𝑥+𝑙𝑤_𝑥(𝑠,𝜔(𝑠,𝑡,𝑥))×exp^𝑡_𝑠2𝑝_𝑥+𝜕_𝑥𝐺∗𝐵₁+𝑣_𝑥+𝑤_𝑥×𝑠^′,𝜔𝑠^′,𝑡,𝑥𝑑𝑠^′𝑛()𝑑𝑠,𝑡,𝑥=𝑛₀(𝜔(0,𝑡,𝑥))×exp^𝑡₀2𝑞_𝑥+𝜕_𝑥𝐺∗𝐵₂+𝑢_𝑥+𝑤_𝑥×((+𝑠,𝜔𝑠,𝑡,𝑥))𝑑𝑠_𝑡0𝑚𝑢_𝑥+𝑙𝑤_𝑥((2𝑞𝑠,𝜔𝑠,𝑡,𝑥))×exp_𝑥+𝜕_𝑥𝐺∗𝐵₂+𝑢_𝑥+𝑤_𝑥×𝑠^′,𝜔𝑠^′,𝑡,𝑥𝑑𝑠^′𝑙()𝑑𝑠,𝑡,𝑥=𝑙₀(𝜔(0,𝑡,𝑥))×exp^𝑡₀2𝑟_𝑥+𝜕_𝑥𝐺∗𝐵₃+𝑢_𝑥+𝑣_𝑥×(𝑠,𝜔(𝑠,𝑡,𝑥))𝑑𝑠+^𝑡₀𝑚𝑢_𝑥+𝑛𝑣_𝑥(𝑠,𝜔(𝑠,𝑡,𝑥))×exp^𝑡_s2𝑟_𝑥+𝜕_𝑥𝐺∗𝐵₃+𝑢_𝑥+𝑣_𝑥×𝑠^′,𝜔𝑠^′,𝑡,𝑥𝑑𝑠^′𝑑𝑠, (16) when (𝑡,𝑥)∈𝐿, 𝑚(𝑡,𝑥)=𝑚_𝑙(𝑒(𝑡,𝑥))×exp^𝑡_{𝑒()𝑡,𝑥}2𝑝_𝑥+𝜕_𝑥𝐺∗𝐵₁+𝑣_𝑥+𝑤_𝑥×((+𝑠,𝜔𝑠,𝑡,𝑥))𝑑𝑠_{𝑡𝑒()𝑡,𝑥}𝑛𝑣_𝑥+𝑙𝑤_𝑥(𝑟,𝜔(𝑟,𝑡,𝑥))×exp^𝑡_s2𝑝_𝑥+𝜕_𝑥𝐺∗𝐵₁+𝑣_𝑥+𝑤_𝑥×𝑠^′,𝜔𝑠^′,𝑡,𝑥𝑑𝑠^′𝑛()𝑑𝑟,𝑡,𝑥=𝑛_𝑙(𝑒(𝑡,𝑥))×exp^𝑡_{𝑒()𝑡,𝑥}2𝑞_𝑥+𝜕_𝑥𝐺∗𝐵₂+𝑢_𝑥+𝑤_𝑥×((+𝑠,𝜔𝑠,𝑡,𝑥))𝑑𝑠_{𝑡𝑒()𝑡,𝑥}𝑚𝑢_𝑥+𝑙𝑤_𝑥(𝑠,𝜔(𝑠,𝑡,𝑥))×exp^𝑡_s2𝑞_𝑥+𝜕_𝑥𝐺∗𝐵₂+𝑢_𝑥+𝑤_𝑥×𝑠^′,𝜔𝑠^′,𝑡,𝑥𝑑𝑠^′𝑙()𝑑𝑠,𝑡,𝑥=𝑙_𝑙(𝑒(𝑡,𝑥))×exp^𝑡_{𝑒()𝑡,𝑥}2𝑟_𝑥+𝜕_𝑥𝐺∗𝐵₃+𝑢_𝑥+𝑣_𝑥×((+𝑠,𝜔𝑠,𝑡,𝑥))𝑑𝑠_{𝑡𝑒()𝑡,𝑥}𝑚𝑢_𝑥+𝑛𝑣_𝑥(𝑠,𝜔(𝑠,𝑡,𝑥))×exp^𝑡_s2𝑟_𝑥+𝜕_𝑥𝐺∗𝐵₃+𝑢_𝑥+𝑣_𝑥×𝑠^′,𝜔𝑠^′,𝑡,𝑥𝑑𝑠^′𝑑𝑠, (17) when (𝑡,𝑥)∈𝑅, 𝑚(𝑡,𝑥)=𝑚_𝑙(𝑒(𝑡,𝑥))×exp^𝑡_{𝑒()𝑡,𝑥}2𝑝_𝑥+𝜕_𝑥𝐺∗𝐵₁+𝑣_𝑥+𝑤_𝑥×((+𝑠,𝜔𝑠,𝑡,𝑥))𝑑𝑠_{𝑡𝑒()𝑡,𝑥}𝑛𝑣_𝑥+𝑙𝑤_𝑥(𝑟,𝜔(𝑟,𝑡,𝑥))×exp^𝑡_s2𝑝_𝑥+𝜕_𝑥𝐺∗𝐵₁+𝑣_𝑥+𝑤_𝑥×𝑠^′,𝜔𝑠^′,𝑡,𝑥𝑑𝑠^′𝑛()𝑑𝑟,𝑡,𝑥=𝑛_𝑙(𝑒(𝑡,𝑥))×exp^𝑡_{𝑒()𝑡,𝑥}2𝑞_𝑥+𝜕_𝑥𝐺∗𝐵₂+𝑢_𝑥+𝑤_𝑥×((+𝑠,𝜔𝑠,𝑡,𝑥))𝑑𝑠_{𝑡𝑒()𝑡,𝑥}𝑚𝑢_𝑥+𝑙𝑤_𝑥(𝑠,𝜔(𝑠,𝑡,𝑥))×exp^𝑡_s2𝑞_𝑥+𝜕_𝑥𝐺∗𝐵₂+𝑢_𝑥+𝑤_𝑥×𝑠^′,𝜔𝑠^′,𝑡,𝑥𝑑𝑠^′𝑙()𝑑𝑠,𝑡,𝑥=𝑙_𝑙(𝑒(𝑡,𝑥))×exp^𝑡_{𝑒()𝑡,𝑥}2𝑟_𝑥+𝜕_𝑥𝐺∗𝐵₃+𝑢_𝑥+𝑣_𝑥×((+𝑠,𝜔𝑠,𝑡,𝑥))𝑑𝑠_{𝑡𝑒()𝑡,𝑥}𝑚𝑢_𝑥+𝑛𝑣_𝑥(𝑠,𝜔(𝑠,𝑡,𝑥))×exp^𝑡_s2𝑟_𝑥+𝜕_𝑥𝐺∗𝐵₃+𝑢_𝑥+𝑣_𝑥×𝑠^′,𝜔𝑠^′,𝑡,𝑥𝑑𝑠^′𝑑𝑠. (18)

Lemma 8. Since (𝑚,𝑛,𝑙)∈𝐿^∞(Ω_𝑇)×𝐿^∞(Ω_𝑇)×𝐿^∞(Ω_𝑇) and satisfies (8), we immediately get that (𝑚,𝑛,𝑙) is the weak solution of (8) and (𝑚,𝑛,𝑙)∈𝑊^1,∞(0,𝑇,𝐻⁻¹(0,1))×𝑊^1,∞(0,𝑇,𝐻⁻¹(0,1))×𝑊^1,∞(0,𝑇,𝐻⁻¹(0,1)). However, the functions 𝑚,𝑛, and 𝑙 satisfy the following estimates: ‖𝑚‖_𝐶⁰_(Ω𝑇)≤max𝑚_0𝐿∞,𝑚_𝑙𝐿∞,𝑚_𝑟𝐿∞+𝑛𝑣_𝑥+𝑙𝑤_{𝑥𝐿∞(0,1)}𝑇×𝑒^{𝑇‖2(𝑝+𝐺∗𝐵1)𝑥+𝑣𝑥+𝑤𝑥‖𝐶0𝑇)(Ω},‖𝑛‖_𝐶⁰_(Ω𝑇)≤max𝑛_0𝐿∞,𝑛_𝑙𝐿∞,𝑛_𝑟𝐿∞+𝑚𝑢_𝑥+𝑙𝑤_{𝑥𝐿∞(0,1)}𝑇×𝑒^{𝑇‖2(𝑞+𝐺∗𝐵2)𝑥+𝑢𝑥+𝑤𝑥‖𝐶0𝑇)(Ω},‖𝑙‖_𝐶⁰_(Ω𝑇)≤max𝑚_0𝐿∞,𝑚_𝑙𝐿∞,𝑚_𝑟L∞+𝑚𝑢_𝑥+𝑛𝑣_{𝑥𝐿∞(0,1)}𝑇×𝑒^{𝑇‖2(𝑟+𝐺∗𝐵3)𝑥+𝑢𝑥+𝑣𝑥‖𝐶0𝑇)(Ω},𝜕_𝑡𝑚_{𝐶0(Ω𝑇)}≤2max𝑚_0𝐿∞,𝑚_𝑙𝐿∞,𝑚_𝑟𝐿∞×𝑝+𝐺∗𝐵_{1𝐿∞((0,𝑇);Lip[0,1])}+‖𝑣‖_𝐿^∞_{((0,𝑇);Lip[0,1])}+‖𝑤‖_𝐿^∞_{((0,𝑇);Lip[0,1])}+‖𝑢‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑢_{𝐿∞(Ω𝑇)}²+‖𝑣‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑣_{𝐿∞(Ω𝑇)}²+‖𝑤‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑤_{𝐿∞(Ω𝑇)}²×1+2𝑝+𝐺∗𝐵_{1𝐿∞((0,𝑇);Lip[0,1])}+‖𝑣‖_𝐿^∞_{((0,𝑇);Lip[0,1])}+‖𝑤‖_𝐿^∞_{((0,𝑇);Lip[0,1])}𝑇×exp2𝑇(𝑝+𝐺∗𝐵₁)_{𝑥𝐿∞(Ω𝑇)}+𝑣_{𝑥𝐿∞(Ω𝑇)}+𝑤_{𝑥𝐿∞(Ω𝑇)},𝜕_𝑡𝑛_{𝐶0(Ω𝑇)}≤2max𝑛_0𝐿∞,𝑛_𝑙𝐿∞,𝑛_𝑟𝐿∞×𝑞+𝐺∗𝐵_{2𝐿∞((0,𝑇);Lip[0,1])}+‖𝑢‖_𝐿^∞_{((0,𝑇);Lip[0,1])}+‖𝑤‖_𝐿^∞_{((0,𝑇);Lip[0,1])}+‖𝑢‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑢_{𝐿∞(Ω𝑇)}²+‖𝑣‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑣_{𝐿∞(Ω𝑇)}²+‖𝑤‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑤_{𝐿∞(Ω𝑇)}²×1+2𝑞+𝐺∗𝐵_{2𝐿∞((0,𝑇);Lip[0,1])}+‖𝑢‖_𝐿^∞_{((0,𝑇);Lip[0,1])}+‖𝑤‖_𝐿^∞_{((0,𝑇);Lip[0,1])}𝑇×exp2𝑇(𝑞+𝐺∗𝐵₂)_{𝑥𝐿∞(Ω𝑇)}+𝑝_{𝑥𝐿∞(Ω𝑇)}+𝑤_{𝑥𝐿∞(Ω𝑇)},𝜕_𝑡𝑙_{𝐶0(Ω𝑇)}≤2max𝑙_0𝐿∞,𝑙_𝑙𝐿∞,𝑙_𝑟𝐿∞×𝑟+𝐺∗𝐵_{3𝐿∞((0,𝑇);Lip[0,1])}+‖𝑢‖_𝐿^∞_{((0,𝑇);Lip[0,1])}+‖𝑣‖_𝐿^∞_{((0,𝑇);Lip[0,1])}+‖𝑢‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑢_{𝐿∞(Ω𝑇)}²+‖𝑣‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑣_{𝐿∞(Ω𝑇)}²+‖𝑤‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑤_{𝐿∞(Ω𝑇)}²×1+2𝑟+𝐺∗𝐵_{3𝐿∞((0,𝑇);Lip[0,1])}+𝑝_{𝐿∞((0,𝑇);Lip[0,1])}+𝑞_{𝐿∞((0,T);Lip[0,1])}𝑇×exp2𝑇𝑟+𝐺∗𝐵_{3𝑥𝐿∞(Ω𝑇)}+𝑢_{𝑥𝐿∞(Ω𝑇)}+𝑣_{𝑥𝐿∞(Ω𝑇)}. (19)

Definition 9. We can define operator 𝐸 and a domain for the system (8) by:  forall𝑝,𝑞,𝑟∈𝐿^∞((0,𝑇);𝐶^1,1([0,1]))∩Lip([0,𝑇];𝐻¹₀(0,1)), 𝐸𝑝=𝑝∈𝐿^∞(0,𝑇);𝐶^1,1[]([0,1])∩Lip0,𝑇;𝐻¹₀(),𝐸𝑞=0,1𝑞∈𝐿^∞(0,𝑇);𝐶^1,1[]([0,1])∩Lip0,𝑇;𝐻¹₀(),𝐸(𝑟)=̃0,1𝑟∈𝐿^∞(0,𝑇);𝐶^1,1[]([0,1])∩Lip0,𝑇;𝐻¹₀(),𝐶0,1_{𝑀0,𝑀1,𝑇}=𝑝,𝑞,𝑟∈𝐿^∞(0,𝑇);𝐶^1,1[]([0,1])∩Lip0,𝑇;𝐻¹₀(0,1)∣‖𝑑‖_𝐿^∞_{((0,𝑇);𝐶}^1,1_[0,1])≤𝑀₀,‖𝑑‖_{Lip((0,𝑇);𝐻}¹_0(0,1))≤𝑀₁, (20) where 𝑑()𝑝()()(),()∈[]×[].𝑡,𝑥=max𝑡,𝑥,𝑞𝑡,𝑥,𝑟𝑡,𝑥𝑡,𝑥0,𝑇0,1 (21)

Obviously 𝐶_{𝑀0,𝑀1,𝑇} is convex and 𝐶_{𝑀0,𝑀1,𝑇} is compact with respect to the norm ‖⋅‖_𝐿^∞_{((0,𝑇);Lip([0,1]))}. We will endow 𝐶_{𝑀0,𝑀1,𝑇} with the norm ‖⋅‖_𝐿^∞_{((0,𝑇);Lip([0,1]))}. There exist positive numbers 𝑀₀,𝑀₁, and 𝑇 such that 𝐸 maps 𝐶_{𝑀0,𝑀1,𝑇} into itself.

Theorem 10. There exists 𝑇>0, and (𝑚,𝑛,𝑙) is a weak solution of (8) with 𝑝,𝑞,𝑟∈𝐿^∞((0,𝑇);𝐶^1,1([0,1]))∩Lip([0,𝑇];𝐻¹₀(0,1)) and 𝑚,𝑛,𝑙∈𝐿^∞(Ω_𝑇). Moreover, any such solution (𝑝,𝑞,𝑟) is in fact in 𝐶⁰([0,𝑇];𝑊^2,𝑃(0,1))∩𝐶¹([0,𝑇];𝑊₀^1,𝑝(0,1)),forall𝑝<+∞. Furthermore, the existence time of a maximal solution 𝑇≥min(𝑇^∗,𝑇), with 𝑇^∗=max_{𝛼>0,𝛽>0}16𝛼ln𝛼−𝐶₁4((1𝛼,1+cosh))𝑇=max_{𝛼>0,𝛽>0}1𝛽6𝛼ln12𝛼²,𝐶₁(1)=maxcosh(1)𝑣sinh^{𝑟𝑖𝐿∞(0,𝑇)}+𝑣_{𝑙𝑖𝐿∞(0,𝑇)},𝑖=1,2,3. (22)

Proof. For 𝑇>0, we consider 𝑚_𝑙,𝑚_𝑟,𝑛_𝑙,𝑛_𝑟,𝑙_𝑙, and 𝑙_𝑟 in 𝐶⁰([0,𝑇]) such that the sets 𝑃_𝑙 and 𝑃_𝑟 have only a finite number of connected components.
Let 𝐶₀=max(‖𝑚_𝑖‖_𝐿∞(0,1),‖𝑛_𝑖‖_{𝐿∞(Γ𝑙)},‖𝑙_𝑖‖_{𝐿∞(Γ𝑟)}), where 𝑖=0,𝑟,𝑙 and 𝐶₁(1)=maxcosh(1)𝑣sinh^{𝑟𝑖𝐿∞(0,𝑇)}+𝑣_{𝑙𝑖𝐿∞(0,𝑇)},𝑖=1,2,3. (23) Now, if 𝑢,𝑣,𝑤∈𝐶_{𝑀0,𝑀1,𝑇} (see (21)), we have 2𝑝+𝐺∗𝐵_1𝑥+𝑣_𝑥+𝑤_{𝑥𝐶∞(Ω𝑇)}≤2𝜕_𝑥𝑢_{𝐶∞(Ω𝑇)}+𝜕_𝑥𝑣_{𝐶∞(Ω𝑇)}+𝜕_𝑥𝑤_{𝐶∞(Ω𝑇)},2𝑞+𝐺∗𝐵_2𝑥+𝑢_𝑥+𝑤_{𝑥𝐶∞(Ω𝑇)}≤2𝜕_𝑥𝑢_{𝐶∞(Ω𝑇)}+𝜕_𝑥𝑣_{𝐶∞(Ω𝑇)}+𝜕_𝑥𝑤_{𝐶∞(Ω𝑇)},2𝑟+𝐺∗𝐵_3𝑥+𝑢_𝑥+𝑣_{𝑥𝐶∞(Ω𝑇)}≤2𝜕_𝑥𝑢_{𝐶∞(Ω𝑇)}+𝜕_𝑥𝑣_{𝐶∞(Ω𝑇)}+𝜕_𝑥𝑤_{𝐶∞(Ω𝑇)}. (24) For all 𝑢,𝑣,𝑤∈𝐿^∞((0,𝑇);𝑊^2,∞(0,1)), we have 𝜕_𝑥𝑢_{𝐿∞(Ω𝑇)}≤2‖𝑢‖_𝐿^∞_(Ω𝑇)𝜕²_𝑥𝑥𝑢_{𝐿∞(Ω𝑇)}≤‖𝑢‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑢_{𝐿∞(Ω𝑇)},𝜕_𝑥𝑣_{𝐿∞(Ω𝑇)}≤2‖𝑣‖_𝐿^∞_(Ω𝑇)𝜕²_𝑥𝑥𝑣_{𝐿∞(Ω𝑇)}≤‖𝑣‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑣_{𝐿∞(Ω𝑇)},𝜕_𝑥𝑤_{𝐿∞(Ω𝑇)}≤2‖𝑤‖_𝐿^∞_(Ω𝑇)𝜕²_𝑥𝑥𝑤_{𝐿∞(Ω𝑇)}≤‖𝑤‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑤_{𝐿∞(Ω𝑇)},𝑛𝑣_𝑥+𝑙𝑤_{𝑥𝐿∞(0,1)}≤‖𝑢‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑢_{𝐿∞(Ω𝑇)}²+‖𝑣‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑣_{𝐿∞(Ω𝑇)}²+‖𝑤‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑤_{𝐿∞(Ω𝑇)}²,𝑚𝑢_𝑥+𝑙𝑤_{𝑥𝐿∞(0,1)}≤‖𝑢‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑢_{𝐿∞(Ω𝑇)}²+‖𝑣‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑣_{𝐿∞(Ω𝑇)}²+‖𝑤‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑤_{𝐿∞(Ω𝑇)}²,𝑚𝑢_𝑥+𝑛𝑣_{𝑥𝐿∞(0,1)}≤‖𝑢‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑢_{𝐿∞(Ω𝑇)}²+‖𝑣‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑣_{𝐿∞(Ω𝑇)}²+‖𝑤‖_𝐿^∞_(Ω𝑇)+𝜕²_𝑥𝑥𝑤_{𝐿∞(Ω𝑇)}². (25) We also define that 𝑑(𝑡,𝑥)=max(𝑝(𝑡,𝑥),𝑞(𝑡,𝑥),𝑟(̃𝑡,𝑥)),(𝑡,𝑥)∈Ω_𝑇. If 𝑝,𝑞,𝑟∈𝐶_{𝑀0,𝑀1,𝑇}, then from Lemmas 2 and 8, we derive that 𝑑_{𝐿∞((0,𝑇);𝐶1,1[0,1])}≤2(1+cosh(1))𝐶₀+12𝑇^∗𝑀₀+𝐶₁²×exp6𝑇^∗𝑀₀+𝐶₁,𝜕_𝑡𝑑_{𝐶0(Ω𝑇)}≤6𝐶₀𝑀₀+𝐶₁+12𝑀₀+2𝐶₁²×1+6𝑀₀+𝐶₁𝑇exp6𝑇𝑀₀+𝐶₁. (26)
Finally, to obtain 𝑝,𝑞,𝑟̃∈𝐶_{𝑀0,𝑀1,𝑇}, it is sufficient to show that 2(1+cosh(1))𝐶₀+12𝑇^∗𝑀₀+𝐶₁²×exp6𝑇^∗𝑀₀+𝐶₁≤𝑀₀,𝑀₀+6𝐶₀𝑀₀+𝐶₁+12𝑀₀+𝐶₁²×1+6𝑇𝑀₀+𝐶₁×exp6𝑇𝑀₀+𝐶₁≤𝑀₁, (27) if we have chosen 𝑇 and 𝑀₀; it is easy to choose 𝑀₁ to satisfy the second inequality. For the above two inequalities, we just choose 𝑀₀ and 𝑀₁ sufficiently large and then 𝑇 close to 0. More precisely: 𝑀₀>2(1+cosh(1))𝐶₀,𝑇^∗≤16𝑀₀+𝐶₁𝑀ln⁰4(1+cosh(1))𝑀₀+𝐶₁,1𝑇≤6𝑀₀+𝐶₁𝑀ln¹−𝑀₀12𝑀₀+𝐶₁². (28)
Maximizing the bound of 𝑇, we can get minimum existence. Then, we get the result announced, where 𝛼=𝑀₀+𝐶₁, 𝛽=𝑀₁−𝑀₀.

Lemma 11. The operator 𝐸∶𝐶_{𝑀0,𝑀1,𝑇}→𝐶_{𝑀0,𝑀1,𝑇} is continuous with respect to ‖⋅‖_𝐿^∞_{((0,𝑇);Lip[0,1])}.

Proof. The proof is omitted here; one can see a similar proof in [8, Proposition 2.4].
Now, we can apply Shauder's fixed point theorem to the operator 𝐸, and we get the result that there exist fixed points 𝑝,𝑞,𝑟 such that 𝐸(𝑝)=𝑝, 𝐸(𝑞)=𝑞, and 𝐸(𝑟)=𝑟, so we know that there exists a wake solution of (9). 𝑝,𝑞,𝑟∈𝐿^∞(0,𝑇);𝐶^1,1([0,1])∩Lip[0,𝑇];𝐻¹₀(0,1). (29)

2.1. Uniqueness

We will prove the weak-strong uniqueness of weak solution of (8) in the following.

Theorem 12. Let (𝑝,𝑚),(𝑞,𝑛),(𝑟,𝑙)∈𝐿^∞((0,𝑇);𝐶^1,1([0,1]))∩Lip([0,𝑇];𝐻¹₀(0,1)×𝐿^∞((0,𝑇);Lip([0,1])) be the weak solution of (7)-(8), then it is unique in 𝐿^∞((0,𝑇);𝐶^1,1([0,1]))×𝐿^∞(Ω_𝑇).

Proof. Define Φ=𝑚−𝑚,  Ψ=𝑛−𝑛,  Υ=𝑙−𝑙̃ and 𝑃=𝑝−𝑝,  𝑄=𝑞−𝑞,  𝐻=𝑟−𝑟̃, then we have 𝑃()(),𝑄()(),H()(),𝑡,⋅=𝐺∗Φ𝑡,⋅𝑡,⋅=𝐺∗Ψ𝑡,⋅𝑡,⋅=𝐺∗Υ𝑡,⋅ (30) where 𝑃,𝑄,𝐻∈Lip([0,𝑇];𝐻¹₀(0,1)), and (Φ,Ψ,Υ)∈𝐿^∞(Ω_𝑇)×𝐿^∞(Ω_𝑇)×𝐿^∞(Ω_𝑇) is the unique weak solution of Φ_𝑡=𝑝+𝐺∗𝐵+𝑣+𝑤Φ_𝑥+2𝑝+𝐺∗𝐵_1𝑥+𝑣_𝑥+𝑤_𝑥Φ+(𝑃+𝑄+𝐻)𝑚_𝑥+2𝑃_𝑥+𝑄_𝑥+𝐻_𝑥𝑚+Ψ𝑣_𝑥+𝑛𝑄_𝑥+Υ𝑤_𝑥+𝑙𝐻̃_𝑥,Ψ_𝑡=𝑞+𝐺∗𝐵+𝑢+𝑤Ψ_𝑥+2𝑞+𝐺∗𝐵_2𝑥+𝑢_𝑥+𝑤_𝑥Ψ+(𝑃+𝑄+𝐻)𝑛_𝑥+𝑃_𝑥+2𝑄_𝑥+𝐻_𝑥𝑛+Φ𝑢_𝑥+𝑚Φ_𝑥+Υ𝑤_𝑥+𝑙𝐻̃_𝑥,Υ_𝑡=(𝑟+𝐺∗𝐵+𝑢+𝑣)Υ_𝑥+2𝑟+𝐺∗𝐵_3𝑥+𝑢_𝑥+𝑣_𝑥Υ+(𝑃+𝑄+𝐻)𝑙_𝑥̃+𝑃_𝑥+𝑄_𝑥+2𝐻_𝑥𝑙̃+Φ𝑢_𝑥+𝑚Φ_𝑥+Ψ𝑣_𝑥+𝑛𝑄_𝑥. (31)
Let 𝑏₁=2𝑝+𝐺∗𝐵_1𝑥+𝑣_𝑥+𝑤_𝑥,𝑏₂=2𝑞+𝐺∗𝐵_2𝑥+𝑢_𝑥+𝑤_𝑥,𝑏₃=2𝑟+𝐺∗𝐵_3𝑥+𝑢_𝑥+𝑣_𝑥,𝑓₁=(𝑃+𝑄+𝐻)𝑚_𝑥+2𝑃_𝑥+𝑄_𝑥+𝐻_𝑥𝑚+Ψ𝑣_𝑥+𝑛𝑄_𝑥+Υ𝑤_𝑥+𝑙𝐻̃_𝑥,𝑓₂=(𝑃+𝑄+𝐻)𝑛_𝑥+𝑃_𝑥+2𝑄_𝑥+𝐻_𝑥𝑛+Φ𝑢_𝑥+𝑚Φ_𝑥+Υ𝑤_𝑥+𝑙𝐻̃_𝑥,𝑓₃=(𝑃+𝑄+𝐻)𝑙_𝑥̃+𝑃_𝑥+𝑄_𝑥+2𝐻_𝑥𝑙̃+Φ𝑢_𝑥+𝑚Φ_𝑥+Ψ𝑣_𝑥+𝑛𝑄_𝑥, (32) with 𝑖₀=0,  𝑖_𝑙=0, and  𝑖_𝑟=0, where 𝑖=Φ,Ψ,Υ.
For (𝑡,𝑥)∈𝑃, we have Φ(𝑡,𝑥)=0,Ψ(𝑡,𝑥)=0, and Υ(𝑡,𝑥)=0.
Then, we get the uniqueness result.
For (𝑡,𝑥)∈𝐼, we have Φ(𝑡,𝑥)=^𝑡₀𝑓₁(𝑠,𝜔(𝑠,𝑡,𝑥))×exp^𝑡_𝑟𝑏₁𝑠^′,𝜔𝑠^′,𝑡,𝑥𝑑𝑠^′Ψ()=𝑑𝑠,𝑡,𝑥^𝑡₀𝑓₂(𝑠,𝜔(𝑠,𝑡,𝑥))×exp^𝑡_𝑟𝑏₂𝑠^′,𝜔𝑠^′,𝑡,𝑥𝑑𝑠^′Υ()=𝑑𝑠,𝑡,𝑥^𝑡₀𝑓₃(𝑠,𝜔(𝑠,𝑡,𝑥))×exp^𝑡_𝑟𝑏₃𝑠^′,𝜔𝑠^′,𝑡,𝑥𝑑𝑠^′𝑑𝑠. (33)
For (𝑡,𝑥)∈𝐿, we have Φ(𝑡,𝑥)=^𝑡_{𝑒()𝑡,𝑥}𝑓₁(𝑠,𝜔(𝑠,𝑡,𝑥))×exp^𝑡_𝑟𝑏₁𝑠^′,𝜔𝑠^′,𝑡,𝑥𝑑𝑠^′Ψ()=𝑑𝑠,𝑡,𝑥^𝑡_{𝑒()𝑡,𝑥}𝑓₂(𝑠,𝜔(𝑠,𝑡,𝑥))×exp^𝑡_𝑟𝑏₂𝑠^′,𝜔𝑠^′,𝑡,𝑥𝑑𝑠^′Υ()=𝑑𝑠,𝑡,𝑥^𝑡_{𝑒()𝑡,𝑥}𝑓₃(𝑠,𝜔(𝑠,𝑡,𝑥))×exp^𝑡_𝑟𝑏₃𝑠^′,𝜔𝑠^′,𝑡,𝑥𝑑𝑠^′𝑑𝑠. (34)
For (𝑡,𝑥)∈𝑅, we have Φ(𝑡,𝑥)=^𝑡_{𝑒()𝑡,𝑥}𝑓₁(𝑠,𝜔(𝑠,𝑡,𝑥))×exp^𝑡_𝑟𝑏₁𝑠^′,𝜔𝑠^′,𝑡,𝑥𝑑𝑠^′Ψ()=𝑑𝑠,𝑡,𝑥^𝑡_{𝑒()𝑡,𝑥}𝑓₂(𝑠,𝜔(𝑠,𝑡,𝑥))×exp^𝑡_𝑟𝑏₂𝑠^′,𝜔𝑠^′,𝑡,𝑥𝑑𝑠^′Υ()=𝑑𝑠,𝑡,𝑥^𝑡_{𝑒()𝑡,𝑥}𝑓₃(𝑠,𝜔(𝑠,𝑡,𝑥))×exp^𝑡_𝑟𝑏₃𝑠^′,𝜔𝑠^′,𝑡,𝑥𝑑𝑠^′𝑑𝑠. (35)
Now since ‖𝑃(𝑡,⋅)‖_𝐿^∞_(0,1)≤5‖Φ(𝑡,⋅)‖_𝐿^∞_(0,1), ‖𝑄(𝑡,⋅)‖_𝐿^∞_(0,1)≤5‖Ψ(𝑡,⋅)‖_𝐿^∞_(0,1), ‖H(𝑡,⋅)‖_𝐿^∞_(0,1)≤5‖Υ(𝑡,⋅)‖_𝐿^∞_(0,1) and 𝑚,𝜕_𝑥𝑚,𝑛,𝜕_𝑥𝑛,𝑙,̃𝜕_𝑥𝑙̃ bounded, we see that for some 𝜆₁>0, 𝜆₂>0,  𝜆₃>0, 𝑓₁(𝑡,⋅)_𝐿∞(0,1)≤𝜆₁‖Φ(𝑡,⋅)‖_𝐿^∞_(0,1)+‖Ψ(𝑡,⋅)‖_𝐿^∞_(0,1)+‖Υ(𝑡,⋅)‖_𝐿^∞_(0,1),𝑓₂(𝑡,⋅)_𝐿∞(0,1)≤𝜆₂‖Φ(𝑡,⋅)‖_𝐿^∞_(0,1)+‖Ψ(𝑡,⋅)‖_𝐿^∞_(0,1)+‖Υ(𝑡,⋅)‖_𝐿^∞_(0,1),𝑓₃(𝑡,⋅)_𝐿∞(0,1)≤𝜆₃‖Φ(𝑡,⋅)‖_𝐿^∞_(0,1)+‖Ψ(𝑡,⋅)‖_𝐿^∞_{(0,1)+‖Υ(𝑡,⋅)‖𝐿∞(0,1)}, (36) since 𝑝,𝑞,𝑟, and 𝐵_𝑖,(𝑖=1,2,3) are bounded, we get that for some 𝜆^′₁>0, 𝜆^′₂>0, 𝜆^′₃>0, ‖Φ(𝑡,⋅)‖_𝐿^∞_(0,1)≤𝜆^′₁^𝑡₀‖Φ(𝑠,⋅)‖_𝐿^∞_(0,1)+‖Ψ(𝑠,⋅)‖_𝐿^∞_(0,1)+‖Υ(𝑠,⋅)‖_𝐿^∞_(0,1)‖Ψ(𝑑𝑠,𝑡,⋅)‖_𝐿^∞_(0,1)≤𝜆^′₂^𝑡₀‖Φ(𝑠,⋅)‖_𝐿^∞_(0,1)+‖Ψ(𝑠,⋅)‖_𝐿^∞_(0,1)+‖Υ(𝑠,⋅)‖_𝐿^∞_(0,1)‖Υ(𝑑𝑠,𝑡,⋅)‖_𝐿^∞_(0,1)≤𝜆^′₃^𝑡₀‖Φ(𝑠,⋅)‖_𝐿^∞_(0,1)+‖Ψ(𝑠,⋅)‖_𝐿^∞_(0,1)+‖Υ(𝑠,⋅)‖_𝐿^∞_(0,1)𝑑𝑠. (37) We can obtain that ‖Φ(𝑡,⋅)‖_𝐿^∞_(0,1)+‖Ψ(𝑡,⋅)‖_𝐿^∞_(0,1)+‖Υ(𝑡,⋅)‖_𝐿^∞_(0,1)≤𝜆^′₁+𝜆^′₂+𝜆^′₃×^𝑡₀‖Φ(𝑠,⋅)‖_𝐿^∞_(0,1)+‖Ψ(𝑠,⋅)‖_𝐿^∞_(0,1)+‖Υ(𝑠,⋅)‖_𝐿^∞_(0,1)𝑑𝑠. (38)
As a result, we get the result of the uniqueness by Gronwall's inequality when (𝑡,𝑥)∈𝐼, (𝑡,𝑥)∈𝐿, (𝑡,𝑥)∈𝑅. Then, we complete the proof of the uniqueness results.

3. Asymptotic Stabilization

3.1. Preliminary Results

The equilibrium state that we want to stabilize is 𝑚=𝑛=𝑙=0, 𝑝=𝑞=𝑟=𝐺∗𝐵_𝑖=0, and 𝑖=1,2,3. 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. However, there is a difficulty in the stabilization problem. We have to prescribe 𝑦_𝑙, and we just need to make a continuous transition at (𝑡,𝑥)=(0,0), and that 𝑦_𝑙 asymptotically converge in time. For convenience, the system (6)–(8) can rewrite in the following 𝜕_𝑡𝑦−𝑎̆(𝑡,𝑥)𝜕_𝑥𝑏̆()𝑓̆(),𝑦()𝑦=𝑡,𝑥𝑦+𝑡,𝑥0,⋅=𝑦₀,𝑦(⋅,0)=𝑦_𝑙,𝑦(1,⋅)=𝑦_𝑟,𝑔̆()(),𝑔̆()=𝑔̆()𝐵̆()𝐵̆()𝑡,𝑥=𝐺∗𝑦𝑡,𝑥𝑡,0𝑡,1=0,𝑡,𝑥=0,𝐺∗𝑡,0=𝑣_𝑙(𝑡),𝐺∗𝐵̆(𝑡,1)=𝑣_𝑟(𝑡), (39) where 𝑏̆()=2𝐵̆𝑡,𝑥𝑝+𝐺∗_𝑥+𝑣_𝑥+𝑤_𝑥002𝐵̆00𝑞+𝐺∗_𝑥+𝑢_𝑥+𝑤_𝑥002𝑟+𝐺∗𝐵̆_𝑥+𝑢_𝑥+𝑣_𝑥,𝑎̆()=̆00̆00𝐵̆,𝑚𝑛𝑙𝑡,𝑥𝑝+𝐺∗𝐵+𝑣+𝑤00𝑢+𝑞+𝐺∗𝐵+𝑤𝑢+𝑣+𝑟+𝐺∗𝑦=,𝑦₀=𝑚₀𝑛₀𝑙₀,𝑓̆(𝑡,𝑥)=𝑛𝑣_𝑥+𝑙𝑤_𝑥𝑚𝑢_𝑥+𝑙𝑤_𝑥𝑚𝑢_𝑥+𝑛𝑣_𝑥=0𝑛𝑙𝑢𝑚0𝑙𝑚𝑛0^𝑥𝑣_𝑥𝑤_𝑥,𝑔̆()=𝑝𝑞𝑟𝑡,𝑥,𝑦_𝑙=𝑚_𝑙𝑛_𝑙𝑙_𝑙,𝑦_𝑟=𝑚_𝑟𝑛_𝑟𝑙_𝑟. (40)

Our feedback law for (3) reads 𝑦∈𝐶⁰↦𝑣([0,1])^𝑙𝑦=𝐴_𝑙𝑦_𝐶0([0,1]),𝑣_𝑟𝑦=𝐴_𝑟𝑦_𝐶0([0,1]),𝜕_𝑡𝑦_𝑙=𝑀𝑦_𝑙, (41) where 𝐴_𝑙>2sinh(1), 𝐴_𝑟>𝐴_𝑙cosh(1)+sinh(2), 𝑀>0, 𝑇>0, and 𝑀=^𝜇000𝜇0_00𝜇,𝜇<0, a symmetric matrix, is the unique matrix solution to the matrix function: 𝑃𝑀+𝑀^𝑇𝑃=−𝑍, (42) for some symmetric positive-definite matrices 𝑃 and 𝑍. Indeed, let 𝑉(𝑡,𝑦_𝑙)=𝑦^𝑇_𝑙𝑃𝑦_𝑙 be the Lyapunov candidate, and that 𝑦_𝑙 asymptotically converges in time is equivalent to that the time derivative of the 𝑉,𝑉̇(𝑡,𝑦_𝑙)=−𝑦^𝑇_𝑙𝑍𝑦_𝑙 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 13. Let 𝑋 be the space of (𝑔,𝑁)∈𝐶⁰([0,𝑇]×[0,1])×𝐶⁰([0,1]) satisfying(1)forall(𝑡,𝑥)∈[0,𝑇]×[0,1],  𝑔(0,𝑥)=𝑦₀(𝑥),  𝑔(𝑡,0)=𝑦₀(0)𝑒^𝑀𝑡,(2)forall𝑡∈[0,𝑇],  ‖𝑔(𝑡,⋅)‖_𝐶0([0,1])≤𝑁(𝑡),(3)𝑁 is nonincreasing, and 𝑁(0)≤‖𝑦₀‖_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}.

Lemma 14. The domain 𝑋 is nonempty, convex, bounded, and closed with respect to the uniform topology.
The proof is elementary and one notices that (𝑦₀(𝑥)𝑒^𝑀𝑡,‖𝑦₀‖_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}𝑒^𝑀𝑡)∈𝑋, so 𝑋 is nonempty.
Now for (𝑦,𝑁)∈𝑋, we define 𝑔^∨ and 𝐺∗𝐵^∨ as the solutions of ∀(𝑡,𝑥)∈Ω_𝑇,𝑔^∨(𝑡,𝑥)=𝐺∗𝑦(𝑡,𝑥_∨),𝑔(𝑡,0)=𝑔^∨(𝑡,1)=0,𝐵^∨(𝑡,𝑥)=0,𝐺∗𝐵^∨(𝑡,0)=𝐴_𝑙𝑁(𝑡),𝐺∗𝐵^∨(𝑡,1)=𝐴_𝑟𝑁(𝑡). (43) One has the following exact formulas: ∀(𝑡,𝑥)∈Ω_𝑇,_∨𝑔(𝑡,𝑥)=−^𝑥₀(𝑥̆)𝑦(𝑥̆)𝑑𝑥̆−(𝑥)sinh𝑥−𝑡,sinhsinh(1)¹₀sinh(𝑥̆−1)𝑦(𝑡,𝑥̆)𝑑𝑥̆,𝐺∗𝐵^∨()=𝑁(𝑡)𝑡,𝑥(1)𝐴sinh^𝑟sinh(𝑥)+𝐴_𝑙sinh(1−𝑥). (44) Therefore, we have the following inequalities: ∀(𝑡,𝑥)∈[0,𝑇]×[0,1_∨],𝑔(((1𝑦(𝑡,𝑥)≤21+cosh))𝑡,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])},𝜕𝑔_𝑥^∨(𝑡,𝑥)≤2cosh(1)𝑦(𝑡,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])},𝜕^2∨_𝑥𝑥𝑔(≤[2((1))]×𝑦(𝑡,𝑥)cosh+1+1𝑡,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])},𝜕_𝑥𝐺∗𝐵^∨≥𝐴(𝑡,𝑥)^𝑟−2cosh(1)𝐴_𝑙(1)𝑁(𝑡),sinh𝐺∗𝐵^∨(𝑡,𝑥)≥𝐴_𝑙𝑁(𝑡). (45)
Let ^∨𝑐(𝑡,𝑥)=𝐺∗𝐵^∨(𝑡,𝑥)+𝑔^∨(𝑡,𝑥), where ^∨𝑐(𝑡,𝑥)=^𝑢𝑣_𝑤(𝑡,𝑥), and in turn those provide ^∨𝑐()≤2((1+(1)𝑡,𝑥1+cosh))cosh(1)𝐴sinh^𝑟+𝐴_𝑙×𝑦(𝑡,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])},𝜕_𝑥^∨𝑐()≤(2)𝑡,𝑥sinh+2𝐴_𝑙cosh(1)−𝐴_𝑟(1)×𝑦(sinh𝑡,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])},𝜕^2∨_𝑥𝑥𝑐()≤2((1(1)𝑡,𝑥1+cosh))+1+cosh(1)𝐴sinh^𝑟+𝐴_𝑙×𝑦(𝑡,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}. (46) Now, if 𝜔 is the flow of ^∨𝑐, 𝜔 is 𝐶¹, and since ^∨𝑐≥0, 𝜔(⋅,𝑡,𝑥) is nondecreasing. This allows us to define the entrance time and then the operator 𝑆 as follows. Let 𝑒(𝑡,𝑥)=min{𝑠∈[0,𝑡]∣𝜔(𝑠,𝑡,𝑥)=0}.
Now, for forall(𝑡,𝑥)∈[0,𝑇]×[0,1], 𝑆(𝑦,𝑁)=(𝑦,𝑁) with the following:(1)if 𝑥≥𝜔(𝑡,0,0), 𝑦(𝑡,𝑥)=𝑦₀(𝜔(0,𝑡,𝑥))exp𝑏^𝑡₀^∨((+𝑟,𝜔𝑟,𝑡,𝑥))𝑑𝑟_𝑡0∨𝑓(𝑟,𝜔(𝑟,𝑡,𝑥))×exp𝑏^𝑡_r^∨𝑟^′,𝜔𝑟^′,𝑡,𝑥𝑑𝑟^′𝑑𝑟, (47)(2)if 𝑥≤𝜔(𝑡,0,0), 𝑦(𝑡,𝑥)=𝑦₀(0)𝑒^{𝑀𝑒(𝑡,𝑥)}exp𝑏^𝑡₀^∨((+𝑟,𝜔𝑟,𝑡,𝑥))𝑑𝑟_{𝑡𝑒()∨𝑡,𝑥}𝑓(𝑟,𝜔(𝑟,𝑡,𝑥))×exp𝑏^𝑡_𝑟^∨𝑟^′,𝜔𝑟^′,𝑡,𝑥𝑑𝑟^′𝑑𝑟, (48)(3)𝑁(𝑡)=‖𝑦(𝑡,⋅)‖_𝐶0([0,1]).

Lemma 15. (1) The operator 𝑆 maps 𝑋 to 𝑋.
(2) The family 𝑆(𝑋) is uniformly bounded and equicontinuous.
(3) 𝑆 is continuous w.r.t. the uniform topology.
The proof is very similar to [10], except for the state 𝑦 here is a three-component vector and the proof is omitted.
Now, we can apply Schauder’s fixed point theorem to 𝑆 and get (𝑦,𝑁) fixed point of 𝑆.

3.2. Stabilization and Global Existence

Theorem 16. For any 𝑦₀∈𝐶⁰([0,1])×𝐶⁰([0,1])×𝐶⁰([0,1]), there exists 𝑦∈𝐶⁰(Ω_𝑇)×𝐶⁰([0,𝑇],𝐶²([0,1])) a weak solution of (39) satisfying ∀𝑥∈[0,1]𝑦(0,𝑥)=𝑦₀(𝑥). (49)
Furthermore, any maximal solution of (39) and (41) is global, and if we let 2((1(1)𝑘=max1+cosh))+1+cosh(1)Asinh^𝑟+A_𝑙,sinh(2)+2𝐴_𝑙cosh(1)−𝐴_𝑟(1),1sinh𝜏=‖𝑀‖₃‖𝑀‖ln²³8𝑘³𝑦₀²_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}, (50) then we have ∀𝑡≥𝜏𝑦(𝑡,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}≤‖𝑀‖₃‖𝑀‖2𝑘1+2𝑘₃𝑒^{‖𝑀‖3(𝑡−𝑥)}. (51)

To finish the proof of Theorem (39), we have to prove the global existence of a maximal solution and the estimate (51).

Proof . First, we rewrite (46) as the following: ∀(𝑡,𝑥)∈Ω_𝑇^∨𝑐(𝑡,𝑥)≤𝑘𝑦(𝑡,⋅)_𝐶0([0,1]),𝜕_𝑥^∨𝑐(𝑡,𝑥)≤𝑘𝑦(𝑡,⋅)_𝐶0([0,1]),𝜕^∨_𝑥𝑥𝑐(𝑡,𝑥)≤𝑘𝑦(𝑡,⋅)_𝐶0([0,1]), (52) where 𝑘=max(2(1+cosh(1))+1+(cosh(1)/sinh(1))(𝐴_𝑟+𝐴_𝑙),(sinh(2)+2𝐴_𝑙cosh(1)−𝐴_𝑟)/sinh(1)).
For 𝑦 is the solution of the transport (39) and it satisfies 𝑦(𝑡,𝑥)=𝑦(𝑠,𝜔(𝑠,𝑡,𝑥))exp𝑏^𝑡₀^∨(𝑟,𝜔(𝑟,𝑡,𝑥))𝑑𝑟+𝑓^𝑡₀^∨(𝑟,𝜔(𝑟,𝑡,𝑥))×exp𝑏^𝑡_𝑟^∨𝑟^′,𝜔𝑟^′,𝑡,𝑥𝑑𝑟^′𝑑𝑟. (53) Combining those facts, we get for 𝑡≥𝑠 the following: 𝑦(≤𝑦((𝑡,𝑥)𝑠,𝜔𝑠,𝑡,𝑥))1+4𝑘²𝑦((𝑡2𝑠,𝜔𝑠,𝑡,𝑥))×exp^𝑡_𝑠𝑘𝑦(𝑟,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}𝑑𝑟. (54) We have also imposed 𝑦(𝑡,0)=𝑦(𝑠,0)𝑒^{𝑀(𝑡−𝑠)} and 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 between time 𝑡 and 𝑠, and we obtain.
For 0≤𝑠≤𝑡, 𝑦(𝑡,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}≤𝑦(𝑠,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}×max𝑒^{‖𝑀‖3(𝑟−𝑥)}1+4𝑘²𝑦(𝑠,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}𝑟×exp2𝑘^𝑡_𝑟𝑦(𝛼,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}𝑑𝛼. (55)
We define 𝑔(𝑟)=𝑒^{‖𝑀‖3(𝑟−𝑠)}1+4𝑘²𝑦(𝑟,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}𝑟×exp2𝑘^𝑡_𝑠𝑦(𝛼,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}𝑑𝛼, (56) and we set 𝑔(𝑟)=𝑔₁(𝑟)+𝑔₂(𝑟), where 𝑔₁(𝑟)=𝑒^{‖𝑀‖3(𝑟−𝑠)}×exp2𝑘^𝑡_𝑟𝑦(𝛼,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])},𝑔₂(𝑟)=𝑒^{‖𝑀‖3(𝑟−𝑠)}4𝑘²𝑦(𝑟,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}𝑟×exp2𝑘^𝑡_𝑟𝑦(𝛼,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}. (57)
Then, we have 𝑔^′(𝑟)=‖𝑀‖₃−2𝑘𝑦(𝑟,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}𝑔₁(𝑟)+1𝑟+‖𝑀‖₃−2𝑘𝑦(𝑟,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}×𝑔₂(𝑟), (58) as long as the quantity ‖𝑦(𝑟,⋅)‖_𝐶0([0,1]) is not equal to zero, it strictly decreases, so if ‖𝑦₀‖_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}>‖𝑀‖₃/2𝑘, for 𝑡 small enough ‖𝑦(𝑡,⋅)‖_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}≥‖𝑀‖₃/2𝑘, and we have the following. 𝑦(𝑡,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}≤𝑦_{0𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}×1+4𝑘²𝑦_{0𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}𝑡e^{‖𝑀‖3𝑡}. (59)
If we define 𝜏=(1/‖𝑀‖₃)      ln(‖𝑀‖²₃/8𝑘³‖𝑦₀‖²_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}), we get that 𝑦(𝜏,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}≤‖𝑀‖₃.2𝑘 (60)
This provides 𝜏≤𝑠≤𝑡, the inequality (which was clear when ‖𝑦₀‖_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}≤‖𝑀‖₃/2𝑘)𝑦(𝑡,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}≤𝑦(𝜏,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}×1+4𝑘²𝑦(𝜏,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}𝑡×exp2𝑘^𝑡_𝑥𝑦(𝑟,⋅)_{𝐶0([0,1])×𝐶0([0,1])×𝐶0([0,1])}≤‖𝑀‖𝑑𝑟₃‖𝑀‖2𝑘1+2𝑘₃𝑒^{‖𝑀‖3(𝑡−x)}. (61)

4. Blow-Up Phenomena

In this section, we present a result with the initial data and boundary profiles under a special condition that ensure strong solutions to following system blow-up in finite time as follows: 𝜕_𝑡𝑚−(𝑢+𝑣+𝑤)(𝑡,𝑥)𝜕_𝑥𝑚=2𝑝+𝐺∗𝐵+𝑣_𝑥+𝑤_𝑥()𝑚+𝑡,𝑥𝑛𝑣_𝑥+𝑙𝑤_𝑥(),𝜕𝑡,𝑥_𝑡𝑛−(𝑢+𝑣+𝑤)(𝑡,𝑥)𝜕_𝑥𝑛=2𝑞+𝐺∗𝐵+𝑢_𝑥+𝑤_𝑥()𝑛+𝑡,𝑥𝑚𝑢_𝑥+𝑙𝑤_𝑥),𝜕(𝑡,𝑥_𝑡𝑙−(𝑢+𝑣+𝑤)(𝑡,𝑥)𝜕_𝑥𝑙=2(𝑟+𝐺∗𝐵)+𝑢_𝑥+𝑣_𝑥()𝑙+𝑡,𝑥𝑚𝑢_𝑥+𝑛𝑣_𝑥(),𝑚()𝑡,𝑥0,⋅=𝑚₀,𝑛(0,⋅)=𝑛₀,𝑙(0,⋅)=𝑙₀[],𝑚(),𝑥∈0,1𝑡,0=𝑚_𝑙=𝑚(𝑡,1)=𝑚_𝑟[],𝑛(),𝑡∈0,𝑇𝑡,0=𝑛_𝑙=𝑛(𝑡,1)=𝑛_𝑟[],𝑙(),𝑡∈0,𝑇𝑡,0=𝑙_𝑙=𝑙(𝑡,1)=𝑙_𝑟,𝑡∈[0,𝑇], (62) where 𝑚=𝑢−𝑢_𝑥𝑥,  𝑛=𝑣−𝑣_𝑥𝑥,  𝑙=𝑤−𝑤_𝑥𝑥. This imply that 𝑢()(),𝑣()(),𝑤()().𝑡,0=𝑢𝑡,1𝑡,0=𝑣𝑡,1𝑡,0=𝑤𝑡,1 (63)

From Definition 3, we can also define 𝜔_𝑡(),()∈[]×[],𝜔()[];=−𝑢+𝑣+𝑤𝑡,𝜔𝑡,𝑦𝑡,𝑥0,𝑇0,10,𝑥1=𝑥,𝑥∈0,1 (64) where 𝑢,𝑣, and 𝑤 denote the solution to (62). Applying classical results in the theory of ordinary differential equations, one can obtain a result on which is crucial in studying blow-up phenomena.

From (62), we obtain that 𝑚_𝑡−𝜕_𝑥(𝑚𝑢+𝑚𝑣+𝑚𝑤)=𝑚𝑢_𝑥+𝑛𝑣_𝑥+𝑙𝑤_𝑥,𝑛_𝑡−𝜕_𝑥(𝑚𝑢+𝑛𝑣+𝑚𝑤)=𝑚𝑢_𝑥+𝑛𝑣_𝑥+𝑙𝑤_𝑥,𝑙_𝑡−𝜕_𝑥(𝑙𝑢+𝑙𝑣+𝑙𝑤)=𝑚𝑢_𝑥+𝑛𝑣_𝑥+𝑙𝑤_𝑥. (65)

Lemma 17. Let 𝑢,𝑣,𝑤∈𝐶([0,𝑇);𝐻^𝑠)∩𝐶¹([0,𝑇);𝐻^𝑠−1),(𝑠≥2), then (64) has a unique solution 𝜔∈𝐶([0,𝑇)×[0,1]). Moreover, the map 𝜔(𝑡,⋅) is an increasing diffeomorphism with 𝜔_𝑥(𝑡,𝑥)=exp−^𝑡₀𝑢_𝑥(𝑠,𝜔(𝑠,𝑥))+𝑣_𝑥(𝑠,𝜔(𝑠,𝑥))+𝑤_𝑥((𝜔𝑠,𝜔𝑠,𝑥))𝑑𝑠>0,_𝑥(0,𝑥)=1,∀𝑥∈[0,1]. (66)

Proof. The proof is omitted here, one can see a similar proof in [12].

Now, we have the following lemma that the potential 𝑚−𝑛,  𝑛−𝑙,  𝑚−𝑙 with compactly supported initial datum 𝑚₀−𝑛₀,  𝑛₀−𝑙₀,  𝑚₀−𝑙₀ also has compact 𝑥 support as long as it exists.

Lemma 18. Assume that 𝑢₀(𝑡,𝑥),𝑣₀(𝑡,𝑥),𝑤₀(𝑡,𝑥)∈𝐻^𝑠×𝐻^𝑠 with 𝑠>(3/2), (𝑢,𝑣,𝑤) is the corresponding solution, if 𝑚₀−𝑛₀,𝑛₀−𝑙₀,𝑚₀−𝑙₀ has compact support, then 𝑚−𝑛,𝑛−𝑙,𝑚−𝑙 also has compact support, moreover, we can obtain that ‖𝑚(𝑡,⋅)−𝑛(𝑡,⋅)‖≤𝑒^3𝐾𝑇𝑚₀(⋅)−𝑛₀(⋅,‖𝑚()()𝑡,⋅−𝑙𝑡,⋅)‖≤𝑒^3𝐾𝑇𝑚₀(⋅)−𝑙₀(⋅,‖𝑛()()𝑡,⋅−𝑙𝑡,⋅)‖≤𝑒^3𝐾𝑇𝑛₀(⋅)−𝑙₀(⋅). (67)

Proof. Since 𝑑𝑚((𝜔𝑑𝑡𝑡,𝜔𝑡,𝑥))_𝑥=𝑚_𝑡+𝑚_𝑥𝜔_𝑡𝜔_𝑥+𝑚𝜔_𝑥𝑡=𝑚_𝑡−𝜕_𝑥(𝑚𝑢+𝑚𝑣+𝑚𝑤)𝜔_𝑥=𝑚𝑢_𝑥+𝑛𝑣_𝑥+𝑙𝑤_𝑥𝜔_𝑥. (68)
Similarly, 𝑑𝑛((𝜔𝑑t𝑡,𝜔𝑡,𝑥))_𝑥=𝑚𝑢_𝑥+𝑛𝑣_𝑥+𝑙𝑤_𝑥𝜔_𝑥,𝑑𝑙((𝜔𝑑𝑡𝑡,𝜔𝑡,𝑥))_𝑥=𝑚𝑢_𝑥+𝑛𝑣_𝑥+𝑙𝑤_𝑥𝜔_𝑥. (69)
So it follows that 𝑑(𝑚((((𝜔𝑑𝑡𝑡,𝜔𝑡,𝑥))−𝑛𝑡,𝜔𝑡,𝑥)))_𝑥𝑑=0,(𝑚((((𝜔𝑑𝑡𝑡,𝜔𝑡,𝑥))−𝑙𝑡,𝜔𝑡,𝑥)))_𝑥𝑑=0,(𝑛((((𝜔𝑑𝑡𝑡,𝜔𝑡,𝑥))−𝑙𝑡,𝜔𝑡,𝑥)))_𝑥=0. (70)
We obtain [𝑚(𝑡,𝜔(𝑡,𝑥))−𝑛(𝑡,𝜔(𝑡,𝑥))]𝜔_𝑥=𝑚₀(𝑥)−𝑛₀(𝑥),[𝑚(𝑡,𝜔(𝑡,𝑥))−𝑙(𝑡,𝜔(𝑡,𝑥))]𝜔_𝑥=𝑚₀(𝑥)−𝑙₀(𝑥),[𝑛(𝑡,𝜔(𝑡,𝑥))−𝑙(𝑡,𝜔(𝑡,𝑥))]𝜔_𝑥=𝑛₀(𝑥)−𝑙₀(𝑥). (71)
From Lemma 17, we have 𝜔_𝑥=exp−^𝑡₀𝑢_𝑥(𝑠,𝜔(𝑠,𝑥))+𝑣_𝑥(𝑠,𝜔(𝑠,𝑥))+𝑤_𝑥(𝑠,𝜔(𝑠,𝑥))𝑑𝑠. (72) If there exist four constants 𝐾,𝐾₁,𝐾₂, and 𝐾₃ such that 𝑢_𝑥≤𝐾₁, 𝑣_𝑥≤𝐾₂, 𝑤_𝑥≤𝐾₃, and 𝐾=max(𝐾₁,𝐾₂,𝐾₃), we can get that ‖𝑚(𝑡,⋅)−𝑛(𝑡,⋅)‖_𝐿^∞=‖𝑚(𝑡,𝜔(𝑡,⋅))−𝑛(𝑡,𝜔(𝑡,⋅))‖_𝐿^∞=exp^𝑡₀𝑢_𝑥(𝑠,𝜔(𝑠,𝑥))+𝑣_𝑥(𝑠,𝜔(𝑠,𝑥))+𝑤_𝑥(𝑠,𝜔(𝑠,𝑥))𝑑𝑠×𝑚₀(𝑥)−𝑛₀(𝑥)𝑑𝑠_𝐿∞≤𝑒^3𝐾𝑇𝑚₀(⋅)−𝑛₀(⋅). (73) Similarly, ‖𝑚(𝑡,⋅)−𝑙(𝑡,⋅)‖≤𝑒^3𝐾𝑇𝑚₀(⋅)−𝑙₀(⋅,‖𝑛()()𝑡,⋅−𝑙𝑡,⋅)‖≤𝑒^3𝐾𝑇𝑛₀(⋅)−𝑙₀(⋅). (74)

Theorem 19. Let 𝑢₀(𝑥),𝑣₀(𝑥), and 𝑤₀(𝑥), 𝑠>3/2 and 𝑢(𝑡,𝑥),𝑣(𝑡,𝑥),𝑤(𝑡,𝑥) be the solution in (62) with time 𝑇. Then, 𝑇 is finite if and only if lim_𝑡→𝑇infinf_[]𝑥∈0,1𝑢_𝑥(𝑥,𝑡)=−∞ (75) or lim_𝑡→𝑇infinf_[]𝑥∈0,1𝑣_𝑥(𝑥,𝑡)=−∞ (76) or lim_𝑡→𝑇infinf_[]𝑥∈0,1𝑤_𝑥(𝑥,𝑡)=−∞. (77)

Proof. Let 𝑢₀(𝑥),𝑣₀(𝑥), and 𝑤₀(𝑥), 𝑠>3/2 and 𝑢(𝑡,𝑥),𝑣(𝑡,𝑥),𝑤(𝑡,𝑥) be the solution (62) with time 𝑇. We know that 𝑢(𝑡,0)=𝑢(𝑡,1),𝑣(𝑡,0)=𝑣(𝑡,1), and 𝑤(𝑡,0)=𝑤(𝑡,1). By the definition of 𝑚,𝑛, and 𝑙, we have ‖𝑚‖²_𝐿2=¹₀𝑢−𝑢_𝑥𝑥²𝑑𝑥=¹₀𝑢²+2𝑢²_𝑥+𝑢²_𝑥𝑥‖𝑛‖𝑑𝑥,_2𝐿2=¹₀𝑣−𝑣_𝑥𝑥²𝑑𝑥=¹₀𝑣²+2𝑣²_𝑥+𝑣²_𝑥𝑥‖𝑙‖𝑑𝑥,_2𝐿2=¹₀𝑤−𝑤_𝑥𝑥²𝑑𝑥=¹₀𝑤²+2𝑤²_𝑥+𝑤²_𝑥𝑥𝑑𝑥. (78)
Hence, ‖𝑢‖²_𝐻2≤‖𝑚‖²_𝐿2≤2‖𝑢‖²_𝐻2,‖𝑣‖²_𝐻2≤‖𝑛‖²_𝐿2≤2‖𝑣‖²_𝐻2,‖𝑤‖²_𝐻2≤‖𝑙‖²_𝐿2≤2‖𝑤‖²_𝐻2.
Multiplying the first equation by 𝑚, the second one by 𝑛, and the third one by 𝑙, after integration by parts and adding up the results, we see that 12𝑑𝑑𝑡¹₀𝑚²+𝑛²+𝑙²=𝑑𝑥₁₀2𝑢_𝑥+𝑣_𝑥+𝑤_𝑥𝑚²+2𝑣_𝑥+𝑢_𝑥+𝑤_𝑥𝑛²+2𝑤_𝑥+𝑢_𝑥+𝑣_𝑥𝑙²+𝑚𝑚_𝑥+𝑛𝑛_𝑥+𝑙𝑙_𝑥×(𝑢+𝑣+𝑤)+𝑚𝑛𝑢_𝑥+𝑣_𝑥+𝑙𝑚𝑢_𝑥+𝑤_𝑥+ln𝑣_𝑥+𝑤_𝑥=𝑑𝑥₁₀2𝑢_𝑥+𝑣_𝑥+𝑤_𝑥−12𝑢_𝑥−12𝑣_𝑥−12𝑤_𝑥𝑚²+2𝑣_𝑥+𝑢_𝑥+𝑤_𝑥−12𝑢_𝑥−12𝑣_𝑥−12𝑤_𝑥𝑛²+2𝑤_𝑥+𝑢_𝑥+𝑣_𝑥−12𝑢_𝑥−12𝑣_𝑥−12𝑤_𝑥𝑙²+𝑚𝑛𝑢_𝑥+𝑣_𝑥+𝑙𝑚𝑢_𝑥+𝑤_𝑥ln𝑣_𝑥+𝑤_𝑥=𝑑𝑥₁₀32𝑢_𝑥+12𝑣_𝑥+12𝑤_𝑥𝑚²+32𝑣_𝑥+12𝑢_𝑥+12𝑤_𝑥𝑛²+32𝑤_𝑥+12𝑢_𝑥+12𝑣_𝑥𝑙²+𝑚𝑛𝑢_𝑥+𝑣_𝑥+𝑙𝑚𝑢_𝑥+𝑤_𝑥+ln𝑣_𝑥+𝑤_𝑥=𝑑𝑥₁₀12𝑢_𝑥+𝑣_𝑥+𝑤_𝑥𝑚²+𝑛²+𝑙²+𝑚²𝑢_𝑥+𝑛²𝑣_𝑥+𝑙²𝑤_𝑥+𝑚𝑛𝑢_𝑥+𝑣_𝑥+𝑙𝑚𝑢_𝑥+𝑤_𝑥+ln𝑣_𝑥+𝑤_𝑥=𝑑𝑥₁₀12𝑢_𝑥+𝑣_𝑥(𝑚+n)²+12𝑢_𝑥+𝑤_𝑥(𝑚+𝑙)²+12𝑣_𝑥+𝑤_𝑥(𝑙+𝑛)²+12𝑢_𝑥𝑚²+12𝑣_𝑥𝑛²+12𝑤_𝑥𝑙²𝑑𝑥 (79) So, we have 𝑑𝑑𝑡¹₀𝑚²+𝑛²+𝑙²≤𝐾𝑑𝑥₁+𝐾₂+𝐾₃×¹₀(𝑚+𝑛)²+(𝑚+𝑙)²+(𝑙+𝑛)²+𝑚²+𝑛²+𝑙²𝐾𝑑𝑥≤5₁+𝐾₂+𝐾₃¹₀𝑚²+𝑛²+𝑙²𝑑𝑥. (80)
By Gronwall's inequality, we get ‖𝑢‖²_𝐻2+‖𝑣‖²_𝐻2+‖𝑤‖²_𝐻2≤¹₀𝑚²+𝑛²+𝑙²𝐾𝑑𝑥≤exp5𝑇₁+𝐾₂+𝐾₃×¹₀𝑚²₀+𝑛²₀+𝑙²₀𝐾𝑑𝑥≤2exp5𝑇₁+𝐾₂+𝐾₃×𝑢₀²_𝐻2+𝑣₀²_𝐻2+𝑤₀²_𝐻2. (81) The above inequality, Soblev’s embedding theorem, ensure that the solution (𝑢(𝑡,x),𝑣(𝑡,𝑥),𝑤(𝑡,𝑥)) cannot blow up in finite time.
On the other hand, if lim_𝑡→𝑇infinf_[]𝑥∈0,1𝑢_𝑥(𝑥,𝑡)=−∞ (82) or lim_𝑡→𝑇infinf_[]𝑥∈0,1𝑣_𝑥(𝑥,𝑡)=−∞ (83) or lim_𝑡→𝑇infinf_[]𝑥∈0,1𝑤_𝑥(𝑥,𝑡)=−∞, (84) then the solution will blow up in finite time.

Acknowledgments

The paper is supported by the National Nature Science Foundation of China (nos. 71073072 and 11171135) and the National Nature Science Foundation of Jiangsu (no. BK 2010329) and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.