The Global Existence of Solutions in Time for a Chemotaxis Model with Two Chemicals

Abstract

This paper concerns the uniform boundedness and global existence of solutions in time for the chemotaxis model with two chemicals. We prove the system has global existence of solutions in time for any dimension 𝑛.

1. Introduction and Statement of Main Result

Chemotaxis is the influence of chemical substances in the environment on the movement of mobile species. Keller and Segel [1] proposed the general chemotaxis system 𝑢_𝑡=∇⋅𝐷₁(𝑢,𝑣)∇𝑢−𝐷₂(),𝑣𝑢,𝑣∇𝑣_𝑡=𝐷_𝑣Δ𝑣−𝑘(𝑣)𝑣+𝑓(𝑢,𝑣), (1) where 𝑢 is the density function of cells (e.g., Dictyostelium discoideum) that are attracted by a chemical substance (e.g., cAMP) produced by themselves and the movement towards a higher concentration of the chemical substance, whose concentration function is 𝑣. 𝐷₁ and 𝐷_𝑣 are the random diffusion rates of cells and the chemical, respectively; 𝐷₂(𝑢,𝑣)∇𝑣 represents the chemotactic flux of cells and 𝐷₂(𝑢,𝑣) is positive for positive 𝑢 and 𝑣 and is called the sensitivity function; and 𝑓(𝑢,𝑣) is the creation rate of the chemical, while 𝑘(𝑣)𝑣 is the degradation rate of the chemical.

The simplest case of (1) is that 𝐷₁,  𝐷_𝑣, and 𝑘 are all positive constants, 𝑓(𝑢,𝑣)=𝑣, and 𝐷₂(𝑢,𝑣)=𝜒𝑢 with 𝜒 being a positive constant. This was called by Childress and Perkus the “minimal model.” When dimension 𝑛=1, solutions exist globally; see [2]. For 𝑛=2, global existence depends on a threshold: when the initial mass lies below the threshold solutions exist globally, while above the threshold solutions blow up in finite time; these results were derived by various authors; see the review article [3, 4]. Many authors have analyzed system (1) for several variants, such as global existence, blow up solutions, and many other results; see [3–5].

Painter et al. [6] proposed a chemotaxis model with two chemicals. They considered a Turing system [7] as a mechanism for providing spatially heterogeneous chemical distributions to which a cell population chemotactically responds. That was the following model: 𝜕𝑤𝐷𝜕𝑡=∇_𝑤∇𝑤−𝑤𝜒₁(𝑢,𝑣)∇𝑢−𝑤𝜒₂(𝑢,𝑣)∇𝑣,𝑥∈Ω,𝑡>0,𝜕𝑢𝐷𝜕𝑡=∇_𝑢⋅∇𝑢+𝑓(𝑢,𝑣),𝑥∈Ω,𝑡>0,𝜕𝑣𝐷𝜕𝑡=∇_𝑣⋅∇𝑣+𝑔(𝑢,𝑣),𝑥∈Ω,𝑡>0,𝜕𝑤=𝜕𝜈𝜕𝑢=𝜕𝜈𝜕𝑣𝑤()𝜕𝜈=0,𝑥∈𝜕Ω,𝑥,0=𝑤₀(𝑥),𝑢(𝑥,0)=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥), (2) where 𝑤 is a cell population, 𝑢 and 𝑣 are chemicals, and the cell population responds chemotactically to both chemical species. 𝜒₁ and 𝜒₂ are the chemotactic sensitivity functions and 𝑓 and 𝑔 define the chemical kinetics. 𝐷_𝑤,  𝐷_𝑢, and 𝐷_𝑣 are taken as constants.

A special form of system (2) is as follows: 𝜕𝑤𝑤𝜕𝑡=𝐷∇∇𝑤−𝑘₁+𝑢²𝑤∇𝑢−𝑘₂+𝑣²∇𝑣,𝑥∈Ω,𝑡>0,𝜕𝑢𝜕𝑡=Δ𝑢+𝛿−𝐾𝑢−𝑢𝑣²,𝑥∈Ω,𝑡>0,𝜕𝑣𝜕𝑡=Δ𝑣+𝐾𝑢+𝑢𝑣²−𝑣,𝑥∈Ω,𝑡>0,𝜕𝑤=𝜕𝜈𝜕𝑢=𝜕𝜈𝜕𝑣𝑤()𝜕𝜈=0,𝑥∈𝜕Ω,𝑡>0,𝑥,0=𝑤₀(𝑥)≥0,𝑢(𝑥,0)=𝑢₀(𝑥)𝑣()≥0,𝑥,0=𝑣₀(𝑥)≥0, (3) where Ω∈ℝ^𝑛 is a bounded domain. The kinetics of (3) are described by a simplified model for the glycolysis reaction [8, 9]. As we know, the global existence of solutions in time of two species such as (1) is investigated by many authors; however, the global existence of solutions in time of three species is studied little. In this paper, we study the global existence of solutions in time of (3); by applying analysis semigroup and energy method we will prove that system (3) has global solutions in time in any dimension 𝑛.

We state the main result of this paper as follows.

Theorem 1. For any 𝑤₀,𝑢₀,𝑣₀∈𝑊^1,𝑝(Ω), 𝑝>𝑛 satisfying 𝑤₀≥0,  𝑢₀≥0,  𝑣₀≥0 on Ω, (3) has a unique positive global solution (𝑤,𝑢,𝑣) such that ()[),𝑊𝑤,𝑢,𝑣∈𝐶0,+∞_1,𝑝(Ω)×𝑊^1,𝑝(Ω)×𝑊^1,𝑝(Ω)∩𝐶^{2+2𝜖,1+𝜖}_loc(Ω×(0,+∞)). (4)

2. The Proof of Theorem 1

Theorem 2. For any 𝑤₀,𝑢₀,𝑣₀∈𝑊^1,𝑝(Ω),  𝑝>𝑛 satisfying 𝑤₀≥0,  𝑢₀≥0,  𝑣₀≥0 on Ω, one has the following conclusions. (i)(3) has a unique solution (𝑤(𝑥,𝑡),𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)) on Ω×[0,𝑇_{(𝑢0,𝑣0)}) with 0<𝑇_{(𝑢0,𝑣0)}≤∞ satisfying (𝑤(⋅,𝑡),𝑢(⋅,𝑡),𝑣(⋅,𝑡))∈𝐶0,𝑇_{(𝑤0,𝑢0,𝑣0)},𝑊^1,𝑝(Ω)×𝑊^1,𝑝(Ω)×𝑊^1,𝑝(Ω)∩𝐶^{2+2𝜖,1+𝜖}_locΩ×0,𝑇_{(𝑤0,𝑢0,𝑣0)},1𝑓𝑜𝑟𝑎𝑛𝑦0<𝜖<4. (5) (ii)Moreover, if, for small 𝛿>0, 𝑡∈[𝛿,𝑇_{(𝑤0,𝑢0,𝑣0)}), ‖(𝑤,𝑢,𝑣)(⋅,𝑡)‖_𝐿^∞_(Ω) is bounded; then 𝑇_{(𝑤0,𝑢0,𝑣0)}=∞; that is, (𝑤,𝑢,𝑣) has global existence, and for any 0≤𝜌≤𝜎≤1, (𝑤,𝑢,𝑣)∈𝐶^𝜌([𝛿,∞),𝐶^2(1−𝜎)(Ω)×𝐶^2(1−𝜎)(Ω)×𝐶^2(1−𝜎)(Ω)).

Proof. Equation (3) can be written as 𝑤𝑢𝑣_𝑡𝐴()∇𝑤𝑢𝑣+0=∇𝑢,𝑣𝛿−𝐾𝑢−𝑢𝑣²𝐾𝑢+𝑢𝑣²,𝐴()∇𝑤𝑢𝑣−𝑣𝑥∈Ω,𝑡>0,𝑢,𝑣=0,𝑥∈𝜕Ω,𝑡>0, (6) where 𝐴()=𝑤𝑢,𝑣𝐷−𝑘₁+𝑢²−𝑤𝑘₂+𝑣²010.001 (7) Since the eigenvalues of 𝐴 are positive, (6) is normally parabolic; then (i) follows from [10]. Note that (6) is also a “triangular system,” so in virtue of [11], we complete the proof of Theorem 2.

In the following, we always assume 𝑤₀,𝑢₀,𝑣₀∈𝑊^1,𝑝(Ω),  𝑝>𝑛 satisfying 𝑤₀≥0, 𝑢₀≥0,  𝑣₀≥0 on Ω.

Lemma 3. For any dimension 𝑛, any solution 𝑢 of (3) has the following estimate: ‖𝑢‖_𝐿²_(Ω)≤𝜇₁,0≤𝑡<𝑇, (8) where 𝑇=𝑇_{(𝑤0,𝑢0,𝑣0)} and 𝜇₁ depends only on ‖𝑢₀‖_𝐿1(Ω), ‖𝑢₀‖_𝐿2(Ω), 𝛿, 𝑘 and |Ω|.

Proof. Integrating the second equation of (3) over Ω, we have 𝑑𝑑𝑡_Ω𝑢𝑑𝑥=𝛿|Ω|−𝐾_Ω𝑢𝑑𝑥−_Ω𝑢𝑣²𝑑𝑥≤−𝐾_Ω𝑢𝑑𝑥+𝛿|Ω|. (9) Integrating (9) with respect to 𝑡, we get _Ω𝑢𝑑𝑥≤𝑒^−𝐾𝑡𝑢_0𝐿1(Ω)+𝛿|Ω|𝐾,thatis‖𝑢‖_𝐿¹_(Ω)≤𝐶,𝐶∼𝑢_0𝐿1(Ω),𝛿,𝐾,|Ω|. (10) Multiplying the second equation of (3) by 𝑢 and integrating with respect to 𝑥 over Ω, we get 𝑑𝑑𝑡_Ω𝑢²𝑑𝑥=−2_Ω|∇𝑢|²𝑑𝑥+2𝛿_Ω𝑢𝑑𝑥−2𝐾_Ω𝑢²𝑑𝑥−2_Ω𝑢²𝑣²𝑑𝑥≤−2𝐾_Ω𝑢²𝑑𝑥+2𝛿‖𝑢‖_𝐿¹_(Ω). (11) Integrating (11) with respect to 𝑡 and together with (10), we obtain ‖𝑢‖_𝐿²_(Ω)≤𝜇₁,𝜇₁∼𝑢_0𝐿1(Ω),𝑢_0𝐿2(Ω),𝛿,𝐾,|Ω|. (12)

Lemma 4. For any dimension 𝑛 and small constants 𝜏₀>0, any solution 𝑢 of (3) has the following estimate: ‖𝑢‖_𝐿^∞_(Ω)≤𝜇₂,𝜏₀≤𝑡<𝑇, (13) where 𝑇=𝑇_{(𝑤0,𝑢0,𝑣0)} and 𝜇₂ is a constant depending on ‖𝑢₀‖_𝐿1(Ω),  ‖𝑢₀‖_𝐿2(Ω),  𝛿, 𝑘, |Ω| and ‖𝑢(⋅,𝜏₀)‖_𝐿∞(Ω).

Proof. Multiplying the second equation of (3) by 𝑢^𝑝−1(𝑝≥2) and integrating with respect to 𝑥 over Ω, we get 1𝑝𝑑𝑑𝑡_Ω𝑢^𝑝4𝑑𝑥=−𝑝−1𝑝²_Ω∇(𝑢^𝑝/2)²𝑑𝑥+𝛿_Ω𝑢^𝑝−1𝑑𝑥−𝐾_Ω𝑢^𝑝𝑑𝑥−_Ω𝑢^𝑝𝑣²𝑑𝑥. (14) Here (14) implies 𝑑𝑑𝑡_Ω𝑢^𝑝4𝑑𝑥+𝑝−1𝑝_Ω∇(𝑢^𝑝/2)²𝑑𝑥≤𝛿𝑝_Ω𝑢^𝑝−1𝑑𝑥−𝐾𝑝_Ω𝑢^𝑝𝑑𝑥. (15) Define 𝑧=𝑢^𝑝/2, by Hölder’s inequality; (15) can be written as 𝑑𝑑𝑡_Ω𝑧²4𝑑𝑥+𝑝−1𝑝_Ω|∇𝑧|²𝑑𝑥≤𝛿𝑝_Ω𝑧^{2(𝑝−1)/𝑝}𝑑𝑥−𝐾𝑝_Ω𝑧²|Ω|𝑑𝑥≤𝛿𝑝^1/𝑝‖𝑧‖_𝐿^{2(𝑝−1)/𝑝}₂. (16) Gagliardo-Nirenberg inequality implies ‖𝑧‖_𝐿²_(Ω)≤𝐶₁‖∇𝑧‖^𝜃_𝐿2(Ω)‖𝑧‖_𝐿^1−𝜃_1(Ω)+𝐶₂‖𝑧‖_𝐿¹_(Ω)𝑛,with𝜃=.𝑛+2 (17) In virtue of (17) and Young’s inequality, we have ‖𝑧‖_𝐿^{2(𝑝−1)/𝑝}₂≤2𝐶²₁‖∇𝑧‖_𝐿^2𝜃_2(Ω)‖𝑧‖_𝐿^2(1−𝜃)_1(Ω)+2𝐶²₂‖𝑧‖²_𝐿1(Ω)^{(𝑝−1)/𝑝}≤2^{(𝑝−1)/𝑝}𝐶₁^{2(𝑝−1)/𝑝}‖∇𝑧‖_𝐿^{2(𝑝−1)𝜃/𝑝}_2(Ω)×‖𝑧‖_𝐿^{2(𝑝−1)(1−𝜃)/𝑝}_1(Ω)+2^{(𝑝−1)/𝑝}𝐶₂^{(2(𝑝−1)/𝑝)}‖𝑧‖_𝐿^{2(𝑝−1)/𝑝}_1(Ω)≤𝑝−1𝜃𝑝𝜀^{𝑝/(𝑝−1)𝜃}‖∇𝑧‖²_𝐿2(Ω)+𝑝−𝑝−1𝜃𝑝𝜀^{−𝑝/(𝑝−(𝑝−1)𝜃)}2^{(𝑝−1)/(𝑝−(𝑝−1)𝜃)}𝐶₁^{2(𝑝−1)/(𝑝−(𝑝−1)𝜃)}‖𝑧‖_𝐿^{2(𝑝−1)(1−𝜃)/(𝑝−(𝑝−1)𝜃)}_1(Ω)+2^{(𝑝−1)/𝑝}𝐶₂^{2(𝑝−1)/𝑝}‖𝑧‖_𝐿^{2(𝑝−1)/𝑝}_1(Ω)≤𝑝−1𝜃𝑝𝜀^{𝑝/(𝑝−1)𝜃}‖∇𝑧‖²_𝐿2(Ω)+𝐶₃𝜀^{−𝑝/(𝑝−(𝑝−1)𝜃)}‖𝑧‖_𝐿^{2(𝑝−1)(1−𝜃)/(𝑝−(𝑝−1)𝜃)}_1(Ω)+𝐶₄‖𝑧‖_𝐿^{2(𝑝−1)/𝑝}_1(Ω), (18) with 𝐶₃=2^(𝑛+2)/2𝐶₁^𝑛+2 and 𝐶₄=2𝐶²₂.
Taking suitable 𝜀 such that 𝛿𝑝|Ω|^1/𝑝((𝑝−1)𝜃/𝑝)𝜀^{𝑝/(𝑝−1)𝜃}=3(𝑝−1)/𝑝, we obtain 𝜀^{−𝑝/(𝑝−(𝑝−1)𝜃)}=3𝛿𝑝|Ω|^1/𝑝𝜃^{(𝑝−1)𝜃/𝑝−𝑝/(𝑝−(𝑝−1)𝜃)}<𝐶₅𝑝^𝑛/2, (19) with 𝐶₅=(𝛿|Ω|^1/𝑝𝜃/3)^𝑛/2.
In view of (16)–(19), we get 𝑑𝑑𝑡_Ω𝑧²𝑑𝑥+𝑝−1𝑝_Ω|∇𝑧|²𝑑𝑥≤𝐶₃𝐶₅𝑝^(𝑛+2)/2𝛿|Ω|^1/𝑝‖𝑧‖_𝐿^{2(𝑝−1)(1−𝜃)/(𝑝−(𝑝−1)𝜃)}_1(Ω)+𝐶₄𝛿𝑝|Ω|^(1/𝑝)‖𝑧‖_𝐿^{2(𝑝−1)/𝑝}_1(Ω)≤𝐶₆𝛿𝑝^(𝑛+2)/2‖𝑧‖_𝐿^{2(𝑝−1)(1−𝜃)/(𝑝−(𝑝−1)𝜃)}_1(Ω)+𝐶₇𝛿𝑝‖𝑧‖_𝐿^{2(𝑝−1)/𝑝}_1(Ω), (20) with 𝐶₆=𝐶₃𝐶₅|Ω|^1/𝑝 and 𝐶₇=𝐶₄|Ω|^1/𝑝.
By Poincaré inequality, there exists a constant 𝜎>0 depending on 𝑛, 𝑝, Ω such that ‖𝑧‖²_𝐿2(Ω)≤1𝜎‖∇𝑧‖²_𝐿2(Ω)+‖𝑧‖²_𝐿1(Ω); (21) we have 𝑝−1𝑝_Ω|∇𝑧|²𝑑𝑥≥𝑝−1𝑝𝜎_Ω𝑧²𝑑𝑥−𝑝−1𝑝‖𝑧‖²_𝐿1(Ω). (22) It follows from (20) and (22) that 𝑑𝑑𝑡_Ω𝑧²𝑑𝑥≤−𝑝−1𝑝𝜎_Ω𝑧²𝑑𝑥+𝐶₆𝛿𝑝^(𝑛+2)/2‖𝑧‖_𝐿^{2(𝑝−1)(1−𝜃)/(𝑝−(𝑝−1)𝜃)}_1(Ω)+𝐶₇𝛿𝑝‖𝑧‖_𝐿^{(2(𝑝−1)/𝑝)}_1(Ω)+𝑝−1𝑝‖𝑧‖²_𝐿1(Ω). (23) After simple calculation, we obtain ‖𝑧‖²_𝐿2(Ω)≤max𝐶₈𝑝^(𝑛+2)/2sup_𝑡≥𝜏0‖𝑧‖_𝐿^{2(𝑝−1)(1−𝜃)/(𝑝−(𝑝−1)𝜃)}_1(Ω)+𝐶₉𝑝sup_𝑡≥𝜏0‖𝑧‖_𝐿^{2(𝑝−1)/𝑝}_1(Ω)+𝐶₁₀sup_𝑡≥𝜏0‖𝑧‖²_𝐿1(Ω),𝑧𝜏₀²_𝐿2(Ω), (24) with 𝐶₈=𝐶₆𝛿/𝜎, 𝐶₉=𝐶₇𝛿/𝜎, and 𝐶₁₀=1/𝜎.
Since 2(𝑝−1)(1−𝜃)/(𝑝−(𝑝−1)𝜃)<2 and 2(𝑝−1)/𝑝<2, we have sup_𝑡≥𝜏0‖𝑧‖²_𝐿2(Ω)≤max𝐶₁₁𝑝^(𝑛+2)/2×maxsup_𝑡≥𝜏0‖𝑧‖²_𝐿1(Ω),1,𝑧(𝜏₀)²_𝐿2(Ω), (25) with 𝐶₁₁=max{𝐶₈,𝐶₉,𝐶₁₀}.
Substituting 𝑧=𝑢^𝑝/2,  𝑝=2^𝑘,  𝑘=1,2,3,… into (25) yields sup_{𝑡≥𝜏0Ω}𝑢^2𝑘𝐶𝑑𝑥≤max₁₁2^{((𝑛+2)/2)𝑘}maxsup_{𝑡≥𝜏0Ω}𝑢^2𝑘−1𝑑𝑥²,𝑢𝜏,1_{02𝑘𝐿∞(Ω)}. (26) Without loss of generality, we can assume sup_{𝑡≥𝜏0Ω}𝑢^2𝑘𝑑𝑥≤𝐶₁₁2^{((𝑛+2)/2)𝑘}sup_{𝑡≥𝜏0Ω}𝑢^2𝑘−1𝑑𝑥²sup_{𝑡≥𝜏0Ω}𝑢^2𝑘𝑑𝑥≤𝐶₁₁2^{((𝑛+2)/2)𝑘20}𝐶₁₁2^{((𝑛+2)/2)(𝑘−1)21}⋯𝐶₁₁2^{((𝑛+2)/2)(𝑘−(𝑘−3))2𝑘−3},𝐶₁₁2^{((𝑛+2)/2)(𝑘−(𝑘−2))2𝑘−2}sup_{𝑡≥𝜏0Ω}𝑢^{2𝑘−(𝑘−1)}𝑑𝑥^2𝑘−1≤𝐶^{2(𝑘−1)−1}₁₁×2^{((𝑛+2)/2)(𝑘+(𝑘−1)21+⋯+4⋅2𝑘−4+3⋅2𝑘−3+2⋅2𝑘−2+2𝑘−1)}×sup_{𝑡≥𝜏0Ω}𝑢²𝑑𝑥^2𝑘−1. (27)(28) Taking 1/2^𝑘 and letting 𝑘→∞ of (28), together with Lemma 3, we have sup_𝑡≥𝜏0‖𝑢‖_𝐿^∞_(Ω)≤𝐶^1/2₁₁2^(𝑛+2)/2sup_{𝑡≥𝜏0Ω}𝑢²𝑑𝑥^1/2≤𝜇₂. (29)

With the notation 𝑤=𝑢+𝑣, (30) we have the following Lemma 5.

Lemma 5. For any dimension 𝑛, 𝑤 has the following estimate: ‖𝑤‖_𝐶¹_(Ω)≤𝜇₃,𝜏₀≤𝑡<𝑇, (31) where 𝑇=𝑇_{(𝑤0,𝑢0,𝑣0)} and 𝜇₃ is a constant depending on ‖𝑢₀‖_𝐿1(Ω),  ‖𝑢₀‖_𝐿2(Ω),   𝛿, 𝑘, |Ω|,   ‖𝑢(⋅,𝜏₀)‖_𝐿∞(Ω), and ‖𝑤(⋅,𝜏₀)‖_{𝑊2,𝑝(Ω)}.

Proof. In view of (3), 𝑤 satisfies the following equation: 𝑑𝑤𝑑𝑡=Δ𝑤+𝛿+𝑢−𝑤; (32) let 𝑋=𝐿^𝑝(Ω), 𝐴=Δ−𝐼 with domain 𝐷=𝑊^2,𝑝(Ω); then 𝐴 generates a linear analysis semigroup on 𝑋 satisfying ‖𝑒^𝐴𝑡‖≤𝐶𝑒^−𝑎𝑡 and ‖𝑒^𝐴𝑡‖_𝑋^𝛼≤𝐶_𝛼𝑡^−𝛼𝑒^−𝑎𝑡 for 0<𝑎<1. Taking 𝑝>𝑛, the fractional space 𝑋^𝛼↪𝐶^𝛾 with 0≤𝛾<2𝛼−(𝑛/𝑝); taking 𝛼=(1/2)+(𝑛/2𝑝) we get 𝛾=1.
In virtue of (32), we obtain 𝑤(⋅,𝑡)=𝑒^{𝐴(𝑡−𝜏0)}𝑤⋅,𝜏₀+^𝑡_𝜏0𝑒^{𝐴(𝑡−𝑠)}((‖𝛿+𝑢⋅,𝑠))𝑑𝑠,𝑤(⋅,𝑡)‖_𝐶¹_(Ω)≤𝐶₁‖𝑤(⋅,𝑡)‖_𝑋^𝛼≤𝐶₁𝑒^{𝐴(𝑡−𝜏0)}𝑤(⋅,𝜏₀)_𝑋𝛼+𝐶₁^𝑡_𝜏0𝑒^{𝐴(𝑡−𝑠)}(𝛿+𝑢(⋅,𝑠))_𝑋𝛼𝑑𝑠≤𝐶₂𝑒^{−𝑎(𝑡−𝜏0)}𝑤(⋅,𝜏₀)_𝑋𝛼+^𝑡_𝜏0𝐶_𝛼(𝑡−𝑠)^−𝛼𝑒^{−𝑎(𝑡−𝑠)}‖𝛿+𝑢(⋅,𝑠)‖_𝐿^𝑝_(Ω)𝑑𝑠≤𝐶₃+𝐶₄sup_{𝜏0≤𝑠≤𝑡}‖𝑢(⋅,𝑡)‖_𝐿^𝑝_(Ω)≤𝜇₃. (33)

By Lemma 4 and (30), one has ‖𝑣‖_𝐿^∞_(Ω)≤‖𝑢‖_𝐿^∞_(Ω)+‖𝑤‖_𝐿^∞_(Ω)≤𝜇₄, (34) where 𝜇₄ is a constant depending on ‖𝑢₀‖_𝐿1(Ω),  ‖𝑢₀‖_𝐿2(Ω),  𝛿, 𝑘, |Ω|,  ‖𝑢(⋅,𝜏₀)‖_𝐿∞(Ω), and ‖𝑤(⋅,𝜏₀)‖_𝑊2,𝑝(Ω).

Similar as the proof of Lemma 5, we can prove ‖𝑣‖_𝐶¹_(Ω)≤𝜇₅, (35) where 𝜇₅ is a constant depending on ‖𝑢₀‖_𝐿1(Ω),  ‖𝑢₀‖_𝐿2(Ω),  𝛿, 𝑘, |Ω|,  ‖𝑢(⋅,𝜏₀)‖_𝐿∞(Ω), ‖𝑤(⋅,𝜏₀)‖_{𝑊2,𝑝(Ω)}, and ‖𝑣(⋅,𝜏₀)‖_{𝑊2,𝑝(Ω)}.

Lemma 5 and (35) yield ‖𝑢‖_𝐶¹_(Ω)≤𝜇₆, (36) where 𝜇₆ is a constant depending on ‖𝑢₀‖_𝐿1(Ω),   ‖𝑢₀‖_𝐿2(Ω),   𝛿,𝑘,|Ω|,   ‖𝑢(⋅,𝜏₀)‖_𝐿∞(Ω),  ‖𝑤(⋅,𝜏₀)‖_{𝑊2,𝑝(Ω)}, and ‖𝑣(⋅,𝜏₀)‖_{𝑊2,𝑝(Ω)}.

Lemma 6. For any dimension 𝑛, any solution 𝜔 of (3) has the following estimate: ‖𝑤‖_𝐿^∞_(Ω)≤𝜇₇,𝜏₀≤𝑡<𝑇, (37) where 𝑇=𝑇_{(𝑤0,𝑢0,𝑣0)} and 𝜇₇ is a constant depending on ‖𝑢₀‖_𝐿1(Ω), ‖𝑢₀‖_𝐿2(Ω), 𝛿, 𝑘, |Ω|, ‖𝑢(⋅,𝜏₀)‖_𝐿∞(Ω), ‖𝑤(⋅,𝜏₀)‖_{𝑊2,𝑝(Ω)}, ‖𝑣(⋅,𝜏₀)‖_{𝑊2,𝑝(Ω)}, ‖𝑤₀(𝑥)‖_𝐿1(Ω), and ‖𝑤(⋅,𝜏₀)‖_𝐿∞(Ω).

Proof. Integrating the first equation of (3) with respect to 𝑥 over Ω and together with the boundary condition, we get _Ω𝑤(𝑥,𝑡)𝑑𝑥=_Ω𝑤₀(𝑥)𝑑𝑥≤𝐶₁. (38) In the following, we will use the inequality as follows: ‖𝑢‖²_𝐿2(Ω)≤𝜀‖∇𝑢‖²_𝐿2(Ω)+𝐶1+𝜀^−𝑛/2‖𝑢‖²_𝐿1(Ω), (39) with 𝐶 depending only on 𝑛 and Ω.
Multiplying the first equation of (3) by 𝑤^𝑠−1(𝑠≥2) and integrating with respect to 𝑥 over Ω imply 1𝑠𝑑𝑑𝑡_Ω𝑤^𝑠𝑑𝑥=−𝐷(𝑠−1)_Ω𝑤^𝑠−2|∇𝑤|²+()𝑑𝑥𝑠−1_Ω𝑤^𝑠−1𝑘₁+𝑢²+()∇𝑢⋅∇𝑤𝑑𝑥𝑠−1_Ω𝑤^𝑠−1𝑘₂+𝑣²≤()∇𝑣⋅∇𝑤𝑑𝑥−4𝐷𝑠−1𝑠²_Ω∇(𝑤^𝑠/2)²𝑑𝑥+𝐶₂(𝑠−1)_Ω𝑤^𝑠−1∇𝑤𝑑𝑥+𝐶₃(𝑠−1)_Ω𝑤^𝑠−1≤()∇𝑤𝑑𝑥−4𝐷𝑠−1𝑠²_Ω∇(𝑤^𝑠/2)²𝑑𝑥+𝐶_4Ω𝑤^𝑠/2∇𝑤^𝑠/2𝑑𝑥. (40) (39) and Hölder’s inequality yield 𝑑𝑑𝑡_Ω𝑤^𝑠()𝑑𝑥≤−4𝐷𝑠−1𝑠_Ω∇(𝑤^𝑠/2)²𝑑𝑥+𝐶₄𝑠_Ω𝑤^𝑠𝑑𝑥^1/2_Ω∇(𝑤^𝑠/2)²𝑑𝑥^1/2≤−2𝐷_Ω∇(𝑤^𝑠/2)²𝑑𝑥+𝐷_Ω∇(𝑤^𝑠/2)²𝐶𝑑𝑥+²⁴𝑠²4𝐷_Ω𝑤^𝑠𝑑𝑥≤−𝐷_Ω∇(𝑤^𝑠/2)²𝑑𝑥+𝐶₅𝑠²_Ω𝑤^𝑠≤𝐷𝑑𝑥𝐶𝜀^−𝑛/2+1𝜀_Ω𝑤^𝑠/2𝑑𝑥²−𝐷𝜀_Ω𝑤^𝑠𝑑𝑥+𝐶₅𝑠²_Ω𝑤^𝑠𝑑𝑥≤−𝐶₅𝑠²_Ω𝑤^𝑠𝑑𝑥+𝐶₆𝑠^𝑛+2_Ω𝑤^𝑠/2𝑑𝑥². (41) For 𝜏₀≤𝑡≤𝑇, by (41), we have 𝑑𝑒𝑑𝑡^{𝐶5𝑠2𝑡}_Ω𝑤^𝑠𝑑𝑥≤𝐶₆𝑠^𝑛+2𝑒^{𝐶5𝑠2𝑡}_Ω𝑤^𝑠/2𝑑𝑥². (42) Integrating (42) with respect to 𝑡 over [𝜏₀,𝑡], we obtain _Ω𝑤^𝑠(𝑥,𝑡)𝑑𝑥≤𝑒^{𝐶5𝑠2(𝜏0−𝑡)}_Ω𝑤^𝑠𝑥,𝜏₀𝑑𝑥+𝐶₇𝑠^𝑛1−𝑒^{𝐶5𝑠2(𝜏0−𝑡)}×sup_{𝜏0≤𝑡≤𝑇Ω}𝑤^𝑠/2𝑑𝑥²≤𝑤𝑥,𝜏₀^𝑠_𝐿∞(Ω)+𝐶₈𝑠^𝑛sup_{𝜏0≤𝑡≤𝑇Ω}𝑤^𝑠/2𝑑𝑥². (43) Let 𝑀(𝑠)=max𝑤𝑥,𝜏_0𝐿∞(Ω),sup_{𝜏0≤𝑡≤𝑇Ω}𝑤^𝑠𝑑𝑥^1/𝑠; (44) then we get 𝑀(𝑠)≤𝐶₉𝑠^𝑛(1/𝑠)𝑀𝑠2,𝑠≥2. (45) Taking 𝑠=2^𝑘,  𝑘=1,2,…, we get 𝑀2^𝑘≤𝐶^1/2𝑘₉2^{𝑘𝑛/2𝑘}𝑀2^𝑘−1≤𝐶^(1/2𝑘₉^)+⋯+(1/2)2^{(𝑘𝑛/2𝑘)+⋯+(𝑛/2)}𝑀(1)≤𝐶𝑀(1). (46) For 𝜏₀<+∞ and (38) one gets ‖𝑤(⋅,𝑡)‖_𝐿^∞_(Ω)≤sup_{𝜏0≤𝑡≤𝑇Ω}𝑤𝑑𝑥≤𝜇₇, (47) where 𝜇₇ is a constant depending on ‖𝑢₀‖_𝐿1(Ω),  ‖𝑢₀‖_𝐿2(Ω), 𝛿, 𝑘, |Ω|,  ‖𝑢(⋅,𝜏₀)‖_𝐿∞(Ω), ‖𝑤(⋅,𝜏₀)‖_{𝑊2,𝑝(Ω)},  ‖𝑣(⋅,𝜏₀)‖_{𝑊2,𝑝(Ω)},  ‖𝑤₀(𝑥)‖_𝐿1(Ω), and ‖𝑤(⋅,𝜏₀)‖_𝐿∞(Ω). Now by Theorem 2(ii) and Lemma 3–Lemma 6, we have proved the Theorem 1.

For 𝐷=𝑘₁=𝑘₂=𝛿=𝐾=1, 𝑇=300000,  Ω=[0,30], 𝑤₀(𝑥)=(1/10000)𝑥³(30−𝑥)²,  𝑢₀(𝑥)=𝑥³(30−𝑥)², and 𝑣₀(𝑥)=𝑥²(30−𝑥)², we have the numerical simulation solutions of (3) as shown in Figure 1.

(a)

(b)

(c)

Figure 1 

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is partially supported by the NNSF of China (71373023), the Beijing Natural Science Foundation (1132003, 1122016, KZ201310028030, KM201311417006, and KM201210017008), and the Zhejiang A&F University Talent Program (2013FR078).