Global Behavior for a Strongly Coupled Predator-Prey Model with One Resource and Two Consumers

Abstract

We consider a strongly coupled predator-prey model with one resource and two consumers, in which the first consumer species feeds on the resource according to the Holling II functional response, while the second consumer species feeds on the resource following the Beddington-DeAngelis functional response, and they compete for the common resource. Using the energy estimates and Gagliardo-Nirenberg-type inequalities, the existence and uniform boundedness of global solutions for the model are proved. Meanwhile, the sufficient conditions for global asymptotic stability of the positive equilibrium for this model are given by constructing a Lyapunov function.

1. Introduction

The principle of competitive exclusion asserts that two or more consumer species cannot coexist indefinitely upon a single limiting resource, which dates back to the pioneering work of Volterra [1] in the 1920s. Subsequently, Ayala [2] in 1969 demonstrated experimentally that two species of Drosophila can coexist upon a single limiting resource. Ayala's experiments have received much attention (see the comprehensive survey by Cantrell and Cosner [3]). Schoener [4] in 1976 found that intraspecific interference among consumers may lead to coexistence of multiple consumer species upon a single resource. To examine more closely the implications of feeding interference among conspecific consumers on consumer-resource dynamics, Cantrell et al. [5] in 2004 proposed the following predator-prey system: 𝑑𝑢𝑢𝑑𝑡=𝑟𝑢1−𝐾−𝑎𝑢𝑣−1+𝑏𝑢𝐴𝑢𝑤,1+𝐵𝑢+𝐷𝑤𝑑𝑣𝑑𝑡=𝑣−𝑙+𝑒𝑢,1+𝑏𝑢𝑑𝑤𝑑𝑡=𝑤−𝐿+𝐸𝑢,1+𝐵𝑢+𝐷𝑤 (1.1) where 𝑟,𝑎,𝑏,𝑒,𝑙,𝐴,𝐵,𝐷,𝐸,𝐿, and 𝐾 are positive constants, 𝑢(𝑡) represents the density of the limiting resource at time 𝑡,𝑣(𝑡) and 𝑤(𝑡) denote two consumers species. The first consumer species feeds upon the resource according to the Holling II functional response, while the second consumer species feeds upon the resource following the Beddington-DeAngelis functional response, and they compete for the common resource. For more details on the backgrounds about this system see [5].

The system (1.1) has a positive equilibrium 𝐸=(𝑢,𝑣,̃𝑤) under the suitable conditions, where 𝑙𝑢=,̃𝑒𝑒−𝑏𝑙𝑣=𝑟𝑢𝑢𝑎𝑙1−𝐾−𝐴),()𝐷𝐸[(𝐸−𝐵𝐿𝑢−𝐿]𝑤=𝐸−𝐵𝐿𝑢−𝐿.𝐷𝐿 (1.2) The Jacobian matrix of the system (1.1) at 𝐸 can be written as 𝐽𝐸=𝑎₁₁𝑎₁₂𝑎₁₃𝑎₂₁𝑎₂₂𝑎₂₃𝑎₃₁𝑎₃₂𝑎₃₃, (1.3) where 𝑎₁₁=𝑢−𝑟𝑢𝐾+𝑎𝑏𝑣̃(1+𝑏𝑢)²+𝐴𝐵𝑤(1+𝐵𝑢+𝐷𝑤)²,𝑎₁₂𝑎𝑢=−𝑢𝑎1+𝑏<0,₁₃𝐴𝑢(𝑢)=−1+𝐵(1+𝐵𝑢+𝐷𝑤)²<0,𝑎₂₁=𝑒𝑣̃(1+𝑏𝑢)²>0,𝑎₃₁=𝐸𝑤(1+𝐷𝑤)(1+𝐵𝑢+𝐷𝑤)²𝑎>0,₃₃𝑢𝑤=−𝐷𝐸(1+𝐵𝑢+𝐷𝑤)²<0,𝑎₂₂=𝑎₂₃=𝑎₃₂=0. (1.4) The following results were proved in [5]:(1)the system (1.1) is dissipative;(2)the positive equilibrium 𝐸 of (1.1) is locally stable if −(𝑎₁₁+𝑎₃₃)>0 and 𝑎₁₁(𝑎₁₁𝑎₃₃−𝑎₁₃𝑎₃₁−𝑎₁₂𝑎₂₁)−𝑎₃₃(𝑎₁₁𝑎₃₃−𝑎₃₁𝑎₃₁)<0; and(3)the positive equilibrium 𝐸 of (1.1) is globally stable if max{𝑏,𝐵}(𝐾−𝑢)<1.

Rescaling the system (1.1) such that 𝑢𝐾𝑎⟼𝑢,𝑣⟼𝑣,𝑤⟼𝑤,𝑟𝑡⟼𝑡,𝑟𝐴⟼𝑎,𝑏𝐾⟼𝑏,𝑟𝑙⟼𝐴,𝐵𝐾⟼𝐵,𝐷⟼𝐷,𝑟()⟼𝑙,𝑒𝐾𝑙𝐿⟼𝑒,𝑟()⟼𝐿,𝐸𝐾𝐿⟼𝐸 (1.5) yields 𝑑𝑢()−𝑑𝑡=𝑢1−𝑢𝑎𝑢𝑣−1+𝑏𝑢𝐴𝑢𝑤,1+𝐵𝑢+𝐷𝑤𝑑𝑣𝑑𝑡=𝑙𝑣−1+𝑒𝑢,1+𝑏𝑢𝑑𝑤𝑑𝑡=𝐿𝑤−1+𝐸𝑢.1+𝐵𝑢+𝐷𝑤 (1.6) The corresponding weakly coupled reaction-diffusion system for (1.6) is as follows: 𝑢_𝑡=𝑑₁Δ𝑢+𝑢(1−𝑢)−𝑎𝑢𝑣−1+𝑏𝑢𝐴𝑢𝑤𝑣1+𝐵𝑢+𝐷𝑤,𝑥∈Ω,𝑡>0,_𝑡=𝑑₂Δ𝑣+𝑙𝑣−1+𝑒𝑢𝑤1+𝑏𝑢,𝑥∈Ω,𝑡>0,_𝑡=𝑑₃Δ𝑤+𝐿𝑤−1+𝐸𝑢𝜕1+𝐵𝑢+𝐷𝑤,𝑥∈Ω,𝑡>0,_𝜈𝑢=𝜕_𝜈𝑣=𝜕_𝜈𝑢()𝑤=0,𝑥∈𝜕Ω,𝑡>0,𝑥,0=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥),𝑤(𝑥,0)=𝑤₀(𝑥),𝑥∈Ω, (1.7) where Ω⊂ℝ^𝑁 is a bounded smooth domain, 𝜈 is the outward unit normal vector of the boundary 𝜕Ω,𝜕_𝜈=𝜕/𝜕𝜈. The constants 𝑑₁,𝑑₂, and 𝑑₃, called diffusion coefficients, are positive, and 𝑢₀(𝑥),𝑣₀(𝑥), and 𝑤₀(𝑥) are nonnegative functions which are not identically zero. The system (1.7) has a constant positive steady-state solution 𝐸^∗=(𝑢^∗,𝑣^∗,𝑤^∗) if and only if 𝑒>𝑏,𝐸−𝐵>𝑒−𝑏,𝐷𝐸(𝑒−𝑏−1)−𝐴(𝑒−𝑏)[(𝐸−𝐵)−(𝑒−𝑏)]>0, (1.8) where 𝑢^∗=1𝑒−𝑏,𝑣^∗=𝑒{𝐷𝐸(𝑒−𝑏−1)−𝐴(𝑒−𝑏)[(𝐸−𝐵)−(𝑒−𝑏)]}𝑎𝐷𝐸(𝑒−𝑏)²,𝑤^∗=(𝐸−𝐵)−(𝑒−𝑏)𝐷().𝑒−𝑏 (1.9)

In [6], Hei and Yu proved the following main results.

(1) The equilibrium (𝑢^∗,𝑣^∗,𝑤^∗) of (1.7) is locally asymptotically stable if (1.8) and 𝐷𝐸²[𝑏(𝑒−𝑏−1)−𝑒]+𝐴(𝑒−𝑏)(𝐵𝑒−𝑏𝐸)[(𝐸−𝐵)−(𝑒−𝑏)]<0 (1.10) hold.

(2) Let 𝑑^∗₂ and 𝑑^∗₃ be fixed positive constants which satisfy 𝑑^∗₂𝜇₁≥𝑙(𝑒−𝑏−1)/(1+𝑏) and 𝑑^∗₃𝜇₁≥𝐿(𝐸−𝐵−1)/(1+𝐵).

Then there exists a positive constant 𝐷₁, such that (1.7) has no nonconstant positive solution if 𝑑₁>𝐷₁, 𝑑₂≥𝑑^∗₂ and 𝑑₃≥𝑑^∗₃, where 0=𝜇₀<𝜇₁<𝜇₂<⋯ are the eigenvalues of the operator −Δ on Ω with the homogeneous Neumann boundary condition.

(3) Let 𝑑_𝑖(𝑖=1,3) be fixed positive constants. Assume that 𝑎^∗₁₁>0, min4𝑑₂−4𝑒𝑙𝑑₁−𝑒𝑑₂4𝑑₂,[𝐷(𝐸−𝐵)−𝐴(𝐸−𝐵−1)]𝐷()>1𝐸−𝐵𝑒−𝑏 (1.11) and (1.8) hold. Furthermore, assume that one of the following conditions is satisfied:(i)𝑎^∗₁₁𝑎^∗₃₃−𝑎^∗₃₁𝑎^∗₁₃<0, 𝜇∈(𝜇_𝑛,𝜇_𝑛+1) for some 𝑛≥1, and the sum 𝜎_𝑛=∑^𝑛_𝑖=1dim𝐸(𝜇_𝑖) is odd;(ii)𝑎^∗₁₁𝑎^∗₃₃−𝑎^∗₃₁𝑎^∗₁₃>0, 𝜇∈(𝜇_𝑘,𝜇_𝑘+1), 𝜇∈(𝜇_𝑛,𝜇_𝑛+1) for some 𝑛≥𝑘≥1, and the sum 𝜎_𝑛=∑^𝑛_𝑖=𝑘+1dim𝐸(𝜇_𝑖) is odd. Then there exists a positive constant 𝐷₂, such that (1.7) has at least one nonconstant positive solution if 𝑑₂≥𝐷₂, where 𝑎^∗₁₁=−𝑢^∗+𝑎𝑏𝑣^∗𝑒²𝑢^∗+𝐴𝐵𝑤^∗𝐸²𝑢^∗,𝑎^∗₃₃=−𝐷𝐸𝐿𝑢^∗𝑤^∗(1+𝐵𝑢^∗+𝐷𝑤^∗)²,𝑎^∗₁₃=−𝐴𝑤^∗𝐸𝑢^∗,𝑎^∗₃₁=𝐸𝐿𝑤^∗1+𝐷𝑤^∗(1+𝐵𝑢^∗+𝐷𝑤^∗)²,𝜇=𝑎^∗₁₁𝑑₃+𝑎^∗₃₃𝑑₁+𝑎^∗₁₁𝑑₃−𝑎^∗₃₃𝑑₁²+4𝑎^∗₃₁𝑎^∗₁₃𝑑₁𝑑₃2𝑑₁𝑑₃,𝑎𝜇=^∗11𝑑₃+𝑎^∗₃₃𝑑₁−𝑎^∗₁₁𝑑₃−𝑎^∗₃₃𝑑₁²+4𝑎^∗₃₁𝑎^∗₁₃𝑑₁𝑑₃2𝑑₁𝑑₃ (1.12) and 𝐸(𝜇_𝑖) is the eigenspace corresponding to 𝜇_𝑖 in 𝐻¹(Ω).

(4) The bifurcation of nonconstant positive solutions for (1.7) was studied.

In recent years, the SKT type cross-diffusion systems have attracted the attention of a great number of investigators and have been successfully developed on the theoretical backgrounds. The above work mainly concentrate on (1) the instability and stability induced by cross-diffusion, and the existence of nonconstant positive steady-state solutions [7–14]; (2) the global existence of strong solutions [15–23]; (3) the global existence of weak solutions based on semidiscretization or finite element approximation [24–30]; and (4) the dynamical behaviors [18, 19, 31, 32], and so forth. The corresponding SKT type cross-diffusion system for (1.7) is as follows: 𝑢_𝑡=Δ𝑑₁𝑢+𝛼₁₁𝑢²+𝛼₁₂𝑢𝑣+𝛼₁₃𝑢𝑤+𝑢(1−𝑢)−𝑎𝑢𝑣−1+𝑏𝑢𝐴𝑢𝑤𝑣1+𝐵𝑢+𝐷𝑤,𝑥∈Ω,𝑡>0,_𝑡=Δ𝑑₂𝑣+𝛼₂₁𝑢𝑣+𝛼₂₂𝑣²+𝛼₂₃𝑣𝑤+𝑙𝑣−1+𝑒𝑢𝑤1+𝑏𝑢,𝑥∈Ω,𝑡>0,_𝑡=Δ𝑑₃𝑤+𝛼₃₁𝑢𝑤+𝛼₃₂𝑣𝑤+𝛼₃₃𝑤²+𝐿𝑤−1+𝐸𝑢𝜕1+𝐵𝑢+𝐷𝑤,𝑥∈Ω,𝑡>0,_𝜈𝑢=𝜕_𝜈𝑣=𝜕_𝜈𝑢()𝑤=0,𝑥∈𝜕Ω,𝑡>0,𝑥,0=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥),𝑤(𝑥,0)=𝑤₀(𝑥),𝑥∈Ω, (1.13) where 𝛼_𝑖𝑗(𝑖,𝑗=1,2,3) are positive constants, 𝛼_𝑖𝑖(𝑖=1,2,3) are referred as self-diffusion pressures, and 𝛼_𝑖𝑗(𝑖,𝑗=1,2,3,𝑖≠𝑗) are cross-diffusion pressures. The self-diffusion implies the movement of individuals from a higher to lower concentration region. Cross-diffusion expresses the population fluxes of one species due to the presence of the other species. The value of cross-diffusion coefficient may be positive, negative, or zero. The positive cross-diffusion coefficient denotes the movement of the species in the direction of lower concentration of another species and negative cross-diffusion coefficient denotes that one species tends to diffuse in the direction of higher concentration of another species (e.g., [33]).

The local existence of solutions for the system (1.13) is an immediate consequence of a series of important papers [34–36] by Amann. Roughly speaking, if 𝑢₀(𝑥),𝑣₀(𝑥), and 𝑤₀(𝑥) in 𝑊¹_𝑝(Ω) with 𝑝>𝑛, then (1.13) has a unique nonnegative solution 𝑢,𝑣,𝑤∈𝐶([0,𝑇),𝑊¹_𝑝(Ω))∩𝐶^∞((0,𝑇),𝐶^∞(Ω)), where 𝑇∈(0,∞] is the maximal existence time for the solution. If the solution (𝑢,𝑣,𝑤) satisfies the estimate sup‖𝑢(⋅,𝑡)‖_𝑊¹_𝑝(Ω),‖𝑣(⋅,𝑡)‖_𝑊¹_𝑝(Ω),‖𝑤(⋅,𝑡)‖_𝑊¹_𝑝(Ω)∶0<𝑡<𝑇<∞, (1.14) then 𝑇=+∞. Moreover, if 𝑢₀(𝑥),𝑣₀(𝑥),𝑤₀(𝑥)∈𝑊²_𝑝(Ω), then 𝑢,𝑣,𝑤∈𝐶([0,∞),𝑊²_𝑝(Ω)).

For the following SKT system 𝑢_𝑡=𝑑₁Δ𝑢()𝑣1+𝛼𝑣+𝛾𝑢+𝑎𝑢1−𝑢−𝑐𝑣,𝑥∈Ω,𝑡>0,_𝑡=𝑑₂Δ)𝑣]()𝜕[(1+𝛿𝑣+𝑏𝑣1−𝑑𝑢−𝑣,𝑥∈Ω,𝑡>0,_𝜈𝑢=𝜕_𝜈𝑢()𝑣=0,𝑥∈𝜕Ω,𝑡>0,𝑥,0=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥),𝑥∈Ω, (P) Yamada in [23] proposed four open problems:(1)the global existence of solutions of (P) in the case 𝛿>0 and the space dimension 𝑁≥6;(2)the global existence in the case 𝛾=0;(3)in order to study the asymptotic behavior of 𝑢,𝑣 as 𝑡→∞ need to establish the uniform boundedness of global solutions; and(4)the global existence of solutions for the following full SKT system: 𝑢_𝑡=𝑑₁Δ𝑢()𝑣1+𝛼𝑣+𝛾𝑢+𝑎𝑢1−𝑢−𝑐𝑣,𝑥∈Ω,𝑡>0,_𝑡=𝑑₂Δ𝑣()𝜕1+𝛽𝑢+𝛿𝑣+𝑏𝑣1−𝑑𝑢−𝑣,𝑥∈Ω,𝑡>0,_𝜈𝑢=𝜕_𝜈𝑢()𝑣=0,𝑥∈𝜕Ω,𝑡>0,𝑥,0=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥),𝑥∈Ω (1.15)with 𝛼,𝛾,𝛽,𝛿>0.

Very few global existence results for (1.13) are known. The main purpose of this paper is to establish the uniform boundedness of global solutions for the system (1.13) in one space dimension. For convenience, we consider the following system: 𝑢_𝑡=𝑑₁𝑢+𝛼₁₁𝑢²+𝛼₁₂𝑢𝑣+𝛼₁₃𝑢𝑤_𝑥𝑥+𝑢(1−𝑢)−𝑎𝑢𝑣−1+𝑏𝑢𝐴𝑢𝑤𝑣1+𝐵𝑢+𝐷𝑤,0<𝑥<1,𝑡>0,_𝑡=𝑑₂𝑣+𝛼₂₁𝑢𝑣+𝛼₂₂𝑣²+𝛼₂₃𝑣𝑤_𝑥𝑥+𝑙𝑣−1+𝑒𝑢𝑤1+𝑏𝑢,0<𝑥<1,𝑡>0,_𝑡=𝑑₃𝑤+𝛼₃₁𝑢𝑤+𝛼₃₂𝑣𝑤+𝛼₃₃𝑤²_𝑥𝑥+𝐿𝑤−1+𝐸𝑢𝑢1+𝐵𝑢+𝐷𝑤,0<𝑥<1,𝑡>0,_𝑥(𝑥,𝑡)=𝑣_𝑥(𝑥,𝑡)=𝑤_𝑥()𝑢()𝑥,𝑡=0,𝑥=0,1,𝑡>0,𝑥,0=𝑢₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥),𝑤(𝑥,0)=𝑤₀(𝑥),0<𝑥<1. (1.16) We firstly investigate the global existence and the uniform boundedness of the solutions for (1.16), then prove the global asymptotic stability of the positive equilibrium (𝑢^∗,𝑣^∗,𝑤^∗) of (1.16) by an important lemma from [37]. The proof is complete and complement to the uniform convergence theorems in papers [38–40].

It is obvious that (𝑢^∗,𝑣^∗,𝑤^∗) is the unique positive equilibrium of the system (1.16) if (1.8) holds.

For simplicity, we denote ‖⋅‖_𝑊^𝑘_𝑝(0,1) by |⋅|_𝑘,𝑝 and ‖⋅‖_𝐿^𝑝_(0,1) by |⋅|_𝑝. Our main results are as follows.

Theorem 1.1. Let 𝑢₀(𝑥),𝑣₀(𝑥),𝑤₀(𝑥)∈𝑊²₂(0,1), (𝑢,𝑣,𝑤) be the unique nonnegative solution of the system (1.16) in the maximal existence interval [0,𝑇). Assume that 8𝛼₁₁𝛼₂₁𝛼₃₁>𝛼₂₁𝛼²₁₃+𝛼²₁₂𝛼₃₁,8𝛼₁₂𝛼₂₂𝛼₃₂>𝛼₃₂𝛼²₂₁+𝛼²₂₃𝛼₁₂,8𝛼₁₃𝛼₂₃𝛼₃₃>𝛼₂₃𝛼²₃₁+𝛼²₃₂𝛼₁₃. (1.17) Then there exist 𝑡₀>0 and positive constants 𝑀^′,𝑀 which depend on 𝑑_𝑖,𝛼_𝑖𝑗(𝑖,𝑗=1,2,3),𝑎,𝑏,𝑒,𝑙,𝐴,𝐵,𝐷,𝐸 and 𝐿, such that sup|𝑢(⋅,𝑡)|_1,2,|𝑣(⋅,𝑡)|_1,2,|𝑤(⋅,𝑡)|_1,2∶𝑡∈𝑡₀,𝑇≤𝑀^′,max𝑢(𝑥,𝑡),𝑣(𝑥,𝑡),𝑤(𝑥,𝑡)∶(𝑥,𝑡)∈[0,1]×𝑡₀,𝑇≤𝑀, (1.18)(1.19) and 𝑇=+∞. Moreover, in the case that 𝑑₁,𝑑₂,𝑑₃≥1,𝑑₂/𝑑₁,𝑑₃/𝑑₁∈[𝑑,𝑑], where 𝑑 and 𝑑 are positive constants, 𝑀^′,𝑀 depend on 𝑑,𝑑, but do not depend on 𝑑₁,𝑑₂,𝑑₃.

Remark 1.2. Since the continuous embedding 𝐻¹(Ω)↪𝐿^∞(Ω) holds only in one space dimension, we can only establish the uniform maximum-norn estimates about time for the solution in one space dimension.

Theorem 1.3. Assume that all conditions in Theorem 1.1 are satisfied. Assume further that 4𝛼𝛽𝑢^∗𝑣^∗𝑤^∗𝑑₁𝑑₂𝑑₃>𝑢^∗𝑀²𝛼𝛼₂₃𝑣^∗+𝛽𝛼₃₂𝑤^∗2𝑑₁+2𝛼₁₁𝑀+𝛼₁₂𝑀+𝛼₁₃𝑀+𝛼𝑣^∗𝑀²𝛼₁₃𝑢^∗+𝛽𝛼₃₁𝑤^∗2𝑑₂+𝛼₂₁𝑀+2𝛼₂₂𝑀+𝛼₂₃𝑀+𝛽𝑤^∗𝑀²𝛼₁₂𝑢^∗+𝛼𝛼₂₁𝑣^∗2𝑑₃+𝛼₃₁𝑀+𝛼₃₂𝑀+2𝛼₃₃𝑀,𝐷𝐸[(𝑒−𝑏)−𝑏(𝑒−𝑏−1)]>𝐴(𝑒−𝑏)[(𝐸−𝐵)−(𝑒−𝑏)](𝐵−𝑏), (1.20)(1.21) and (1.8) hold, where 𝛼=𝑎(1+𝑏𝑢^∗)/𝑒𝑙,𝛽=𝐴(1+𝐵𝑢^∗)/𝐸𝐿(1+𝐷𝑤^∗),𝑀 is given by (1.19). Then the unique positive equilibrium (𝑢^∗,𝑣^∗,𝑤^∗) of (1.16) is globally asymptotically stable.

Remark 1.4. The system (1.16) has no nonconstant positive steady-state solution if all conditions of Theorem 1.3 hold.

Examples. The following two examples satisfy all conditions of Theorem 1.3:𝛼₁₁=14,𝛼₁₂=1,𝛼₁₃𝛼=1,₂₁=52,𝛼₂₂=1,𝛼₂₃=14,𝛼₃₁=1,𝛼₃₂=1,𝛼₃₃1=2,𝑏=1,𝑒=3,𝐴=1,𝐵=22,𝐷=3𝑑,𝐸=3,₁,𝑑₂,𝑑₃𝛼≫1;₁₁=2,𝛼₁₂=1,𝛼₁₃𝛼=2,₂₁=32,𝛼₂₂=12,𝛼₂₃𝛼=1,₃₁=2,𝛼₃₂=2,𝛼₃₃1=3,𝑏=25,𝑒=22,𝐴=3,1𝐵=23,𝐷=4𝑑,𝐸=4,₁,𝑑₂,𝑑₃≫1. (1.22)

2. Global Solutions

In order to establish the uniform 𝑊¹₂-estimates of the solutions for the system (1.16), the following Gagliardo-Nirenberg-type inequalities and the corresponding corollary play important roles (see [38, 41]).

Theorem 2.1. Let Ω⊂ℝ^𝑛 be a bounded domain with 𝜕Ω∈𝐶^𝑚. For every function 𝑢∈𝑊^𝑚_𝑟(Ω),1≤𝑞,𝑟≤∞, the derivative 𝐷^𝑗𝑢(0≤𝑗<𝑚) satisfies the inequality 𝐷^𝑗𝑢_𝑝≤𝐶𝐷^𝑚𝑢^𝑎_𝑟|𝑢|_𝑞^1−𝑎+|𝑢|_𝑞, (2.1) provided one of the following three conditions is satisfied: (1)  𝑟≤𝑞, (2)  0<𝑛(𝑟−𝑞)/(𝑚𝑟𝑞)<1, or (3)  𝑛(𝑟−𝑞)/(𝑚𝑟𝑞)=1 and 𝑚−𝑛/𝑞 is not a nonnegative integer, where 1/𝑝=𝑗/𝑛+𝑎(1/𝑟−𝑚/𝑛)+(1−𝑎)/𝑞 for all 𝑎∈[𝑗/𝑚,1), and the positive constant 𝐶 depends on 𝑛,𝑚,𝑗,𝑞,𝑟,𝑎.

Corollary 2.2. There exists a universal constant 𝐶 such that |𝑢|₂≤𝐶𝑢_𝑥2^1/3|𝑢|₁^2/3+|𝑢|₁,∀𝑢∈𝑊¹₂(0,1),|𝑢|₄≤𝐶𝑢_𝑥2^1/2|𝑢|₁^1/2+|𝑢|₁,∀𝑢∈𝑊¹₂(0,1),|𝑢|_5/2≤𝐶𝑢_𝑥2^2/5|𝑢|₁^3/5+|𝑢|₁,∀𝑢∈𝑊¹₂(0,1),𝑢_𝑥2≤𝐶𝑢_𝑥𝑥2^3/5|𝑢|₁^2/5+|𝑢|₁,∀𝑢∈𝑊²₂(0,1). (2.2)(2.3)(2.4)(2.5)
Throughout this paper, we always denote that 𝐶 is a Sobolev embedding constant or other kind of universal constant, 𝐴_𝑗,𝐵_𝑗,𝐶_𝑗 are some positive constants which depend only on 𝛼_𝑖𝑗(𝑖,𝑗=1,2,3),𝑎,𝑏,𝑒,𝑙,𝐴,𝐵,𝐷,𝐸 and 𝐿,𝐾_𝑗 are positive constants depending on 𝑑_𝑖,𝛼_𝑖𝑗(𝑖,𝑗=1,2,3),𝑎,𝑏,𝑒,𝑙,𝐴,𝐵,𝐷,𝐸 and 𝐿. When 𝑑₁,𝑑₂,𝑑₃≥1,𝑑₂/𝑑₁,𝑑₃/𝑑₁∈[𝑑,𝑑], 𝐾_𝑗 depend on 𝑑,𝑑, but do not depend on 𝑑₁,𝑑₂,𝑑₃.

Proof of Theorem 1.1. Taking integration of the three equations in (1.16) over (0,1), respectively, and combining the three integration equalities linearly, we have 𝑑𝑑𝑡¹₀[(𝑒𝑙+𝐸𝐿)𝑢+𝑎𝑣+𝐴𝑤]𝑑𝑥≤¹₀[(𝑒𝑙+𝐸𝐿)𝑢(1−𝑢)−𝑎𝑙𝑣−𝐴𝐿𝑤]𝑑𝑥. (2.6) It follows from the Young inequality and the Hölder inequality that 𝑑𝑑𝑡¹₀[(𝑒𝑙+𝐸𝐿)𝑢+𝑎𝑣+𝐴𝑤]𝑑𝑥≤𝐶₁−𝑘¹₀[(𝑒𝑙+𝐸𝐿)𝑢+𝑎𝑣+𝐴𝑤]𝑑𝑥, (2.7) where 𝑘=min{𝑙,𝐿},𝐶₁=(𝑘+1)²(𝑒𝑙+𝐸𝐿)/4. So there exist positive constants 𝑀₀ and 𝜏₀ depending on 𝑎,𝑏,𝑒,𝑙,𝐴,𝐵,𝐷,𝐸, and 𝐿, such that ¹₀𝑢𝑑𝑥,¹₀𝑣𝑑𝑥,¹₀𝑤𝑑𝑥≤𝑀₀,𝑡≥𝜏₀. (2.8) Moreover, there exists a positive constant 𝑀^′₀ which depends on 𝑎,𝑏,𝑒,𝑙,𝐴,𝐵,𝐷,𝐸,𝐿 and 𝐿¹-norm of 𝑢₀,𝑣₀,𝑤₀, such that ¹₀𝑢𝑑𝑥,¹₀𝑣𝑑𝑥,¹₀𝑤𝑑𝑥≤𝑀^′₀,𝑡≥0. (2.8)^′
Multiplying the first three equations in the system (1.16) by 𝑢,𝑣,𝑤, respectively, and integrating over (0,1), we have 12𝑑𝑑𝑡¹₀𝑢²𝑑𝑥≤−𝑑₁¹₀𝑢²_𝑥𝑑𝑥−¹₀2𝛼₁₁𝑢+𝛼₁₂𝑣+𝛼₁₃𝑤𝑢²_𝑥+𝛼₁₂𝑣_𝑥+𝛼₁₃𝑤_𝑥𝑢𝑢_𝑥+𝑑𝑥₁₀𝑢²1𝑑𝑥,2𝑑𝑑𝑡¹₀𝑣²𝑑𝑥≤−𝑑₂¹₀𝑣²_𝑥𝑑𝑥−¹₀𝛼₂₁𝑢+2𝛼₂₂𝑣+𝛼₂₃𝑤𝑣²_𝑥+𝛼₂₁𝑢_𝑥+𝛼₂₃𝑤_𝑥𝑣𝑣_𝑥+𝑑𝑥𝑒𝑙𝑏¹₀𝑣²1𝑑𝑥,2𝑑𝑑𝑡¹₀𝑤²𝑑𝑥≤−𝑑₃¹₀𝑤²_𝑥𝑑𝑥−¹₀𝛼₃₁𝑢+𝛼₃₂𝑣+2𝛼₃₃𝑤𝑤²_𝑥+𝛼₃₁𝑢_𝑥+𝛼₃₂𝑣_𝑥𝑤𝑤_𝑥+𝑑𝑥𝐸𝐿𝐵¹₀𝑤²𝑑𝑥, (2.9) from which it follows that 12𝑑𝑑𝑡¹₀𝑢²+𝑣²+𝑤²𝑑𝑥≤−𝑑¹₀𝑢²_𝑥+𝑣²_𝑥+𝑤²_𝑥𝑑𝑥+¹₀𝑢²𝑑𝑥+𝑒𝑙𝑏¹₀𝑣²𝑑𝑥+𝐸𝐿𝐵¹₀𝑤²𝑑𝑥−¹₀𝑞𝑢_𝑥,𝑣_𝑥,𝑤_𝑥𝑑𝑥≤−𝑑¹₀𝑢²_𝑥+𝑣²_𝑥+𝑤²_𝑥𝑑𝑥+1+𝑒𝑙𝑏+𝐸𝐿𝐵¹₀𝑢²+𝑣²+𝑤²𝑑𝑥−¹₀𝑞𝑢_𝑥,𝑣_𝑥,𝑤_𝑥𝑑𝑥, (2.10) where 𝑑=min{𝑑₁,𝑑₂,𝑑₃}, 𝑞𝑢_𝑥,𝑣_𝑥,𝑤_𝑥=2𝛼₁₁𝑢+𝛼₁₂𝑣+𝛼₁₃𝑤𝑢²_𝑥+𝛼₁₂𝑢+𝛼₂₁𝑣𝑢_𝑥𝑣_𝑥+𝛼₂₁𝑢+2𝛼₂₂𝑣+𝛼₂₃𝑤𝑣²_𝑥+𝛼₁₃𝑢+𝛼₃₁𝑤𝑢_𝑥𝑤_𝑥+𝛼₃₁𝑢+𝛼₃₂𝑣+2𝛼₃₃𝑤𝑤²_𝑥+𝛼₂₃𝑣+𝛼₃₂𝑤𝑣_𝑥𝑤_𝑥. (2.11) It is obvious that 𝑞(𝑢_𝑥,𝑣_𝑥,𝑤_𝑥) is a positive definite quadratic form of 𝑢_𝑥,𝑣_𝑥,𝑤_𝑥 if (1.17) holds. So (1.17) implies that 12𝑑𝑑𝑡¹₀𝑢²+𝑣²+𝑤²𝑑𝑥≤−𝑑¹₀𝑢²_𝑥+𝑣²_𝑥+𝑤²_𝑥𝑑𝑥+1+𝑒𝑙𝑏+𝐸𝐿𝐵¹₀𝑢²+𝑣²+𝑤²𝑑𝑥. (2.12)
Now, we proceed in the following two cases.
(i) One has  𝑡≥𝜏₀. The inequality (2.2) implies that |𝑢|⁶₂≤𝐶(|𝑢_𝑥|²₂|𝑢|⁴₁+|𝑢|⁶₁)≤𝐶𝑀⁴₀(|𝑢_𝑥|²₂+𝑀²₀). So we have ∫¹₀𝑢²_𝑥𝑑𝑥≥(1/𝐶𝑀⁴₀)(∫¹₀𝑢²𝑑𝑥)³−𝑀²₀, and −¹₀𝑢²_𝑥+𝑣²_𝑥+𝑤²_𝑥1𝑑𝑥≤−9𝐶𝑀⁴₀¹₀𝑢²+𝑣²+𝑤²𝑑𝑥³+3𝑀²₀. (2.13) It follows from (2.12) and (2.13) that 12𝑑𝑑𝑡¹₀𝑢²+𝑣²+𝑤²𝑑𝑥≤𝑑−𝐶₂¹₀𝑢²+𝑣²+𝑤²𝑑𝑥³+3𝑀²₀+1𝑑1+𝑒𝑙𝑏+𝐸𝐿𝐵¹₀𝑢²+𝑣²+𝑤²𝑑𝑥. (2.14) This means that there exist positive constants 𝜏₁ and 𝑀₁ depending on 𝑑_𝑖(𝑖=1,2,3),𝑎,𝑏,𝑒,𝑙,𝐴,𝐵,𝐷,𝐸, and 𝐿, such that ¹₀𝑢²𝑑𝑥,¹₀𝑣²𝑑𝑥,¹₀𝑤²𝑑𝑥≤𝑀₁,𝑡≥𝜏₁. (2.15) When 𝑑≥1,𝑀₁ is independent of 𝑑 because the zero point of the right-hand side in (2.14) can be estimated by positive constants independent of 𝑑.
(ii) One has  𝑡≥0. Repeating estimates in (i) by (2.8)^′, we can obtain that there exists a positive constant 𝑀^′₁ depending on 𝑑_𝑖(𝑖=1,2,3),𝑎,𝑏,𝑒,𝑙,𝐴,𝐵,𝐷,𝐸,𝐿 and the 𝐿¹,𝐿²-norm of 𝑢₀,𝑣₀,𝑤₀, such that ¹₀𝑢²𝑑𝑥,¹₀𝑣²𝑑𝑥,¹₀𝑤²𝑑𝑥≤𝑀^′₁,𝑡≥0. (2.15)^′ When 𝑑≥1,𝑀^′₁ is independent of 𝑑.
To estimate |𝑢_𝑥|₂,|𝑣_𝑥|₂,|𝑤_𝑥|₂, we introduce the following scaling: 𝑢𝑢=𝑑₁,̂𝑣𝑣=𝑑₁,𝑤𝑤=𝑑₁,𝑡̂=𝑑₁𝑡. (2.16) Denoting 𝜉=𝑑₂/𝑑₁,𝜂=𝑑₃/𝑑₁, and using 𝑢,𝑣,𝑤,𝑡 instead of 𝑢,𝑣,̂𝑤,𝑡̂, respectively, then the system (1.16) reduces to 𝑢_𝑡=𝑃_𝑥𝑥()𝑣+𝑓𝑢,𝑣,𝑤,0<𝑥<1,𝑡>0,_𝑡=𝑄_𝑥𝑥()𝑤+𝑔𝑢,𝑣,𝑤,0<𝑥<1,𝑡>0,_𝑡=𝑅_𝑥𝑥()𝑢+ℎ𝑢,𝑣,𝑤,0<𝑥<1,𝑡>0,_𝑥(𝑥,𝑡)=𝑣_𝑥(𝑥,𝑡)=𝑤_𝑥()𝑢()=𝑢𝑥,𝑡=0,𝑥=0,1,𝑡>0,𝑥,0₀(𝑥),𝑣(𝑥,0)=𝑣₀(𝑥),𝑤(𝑥,0)=𝑤₀(𝑥),0<𝑥<1, (2.17) where 𝑃=𝑢+𝛼₁₁𝑢²+𝛼₁₂𝑢𝑣+𝛼₁₃𝑢𝑤,𝑄=𝜉𝑣+𝛼₂₁𝑢𝑣+𝛼₂₂𝑣²+𝛼₂₃𝑣𝑤,𝑅=𝜂𝑤+𝛼₃₁𝑢𝑤+𝛼₃₂𝑣𝑤+𝛼₃₃𝑤²𝑑,𝑓(𝑢,𝑣,𝑤)=₁₋₁𝑢−𝑢²−(𝑎𝑢𝑣/(1+𝑏𝑑₁𝑢))−(𝐴𝑢𝑤/(1+𝐵𝑑₁𝑢+𝐷𝑑₁𝑤)),𝑔(𝑢,𝑣,𝑤)=𝑙𝑣(−𝑑₁⁻¹+(𝑒𝑢/(1+𝑏𝑑₁𝑢))),ℎ(𝑢,𝑣,𝑤)=𝐿𝑤(−𝑑₁⁻¹+(𝐸𝑢/(1+𝐵𝑑₁𝑢+𝐷𝑑₁𝑤))).
We still proceed in the following two cases.(i) One has 𝑡≥𝜏^∗₁=𝑑₁𝜏₁. It is not hard to verify that ¹₀𝑢𝑑𝑥,¹₀𝑣𝑑𝑥,¹₀𝑤𝑑𝑥≤𝑀₀𝑑₁⁻¹,₁₀𝑢²𝑑𝑥,¹₀𝑣²𝑑𝑥,¹₀𝑤²𝑑𝑥≤𝑀₁𝑑₁⁻²,|𝑃|₁,|𝑄|₁,|𝑅|₁≤𝐴₁𝐾₁𝑑₁⁻¹, (2.18)where 𝐾₁=(1+𝜉+𝜂)+𝑀₁𝑑₁⁻¹,𝐴₁=max{𝑀₀,𝛼₁₁+𝛼₁₂+𝛼₁₃,𝛼₂₁+𝛼₂₂+𝛼₂₃,𝛼₃₁+𝛼₃₂+𝛼₃₃}.
Multiplying the first three equations in (2.17) by 𝑃_𝑡,𝑄_𝑡,𝑅_𝑡, integrating them over the domain (0,1), respectively, and then adding up the three integration equalities, we have 12𝑦^′(𝑡)=−¹₀𝑢²_𝑡𝑑𝑥−𝜉¹₀𝑣²_𝑡𝑑𝑥−𝜂¹₀𝑤²_𝑡𝑑𝑥−¹₀𝑞𝑢_𝑡,𝑣_𝑡,𝑤_𝑡+𝑑𝑥₁₀1+2𝛼₁₁𝑢+𝛼₁₂𝑣+𝛼₁₃𝑤𝑢_𝑡𝑓+𝛼₁₂𝑢𝑣_𝑡𝑓+𝛼₁₃𝑢𝑤_𝑡𝑓+𝑑𝑥₁₀𝜉+𝛼₂₁𝑢+2𝛼₂₂𝑣+𝛼₂₃𝑤𝑣_𝑡𝑔+𝛼₂₁𝑣𝑢_𝑡𝑔+𝛼₂₃𝑣𝑤_𝑡𝑔+𝑑𝑥₁₀𝜂+𝛼₃₁𝑢+𝛼₃₂𝑣+2𝛼₃₃𝑤𝑤_𝑡ℎ+𝛼₃₁𝑤𝑢_𝑡ℎ+𝛼₃₂𝑤𝑣_𝑡ℎ𝑑𝑥, (2.19) where 𝑦(𝑡)=∫¹₀(𝑃²_𝑥+𝑄²_𝑥+𝑅²_𝑥)𝑑𝑥. Notice by (1.17) that there exists a positive constant 𝐶₃ depending only on 𝛼_𝑖𝑗(𝑖,𝑗=1,2,3), such that 𝑞𝑢_𝑡,𝑣_𝑡,𝑤_𝑡≥𝐶₃(𝑢+𝑣+𝑤)𝑢²_𝑡+𝑣²_𝑡+𝑤²_𝑡. (2.20) Thus, 12𝑦^′(𝑡)≤−¹₀𝑢²_𝑡𝑑𝑥−𝜉¹₀𝑣²_𝑡𝑑𝑥−𝜂¹₀𝑤²_𝑡𝑑𝑥−𝐶₃¹₀(𝑢+𝑣+𝑤)𝑢²_𝑡+𝑣²_𝑡+𝑤²_𝑡+𝑑𝑥₁₀1+2𝛼₁₁𝑢+𝛼₁₂𝑣+𝛼₁₃𝑤𝑢_𝑡𝑓+𝛼₁₂𝑢𝑣_𝑡𝑓+𝛼₁₃𝑢𝑤_𝑡𝑓+𝑑𝑥₁₀𝜉+𝛼₂₁𝑢+2𝛼₂₂𝑣+𝛼₂₃𝑤𝑣_𝑡𝑔+𝛼₂₁𝑣𝑢_𝑡𝑔+𝛼₂₃𝑣𝑤_𝑡𝑔+𝑑𝑥₁₀𝜂+𝛼₃₁𝑢+𝛼₃₂𝑣+2𝛼₃₃𝑤𝑤_𝑡ℎ+𝛼₃₁𝑤𝑢_𝑡ℎ+𝛼₃₂𝑤𝑣_𝑡ℎ𝑑𝑥. (2.21)
Using the Young inequality, Hölder inequality, and (2.18), we can obtain the following estimates: ¹₀𝑢⁴𝑑𝑥≤¹₀𝑢²𝑑𝑥^1/31₀𝑢⁵𝑑𝑥^2/3≤𝑀₁^1/3𝑑₁^−2/31₀𝑢⁵𝑑𝑥^2/3,₁₀𝑢𝑣²𝑑𝑥≤¹₀𝑢²𝑑𝑥^1/21₀𝑣²𝑑𝑥^1/61₀𝑣⁵𝑑𝑥^1/3≤𝑀₁^2/3𝑑₁^−4/31₀𝑣⁵𝑑𝑥^1/3,₁₀𝑢³𝑑𝑥≤¹₀𝑢²𝑑𝑥^2/31₀𝑢⁵𝑑𝑥^1/3≤𝑀₁^2/3𝑑₁^−4/31₀𝑢⁵𝑑𝑥^1/3,₁₀𝑢⁴4𝑣𝑑𝑥≤5¹₀𝑢⁵1𝑑𝑥+5¹₀𝑣⁵4𝑑𝑥≤5¹₀𝑢⁵+𝑣⁵𝑑𝑥. (2.22)
Applying the above estimates and Gagliardo-Nirenberg-type inequalities to the terms on the right-hand side of (2.21), we have−¹₀𝑢²_𝑡1𝑑𝑥≤−2¹₀𝑃²_𝑥𝑥𝑑𝑥+¹₀𝑓²𝑑𝑥,−𝜉¹₀𝑣²_𝑡𝜉𝑑𝑥≤−2¹₀𝑄²_𝑥𝑥𝑑𝑥+𝜉¹₀𝑔²𝑑𝑥,−𝜂¹₀𝑤²_𝑡𝜂𝑑𝑥≤−2¹₀𝑅²_𝑥𝑥𝑑𝑥+𝜂¹₀ℎ²𝑑𝑥,₁₀𝑓²𝑑𝑥≤¹₀𝑑₁⁻²𝑢²+𝑢⁴+𝑎²𝑣²𝑏𝑑₁²+𝐴²𝑤²𝐵𝑑₁²+2𝑎𝑢²𝑣𝑏𝑑₁+2𝐴𝑢²𝑤𝐵𝑑₁+2𝑎𝐴𝑣𝑤𝑏𝐵𝑑²₁≤𝑎𝑑𝑥1+2𝑏²𝐴+2𝐵²𝑀₁𝑑₁⁻⁴+𝑀₁^1/3𝑑₁^−2/31₀𝑢⁵𝑑𝑥^2/3𝑎+2𝑏+𝐴𝐵𝑀₁^2/3𝑑₁^−7/31₀𝑢⁵𝑑𝑥^1/3,𝜉¹₀𝑔²𝑑𝑥≤𝜉¹₀𝑙²𝑑₁⁻²𝑣²+𝑒²𝑙²𝑣²𝑏𝑑₁²𝑑𝑥≤𝜉𝑙²𝑒1+𝑏²𝑀₁𝑑₁⁻⁴,𝜂¹₀ℎ²𝑑𝑥≤𝜂¹₀𝐿²𝑑₁⁻²𝑤²+𝐸²𝐿²𝑤²𝐵𝑑₁²𝑑𝑥≤𝜂𝐿²𝐸1+𝐵²𝑀₁𝑑₁⁻⁴,−¹₀𝑢²_𝑡𝑑𝑥−𝜉¹₀𝑣²_𝑡𝑑𝑥−𝜂¹₀𝑤²_𝑡1𝑑𝑥≤−2¹₀𝑃²_𝑥𝑥𝜉𝑑𝑥−2¹₀𝑄²_𝑥𝑥𝜂𝑑𝑥−2¹₀𝑅²_𝑥𝑥+𝑎𝑑𝑥1+2𝑏²+𝐴𝐵²+𝜉𝑙²𝑒1+𝑏²+𝜂𝐿²𝐸1+𝐵²𝑀₁𝑑₁⁻⁴+𝑀₁^1/3𝑑₁^−2/31₀𝑢⁵𝑑𝑥^2/3𝑎+2𝑏+𝐴𝐵𝑀₁^2/3𝑑₁^−7/31₀𝑢⁵𝑑𝑥^1/3. (2.23)(2.24) Similarly, we can obtain ¹₀𝑢_𝑡𝑓𝑑𝑥≤¹₀𝑢_𝑡𝑑₁⁻¹𝑢+𝑢²+𝑎𝑏𝑑₁𝐴𝑣+𝐵𝑑₁𝑤≤𝑑𝑑𝑥₁₋₂2𝜀¹₀𝜀𝑢𝑑𝑥+2¹₀𝑢𝑢²_𝑡1𝑑𝑥+2𝜀¹₀𝑢³𝜀𝑑𝑥+2¹₀𝑢𝑢²_𝑡+𝑑𝑥𝑎/𝑏𝑑₁²2𝜀¹₀𝜀𝑣𝑑𝑥+2¹₀𝑣𝑢²_𝑡𝑑𝑥+𝐴/𝐵𝑑₁²2𝜀¹₀𝜀𝑤𝑑𝑥+2¹₀𝑤𝑢²_𝑡≤1𝑑𝑥𝑎2𝜀1+𝑏²+𝐴𝐵²𝑀₀𝑑₁⁻³+1𝑀2𝜀^12/3𝑑₁^−4/31₀𝑢⁵𝑑𝑥^1/3+𝜀¹₀𝑢𝑢²_𝑡𝜀𝑑𝑥+2¹₀𝑣𝑢²_𝑡𝜀𝑑𝑥+2¹₀𝑤𝑢²_𝑡𝑑𝑥,2𝛼₁₁¹₀𝑢𝑢_𝑡𝑓𝑑𝑥≤2𝛼₁₁¹₀𝑢𝑢_𝑡𝑑₁⁻¹𝑢+𝑢²+𝑎𝑏𝑑₁𝐴𝑣+𝐵𝑑₁𝑤≤𝛼𝑑𝑥₂₁₁𝑑₁⁻²𝜀¹₀𝑢³𝑑𝑥+𝜀¹₀𝑢𝑢²_𝑡𝛼𝑑𝑥+²¹¹𝜀¹₀𝑢⁵𝑑𝑥+𝜀¹₀𝑢𝑢²_𝑡+𝛼𝑑𝑥₂₁₁𝑎²𝜀𝑏²𝑑²₁¹⁰𝑢²𝑣𝑑𝑥+𝜀¹₀𝑣𝑢²_𝑡𝛼𝑑𝑥+²¹¹𝐴²𝜀𝐵²𝑑²₁¹⁰𝑢²𝑤𝑑𝑥+𝜀¹₀𝑤𝑢²_𝑡≤𝛼𝑑𝑥₂₁₁𝜀𝑎1+𝑏²+𝐴𝐵²𝑀₁^2/3𝑑₁^−10/31₀𝑢⁵𝑑𝑥^1/3+𝛼²₁₁𝜀¹₀𝑢⁵𝑑𝑥+2𝜀¹₀𝑢𝑢²_𝑡𝑑𝑥+𝜀¹₀𝑣𝑢²_𝑡𝑑𝑥+𝜀¹₀𝑤𝑢²_𝑡𝛼𝑑𝑥,₁₂₁₀𝑣𝑢_𝑡𝑓𝑑𝑥≤𝛼₁₂¹₀𝑣𝑢_𝑡𝑑₁⁻¹𝑢+𝑢²+𝑎𝑏𝑑₁𝐴𝑣+𝐵𝑑₁𝑤≤𝛼𝑑𝑥₂₁₂𝑑₁⁻²2𝜀¹₀𝑢𝑣²𝜀𝑑𝑥+2¹₀𝑢𝑢²_𝑡𝛼𝑑𝑥+²¹²2𝜀¹₀𝑢⁴𝜀𝑣𝑑𝑥+2¹₀𝑣𝑢²_𝑡+𝛼𝑑𝑥₂₁₂𝑎²2𝜀𝑏²𝑑²₁¹⁰𝑣³𝜀𝑑𝑥+2¹₀𝑣𝑢²_𝑡𝛼𝑑𝑥+²¹²𝐴²2𝜀𝐵²𝑑²₁¹⁰𝑣²𝜀𝑤𝑑𝑥+2¹₀𝑤𝑢²_𝑡≤𝛼𝑑𝑥₂₁₂𝑎2𝜀1+𝑏²+𝐴𝐵²𝑀₁^2/3𝑑₁^−10/31₀𝑣⁵𝑑𝑥^1/3+2𝛼²₁₂5𝜀¹₀𝑢⁵+𝑣⁵+𝜀𝑑𝑥2¹₀𝑢𝑢²_𝑡𝑑𝑥+𝜀¹₀𝑣𝑢²_𝑡𝜀𝑑𝑥+2¹₀𝑤𝑢²_𝑡𝛼𝑑𝑥,₁₃₁₀𝑤𝑢_𝑡𝑓𝑑𝑥≤𝛼₁₃¹₀𝑤𝑢_𝑡𝑑₁⁻¹𝑢+𝑢²+𝑎𝑏𝑑₁𝐴𝑣+𝐵𝑑₁𝑤≤𝛼𝑑𝑥₂₁₃𝑑₁⁻²2𝜀¹₀𝑢𝑤²𝜀𝑑𝑥+2¹₀𝑢𝑢²_𝑡𝛼𝑑𝑥+²¹³2𝜀¹₀𝑢⁴𝜀𝑤𝑑𝑥+2¹₀𝑤𝑢²_𝑡+𝛼𝑑𝑥₂₁₃𝑎²2𝜀𝑏²𝑑²₁¹⁰𝑤²𝜀𝑣𝑑𝑥+2¹₀𝑣𝑢²_𝑡𝛼𝑑𝑥+²¹³𝐴²2𝜀𝐵²𝑑²₁¹⁰𝑤³𝜀𝑑𝑥+2¹₀𝑤𝑢²_𝑡≤𝛼𝑑𝑥₂₁₃𝑎2𝜀1+𝑏²+𝐴𝐵²𝑀₁^2/3𝑑₁^−10/31₀𝑤⁵𝑑𝑥^1/3+2𝛼²₁₃5𝜀¹₀𝑢⁵+𝑤⁵+𝜀𝑑𝑥2¹₀𝑢𝑢²_𝑡𝑑𝑥+𝜀¹₀𝑤𝑢²_𝑡𝜀𝑑𝑥+2¹₀𝑣𝑢²_𝑡𝛼𝑑𝑥₁₂₁₀𝑢𝑣_𝑡𝑓𝑑𝑥≤𝛼₁₂¹₀𝑢𝑣_𝑡𝑑₁⁻¹𝑢+𝑢²+𝑎𝑏𝑑₁𝐴𝑣+𝐵𝑑₁𝑤≤𝛼𝑑𝑥₂₁₂𝑑₁⁻²2𝜀¹₀𝑢³𝜀𝑑𝑥+2¹₀𝑢𝑣²_𝑡𝛼𝑑𝑥+²¹²2𝜀¹₀𝑢⁵𝜀𝑑𝑥+2¹₀𝑢𝑣²_𝑡+𝛼𝑑𝑥₂₁₂𝑎²2𝜀𝑏²𝑑²₁¹⁰𝑢²𝜀𝑣𝑑𝑥+2¹₀𝑣𝑣²_𝑡𝛼𝑑𝑥+²¹²𝐴²2𝜀𝐵²𝑑²₁¹⁰𝑢²𝜀𝑤𝑑𝑥+2¹₀𝑤𝑣²_𝑡≤𝛼𝑑𝑥₂₁₂𝑎2𝜀1+𝑏²+𝐴𝐵²𝑀₁^2/3𝑑₁^−10/31₀𝑢⁵𝑑𝑥^1/3+𝛼²₁₂2𝜀¹₀𝑢⁵𝑑𝑥+𝜀¹₀𝑢𝑣²_𝑡𝜀𝑑𝑥+2¹₀𝑣𝑣²_𝑡𝜀𝑑𝑥+2¹₀𝑤𝑣²_𝑡𝛼𝑑𝑥,₁₃₁₀𝑢𝑤_𝑡𝑓𝑑𝑥≤𝛼₁₃¹₀𝑢𝑤_𝑡𝑑₁⁻¹𝑢+𝑢²+𝑎𝑏𝑑₁𝐴𝑣+𝐵𝑑₁𝑤≤𝛼𝑑𝑥₂₁₃𝑑₁⁻²2𝜀¹₀𝑢³𝜀𝑑𝑥+2¹₀𝑢𝑤²_𝑡𝛼𝑑𝑥+²¹³2𝜀¹₀𝑢⁵𝜀𝑑𝑥+2¹₀𝑢𝑤²_𝑡+𝛼𝑑𝑥₂₁₃𝑎²2𝜀𝑏²𝑑²₁¹⁰𝑢²𝜀𝑣𝑑𝑥+2¹₀𝑣𝑤²_𝑡𝛼𝑑𝑥+²¹³𝐴²2𝜀𝐵²𝑑²₁¹⁰𝑢²𝜀𝑤𝑑𝑥+2¹₀𝑤𝑤²_𝑡≤𝛼𝑑𝑥,₂₁₃𝑎2𝜀1+𝑏²+𝐴𝐵²𝑀₁^2/3𝑑₁^−10/31₀𝑢⁵𝑑𝑥^1/3+𝛼²₁₃2𝜀¹₀𝑢⁵𝑑𝑥+𝜀¹₀𝑢𝑤²_𝑡𝜀𝑑𝑥+2¹₀𝑣𝑤²_𝑡𝜀𝑑𝑥+2¹₀𝑤𝑤²_𝑡𝜉𝑑𝑥,₁₀𝑣_𝑡𝑔𝑑𝑥≤𝜉¹₀𝑣_𝑡𝑙𝑣𝑑₁⁻¹+𝑒𝑏𝑑₁𝜉𝑑𝑥≤²𝑙²𝑑₁⁻²(1+(𝑒/𝑏))²2𝜀¹₀𝜀𝑣𝑑𝑥+2¹₀𝑣𝑣²_𝑡≤𝜉𝑑𝑥₂𝑙²(1+(𝑒/𝑏))²𝑀2𝜀⁰𝑑₁⁻³+𝜀2¹₀𝑣𝑣²_𝑡𝛼𝑑𝑥,₂₁₁₀𝑢𝑣_𝑡𝑔𝑑𝑥≤𝛼₂₁¹₀𝑢𝑣_𝑡𝑙𝑣𝑑₁⁻¹+𝑒𝑏𝑑₁≤𝛼𝑑𝑥₂₂₁𝑙²𝑑₁⁻²(1+(𝑒/𝑏))²2𝜀¹₀𝑢²𝜀𝑣𝑑𝑥+2¹₀𝑣𝑣²_𝑡≤𝛼𝑑𝑥₂₂₁𝑙²(1+(𝑒/𝑏))²𝑀2𝜀^12/3𝑑₁^−10/31₀𝑢⁵𝑑𝑥^1/3+𝜀2¹₀𝑣𝑣²_𝑡𝑑𝑥,2𝛼₂₂¹₀𝑣𝑣_𝑡𝑔𝑑𝑥≤2𝛼₂₂¹₀𝑣²𝑣_𝑡𝑙𝑑₁⁻¹+𝑒𝑏𝑑₁≤𝛼𝑑𝑥₂₂₂𝑙²𝑑₁⁻²(1+(𝑒/𝑏))²𝜀¹₀𝑣³𝑑𝑥+𝜀¹₀𝑣𝑣²_𝑡≤𝛼𝑑𝑥₂₂₁𝑙²(1+(𝑒/𝑏))²𝜀𝑀₁^2/3𝑑₁^−10/31₀𝑣⁵𝑑𝑥^1/3+𝜀¹₀𝑣𝑣²_𝑡𝛼𝑑𝑥,₂₃₁₀𝑤𝑣_𝑡𝑔𝑑𝑥≤𝛼₂₃¹₀𝑤𝑣_𝑡𝑙𝑣𝑑₁⁻¹+𝑒𝑏𝑑₁≤𝛼𝑑𝑥₂₂₃𝑙²𝑑₁⁻²(1+(𝑒/𝑏))²2𝜀¹₀𝑣²𝜀𝑤𝑑𝑥+2¹₀𝑤𝑣²_𝑡≤𝛼𝑑𝑥₂₂₃𝑙²(1+(𝑒/𝑏))²𝑀2𝜀^12/3𝑑₁^−10/31₀𝑣⁵𝑑𝑥^1/3+𝜀2¹₀𝑤𝑣²_𝑡𝛼𝑑𝑥,₂₁₁₀𝑣𝑢_𝑡𝑔𝑑𝑥≤𝛼₂₁¹₀𝑣²𝑣_𝑡𝑙𝑑₁⁻¹+𝑒𝑏𝑑₁≤𝛼𝑑𝑥₂₂₁𝑙²𝑑₁⁻²(1+(𝑒/𝑏))²2𝜀¹₀𝑣³𝜀𝑑𝑥+2¹₀𝑣𝑢²_𝑡≤𝛼𝑑𝑥₂₂₁𝑙²(1+(𝑒/𝑏))²𝑀2𝜀^12/3𝑑₁^−10/31₀𝑣⁵𝑑𝑥^1/3+𝜀2¹₀𝑣𝑢²_𝑡𝛼𝑑𝑥,₂₃₁₀𝑣𝑤_𝑡𝑔𝑑𝑥≤𝛼₂₃¹₀𝑣²𝑤_𝑡𝑙𝑑₁⁻¹+𝑒𝑏𝑑₁≤𝛼𝑑𝑥₂₂₃𝑙²𝑑₁⁻²(1+(𝑒/𝑏))²2𝜀¹₀𝑣³𝜀𝑑𝑥+2¹₀𝑣𝑤²_𝑡≤𝛼𝑑𝑥₂₂₃𝑙²(1+(𝑒/𝑏))²𝑀2𝜀^12/3𝑑₁^−10/31₀𝑣⁵𝑑𝑥^1/3+𝜀2¹₀𝑣𝑤²_𝑡𝜂𝑑𝑥,₁₀𝑤_𝑡ℎ𝑑𝑥≤𝜂¹₀𝑤_𝑡𝐿𝑤𝑑₁⁻¹+𝐸𝐵𝑑₁≤𝜂𝑑𝑥₂𝐿²𝑑₁⁻²(1+(𝐸/𝐵))²2𝜀¹₀𝜀𝑤𝑑𝑥+2¹₀𝑤𝑤²_𝑡≤𝜂𝑑𝑥₂𝐿²(1+(𝐸/𝐵))²𝑀2𝜀⁰𝑑₁⁻³+𝜀2¹₀𝑤𝑤²_𝑡𝛼𝑑𝑥,₃₁₁₀𝑢𝑤_𝑡ℎ𝑑𝑥≤𝛼₃₁¹₀𝑢𝑤_𝑡𝐿𝑤𝑑₁⁻¹+𝐸𝐵𝑑₁≤𝛼𝑑𝑥₂₃₁𝐿²𝑑₁⁻²(1+(𝐸/𝐵))²2𝜀¹₀𝑤²𝜀𝑢𝑑𝑥+2¹₀𝑢𝑤²_𝑡≤𝛼𝑑𝑥₂₃₁𝐿²(1+(𝐸/𝐵))²𝑀2𝜀^12/3𝑑₁^−10/31₀𝑤⁵𝑑𝑥^1/3+𝜀2¹₀𝑢𝑤²_𝑡𝛼𝑑𝑥,₃₂₁₀𝑣𝑤_𝑡ℎ𝑑𝑥≤𝛼₃₂¹₀𝑣𝑤_𝑡𝐿𝑤𝑑₁⁻¹+𝐸𝐵𝑑₁≤𝛼𝑑𝑥₂₃₂𝐿²𝑑₁⁻²(1+(𝐸/𝐵))²2𝜀¹₀𝑤²𝜀𝑣𝑑𝑥+2¹₀𝑣𝑤²_𝑡≤𝛼𝑑𝑥₂₃₂𝐿²(1+(𝐸/𝐵))²𝑀2𝜀^12/3𝑑₁^−10/31₀𝑤⁵𝑑𝑥^1/3+𝜀2¹₀𝑣𝑤²_𝑡𝑑𝑥,2𝛼₃₃¹₀𝑤𝑤_𝑡ℎ𝑑𝑥≤2𝛼₃₃¹₀𝑤²𝑤_𝑡𝐿𝑑₁⁻¹+𝐸𝐵𝑑₁≤𝛼𝑑𝑥₂₃₃𝐿²𝑑₁⁻²(1+(𝐸/𝐵))²𝜀¹₀𝑤³𝑑𝑥+𝜀¹₀𝑤𝑤²_𝑡≤𝛼𝑑𝑥₂₃₃𝐿²(1+(𝐸/𝐵))²𝜀𝑀₁^2/3𝑑₁^−10/31₀𝑤⁵𝑑𝑥^1/3+𝜀¹₀𝑤𝑤²_𝑡𝛼𝑑𝑥,₃₁₁₀𝑤𝑢_𝑡ℎ𝑑𝑥≤𝛼₃₁¹₀𝑤²𝑢_𝑡𝐿𝑑₁⁻¹+𝐸𝐵𝑑₁≤𝛼𝑑𝑥₂₃₁𝐿²𝑑₁⁻²(1+(𝐸/𝐵))²2𝜀¹₀𝑤³𝜀𝑑𝑥+2¹₀𝑤𝑢²_𝑡≤𝛼𝑑𝑥₂₃₁𝐿²(1+(𝐸/𝐵))²𝑀2𝜀^12/3𝑑₁^−10/31₀𝑣⁵𝑑𝑥^1/3+𝜀2¹₀𝑤𝑢²_𝑡𝛼𝑑𝑥,₃₂₁₀𝑤𝑣_𝑡ℎ𝑑𝑥≤𝛼₃₂¹₀𝑤²𝑣_𝑡𝐿𝑑₁⁻¹+𝐸𝐵𝑑₁≤𝛼𝑑𝑥₂₃₂𝐿²𝑑₁⁻²(1+(𝐸/𝐵))²2𝜀¹₀𝑤³𝜀𝑑𝑥+2¹₀𝑤𝑣²_𝑡≤𝛼𝑑𝑥₂₃₂𝐿²(1+(𝐸/𝐵))²𝑀2𝜀^12/3𝑑₁^−10/31₀𝑤⁵𝑑𝑥^1/3+𝜀2¹₀𝑤𝑣²_𝑡𝑑𝑥. (2.25)
By the above inequalities and the condition (1.17), we have ¹₀1+2𝛼₁₁𝑢+𝛼₁₂𝑣+𝛼₁₃𝑤𝑢_𝑡𝑓+𝛼₁₂𝑢𝑣_𝑡𝑓+𝛼₁₃𝑢𝑤_𝑡𝑓+𝑑𝑥₁₀𝜉+𝛼₂₁𝑢+2𝛼₂₂𝑣+𝛼₂₃𝑤𝑣_𝑡𝑔+𝛼₂₁𝑣𝑢_𝑡𝑔+𝛼₂₃𝑣𝑤_𝑡𝑔+𝑑𝑥₁₀𝜂+𝛼₃₁𝑢+𝛼₃₂𝑣+2𝛼₃₃𝑤𝑤_𝑡ℎ+𝛼₃₁𝑤𝑢_𝑡ℎ+𝛼₃₂𝑤𝑣_𝑡ℎ≤𝑑𝑥𝜆𝜀¹₀(𝑢+𝑣+𝑤)𝑢²_𝑡+𝑣²_𝑡+𝑤²_𝑡𝐶𝑑𝑥+⁴𝜀1+𝜉²+𝜂²𝑀₀𝑑₁⁻³+𝐶₅𝜀1+𝑑₁⁻²𝑀₁^2/3𝑑₁^−4/31₀𝑢⁵+𝑣⁵+𝑤⁵𝑑𝑥^1/3+𝐶₆𝜀¹₀𝑢⁵+𝑣⁵+𝑤⁵𝑑𝑥, (2.26) where 𝜆 is a constant depending only on 𝜀(𝛼_𝑖𝑗)(𝑖,𝑗=1,2,3). Choose a small enough positive number 𝜀 which depends on 𝛼_𝑖𝑗(𝑖,𝑗=1,2,3),𝑎,𝑏,𝑒,𝑙,𝐴,𝐵,𝐷,𝐸, and 𝐿, such that 𝜆𝜀<𝐶₃. Substituting inequalities (2.24) and (2.26) into (2.21), one can obtain 12𝑦^′(𝑡)1≤−2¹₀𝑃²_𝑥𝑥𝜉𝑑𝑥−2¹₀𝑄²_𝑥𝑥𝜂𝑑𝑥−2¹₀𝑅²_𝑥𝑥𝑑𝑥+𝐵₁𝐾₂𝑑₁⁻³+𝐵₂𝑌+𝐵₃𝐾₃𝑑₁^−2/3𝑌^2/3+𝐵₄𝐾₄𝑑₁^−4/3𝑌^1/3, (2.27) where 𝑌=∫¹₀(𝑢⁵+𝑣⁵+𝑤⁵)𝑑𝑥, 𝐾₂=(1+𝜉²+𝜂²)𝑀₀+(1+𝜉+𝜂)𝑀₁𝑑₁⁻¹, 𝐾₃=𝑀₁^1/3, 𝐾₄=𝑀₁^2/3(1+𝑑₁⁻²).
Note that 𝑃≥𝛼₁₁𝑢²,𝑄≥𝛼₂₂𝑣²,𝑅≥𝛼₃₃𝑤². (2.28) It follows from (2.18) and (2.4) to functions 𝑃,𝑄,𝑅 that 𝑌≤𝐵₅¹₀𝑃^5/2+𝑄^5/2+𝑅^5/2𝑑𝑥≤𝐵₆𝐾₁^3/2𝑑₁^−3/2𝑦^1/2+𝐵₆𝐾₁^5/2𝑑₁^−5/2,𝑌^1/3≤𝐵₇𝐾₁^1/2𝑑₁^−1/2𝑦^1/6+𝐵₇𝐾₁^5/6𝑑₁^−5/6,𝑌^2/3≤𝐵₈𝐾₁𝑑₁⁻¹𝑦^1/3+𝐵₈𝐾₁^5/3𝑑₁^−5/3. (2.29) Moreover, one can obtain by (2.5) and (2.18) that −12¹₀𝑃²_𝑥𝑥𝜉𝑑𝑥−2¹₀𝑄²_𝑥𝑥𝜂𝑑𝑥−2¹₀𝑅²_𝑥𝑥𝑑𝑥≤−𝐵₉min1,𝜉,𝜂𝐾₁^−4/3𝑑₁^4/3𝑦^5/3+1+𝜉+𝜂𝐾²₁𝑑₁⁻². (2.30) Combining (2.27),(2.29), and (2.30), we have 12𝑦^′(𝑡)≤−𝐴₁min1,𝜉,𝜂𝐾₁^−4/3𝑑₁^4/3𝑦^5/3+𝐴₂1+𝜉+𝜂𝐾²₁𝑑₁⁻²+𝐾₂𝑑₁⁻³+𝐾₁^5/2𝑑₁^−5/2+𝐾₁^5/3𝐾₃𝑑₁^−7/3+𝐾₁^5/6𝐾₄𝑑₁^−13/6+𝐴₃𝐾₁^3/2𝑑₁^−3/2𝑦^1/2+𝐴₄𝐾₁𝐾₃𝑑₁^−5/3𝑦^1/3+𝐴₅𝐾₁^1/2𝐾₄𝑑₁^−11/6𝑦^1/6. (2.31) Multiplying inequality (2.31) by 𝑑²₁, we have 12𝑦^′(𝑡)≤−𝐴₁min1,𝜉,𝜂𝐾₁^−4/3𝑦^5/3+𝐴₂1+𝜉+𝜂𝐾²₁+𝐾₂𝑑₁⁻¹+𝐾₁^5/2𝑑₁^−1/2+𝐾₁^5/3𝐾₃𝑑₁^−1/3+𝐾₁^5/6𝐾₄𝑑₁^−1/6+𝐴₃𝐾₁^3/2𝑑₁^−1/2𝑦^1/2+𝐴₄𝐾₁𝐾₃𝑑₁^−1/3𝑦^1/3+𝐴₅𝐾₁^1/2𝐾₄𝑑₁^−1/6𝑦^1/6, (2.32)where 𝑦=∫¹₀[(𝑑₁𝑃_𝑥)²+(𝑑₁𝑄_𝑥)²+(𝑑₁𝑅_𝑥)²]𝑑𝑥. The inequality (2.32) implies that there exist 𝜏₂>0 and positive constant 𝑀₂ depending on 𝑑_𝑖,𝛼_𝑖𝑗(𝑖,𝑗=1,2,3),𝑎,𝑏,𝑒,𝑙,𝐴,𝐵,𝐷,𝐸, and 𝐿, such that ¹₀𝑑₁𝑃_𝑥²𝑑𝑥,¹₀𝑑₁𝑄_𝑥²𝑑𝑥,¹₀𝑑₁𝑅_𝑥²𝑑𝑥≤𝑀₂,𝑡≥𝜏₂. (2.33) In the case that 𝑑₁,𝑑₂,𝑑₃≥1, 𝜉,𝜂∈[𝑑,𝑑], the coefficients of inequality (2.31) can be estimated by some constants which depend on 𝑑,𝑑, but do not depend on 𝑑₁,𝑑₂,𝑑₃. So 𝑀₂ depends on 𝛼_𝑖𝑗(𝑖,𝑗=1,2,3),𝑎,𝑏,𝑒,𝑙,𝐴,𝐵,𝐷,𝐸, 𝐿,𝑑, and 𝑑, but it is irrelevant to 𝑑₁,𝑑₂,𝑑₃, when 𝑑₁,𝑑₂,𝑑₃≥1 and 𝜉,𝜂∈[𝑑,𝑑]. Since 𝑃_𝑥𝑄_𝑥𝑅_𝑥=𝑃_𝑢𝑃_𝑣𝑃_𝑤𝑄_𝑢𝑄_𝑣𝑄_𝑤𝑅_𝑢𝑅_𝑣𝑅_𝑤𝑢_𝑥𝑣_𝑥𝑤_𝑥, (2.34) we can transform the formulations of 𝑢_𝑥,𝑣_𝑥,𝑤_𝑥 into fraction representations, then distribute the denominators of the absolute value of the fractions to the numerators item and enlarge the term concerning with 𝑢_𝑥,𝑣_𝑥 or 𝑤_𝑥 to obtain 𝑑₁𝑢_𝑥+𝑑₁𝑣_𝑥+𝑑₁𝑤_𝑥≤𝐶^′𝑑₁𝑃_𝑥+𝑑₁𝑄_𝑥+𝑑₁𝑅_𝑥,0<𝑥<1,𝑡>0, (2.35) where 𝐶^′ is a constant depending only on 𝜉,𝜂,𝛼_𝑖𝑗(𝑖,𝑗=1,2,3). After scaling back and contacting estimates (2.33) and (2.35), there exist positive constant 𝑀₂ depending on 𝑑_𝑖,𝛼_𝑖𝑗(𝑖,𝑗=1,2,3),𝑎,𝑏,𝑒,𝑙,𝐴,𝐵,𝐷,𝐸,𝐿, and 𝜏₂>0, such that ¹₀𝑢²_𝑥𝑑𝑥,¹₀𝑣²_𝑥𝑑𝑥,¹₀𝑤²_𝑥𝑑𝑥≤𝑀₂,𝑡≥𝜏₂. (2.36) When 𝑑₁,𝑑₂,𝑑₃≥1 and 𝜉,𝜂∈[𝑑,𝑑], 𝑀₂ is independent of 𝑑₁,𝑑₂,𝑑₃.(ii) One has 𝑡≥0. Modifying the dependency of the coefficients in inequalities (2.17)–(2.22), namely, replacing 𝑀₀,𝑀₁ with 𝑀^′₀,𝑀^′₁, there exists a positive constant 𝑀^′₂ depending on 𝑑_𝑖,𝛼_𝑖𝑗(𝑖,𝑗=1,2,3),𝑎,𝑏,𝑒,𝑙,𝐴,𝐵, D,𝐸,𝐿, and the 𝑊¹₂-norm of 𝑢₀,𝑣₀,𝑤₀, such that ¹₀𝑢²_𝑥𝑑𝑥,¹₀𝑣²_𝑥𝑑𝑥,¹₀𝑤²_𝑥𝑑𝑥≤𝑀^′₂,𝑡≥0. (2.36)^′Furthermore, in the case that 𝑑₁,𝑑₂,𝑑₃≥1, 𝜉,𝜂∈[𝑑,𝑑], 𝑀^′₂ depends on 𝑑,𝑑, but does not depend on 𝑑₁,𝑑₂,𝑑₃.
Summarizing estimates (2.8), (2.15), (2.36) and Sobolev embedding theorem, there exist positive constants 𝑀,𝑀^′ depending only on 𝑑_𝑖,𝛼_𝑖𝑗(𝑖,𝑗=1,2,3),𝑎,𝑏,𝑒,𝑙,𝐴,𝐵,𝐷,𝐸, and 𝐿, such that (1.18) and (1.19) hold. In particular, 𝑀,𝑀^′ depend only on 𝛼_𝑖𝑗(𝑖,𝑗=1,2,3),𝑎,𝑏,𝑒,𝑙,𝐴,𝐵,𝐷,𝐸,𝐿,𝑑, and 𝑑, but do not depend on 𝑑₁,𝑑₂,𝑑₃, when 𝑑₁,𝑑₂,𝑑₃≥1 and 𝜉,𝜂∈[𝑑,𝑑].
Similarly, according to (2.8)^′, (2.15)^′, (2.36)^′, there exists a positive constant 𝑀^′′ depending on 𝑑_𝑖,𝛼_𝑖𝑗(𝑖,𝑗=1,2,3),𝑎,𝑏,𝑒,𝑙,𝐴,𝐵,𝐷,𝐸,𝐿 and the initial functions 𝑢₀,𝑣₀,𝑤₀, such that |𝑢(⋅,𝑡)|_1,2,|𝑣(⋅,𝑡)|_1,2,|𝑤(⋅,𝑡)|_1,2≤𝑀^′′,𝑡≥0. (2.37) Further, in the case that 𝑑₁,𝑑₂,𝑑₃≥1, 𝜉,𝜂∈[𝑑,𝑑], 𝑀^′′ depends only on 𝑑,𝑑, but do not depend on 𝑑₁,𝑑₂,𝑑₃. Thus, 𝑇=+∞. This completes the proof of Theorem 1.1.

3. Global Stability

In order to obtain the uniform convergence of the solution for the system (1.16), we recall the following result which can be found in [37, 42].

Lemma 3.1. Let 𝑎 and 𝑏 be positive constants. Assume that 𝜑,𝜓∈𝐶¹([𝑎,+∞)),𝜓(𝑡)≥0 and 𝜑 is bounded from below. If 𝜑^′(𝑡)≤−𝑏𝜓(𝑡) and 𝜓^′(𝑡) is bounded from above in [𝑎,+∞), then lim_𝑡→∞𝜓(𝑡)=0.

Proof of Theorem 1.3. Let (𝑢,𝑣,𝑤) be a solution for the system (1.16) with initial functions 𝑢₀(𝑥),𝑣₀(𝑥),𝑤₀(𝑥)≥(≢)0. From the strong maximum principle for parabolic equations, it is not hard to verify that 𝑢,𝑣,𝑤>0 for 𝑡>0. Define the function 𝐻(𝑢,𝑣,𝑤)=¹₀𝑢−𝑢^∗−𝑢^∗𝑢ln𝑢^∗𝑑𝑥+𝛼¹₀𝑣−𝑣^∗−𝑣^∗𝑣ln𝑣^∗𝑑𝑥+𝛽¹₀𝑤−𝑤^∗−𝑤^∗𝑤ln𝑤^∗𝑑𝑥. (3.1) Then the time derivative of 𝐻(𝑢,𝑣,𝑤) for the system (1.16) satisfies 𝑑𝐻𝑑𝑡=¹₀𝑢−𝑢^∗𝑢𝑢_𝑡𝑑𝑥+𝛼¹₀𝑣−𝑣^∗𝑣𝑣_𝑡𝑑𝑥+𝛽¹₀𝑤−𝑤^∗𝑤𝑤_𝑡𝑑𝑥=−¹₀𝑢^∗𝑢²𝑑₁+2𝛼₁₁𝑢+𝛼₁₂𝑣+𝛼₁₃𝑤𝑢²_𝑥+𝑢^∗𝑢𝛼₁₂𝑣_𝑥+𝛼₁₃𝑤_𝑥𝑢_𝑥+𝛼𝑣^∗𝑣²𝑑₂+𝛼₂₁𝑢+2𝛼₂₂𝑣+𝛼₂₃𝑤𝑣²_𝑥+𝛼𝑣^∗𝑣𝛼₂₁𝑢_𝑥+𝛼₂₃𝑤_𝑥𝑣_𝑥+𝛽𝑤^∗𝑤²𝑑₃+𝛼₃₁𝑢+𝛼₃₂𝑣+2𝛼₃₃𝑤𝑤²_𝑥+𝛽𝑤^∗𝑤𝛼₃₁𝑢_𝑥+𝛼₁₃𝑣_𝑥𝑤_𝑥+𝑑𝑥₁₀𝑢−𝑢^∗1−𝑢−𝑎𝑣−1+𝑏𝑢𝐴𝑤1+𝐵𝑢+𝐷𝑤𝑑𝑥+𝛼¹₀𝑣−𝑣^∗𝑚−1+𝑒𝑢1+𝑏𝑢𝑑𝑥+𝛽¹₀𝑤−𝑤^∗𝑀−1+𝐸𝑢1+𝐵𝑢+𝐷𝑤𝑑𝑥. (3.2)
The first integrand in the right hand of (3.2) is positive definite if 4𝛼𝛽𝑢^∗𝑣^∗𝑤^∗𝑑₁+2𝛼₁₁𝑢+𝛼₁₂𝑣+𝛼₁₃𝑤𝑑₂+𝛼₂₁𝑢+2𝛼₂₂𝑣+𝛼₂₃𝑤𝑑₃+𝛼₃₁𝑢+𝛼₃₂𝑣+2𝛼₃₃𝑤>𝑢^∗𝛼𝛼₂₃𝑣^∗𝑤+𝛽𝛼₃₂𝑤^∗𝑣²𝑑₁+2𝛼₁₁𝑢+𝛼₁₂𝑣+𝛼₁₃𝑤+𝛼𝑣^∗𝛼₁₃𝑢^∗𝑤+𝛽𝛼₃₁𝑤^∗𝑢²𝑑₂+𝛼₂₁𝑢+2𝛼₂₂𝑣+𝛼₂₃𝑤+𝛽𝑤^∗𝛼₁₂𝑢^∗𝑣+𝛼𝛼₂₁𝑣^∗𝑢²𝑑₃+𝛼₃₁𝑢+𝛼₃₂𝑣+2𝛼₃₃𝑤. (3.3) By the maximum-norm estimate in Theorem 1.1, the condition (1.20) implies (3.3). Therefore, we have 𝑑𝐻𝑑𝑡≤−¹₀1−𝑎𝑏𝑣^∗(1+𝑏𝑢^∗)−)(1+𝑏𝑢𝐴𝐵𝑤^∗(1+𝐵𝑢^∗+𝐷𝑤^∗)(1+𝐵𝑢+𝐷𝑤)𝑢−𝑢^∗2𝑑𝑥−𝛽¹₀𝑀𝐷𝐸𝑢^∗(1+𝐵𝑢^∗+𝐷𝑤^∗)(1+𝐵𝑢+𝐷𝑤)𝑤−𝑤^∗2𝑑𝑥≤−¹₀1−𝑎𝑏𝑣^∗1+𝑏𝑢^∗−𝐴𝐵𝑤^∗1+𝐵𝑢^∗+𝐷𝑤^∗𝑢−𝑢^∗2𝑑𝑥−𝛽¹₀𝑀𝐷𝐸𝑢^∗1+𝐵𝑢^∗+𝐷𝑤^∗𝑤−𝑤^∗2𝑑𝑥≜−𝑙₁¹₀𝑢−𝑢^∗2𝑑𝑥−𝛽𝑙₂¹₀𝑤−𝑤^∗2𝑑𝑥, (3.4) where 𝑙₁=1−𝑎𝑏𝑣^∗1+𝑏𝑢^∗−𝐴𝐵𝑤^∗1+𝐵𝑢^∗+𝐷𝑤^∗,𝑙₂=𝑀𝐷𝐸𝑢^∗1+𝐵𝑢^∗+𝐷𝑤^∗>0. (3.5) The condition (1.21) implies 𝑙₁>0. Using the similar argument in the proof of Theorem 4.2 in [42], by the maximum-norm estimate in Theorem 1.1 and some tedious calculations, we can prove lim_𝑡→∞¹₀𝑢−𝑢^∗2𝑑𝑥=lim_𝑡→∞¹₀𝑣−𝑣^∗2𝑑𝑥=lim_𝑡→∞¹₀𝑤−𝑤^∗2𝑑𝑥=0. (3.6)
It follows from (3.6) and Gagliardo-Nirenberg-type inequality |𝑢|_∞≤𝐶|𝑢|^1/2_1,2|𝑢|₂^1/2 that (𝑢,𝑣,𝑤) converges uniformly on (𝑢^∗,𝑣^∗,𝑤^∗). By the fact that 𝐻(𝑢,𝑣,𝑤) is decreasing for 𝑡≥0, it is obvious that (𝑢^∗,𝑣^∗,𝑤^∗) is globally asymptotically stable. So the proof of Theorem 1.3 is completed.

Acknowledgments

This work is supported by the China National Natural Science Foundation (no. 11061031; 11161041), the Fundamental Research Funds for the Gansu University, NWNU-KJCXGC-03-61 and the Fundamental Research Funds for the Central Universities (no. zyz2012074).