Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 702603 | 12 pages | https://doi.org/10.1155/2012/702603

Global Existence of Classical Solutions to a Three-Species Predator-Prey Model with Two Prey-Taxes

Academic Editor: Alain Miranville
Received08 Sep 2011
Accepted23 Nov 2011
Published11 Jan 2012

Abstract

We are concerned with three-species predator-prey model including two prey-taxes and Holling type II functional response under no flux boundary condition. By applying the contraction mapping principle, the parabolic Schauder estimates, and parabolic 𝐿𝑝 estimates, we prove that there exists a unique global classical solution of this system.

1. Introduction

In addition to random diffusion of the predator and the prey, the spatial-temporal variations of the predators’ velocity are directed by prey gradient. Several field studies measuring characteristics of individual movement confirm the basis of the hypothesis about the dependence of acceleration on a stimulus [1]. Understanding spatial and temporal behaviors of interacting species in ecological system is a central problem in population ecology. Various types of mathematical models have been proposed to study problem of predator-prey. Recently, the appearance of prey-taxis in relation to ecological interactions of species was studied by many scholars, ecologists, and mathematicians [2–5].

In [2] the authors proved the existence and uniqueness of weak solutions to the two-species predator-prey model with one prey-taxis. In [3], the author extended the results of [2] to an π‘›Γ—π‘š reaction-diffusion-taxis system. In [4], the author proved the existence and uniqueness of classical solutions to this model. In this paper, we deal with three-species predator-prey model with two prey-taxes including Holling type II functional response as follows: πœ•π‘’1πœ•π‘‘βˆ’π‘‘1Δ𝑒1𝛽+βˆ‡β‹…1𝑒1βˆ‡π‘’2𝛽+βˆ‡β‹…2𝑒1βˆ‡π‘’3ξ€Έ=βˆ’π‘Žπ‘’1+𝑒2𝑐2𝑒1𝑒2π‘š2+𝑏2𝑒2+𝑒3𝑐3𝑒1𝑒3π‘š3+𝑏3𝑒3in(0,𝑇)Γ—Ξ©,πœ•π‘’2πœ•π‘‘βˆ’π‘‘2Δ𝑒2=π‘Ÿ2𝑒1βˆ’2𝐾2𝑒2βˆ’π‘2𝑒1𝑒2π‘š2+𝑏2𝑒2in(0,𝑇)Γ—Ξ©,πœ•π‘’3πœ•π‘‘βˆ’π‘‘3Δ𝑒3=π‘Ÿ3𝑒1βˆ’3𝐾3𝑒2βˆ’π‘3𝑒1𝑒3π‘š3+𝑏3𝑒3in(0,𝑇)Γ—Ξ©,πœ•π‘’1=πœ•πœˆπœ•π‘’2=πœ•πœˆπœ•π‘’3ξ€·π‘’πœ•πœˆ=0on(0,𝑇)Γ—πœ•Ξ©,1(0,π‘₯),𝑒2(0,π‘₯),𝑒3ξ€Έ=𝑒(0,π‘₯)10(π‘₯),𝑒20(π‘₯),𝑒30ξ€Έ(π‘₯)β‰₯(0,0)inΞ©,(1.1) where Ξ© is a bounded domain in 𝑅𝑁(𝑁β‰₯1 is an integer) with a smooth boundary πœ•Ξ©; 𝑒1 and 𝑒𝑖(𝑖=2,3) represent the densities of the predator and prey, respectively; the positive constants 𝑑1, 𝑑2, and 𝑑3 are the diffusion coefficient of the corresponding species; the positive constants π‘Ž,𝐾𝑖,π‘Ÿπ‘–,π‘šπ‘–,𝑒𝑖,π‘šπ‘–/𝑐𝑖,𝑏𝑖/𝑐𝑖,π‘šπ‘–/𝑏𝑖(𝑖=2,3) represent the death rate of the predator, the carrying capacity of prey, the prey intrinsic growth rate, the half-saturation constant, the conversion rate, the time spent by a predator to catch a prey, the manipulation time which is a saturation effect for large densities of prey, the density of prey necessary to achieve one-half the rate, respectively; the predators are attracted by the preys, and the positive constant 𝛽𝑖(𝑖=1,2) denotes their prey-tactic sensitivity. The parts 𝛽1𝑒1βˆ‡π‘’2 and 𝛽2𝑒1βˆ‡π‘’3 of the flux are directed toward the increasing population density of 𝑒2 and 𝑒3, respectively. In this way, the predators move in the direction of higher concentration of the prey species.

The aim of this paper is to prove that there is a unique classical solution to the model (1.1). It is difficult to deal with the two prey-taxes terms. To get our goal we employ the techniques developed by [6, 7] to investigate.

Throughout this paper we assume that𝛽1=0,𝛽2=0,for𝑒1β‰₯𝑒1π‘š.(1.2) The assumptions that 𝛽1=0 for 𝑒1β‰₯𝑒1π‘š and 𝛽2=0 for 𝑒1β‰₯𝑒1π‘š have a clear biological interpretation [2]: the predators stop to accumulate at given point of Ξ© after their density attains a certain threshold value 𝑒1π‘š and the prey-tactic sensitivity 𝛽1 and 𝛽2 vanishes identically when 𝑒1β‰₯𝑒1π‘š.

Throughout this paper we also assume that𝑒20,𝑒30≀𝐾,πœ•Ξ©βˆˆπΆ2+𝛼,𝑒10(π‘₯),𝑒20(π‘₯),𝑒30(π‘₯)∈𝐢2+𝛼Ω,where0<𝛼<1,πœ•π‘’10=πœ•πœˆπœ•π‘’20=πœ•πœˆπœ•π‘’30πœ•πœˆ=0,onπœ•Ξ©.(1.3) Denote by πΆπ‘š+𝛼,𝛽π‘₯,𝑑(𝑄𝑇) (π‘šβ‰₯0 is integer, 0<𝛼<1,0<𝛽<1) the space of function 𝑒(π‘₯,𝑑) with finite norm [8]:β€–π‘’β€–πΆπ‘š+𝛼,𝛽π‘₯,𝑑(𝑄𝑇)=π‘šξ“||𝑙||=0sup𝑄𝑇||𝐷𝑙π‘₯𝑒||+𝐷𝑙π‘₯𝑒(𝛼)π‘₯,𝑄𝑇+𝐷lπ‘₯𝑒(𝛽)𝑑,𝑄𝑇,(1.4) whereβŸ¨π‘€βŸ©(𝛼)π‘₯,𝑄𝑇=(π‘₯,𝑑),(𝑦,𝑑)βˆˆπ‘„π‘‡||||𝑀(π‘₯,𝑑)βˆ’π‘€(𝑦,𝑑)||||π‘₯βˆ’π‘¦π›Ό,βŸ¨π‘€βŸ©(𝛽)𝑑,𝑄𝑇=(π‘₯,𝑑),(π‘₯,𝜏)βˆˆπ‘„π‘‡||||𝑀(π‘₯,𝑑)βˆ’π‘€(π‘₯,𝜏)|π‘‘βˆ’πœ|𝛽.(1.5) We denote by 𝐢2+𝛼,1+𝛽π‘₯,𝑑(𝑄𝑇) the space of functions 𝑒(π‘₯,𝑑) with norm‖𝑒‖𝐢2+𝛼,𝛽π‘₯,𝑑(𝑄𝑇)+‖‖𝑒𝑑‖‖𝐢𝛼,𝛽π‘₯,𝑑(𝑄𝑇).(1.6)

The main result of this paper is as follows.

Theorem 1.1. Under assumptions (1.2) and (1.3), for any given 𝑇>0 there exists a unique solution 𝐔=(𝑒1,𝑒2,𝑒3)∈𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇) of the system (1.1), where 𝑄𝑇=(0,𝑇)Γ—Ξ©. Moreover, 𝑒1(π‘₯,𝑑)β‰₯0,0≀𝑒2(π‘₯,𝑑)≀𝐾2,0≀𝑒3(π‘₯,𝑑)≀𝐾3,(1.7) for any π‘₯∈Ω and 𝑑>0.

This paper is organized as follows. In Section 2, we present some preliminary lemmas that will be used in proving later theorem. In Section 3, we prove local existence and uniqueness to system (1.1). In Section 4, we prove global existence to system (1.1).

2. Some Preliminaries

For the convenience of notations, in what follows we denote various constants which depend on 𝑇 by 𝑁, while we denote various constants which are independent of 𝑇 by 𝑁0.

Lemma 2.1. Let (𝑒,π‘₯)∈𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇). Then ‖𝑒(π‘₯,𝑑)βˆ’π‘’(π‘₯,0)‖𝐢1+𝛼,𝛼/2(𝑄𝑇)≀𝑁0πœ‚(𝑇)‖𝑒‖𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇),(2.1) where πœ‚(𝑇)=max{𝑇𝛼/2,𝑇(1βˆ’π›Ό)/2}.

Proof. Using the definition of HΓΆlder norm, we have ||||𝑒(π‘₯,𝑑)βˆ’π‘’(π‘₯,0)|𝑑|(1+𝛼)/2≀||𝐷𝑑𝑒||β‹…||𝑇||(1βˆ’π›Ό)/2,[]𝑒(π‘₯,𝑑)βˆ’π‘’(π‘₯,0)𝐢1+𝛼,0(𝑄𝑇)≀𝑁0‖‖𝐷2π‘₯𝑒(π‘₯,𝑑)βˆ’π·2π‘₯‖‖𝑒(π‘₯,0)𝐿∞(𝑄𝑇)≀𝑁0𝐷2π‘₯𝑒𝐢0,𝛼/2(𝑄𝑇)β‹…||𝑇||𝛼/2≀𝑁0[𝑒]𝐢2,𝛼/2(𝑄𝑇)β‹…||𝑇||𝛼/2,(2.2) which yields that ‖𝑒(π‘₯,𝑑)βˆ’π‘’(π‘₯,0)‖𝐢1+𝛼,(1+𝛼)/2(𝑄𝑇)≀𝑁0πœ‚β€–π‘’β€–πΆ2+𝛼,(1+𝛼)/2(𝑄𝑇).(2.3) Therefore, ‖𝑒(π‘₯,𝑑)βˆ’π‘’(π‘₯,0)‖𝐢1+𝛼,𝛼/2(𝑄𝑇)≀𝑁0πœ‚β€–π‘’β€–πΆ2+𝛼,1+(𝛼/2)(𝑄𝑇).(2.4)

We now consider the following nonlinear parabolic problem:πœ•π‘’1πœ•π‘‘βˆ’π‘‘1Δ𝑒1𝛽+βˆ‡β‹…1𝑒1βˆ‡π‘’2𝛽+βˆ‡β‹…2𝑒1βˆ‡π‘’3ξ€Έ=𝑒1𝑓in(0,𝑇)Γ—Ξ©,πœ•π‘’1π‘’πœ•πœˆ=0on(0,𝑇)Γ—πœ•Ξ©,1(0,π‘₯)=𝑒10(π‘₯)inΞ©.(2.5) By the parabolic maximum principle, we have 𝑒1(π‘₯,𝑑)β‰₯0.

Lemma 2.2. Let 𝑒2(π‘₯,𝑑),𝑒3(π‘₯,𝑑)∈𝐢2+𝛼,1+(𝛼/2)𝑄𝑇,𝑓(π‘₯,𝑑)βˆˆπΆπ›Ό,𝛼/2𝑄𝑇,‖‖𝑒2‖‖𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇),‖‖𝑒3‖‖𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇),‖𝑓‖𝐢𝛼,𝛼/2(𝑄𝑇)≀𝑁0.(2.6) Then, under assumptions (1.2) and (1.3), there exists a unique nonnegative solution 𝑒1(π‘₯,𝑑)∈𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇) of the nonlinear problem (2.5) for small 𝑇>0 which depends on ‖𝑒10(π‘₯)‖𝐢2+𝛼(Ξ©).

Proof. This proof is similar to that of Lemma  2.1 in [4]. For reader’s convenience we include the proof here. We will prove by a fixed point argument. Let us introduce the Banach space 𝑋 of function 𝑒1 with norm ‖𝑒1‖𝐢1+𝛼,𝛼/2(𝑄𝑇)(0<𝑇<1) and a subset 𝑋𝐴={𝑒1βˆˆπ‘‹βˆΆπ‘’1(π‘₯,0)=𝑒10(π‘₯)and‖𝑒1‖𝐢1+𝛼,𝛼/2(𝑄𝑇)≀𝐴}, where 𝐴=‖𝑒10(π‘₯)‖𝐢2+𝛼(Ξ©)+‖𝑒20(π‘₯)‖𝐢2+𝛼(Ξ©)+‖𝑒30(π‘₯)‖𝐢2+𝛼(Ξ©)+1. For any 𝑒1βˆˆπ‘‹π΄, we define a corresponding function ̃𝑒1=𝐹𝑒1, where ̃𝑒1 satisfies the equations πœ•Μƒπ‘’1πœ•π‘‘βˆ’π‘‘1Δ̃𝑒1βˆ’Μƒπ‘’1𝑓=βˆ’π›½1βˆ‡π‘’1β‹…βˆ‡π‘’2βˆ’π›½2βˆ‡π‘’1β‹…βˆ‡π‘’3βˆ’π›½1𝑒1Δ𝑒2βˆ’π›½2𝑒1Δ𝑒3in(0,𝑇)Γ—Ξ©,πœ•Μƒπ‘’1πœ•πœˆ=0on(0,𝑇)Γ—πœ•Ξ©,̃𝑒1(0,π‘₯)=𝑒10(π‘₯)inΞ©.(2.7) By 𝑒1βˆˆπ‘‹π΄, we have β„Ž1β‰œβˆ’π›½1βˆ‡π‘’1β‹…βˆ‡π‘’2βˆ’π›½2βˆ‡π‘’1β‹…βˆ‡π‘’3βˆ’π›½1𝑒1Δ𝑒2βˆ’π›½2𝑒1Δ𝑒3βˆˆπΆπ›Ό,𝛼/2𝑄𝑇.(2.8) By the parabolic Schauder theory, this yields that there exists a unique solution ̃𝑒 to (2.7) and ‖‖̃𝑒1‖‖𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇)≀‖‖̃𝑒1|𝑑=0‖‖𝐢2+𝛼(Ξ©)+𝑀1(𝐴)≀𝐴+𝑀1(𝐴)β‰œπ‘€2(𝐴),(2.9) where 𝑀2(A) is some constant which depends only on 𝐴. For any function ̃𝑒1(π‘₯,𝑑), by Lemma 2.1 and combining (2.9), if 𝑇 is sufficiently small (𝑇 depends only on 𝐴), then we have ‖‖̃𝑒1β€–β€–(π‘₯,𝑑)𝐢1+𝛼,𝛼/2(𝑄𝑇)≀‖‖̃𝑒1β€–β€–(π‘₯,0)𝐢1+𝛼,𝛼/2(𝑄T)+𝑁0β€–β€–πœ‚(𝑇)̃𝑒1‖‖𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇)≀‖‖̃𝑒10β€–β€–(π‘₯)𝐢2+𝛼+1≀𝐴.(2.10) Therefore, ̃𝑒1(π‘₯,𝑑)βˆˆπ‘‹π΄ and 𝐹 maps 𝑋𝐴 into itself. We now prove that 𝐹 is contractive. Take 𝑒11,𝑒12 in 𝑋𝐴, and set ̃𝑒11=𝐹𝑒11,̃𝑒12=𝐹𝑒12,̃𝑣=̃𝑒11βˆ’Μƒπ‘’12. Then, it follows from (2.7) that ̃𝑣 solves the following systems: πœ•Μƒπ‘£πœ•π‘‘βˆ’π‘‘1Ξ”ΜƒΜƒπ‘£βˆ’π‘£π‘“=β„Ž2πœ•Μƒπ‘£in(0,𝑇)Γ—Ξ©,Μƒπœ•πœˆ=0on(0,𝑇)Γ—πœ•Ξ©,𝑣(0,π‘₯)=0inΞ©,(2.11) where β„Ž2β‰œβˆ’π›½1ξ€·βˆ‡π‘’11βˆ’βˆ‡π‘’12ξ€Έβ‹…βˆ‡π‘’2βˆ’π›½2ξ€·βˆ‡π‘’11βˆ’βˆ‡π‘’12ξ€Έβ‹…βˆ‡π‘’3βˆ’π›½1𝑒11βˆ’π‘’12Δ𝑒2βˆ’π›½2𝑒11βˆ’π‘’12Δ𝑒3.(2.12) By 𝑒11,𝑒12βˆˆπ‘‹π΄ and conditions of Lemma 2.2, it is easy to check that β€–β€–β„Ž2β€–β€–πΏβˆž(𝑄𝑇)≀‖‖𝛽1ξ€·βˆ‡π‘’11βˆ’βˆ‡π‘’12ξ€Έβ‹…βˆ‡π‘’2‖‖𝐢0(𝑄𝑇)+‖‖𝛽2ξ€·βˆ‡π‘’11βˆ’βˆ‡π‘’12ξ€Έβ‹…βˆ‡π‘’3‖‖𝐢0(𝑄𝑇)+‖‖𝛽1𝑒11βˆ’π‘’12Δ𝑒2‖‖𝐢0(𝑄𝑇)+‖‖𝛽2𝑒11βˆ’π‘’12Δ𝑒3‖‖𝐢0(𝑄𝑇)≀𝑁0‖‖𝑒11βˆ’π‘’12‖‖𝐢1,0+𝑁0‖‖𝑒11βˆ’π‘’12‖‖𝐢0(𝑄𝑇)≀𝑁0‖‖𝑒11βˆ’π‘’12‖‖𝐢1,0(𝑄𝑇).(2.13) Using the assumption ‖𝑓‖𝛼,𝛼/2≀𝑁0 and the 𝐿𝑝-estimate, we have β€–π‘“β€–πΏβˆž(𝑄𝑇)≀‖𝑓‖𝐢𝛼,𝛼/2≀𝑁0,‖̃𝑣‖𝑀𝑝2,1≀𝑁0β€–β€–β„Ž2β€–β€–πΏβˆž(𝑄𝑇).(2.14) For any 𝑝β‰₯1, by using Sobolev embedding π‘Šπ‘2,1(𝑄𝑇)β†ͺ𝐢1+𝛾,(1+𝛾)/2(𝑄𝑇) (𝛾=1βˆ’(5/𝑝)>𝛼 if we take 𝑝 sufficiently large), we have ‖̃𝑣‖𝐢1+𝛾,(1+𝛾)/2(𝑄𝑇)≀𝑁0β€–β€–β„Ž2β€–β€–πΏβˆž(𝑄𝑇)≀𝑁0‖‖𝑒11βˆ’π‘’12‖‖𝐢1,0(𝑄𝑇).(2.15) Then, noting 𝛾>𝛼, we can easily check that [4] ‖̃𝑣‖𝐢1+𝛼,𝛼/2(𝑄𝑇)≀𝑁0𝑇𝛼/2‖‖𝑒11βˆ’π‘’12‖‖𝐢1+𝛼,𝛼/2(𝑄𝑇).(2.16) Taking 𝑇 small such that 𝑁0𝑇𝛼/2<1/2, we conclude from (2.16) that 𝐹 is contractive in 𝑋𝐴. Therefore 𝐹 has a unique fixed point 𝑒1, which is the unique solution to (2.5). Moreover, we can raise the regularity of 𝑒1 to 𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇) by using the parabolic Schauder estimates.

3. Local Existence and Uniqueness of Solutions

In this section, we will prove Theorem 3.1 which show that system (1.1) has a unique solution 𝐔(π‘₯,𝑑)=(𝑒1,𝑒2,𝑒3)∈𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇) as done in [6, 7].

Theorem 3.1. Assume that (1.2) and (1.3) hold, then there exists a unique solution 𝐔(π‘₯,𝑑)=(𝑒1,𝑒2,𝑒3)∈𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇) of the system (1.2) for small 𝑇>0 which depends on ‖‖𝐔0β€–β€–(π‘₯)𝐢2+𝛼(Ξ©)β‰œβ€–β€–π‘’10β€–β€–(π‘₯)𝐢2+𝛼(Ξ©)+‖‖𝑒20β€–β€–(π‘₯)𝐢2+𝛼(Ξ©)+‖‖𝑒30β€–β€–(π‘₯)𝐢2+𝛼(Ξ©).(3.1) Furthermore, 𝑒1(π‘₯,𝑑)β‰₯0,𝑒2(π‘₯,𝑑)β‰₯0,𝑒3(π‘₯,𝑑)β‰₯0.

Proof. We will prove the local existence by a fixed point argument again. Introducing the Banach space 𝑋 of the function 𝐔, we define the norm ‖𝐔‖𝐢𝛼,𝛼/2(𝑄𝑇)=‖‖𝑒1‖‖𝐢𝛼,𝛼/2(𝑄𝑇)+‖‖𝑒2‖‖𝐢𝛼,𝛼/2(𝑄𝑇)+‖‖𝑒3‖‖𝐢𝛼,𝛼/2(𝑄𝑇)(0<𝑇<1),(3.2) and a subset 𝑋𝐴=ξ€½π”βˆˆπ‘‹βˆΆπ‘’1,𝑒2,𝑒3β‰₯0,‖𝐔‖𝐢𝛼,𝛼/2(𝑄𝑇)ξ€Ύ,≀𝐴(3.3) where 𝐔𝑒(π‘₯,0)=10(π‘₯),𝑒20(π‘₯),𝑒30ξ€Έ,‖‖𝑒(π‘₯)𝐴=10β€–β€–(π‘₯)𝐢2+𝛼(Ξ©)+‖‖𝑒20β€–β€–(π‘₯)𝐢2+𝛼(Ξ©)+‖‖𝑒30β€–β€–(π‘₯)𝐢2+𝛼(Ξ©)+1.(3.4) For any π”βˆˆπ‘‹π΄, we define correspondingly function 𝐔=𝐇𝐔 by 𝐔=(𝑒1,𝑒2,𝑒3), where 𝐔 satisfies the equations πœ•π‘’2πœ•π‘‘βˆ’π‘‘2Δ𝑒2=ξ‚Έπ‘Ÿ2𝑒1βˆ’2𝐾2ξ‚Άβˆ’π‘2𝑒1π‘š2+𝑏2𝑒2𝑒2πœ•in(0,𝑇)Γ—Ξ©,𝑒3πœ•π‘‘βˆ’π‘‘3Δ𝑒3=ξ‚Έπ‘Ÿ3𝑒1βˆ’3𝐾3ξ‚Άβˆ’π‘3𝑒1π‘š3+𝑏3𝑒3𝑒3πœ•in(0,𝑇)Γ—Ξ©,𝑒2=πœ•πœ•πœˆπ‘’3πœ•πœˆ=0on(0,𝑇)Γ—πœ•Ξ©,𝑒2(π‘₯,0)=𝑒20(π‘₯),𝑒3(π‘₯,0)=𝑒30πœ•(π‘₯),π‘₯∈Ω,(3.5)𝑒1πœ•π‘‘βˆ’π‘‘1Δ𝑒1𝛽+βˆ‡β‹…1𝑒1βˆ‡π‘’2𝛽+βˆ‡β‹…2𝑒1βˆ‡π‘’3ξ€Έ=βˆ’π‘Žπ‘’1+𝑒2𝑐2𝑒1𝑒2π‘š2+𝑏2𝑒2+𝑒3𝑐3𝑒1𝑒3π‘š3+𝑏3𝑒3πœ•in(0,T)Γ—Ξ©,𝑒1πœ•πœˆ=0on(0,𝑇)Γ—πœ•Ξ©,𝑒1(π‘₯,0)=𝑒10(π‘₯),π‘₯∈Ω.(3.6) By (3.5), (𝑒1,𝑒2,𝑒3)βˆˆπ‘‹π΄, assumption (1.3), and the parabolic Schauder theory, we have that there exists a unique solution 𝑒2,𝑒3 to (3.5) and ‖‖𝑒2‖‖𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇)≀‖‖𝑒2|𝑑=0‖‖𝐢2+𝛼+𝑀3(𝐴)≀𝐴+𝑀3(𝐴)β‰œπ‘€4(𝐴).(3.7) Similarly, ‖‖𝑒3‖‖𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇)≀‖‖𝑒3|𝑑=0‖‖𝐢2+𝛼+𝑀5(𝐴)≀𝐴+𝑀5(𝐴)β‰œπ‘€6(𝐴).(3.8) Moreover, by parabolic maximum principle, we have 𝑒2(π‘₯,𝑑)β‰₯0in𝑄𝑇,𝑒3(π‘₯,𝑑)β‰₯0in𝑄𝑇.(3.9) Similarly, by using Lemma 2.2, from (3.6) we can conclude that there exists a unique solution 𝑒1 satisfying ‖‖𝑒1‖‖𝐢2+𝛼,1+(𝛼/2)𝑄𝑇≀𝑀7(𝐴),(3.10) and by parabolic maximum principle we have 𝑒1(π‘₯,𝑑)β‰₯0 in 𝑄𝑇. From (3.7), (3.8), and (3.10), we have ‖‖𝐔‖‖𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇)≀𝑀8(𝐴).(3.11) For any function 𝐔(π‘₯,𝑑), using Lemma 2.1 we get ‖‖𝐔(π‘₯,𝑑)βˆ’β€–β€–π”(π‘₯,0)𝐢𝛼,𝛼/2(𝑄𝑇)≀𝑁0πœ‚(𝑇)‖𝐔‖𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇).(3.12) From (3.11) and (3.12), if 𝑇 is sufficiently small we have ‖‖‖‖𝐔(π‘₯,𝑑)𝐢𝛼,𝛼/2(𝑄𝑇)≀‖‖‖‖𝐔(π‘₯,0)𝐢𝛼,𝛼/2+𝑁0πœ‚(𝑇)𝑀8≀‖‖𝐔(𝐴)0β€–β€–(π‘₯)𝐢2+𝛼(Ξ©)+1≑𝐴,(3.13) which yields π”βˆˆπ‘‹π΄. Therefore, 𝐻 maps 𝑋𝐴 into itself.
Next, we can prove that 𝐻 is contractive as done in the proof of Lemma 2.2 in 𝑋𝐴 if we take 𝑇 sufficiently small. By the contraction mapping theorem 𝐻 has a unique fixed point 𝐔, which is the unique solution of (1.1). Moreover, we can raise the regularity of 𝐔 to 𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇) by using the parabolic Schauder estimates.

4. Global Existence

First we establish some a priori estimates to (1.1).

Lemma 4.1. Suppose that 𝐔=(𝑒1,𝑒2,𝑒3)∈𝐢2,1(𝑄𝑇) is a solution to the system (1.1), then there holds 𝑒1β‰₯0,0≀𝑒2≀𝐾2,0≀𝑒3≀𝐾3.(4.1)

Proof. It follows from (1.1) that πœ•π‘’1πœ•π‘‘βˆ’π‘‘1Δ𝑒1+𝛽1βˆ‡π‘’2+𝛽2βˆ‡π‘’3ξ€Έβ‹…βˆ‡π‘’1+𝛽1Δ𝑒2+𝛽2Δ𝑒3𝑒+π‘Žβˆ’2𝑐2𝑒2π‘š2+𝑏2𝑒2βˆ’π‘’3𝑐3𝑒3π‘š3+𝑏3𝑒3𝑒1=0in(0,𝑇)Γ—Ξ©,πœ•π‘’1π‘’πœ•πœˆ=0on(0,𝑇)Γ—πœ•Ξ©,1(0,π‘₯)=𝑒10(π‘₯)β‰₯0inΞ©.(4.2) Obviously, 𝑒1≑0 is a subsolution to (4.2). Using the maximum principle, we get 𝑒1β‰₯0. Similarly, we have 𝑒2β‰₯0 and 𝑒3β‰₯0.
On the other hand, it follows from model (1.1) thatπœ•π‘’2πœ•π‘‘βˆ’π‘‘2Δ𝑒2βˆ’π‘Ÿ2𝑒1βˆ’2𝐾2𝑒2+𝑐2𝑒1𝑒2π‘š2+𝑏2𝑒2𝑐=0≀2𝐾2𝑒1π‘š2+𝑏2𝐾2in(0,𝑇)Γ—Ξ©,πœ•π‘’2π‘’πœ•πœˆ=0on(0,𝑇)Γ—πœ•Ξ©,2(0,π‘₯)=𝑒20(π‘₯)inΞ©,(4.3) which implies that 𝐾2 is a subsolution to problem (4.3). Hence we have 0≀𝑒2(π‘₯,𝑑)≀𝐾2. Similarly, we get 0≀𝑒3(π‘₯,𝑑)≀𝐾3. This completes the proof of Lemma 4.1.

Lemma 4.2. Suppose that 𝐔=(𝑒1,𝑒2,𝑒3)∈𝐢2,1(𝑄𝑇) is a solution to the system (1.1), then for any 𝑝>1 there holds ‖‖𝑒1‖‖𝐿𝑝(𝑄𝑇)‖‖𝑒≀𝑁,2‖‖𝐿𝑝(𝑄𝑇)‖‖𝑒≀𝑁,2‖‖𝐿𝑝(𝑄𝑇)≀𝑁.(4.4)

Proof. Multiplying the first equation of (1.1) by 𝑒1π‘βˆ’1, integrating over 𝑄𝑇, using the no-flux boundary condition, and noting 𝑒1β‰₯0, we get 1π‘ξ€œΞ©π‘’π‘1(1π‘₯,𝑑)𝑑π‘₯βˆ’π‘ξ€œΞ©π‘’π‘0(π‘₯,𝑑)𝑑π‘₯+(π‘βˆ’1)𝑑1ξ€œπ‘‘0ξ€œΞ©π‘’1π‘βˆ’2||βˆ‡π‘’1||2ξ€œπ‘‘π‘₯𝑑𝑑≀(π‘βˆ’1)𝑑0ξ€œΞ©π›½1𝑒1π‘βˆ’1βˆ‡π‘’1β‹…βˆ‡π‘’2ξ€œπ‘‘π‘₯𝑑𝑑+(π‘βˆ’1)𝑑0ξ€œΞ©π›½2𝑒1π‘βˆ’1βˆ‡π‘’1β‹…βˆ‡π‘’3+𝑒𝑑π‘₯𝑑𝑑2𝑐2𝑏2ξ€œπ‘‘0ξ€œΞ©π‘’π‘1𝑒𝑑π‘₯𝑑𝑑+3𝑐3𝑏3ξ€œπ‘‘0ξ€œΞ©π‘’π‘1𝑑π‘₯𝑑𝑑.(4.5) For 𝑒1β‰₯𝑒1π‘š, we get 1π‘ξ€œΞ©π‘’π‘1(1π‘₯,𝑑)𝑑π‘₯βˆ’π‘ξ€œΞ©π‘’π‘0(π‘₯,𝑑)𝑑π‘₯+(π‘βˆ’1)𝑑1ξ€œπ‘‘0ξ€œΞ©π‘’1π‘βˆ’2||βˆ‡π‘’1||2≀𝑒𝑑π‘₯𝑑𝑑2𝑐2𝑏2+𝑒3𝑐3𝑏3ξ‚Άξ€œπ‘‘0ξ€œΞ©π‘’π‘1𝑑π‘₯𝑑𝑑.(4.6) Therefore ξ€œΞ©π‘’π‘1(π‘₯,𝑑)𝑑𝑑≀𝑁0+𝑁0ξ€œπ‘‘0ξ€œΞ©π‘’π‘1𝑑π‘₯𝑑𝑑.(4.7) Using Gronwall’s Lemma, we have ξ€œπ‘‘0ξ€œΞ©π‘’π‘1(π‘₯,𝑑)𝑑𝑑≀𝑁.(4.8) Therefore, for 𝑒1<𝑒1π‘š, we have ξ€œπ‘‘0ξ€œΞ©π‘’π‘1(ξ€œπ‘₯,𝑑)𝑑𝑑≀𝑑0ξ€œΞ©π‘’π‘1π‘š(π‘₯,𝑑)𝑑𝑑≀𝑁.(4.9) Obviously, we have ξ€œπ‘‘0ξ€œΞ©π‘’π‘2(ξ€œπ‘₯,𝑑)𝑑t≀𝑑0ξ€œΞ©πΎπ‘2(ξ€œπ‘₯,𝑑)𝑑𝑑≀𝑁,𝑑0ξ€œΞ©π‘’π‘3ξ€œ(π‘₯,𝑑)𝑑𝑑≀𝑑0ξ€œΞ©πΎπ‘3(π‘₯,𝑑)𝑑𝑑≀𝑁.(4.10) This completes the proof of Lemma 4.2.

Lemma 4.3. Suppose that 𝐔=(𝑒1,𝑒2,𝑒3)∈𝐢2,1(Q𝑇) is a solution to the system (1.1), then for any 𝑝>5 there holds ‖‖𝑒1‖‖𝑀𝑝2,1(𝑄𝑇)‖‖𝑒≀𝑁,2‖‖𝑀𝑝2,1(𝑄𝑇)‖‖𝑒≀𝑁,3‖‖𝑀𝑝2,1(𝑄𝑇)≀𝑁.(4.11)

Proof. Note that the second equation of (1.1) can be rewritten as follows: πœ•π‘’2πœ•π‘‘βˆ’π‘‘2Δ𝑒2βˆ’ξ‚΅π‘Ÿ2βˆ’π‘Ÿ2𝐾2𝑒2βˆ’π‘2𝑒1π‘š2+𝑏2𝑒2𝑒2=0,(4.12) where β€–π‘Ÿ2βˆ’(π‘Ÿ2/𝐾2)𝑒2βˆ’(𝑐2𝑒1/(π‘š2+𝑏2𝑒2))‖𝐿𝑝(𝑄𝑇)≀𝑁.
By the parabolic 𝐿𝑝-estimate, we have‖‖𝑒2‖‖𝑀𝑝2,1(𝑄𝑇)≀𝑁.(4.13) Using the Sobolev embedding theorem (taking 𝑝>5), we get β€–β€–βˆ‡π‘’2β€–β€–πΏβˆž(𝑄𝑇)≀𝑁.(4.14) Similarly, we can obtain ‖‖𝑒3‖‖𝑀𝑝2,1(𝑄𝑇)‖‖≀𝑁,βˆ‡π‘’3β€–β€–πΏβˆž(𝑄𝑇)≀𝑁.(4.15) It follows from the first equation of (1.1) that πœ•π‘’1πœ•π‘‘βˆ’π‘‘1Δ𝑒1+𝛽1βˆ‡π‘’2+𝛽2βˆ‡π‘’3ξ€Έβ‹…βˆ‡π‘’1𝛽=βˆ’1Δ𝑒2+𝛽2Δ𝑒3𝑒+π‘Žβˆ’2𝑐2𝑒2π‘š2+𝑏2𝑒2βˆ’π‘’3𝑐3𝑒3π‘š3+𝑏3𝑒3𝑒1in(0,𝑇)Γ—Ξ©,πœ•π‘’1π‘’πœ•πœˆ=0on(0,𝑇)Γ—πœ•Ξ©,1(0,π‘₯)=𝑒10(π‘₯)β‰₯0inΞ©,(4.16) where β€–β€–β€–βˆ’ξ‚΅π›½1Δ𝑒2+𝛽2Δ𝑒3𝑒+π‘Žβˆ’2𝑐2𝑒2π‘š2+𝑏2𝑒2βˆ’π‘’3𝑐3𝑒3π‘š3+𝑏3𝑒3𝑒1‖‖‖𝐿𝑝(𝑄𝑇)≀𝑁.(4.17) Using the parabolic 𝐿𝑝-estimates again, we have ‖‖𝑒1‖‖𝑀𝑝2,1(𝑄𝑇)≀𝑁.(4.18) This completes the proof of Lemma 4.3.

Lemma 4.4. Suppose that 𝐔=(𝑒1,𝑒2,𝑒3)∈𝐢2,1(𝑄𝑇) is a solution to the system (1.1), then there holds ‖‖𝑒1‖‖𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇)‖‖𝑒≀𝑁,2‖‖𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇)‖‖𝑒≀𝑁,3‖‖𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇)≀𝑁.(4.19)

Proof. Using the Sobolev embedding theorem (taking 𝑝>5) and Lemma 4.3, we have ‖‖𝑒1‖‖𝐢𝛼,𝛼/2(𝑄𝑇)‖‖𝑒≀𝑁,2‖‖𝐢𝛼,𝛼/2(𝑄𝑇)‖‖𝑒≀𝑁,3‖‖𝐢𝛼,𝛼/2(𝑄𝑇)≀𝑁.(4.20) Using (4.20) and the Schauder estimates to the second and third equation of model (1.1), we have ‖‖𝑒2‖‖𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇)‖‖𝑒≀𝑁,3‖‖𝐢2+𝛼,1+(𝛼/2)(Q𝑇)≀𝑁.(4.21) Applying the parabolic Schauder estimate to (4.16) and using (4.21), we have ‖‖𝑒1‖‖𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇)≀𝑁.(4.22) This completes the proof of Lemma 4.4.

Therefore, we can extend the local solution established in Theorem 3.1 to all 𝑑>0, as done in [6, 7]. Namely, we have the following.

Theorem 4.5. Under assumptions (1.2) and (1.3), there exists a unique solution 𝐔=(𝑒1,𝑒2,𝑒3)∈𝐢2+𝛼,1+(𝛼/2)(𝑄𝑇) of the system (1.2) for any given 𝑇>0. Moreover, 𝑒1(π‘₯,𝑑)β‰₯0,0≀𝑒2≀𝐾2,0≀𝑒3≀𝐾3.(4.23) for any π‘₯∈Ω and 𝑑>0.

Acknowledgments

The authors are grateful to the referees for their helpful comments and suggestions. This work is supported by the Fundamental Research Funds for the Central Universities (no. XDJK2009C152).

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Copyright Β© 2012 Chenglin Li and Zhangguo Hong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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