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 šæš‘ 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 predatorsā€™ 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 [2ā€“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 š‘›Ć—š‘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: šœ•š‘¢1šœ•š‘”āˆ’š‘‘1Ī”š‘¢1ī€·š›½+āˆ‡ā‹…1š‘¢1āˆ‡š‘¢2ī€øī€·š›½+āˆ‡ā‹…2š‘¢1āˆ‡š‘¢3ī€ø=āˆ’š‘Žš‘¢1+š‘’2š‘2š‘¢1š‘¢2š‘š2+š‘2š‘¢2+š‘’3š‘3š‘¢1š‘¢3š‘š3+š‘3š‘¢3in(0,š‘‡)ƗĪ©,šœ•š‘¢2šœ•š‘”āˆ’š‘‘2Ī”š‘¢2=š‘Ÿ2ī‚µš‘¢1āˆ’2š¾2ī‚¶š‘¢2āˆ’š‘2š‘¢1š‘¢2š‘š2+š‘2š‘¢2in(0,š‘‡)ƗĪ©,šœ•š‘¢3šœ•š‘”āˆ’š‘‘3Ī”š‘¢3=š‘Ÿ3ī‚µš‘¢1āˆ’3š¾3ī‚¶š‘¢2āˆ’š‘3š‘¢1š‘¢3š‘š3+š‘3š‘¢3in(0,š‘‡)ƗĪ©,šœ•š‘¢1=šœ•šœˆšœ•š‘¢2=šœ•šœˆšœ•š‘¢3ī€·š‘¢šœ•šœˆ=0on(0,š‘‡)Ɨšœ•Ī©,1(0,š‘„),š‘¢2(0,š‘„),š‘¢3ī€ø=ī€·š‘¢(0,š‘„)10(š‘„),š‘¢20(š‘„),š‘¢30ī€ø(š‘„)ā‰„(0,0)inĪ©,(1.1) where Ī© is a bounded domain in š‘…š‘(š‘ā‰„1 is an integer) with a smooth boundary šœ•Ī©; š‘¢1 and š‘¢š‘–(š‘–=2,3) represent the densities of the predator and prey, respectively; the positive constants š‘‘1, š‘‘2, and š‘‘3 are the diffusion coefficient of the corresponding species; the positive constants š‘Ž,š¾š‘–,š‘Ÿš‘–,š‘šš‘–,š‘’š‘–,š‘šš‘–/š‘š‘–,š‘š‘–/š‘š‘–,š‘šš‘–/š‘š‘–(š‘–=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 š›½š‘–(š‘–=1,2) denotes their prey-tactic sensitivity. The parts š›½1š‘¢1āˆ‡š‘¢2 and š›½2š‘¢1āˆ‡š‘¢3 of the flux are directed toward the increasing population density of š‘¢2 and š‘¢3, 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 thatš›½1=0,š›½2=0,forš‘¢1ā‰„š‘¢1š‘š.(1.2) The assumptions that š›½1=0 for š‘¢1ā‰„š‘¢1š‘š and š›½2=0 for š‘¢1ā‰„š‘¢1š‘š have a clear biological interpretation [2]: the predators stop to accumulate at given point of Ī© after their density attains a certain threshold value š‘¢1š‘š and the prey-tactic sensitivity š›½1 and š›½2 vanishes identically when š‘¢1ā‰„š‘¢1š‘š.

Throughout this paper we also assume thatš‘¢20,š‘¢30ā‰¤š¾,šœ•Ī©āˆˆš¶2+š›¼,š‘¢10(š‘„),š‘¢20(š‘„),š‘¢30(š‘„)āˆˆš¶2+š›¼ī‚€Ī©ī‚,where0<š›¼<1,šœ•š‘¢10=šœ•šœˆšœ•š‘¢20=šœ•šœˆšœ•š‘¢30šœ•šœˆ=0,onšœ•Ī©.(1.3) Denote by š¶š‘š+š›¼,š›½š‘„,š‘”(š‘„š‘‡) (š‘šā‰„0 is integer, 0<š›¼<1,0<š›½<1) the space of function š‘¢(š‘„,š‘”) with finite norm [8]:ā€–š‘¢ā€–š¶š‘š+š›¼,š›½š‘„,š‘”(š‘„š‘‡)=š‘šī“||š‘™||=0īƒ¬supš‘„š‘‡||š·š‘™š‘„š‘¢||+ī«š·š‘™š‘„š‘¢ī¬(š›¼)š‘„,š‘„š‘‡+ī«š·lš‘„š‘¢ī¬(š›½)š‘”,š‘„š‘‡īƒ­,(1.4) whereāŸØš‘¤āŸ©(š›¼)š‘„,š‘„š‘‡=ī“(š‘„,š‘”),(š‘¦,š‘”)āˆˆš‘„š‘‡||||š‘¤(š‘„,š‘”)āˆ’š‘¤(š‘¦,š‘”)||||š‘„āˆ’š‘¦š›¼,āŸØš‘¤āŸ©(š›½)š‘”,š‘„š‘‡=ī“(š‘„,š‘”),(š‘„,šœ)āˆˆš‘„š‘‡||||š‘¤(š‘„,š‘”)āˆ’š‘¤(š‘„,šœ)|š‘”āˆ’šœ|š›½.(1.5) We denote by š¶2+š›¼,1+š›½š‘„,š‘”(š‘„š‘‡) the space of functions š‘¢(š‘„,š‘”) with normā€–š‘¢ā€–š¶2+š›¼,š›½š‘„,š‘”(š‘„š‘‡)+ā€–ā€–š‘¢š‘”ā€–ā€–š¶š›¼,š›½š‘„,š‘”(š‘„š‘‡).(1.6)

The main result of this paper is as follows.

Theorem 1.1. Under assumptions (1.2) and (1.3), for any given š‘‡>0 there exists a unique solution š”=(š‘¢1,š‘¢2,š‘¢3)āˆˆš¶2+š›¼,1+(š›¼/2)(š‘„š‘‡) of the system (1.1), where š‘„š‘‡=(0,š‘‡)ƗĪ©. Moreover, š‘¢1(š‘„,š‘”)ā‰„0,0ā‰¤š‘¢2(š‘„,š‘”)ā‰¤š¾2,0ā‰¤š‘¢3(š‘„,š‘”)ā‰¤š¾3,(1.7) for any š‘„āˆˆĪ© and š‘”>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 š‘‡ by š‘, while we denote various constants which are independent of š‘‡ by š‘0.

Lemma 2.1. Let (š‘¢,š‘„)āˆˆš¶2+š›¼,1+(š›¼/2)(š‘„š‘‡). Then ā€–š‘¢(š‘„,š‘”)āˆ’š‘¢(š‘„,0)ā€–š¶1+š›¼,š›¼/2(š‘„š‘‡)ā‰¤š‘0šœ‚(š‘‡)ā€–š‘¢ā€–š¶2+š›¼,1+(š›¼/2)(š‘„š‘‡),(2.1) where šœ‚(š‘‡)=max{š‘‡š›¼/2,š‘‡(1āˆ’š›¼)/2}.

Proof. Using the definition of Hƶlder norm, we have ||||š‘¢(š‘„,š‘”)āˆ’š‘¢(š‘„,0)|š‘”|(1+š›¼)/2ā‰¤||š·š‘”š‘¢||ā‹…||š‘‡||(1āˆ’š›¼)/2,[]š‘¢(š‘„,š‘”)āˆ’š‘¢(š‘„,0)š¶1+š›¼,0(š‘„š‘‡)ā‰¤š‘0ā€–ā€–š·2š‘„š‘¢(š‘„,š‘”)āˆ’š·2š‘„ā€–ā€–š‘¢(š‘„,0)šæāˆž(š‘„š‘‡)ā‰¤š‘0ī€ŗš·2š‘„š‘¢ī€»š¶0,š›¼/2(š‘„š‘‡)ā‹…||š‘‡||š›¼/2ā‰¤š‘0[š‘¢]š¶2,š›¼/2(š‘„š‘‡)ā‹…||š‘‡||š›¼/2,(2.2) which yields that ā€–š‘¢(š‘„,š‘”)āˆ’š‘¢(š‘„,0)ā€–š¶1+š›¼,(1+š›¼)/2(š‘„š‘‡)ā‰¤š‘0šœ‚ā€–š‘¢ā€–š¶2+š›¼,(1+š›¼)/2(š‘„š‘‡).(2.3) Therefore, ā€–š‘¢(š‘„,š‘”)āˆ’š‘¢(š‘„,0)ā€–š¶1+š›¼,š›¼/2(š‘„š‘‡)ā‰¤š‘0šœ‚ā€–š‘¢ā€–š¶2+š›¼,1+(š›¼/2)(š‘„š‘‡).(2.4)

We now consider the following nonlinear parabolic problem:šœ•š‘¢1šœ•š‘”āˆ’š‘‘1Ī”š‘¢1ī€·š›½+āˆ‡ā‹…1š‘¢1āˆ‡š‘¢2ī€øī€·š›½+āˆ‡ā‹…2š‘¢1āˆ‡š‘¢3ī€ø=š‘¢1š‘“in(0,š‘‡)ƗĪ©,šœ•š‘¢1š‘¢šœ•šœˆ=0on(0,š‘‡)Ɨšœ•Ī©,1(0,š‘„)=š‘¢10(š‘„)inĪ©.(2.5) By the parabolic maximum principle, we have š‘¢1(š‘„,š‘”)ā‰„0.

Lemma 2.2. Let š‘¢2(š‘„,š‘”),š‘¢3(š‘„,š‘”)āˆˆš¶2+š›¼,1+(š›¼/2)ī€·š‘„š‘‡ī€ø,š‘“(š‘„,š‘”)āˆˆš¶š›¼,š›¼/2ī€·š‘„š‘‡ī€ø,ā€–ā€–š‘¢2ā€–ā€–š¶2+š›¼,1+(š›¼/2)(š‘„š‘‡),ā€–ā€–š‘¢3ā€–ā€–š¶2+š›¼,1+(š›¼/2)(š‘„š‘‡),ā€–š‘“ā€–š¶š›¼,š›¼/2(š‘„š‘‡)ā‰¤š‘0.(2.6) Then, under assumptions (1.2) and (1.3), there exists a unique nonnegative solution š‘¢1(š‘„,š‘”)āˆˆš¶2+š›¼,1+(š›¼/2)(š‘„š‘‡) of the nonlinear problem (2.5) for small š‘‡>0 which depends on ā€–š‘¢10(š‘„)ā€–š¶2+š›¼(Ī©).

Proof. This proof is similar to that of Lemmaā€‰ā€‰2.1 in [4]. For readerā€™s convenience we include the proof here. We will prove by a fixed point argument. Let us introduce the Banach space š‘‹ of function š‘¢1 with norm ā€–š‘¢1ā€–š¶1+š›¼,š›¼/2(š‘„š‘‡)(0<š‘‡<1) and a subset š‘‹š“={š‘¢1āˆˆš‘‹āˆ¶š‘¢1(š‘„,0)=š‘¢10(š‘„)andā€–š‘¢1ā€–š¶1+š›¼,š›¼/2(š‘„š‘‡)ā‰¤š“}, where š“=ā€–š‘¢10(š‘„)ā€–š¶2+š›¼(Ī©)+ā€–š‘¢20(š‘„)ā€–š¶2+š›¼(Ī©)+ā€–š‘¢30(š‘„)ā€–š¶2+š›¼(Ī©)+1. For any š‘¢1āˆˆš‘‹š“, we define a corresponding function Ģƒš‘¢1=š¹š‘¢1, where Ģƒš‘¢1 satisfies the equations šœ•Ģƒš‘¢1šœ•š‘”āˆ’š‘‘1Ī”Ģƒš‘¢1āˆ’Ģƒš‘¢1š‘“=āˆ’š›½1āˆ‡š‘¢1ā‹…āˆ‡š‘¢2āˆ’š›½2āˆ‡š‘¢1ā‹…āˆ‡š‘¢3āˆ’š›½1š‘¢1Ī”š‘¢2āˆ’š›½2š‘¢1Ī”š‘¢3in(0,š‘‡)ƗĪ©,šœ•Ģƒš‘¢1šœ•šœˆ=0on(0,š‘‡)Ɨšœ•Ī©,Ģƒš‘¢1(0,š‘„)=š‘¢10(š‘„)inĪ©.(2.7) By š‘¢1āˆˆš‘‹š“, we have ā„Ž1ā‰œāˆ’š›½1āˆ‡š‘¢1ā‹…āˆ‡š‘¢2āˆ’š›½2āˆ‡š‘¢1ā‹…āˆ‡š‘¢3āˆ’š›½1š‘¢1Ī”š‘¢2āˆ’š›½2š‘¢1Ī”š‘¢3āˆˆš¶š›¼,š›¼/2ī€·š‘„š‘‡ī€ø.(2.8) By the parabolic Schauder theory, this yields that there exists a unique solution Ģƒš‘¢ to (2.7) and ā€–ā€–Ģƒš‘¢1ā€–ā€–š¶2+š›¼,1+(š›¼/2)(š‘„š‘‡)ā‰¤ā€–ā€–Ģƒš‘¢1|š‘”=0ā€–ā€–š¶2+š›¼(Ī©)+š‘€1(š“)ā‰¤š“+š‘€1(š“)ā‰œš‘€2(š“),(2.9) where š‘€2(A) is some constant which depends only on š“. For any function Ģƒš‘¢1(š‘„,š‘”), by Lemma 2.1 and combining (2.9), if š‘‡ is sufficiently small (š‘‡ depends only on š“), then we have ā€–ā€–Ģƒš‘¢1ā€–ā€–(š‘„,š‘”)š¶1+š›¼,š›¼/2(š‘„š‘‡)ā‰¤ā€–ā€–Ģƒš‘¢1ā€–ā€–(š‘„,0)š¶1+š›¼,š›¼/2(š‘„T)+š‘0ā€–ā€–šœ‚(š‘‡)Ģƒš‘¢1ā€–ā€–š¶2+š›¼,1+(š›¼/2)(š‘„š‘‡)ā‰¤ā€–ā€–Ģƒš‘¢10ā€–ā€–(š‘„)š¶2+š›¼+1ā‰¤š“.(2.10) Therefore, Ģƒš‘¢1(š‘„,š‘”)āˆˆš‘‹š“ and š¹ maps š‘‹š“ into itself. We now prove that š¹ is contractive. Take š‘¢11,š‘¢12 in š‘‹š“, and set Ģƒš‘¢11=š¹š‘¢11,Ģƒš‘¢12=š¹š‘¢12,Ģƒš‘£=Ģƒš‘¢11āˆ’Ģƒš‘¢12. Then, it follows from (2.7) that Ģƒš‘£ solves the following systems: šœ•Ģƒš‘£šœ•š‘”āˆ’š‘‘1Ī”ĢƒĢƒš‘£āˆ’š‘£š‘“=ā„Ž2šœ•Ģƒš‘£in(0,š‘‡)ƗĪ©,Ģƒšœ•šœˆ=0on(0,š‘‡)Ɨšœ•Ī©,š‘£(0,š‘„)=0inĪ©,(2.11) where ā„Ž2ā‰œāˆ’š›½1ī€·āˆ‡š‘¢11āˆ’āˆ‡š‘¢12ī€øā‹…āˆ‡š‘¢2āˆ’š›½2ī€·āˆ‡š‘¢11āˆ’āˆ‡š‘¢12ī€øā‹…āˆ‡š‘¢3āˆ’š›½1ī€·š‘¢11āˆ’š‘¢12ī€øĪ”š‘¢2āˆ’š›½2ī€·š‘¢11āˆ’š‘¢12ī€øĪ”š‘¢3.(2.12) By š‘¢11,š‘¢12āˆˆš‘‹š“ and conditions of Lemma 2.2, it is easy to check that ā€–ā€–ā„Ž2ā€–ā€–šæāˆž(š‘„š‘‡)ā‰¤ā€–ā€–š›½1ī€·āˆ‡š‘¢11āˆ’āˆ‡š‘¢12ī€øā‹…āˆ‡š‘¢2ā€–ā€–š¶0(š‘„š‘‡)+ā€–ā€–š›½2ī€·āˆ‡š‘¢11āˆ’āˆ‡š‘¢12ī€øā‹…āˆ‡š‘¢3ā€–ā€–š¶0(š‘„š‘‡)+ā€–ā€–š›½1ī€·š‘¢11āˆ’š‘¢12ī€øĪ”š‘¢2ā€–ā€–š¶0(š‘„š‘‡)+ā€–ā€–š›½2ī€·š‘¢11āˆ’š‘¢12ī€øĪ”š‘¢3ā€–ā€–š¶0(š‘„š‘‡)ā‰¤š‘0ā€–ā€–š‘¢11āˆ’š‘¢12ā€–ā€–š¶1,0+š‘0ā€–ā€–š‘¢11āˆ’š‘¢12ā€–ā€–š¶0(š‘„š‘‡)ā‰¤š‘0ā€–ā€–š‘¢11āˆ’š‘¢12ā€–ā€–š¶1,0(š‘„š‘‡).(2.13) Using the assumption ā€–š‘“ā€–š›¼,š›¼/2ā‰¤š‘0 and the šæš‘-estimate, we have ā€–š‘“ā€–šæāˆž(š‘„š‘‡)ā‰¤ā€–š‘“ā€–š¶š›¼,š›¼/2ā‰¤š‘0,ā€–Ģƒš‘£ā€–š‘¤š‘2,1ā‰¤š‘0ā€–ā€–ā„Ž2ā€–ā€–šæāˆž(š‘„š‘‡).(2.14) For any š‘ā‰„1, by using Sobolev embedding š‘Šš‘2,1(š‘„š‘‡)ā†Ŗš¶1+š›¾,(1+š›¾)/2(š‘„š‘‡) (š›¾=1āˆ’(5/š‘)>š›¼ if we take š‘ sufficiently large), we have ā€–Ģƒš‘£ā€–š¶1+š›¾,(1+š›¾)/2(š‘„š‘‡)ā‰¤š‘0ā€–ā€–ā„Ž2ā€–ā€–šæāˆž(š‘„š‘‡)ā‰¤š‘0ā€–ā€–š‘¢11āˆ’š‘¢12ā€–ā€–š¶1,0(š‘„š‘‡).(2.15) Then, noting š›¾>š›¼, we can easily check that [4] ā€–Ģƒš‘£ā€–š¶1+š›¼,š›¼/2(š‘„š‘‡)ā‰¤š‘0š‘‡š›¼/2ā€–ā€–š‘¢11āˆ’š‘¢12ā€–ā€–š¶1+š›¼,š›¼/2(š‘„š‘‡).(2.16) Taking š‘‡ small such that š‘0š‘‡š›¼/2<1/2, we conclude from (2.16) that š¹ is contractive in š‘‹š“. Therefore š¹ has a unique fixed point š‘¢1, which is the unique solution to (2.5). Moreover, we can raise the regularity of š‘¢1 to š¶2+š›¼,1+(š›¼/2)(š‘„š‘‡) 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 š”(š‘„,š‘”)=(š‘¢1,š‘¢2,š‘¢3)āˆˆš¶2+š›¼,1+(š›¼/2)(š‘„š‘‡) as done in [6, 7].

Theorem 3.1. Assume that (1.2) and (1.3) hold, then there exists a unique solution š”(š‘„,š‘”)=(š‘¢1,š‘¢2,š‘¢3)āˆˆš¶2+š›¼,1+(š›¼/2)(š‘„š‘‡) of the system (1.2) for small š‘‡>0 which depends on ā€–ā€–š”0ā€–ā€–(š‘„)š¶2+š›¼(Ī©)ā‰œā€–ā€–š‘¢10ā€–ā€–(š‘„)š¶2+š›¼(Ī©)+ā€–ā€–š‘¢20ā€–ā€–(š‘„)š¶2+š›¼(Ī©)+ā€–ā€–š‘¢30ā€–ā€–(š‘„)š¶2+š›¼(Ī©).(3.1) Furthermore, š‘¢1(š‘„,š‘”)ā‰„0,š‘¢2(š‘„,š‘”)ā‰„0,š‘¢3(š‘„,š‘”)ā‰„0.

Proof. We will prove the local existence by a fixed point argument again. Introducing the Banach space š‘‹ of the function š”, we define the norm ā€–š”ā€–š¶š›¼,š›¼/2(š‘„š‘‡)=ā€–ā€–š‘¢1ā€–ā€–š¶š›¼,š›¼/2(š‘„š‘‡)+ā€–ā€–š‘¢2ā€–ā€–š¶š›¼,š›¼/2(š‘„š‘‡)+ā€–ā€–š‘¢3ā€–ā€–š¶š›¼,š›¼/2(š‘„š‘‡)(0<š‘‡<1),(3.2) and a subset š‘‹š“=ī€½š”āˆˆš‘‹āˆ¶š‘¢1,š‘¢2,š‘¢3ā‰„0,ā€–š”ā€–š¶š›¼,š›¼/2(š‘„š‘‡)ī€¾,ā‰¤š“(3.3) where š”ī€·š‘¢(š‘„,0)=10(š‘„),š‘¢20(š‘„),š‘¢30ī€ø,ā€–ā€–š‘¢(š‘„)š“=10ā€–ā€–(š‘„)š¶2+š›¼(Ī©)+ā€–ā€–š‘¢20ā€–ā€–(š‘„)š¶2+š›¼(Ī©)+ā€–ā€–š‘¢30ā€–ā€–(š‘„)š¶2+š›¼(Ī©)+1.(3.4) For any š”āˆˆš‘‹š“, we define correspondingly function š”=š‡š” by š”=(š‘¢1,š‘¢2,š‘¢3), where š” satisfies the equations šœ•š‘¢2šœ•š‘”āˆ’š‘‘2Ī”š‘¢2=ī‚øš‘Ÿ2ī‚µš‘¢1āˆ’2š¾2ī‚¶āˆ’š‘2š‘¢1š‘š2+š‘2š‘¢2ī‚¹š‘¢2šœ•in(0,š‘‡)ƗĪ©,š‘¢3šœ•š‘”āˆ’š‘‘3Ī”š‘¢3=ī‚øš‘Ÿ3ī‚µš‘¢1āˆ’3š¾3ī‚¶āˆ’š‘3š‘¢1š‘š3+š‘3š‘¢3ī‚¹š‘¢3šœ•in(0,š‘‡)ƗĪ©,š‘¢2=šœ•šœ•šœˆš‘¢3šœ•šœˆ=0on(0,š‘‡)Ɨšœ•Ī©,š‘¢2(š‘„,0)=š‘¢20(š‘„),š‘¢3(š‘„,0)=š‘¢30šœ•(š‘„),š‘„āˆˆĪ©,(3.5)š‘¢1šœ•š‘”āˆ’š‘‘1Ī”š‘¢1ī€·š›½+āˆ‡ā‹…1š‘¢1āˆ‡š‘¢2ī€øī€·š›½+āˆ‡ā‹…2š‘¢1āˆ‡š‘¢3ī€ø=āˆ’š‘Žš‘¢1+š‘’2š‘2š‘¢1š‘¢2š‘š2+š‘2š‘¢2+š‘’3š‘3š‘¢1š‘¢3š‘š3+š‘3š‘¢3šœ•in(0,T)ƗĪ©,š‘¢1šœ•šœˆ=0on(0,š‘‡)Ɨšœ•Ī©,š‘¢1(š‘„,0)=š‘¢10(š‘„),š‘„āˆˆĪ©.(3.6) By (3.5), (š‘¢1,š‘¢2,š‘¢3)āˆˆš‘‹š“, assumption (1.3), and the parabolic Schauder theory, we have that there exists a unique solution š‘¢2,š‘¢3 to (3.5) and ā€–ā€–š‘¢2ā€–ā€–š¶2+š›¼,1+(š›¼/2)(š‘„š‘‡)ā‰¤ā€–ā€–š‘¢2|š‘”=0ā€–ā€–š¶2+š›¼+š‘€3(š“)ā‰¤š“+š‘€3(š“)ā‰œš‘€4(š“).(3.7) Similarly, ā€–ā€–š‘¢3ā€–ā€–š¶2+š›¼,1+(š›¼/2)(š‘„š‘‡)ā‰¤ā€–ā€–š‘¢3|š‘”=0ā€–ā€–š¶2+š›¼+š‘€5(š“)ā‰¤š“+š‘€5(š“)ā‰œš‘€6(š“).(3.8) Moreover, by parabolic maximum principle, we have š‘¢2(š‘„,š‘”)ā‰„0inš‘„š‘‡,š‘¢3(š‘„,š‘”)ā‰„0inš‘„š‘‡.(3.9) Similarly, by using Lemma 2.2, from (3.6) we can conclude that there exists a unique solution š‘¢1 satisfying ā€–ā€–š‘¢1ā€–ā€–š¶2+š›¼,1+(š›¼/2)š‘„š‘‡ā‰¤š‘€7(š“),(3.10) and by parabolic maximum principle we have š‘¢1(š‘„,š‘”)ā‰„0 in š‘„š‘‡. From (3.7), (3.8), and (3.10), we have ā€–ā€–š”ā€–ā€–š¶2+š›¼,1+(š›¼/2)(š‘„š‘‡)ā‰¤š‘€8(š“).(3.11) For any function š”(š‘„,š‘”), using Lemma 2.1 we get ā€–ā€–š”(š‘„,š‘”)āˆ’ā€–ā€–š”(š‘„,0)š¶š›¼,š›¼/2(š‘„š‘‡)ā‰¤š‘0šœ‚(š‘‡)ā€–š”ā€–š¶2+š›¼,1+(š›¼/2)(š‘„š‘‡).(3.12) From (3.11) and (3.12), if š‘‡ is sufficiently small we have ā€–ā€–ā€–ā€–š”(š‘„,š‘”)š¶š›¼,š›¼/2(š‘„š‘‡)ā‰¤ā€–ā€–ā€–ā€–š”(š‘„,0)š¶š›¼,š›¼/2+š‘0šœ‚(š‘‡)š‘€8ā‰¤ā€–ā€–š”(š“)0ā€–ā€–(š‘„)š¶2+š›¼(Ī©)+1ā‰”š“,(3.13) which yields š”āˆˆš‘‹š“. Therefore, š» maps š‘‹š“ into itself.
Next, we can prove that š» is contractive as done in the proof of Lemma 2.2 in š‘‹š“ if we take š‘‡ sufficiently small. By the contraction mapping theorem š» has a unique fixed point š”, which is the unique solution of (1.1). Moreover, we can raise the regularity of š” to š¶2+š›¼,1+(š›¼/2)(š‘„š‘‡) by using the parabolic Schauder estimates.

4. Global Existence

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

Lemma 4.1. Suppose that š”=(š‘¢1,š‘¢2,š‘¢3)āˆˆš¶2,1(š‘„š‘‡) is a solution to the system (1.1), then there holds š‘¢1ā‰„0,0ā‰¤š‘¢2ā‰¤š¾2,0ā‰¤š‘¢3ā‰¤š¾3.(4.1)

Proof. It follows from (1.1) that šœ•š‘¢1šœ•š‘”āˆ’š‘‘1Ī”š‘¢1+ī€·š›½1āˆ‡š‘¢2+š›½2āˆ‡š‘¢3ī€øā‹…āˆ‡š‘¢1+ī‚µš›½1Ī”š‘¢2+š›½2Ī”š‘¢3š‘’+š‘Žāˆ’2š‘2š‘¢2š‘š2+š‘2š‘¢2āˆ’š‘’3š‘3š‘¢3š‘š3+š‘3š‘¢3ī‚¶š‘¢1=0in(0,š‘‡)ƗĪ©,šœ•š‘¢1š‘¢šœ•šœˆ=0on(0,š‘‡)Ɨšœ•Ī©,1(0,š‘„)=š‘¢10(š‘„)ā‰„0inĪ©.(4.2) Obviously, š‘¢1ā‰”0 is a subsolution to (4.2). Using the maximum principle, we get š‘¢1ā‰„0. Similarly, we have š‘¢2ā‰„0 and š‘¢3ā‰„0.
On the other hand, it follows from model (1.1) thatšœ•š‘¢2šœ•š‘”āˆ’š‘‘2Ī”š‘¢2āˆ’š‘Ÿ2ī‚µš‘¢1āˆ’2š¾2ī‚¶š‘¢2+š‘2š‘¢1š‘¢2š‘š2+š‘2š‘¢2š‘=0ā‰¤2š¾2š‘¢1š‘š2+š‘2š¾2in(0,š‘‡)ƗĪ©,šœ•š‘¢2š‘¢šœ•šœˆ=0on(0,š‘‡)Ɨšœ•Ī©,2(0,š‘„)=š‘¢20(š‘„)inĪ©,(4.3) which implies that š¾2 is a subsolution to problem (4.3). Hence we have 0ā‰¤š‘¢2(š‘„,š‘”)ā‰¤š¾2. Similarly, we get 0ā‰¤š‘¢3(š‘„,š‘”)ā‰¤š¾3. This completes the proof of Lemma 4.1.

Lemma 4.2. Suppose that š”=(š‘¢1,š‘¢2,š‘¢3)āˆˆš¶2,1(š‘„š‘‡) is a solution to the system (1.1), then for any š‘>1 there holds ā€–ā€–š‘¢1ā€–ā€–šæš‘(š‘„š‘‡)ā€–ā€–š‘¢ā‰¤š‘,2ā€–ā€–šæš‘(š‘„š‘‡)ā€–ā€–š‘¢ā‰¤š‘,2ā€–ā€–šæš‘(š‘„š‘‡)ā‰¤š‘.(4.4)

Proof. Multiplying the first equation of (1.1) by š‘¢1š‘āˆ’1, integrating over š‘„š‘‡, using the no-flux boundary condition, and noting š‘¢1ā‰„0, we get 1š‘ī€œĪ©š‘¢š‘1(1š‘„,š‘”)š‘‘š‘„āˆ’š‘ī€œĪ©š‘¢š‘0(š‘„,š‘”)š‘‘š‘„+(š‘āˆ’1)š‘‘1ī€œš‘”0ī€œĪ©š‘¢1š‘āˆ’2||āˆ‡š‘¢1||2ī€œš‘‘š‘„š‘‘š‘”ā‰¤(š‘āˆ’1)š‘”0ī€œĪ©š›½1š‘¢1š‘āˆ’1āˆ‡š‘¢1ā‹…āˆ‡š‘¢2ī€œš‘‘š‘„š‘‘š‘”+(š‘āˆ’1)š‘”0ī€œĪ©š›½2š‘¢1š‘āˆ’1āˆ‡š‘¢1ā‹…āˆ‡š‘¢3+š‘’š‘‘š‘„š‘‘š‘”2š‘2š‘2ī€œš‘”0ī€œĪ©š‘¢š‘1š‘’š‘‘š‘„š‘‘š‘”+3š‘3š‘3ī€œš‘”0ī€œĪ©š‘¢š‘1š‘‘š‘„š‘‘š‘”.(4.5) For š‘¢1ā‰„š‘¢1š‘š, we get 1š‘ī€œĪ©š‘¢š‘1(1š‘„,š‘”)š‘‘š‘„āˆ’š‘ī€œĪ©š‘¢š‘0(š‘„,š‘”)š‘‘š‘„+(š‘āˆ’1)š‘‘1ī€œš‘”0ī€œĪ©š‘¢1š‘āˆ’2||āˆ‡š‘¢1||2ā‰¤ī‚µš‘’š‘‘š‘„š‘‘š‘”2š‘2š‘2+š‘’3š‘3š‘3ī‚¶ī€œš‘”0ī€œĪ©š‘¢š‘1š‘‘š‘„š‘‘š‘”.(4.6) Therefore ī€œĪ©š‘¢š‘1(š‘„,š‘”)š‘‘š‘”ā‰¤š‘0+š‘0ī€œš‘”0ī€œĪ©š‘¢š‘1š‘‘š‘„š‘‘š‘”.(4.7) Using Gronwallā€™s Lemma, we have ī€œš‘”0ī€œĪ©š‘¢š‘1(š‘„,š‘”)š‘‘š‘”ā‰¤š‘.(4.8) Therefore, for š‘¢1<š‘¢1š‘š, we have ī€œš‘”0ī€œĪ©š‘¢š‘1(ī€œš‘„,š‘”)š‘‘š‘”ā‰¤š‘”0ī€œĪ©š‘¢š‘1š‘š(š‘„,š‘”)š‘‘š‘”ā‰¤š‘.(4.9) Obviously, we have ī€œš‘”0ī€œĪ©š‘¢š‘2(ī€œš‘„,š‘”)š‘‘tā‰¤š‘”0ī€œĪ©š¾š‘2(ī€œš‘„,š‘”)š‘‘š‘”ā‰¤š‘,š‘”0ī€œĪ©š‘¢š‘3ī€œ(š‘„,š‘”)š‘‘š‘”ā‰¤š‘”0ī€œĪ©š¾š‘3(š‘„,š‘”)š‘‘š‘”ā‰¤š‘.(4.10) This completes the proof of Lemma 4.2.

Lemma 4.3. Suppose that š”=(š‘¢1,š‘¢2,š‘¢3)āˆˆš¶2,1(Qš‘‡) is a solution to the system (1.1), then for any š‘>5 there holds ā€–ā€–š‘¢1ā€–ā€–š‘¤š‘2,1(š‘„š‘‡)ā€–ā€–š‘¢ā‰¤š‘,2ā€–ā€–š‘¤š‘2,1(š‘„š‘‡)ā€–ā€–š‘¢ā‰¤š‘,3ā€–ā€–š‘¤š‘2,1(š‘„š‘‡)ā‰¤š‘.(4.11)

Proof. Note that the second equation of (1.1) can be rewritten as follows: šœ•š‘¢2šœ•š‘”āˆ’š‘‘2Ī”š‘¢2āˆ’ī‚µš‘Ÿ2āˆ’š‘Ÿ2š¾2š‘¢2āˆ’š‘2š‘¢1š‘š2+š‘2š‘¢2ī‚¶š‘¢2=0,(4.12) where ā€–š‘Ÿ2āˆ’(š‘Ÿ2/š¾2)š‘¢2āˆ’(š‘2š‘¢1/(š‘š2+š‘2š‘¢2))ā€–šæš‘(š‘„š‘‡)ā‰¤š‘.
By the parabolic šæš‘-estimate, we haveā€–ā€–š‘¢2ā€–ā€–š‘¤š‘2,1(š‘„š‘‡)ā‰¤š‘.(4.13) Using the Sobolev embedding theorem (taking š‘>5), we get ā€–ā€–āˆ‡š‘¢2ā€–ā€–šæāˆž(š‘„š‘‡)ā‰¤š‘.(4.14) Similarly, we can obtain ā€–ā€–š‘¢3ā€–ā€–š‘¤š‘2,1(š‘„š‘‡)ā€–ā€–ā‰¤š‘,āˆ‡š‘¢3ā€–ā€–šæāˆž(š‘„š‘‡)ā‰¤š‘.(4.15) It follows from the first equation of (1.1) that šœ•š‘¢1šœ•š‘”āˆ’š‘‘1Ī”š‘¢1+ī€·š›½1āˆ‡š‘¢2+š›½2āˆ‡š‘¢3ī€øā‹…āˆ‡š‘¢1ī‚µš›½=āˆ’1Ī”š‘¢2+š›½2Ī”š‘¢3š‘’+š‘Žāˆ’2š‘2š‘¢2š‘š2+š‘2š‘¢2āˆ’š‘’3š‘3š‘¢3š‘š3+š‘3š‘¢3ī‚¶š‘¢1in(0,š‘‡)ƗĪ©,šœ•š‘¢1š‘¢šœ•šœˆ=0on(0,š‘‡)Ɨšœ•Ī©,1(0,š‘„)=š‘¢10(š‘„)ā‰„0inĪ©,(4.16) where ā€–ā€–ā€–āˆ’ī‚µš›½1Ī”š‘¢2+š›½2Ī”š‘¢3š‘’+š‘Žāˆ’2š‘2š‘¢2š‘š2+š‘2š‘¢2āˆ’š‘’3š‘3š‘¢3š‘š3+š‘3š‘¢3ī‚¶š‘¢1ā€–ā€–ā€–šæš‘(š‘„š‘‡)ā‰¤š‘.(4.17) Using the parabolic šæš‘-estimates again, we have ā€–ā€–š‘¢1ā€–ā€–š‘¤š‘2,1(š‘„š‘‡)ā‰¤š‘.(4.18) This completes the proof of Lemma 4.3.

Lemma 4.4. Suppose that š”=(š‘¢1,š‘¢2,š‘¢3)āˆˆš¶2,1(š‘„š‘‡) is a solution to the system (1.1), then there holds ā€–ā€–š‘¢1ā€–ā€–š¶2+š›¼,1+(š›¼/2)(š‘„š‘‡)ā€–ā€–š‘¢ā‰¤š‘,2ā€–ā€–š¶2+š›¼,1+(š›¼/2)(š‘„š‘‡)ā€–ā€–š‘¢ā‰¤š‘,3ā€–ā€–š¶2+š›¼,1+(š›¼/2)(š‘„š‘‡)ā‰¤š‘.(4.19)

Proof. Using the Sobolev embedding theorem (taking š‘>5) and Lemma 4.3, we have ā€–ā€–š‘¢1ā€–ā€–š¶š›¼,š›¼/2(š‘„š‘‡)ā€–ā€–š‘¢ā‰¤š‘,2ā€–ā€–š¶š›¼,š›¼/2(š‘„š‘‡)ā€–ā€–š‘¢ā‰¤š‘,3ā€–ā€–š¶š›¼,š›¼/2(š‘„š‘‡)ā‰¤š‘.(4.20) Using (4.20) and the Schauder estimates to the second and third equation of model (1.1), we have ā€–ā€–š‘¢2ā€–ā€–š¶2+š›¼,1+(š›¼/2)(š‘„š‘‡)ā€–ā€–š‘¢ā‰¤š‘,3ā€–ā€–š¶2+š›¼,1+(š›¼/2)(Qš‘‡)ā‰¤š‘.(4.21) Applying the parabolic Schauder estimate to (4.16) and using (4.21), we have ā€–ā€–š‘¢1ā€–ā€–š¶2+š›¼,1+(š›¼/2)(š‘„š‘‡)ā‰¤š‘.(4.22) This completes the proof of Lemma 4.4.

Therefore, we can extend the local solution established in Theorem 3.1 to all š‘”>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 š”=(š‘¢1,š‘¢2,š‘¢3)āˆˆš¶2+š›¼,1+(š›¼/2)(š‘„š‘‡) of the system (1.2) for any given š‘‡>0. Moreover, š‘¢1(š‘„,š‘”)ā‰„0,0ā‰¤š‘¢2ā‰¤š¾2,0ā‰¤š‘¢3ā‰¤š¾3.(4.23) for any š‘„āˆˆĪ© and š‘”>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).