Existence of Solutions for a Quasilinear Reaction Diffusion System

Abstract

The degenerate reaction diffusion system has been applied to a variety of physical and engineering problems. This paper is extended the existence of solutions from the quasimonotone reaction functions (e.g., inhibitor-inhibitor mechanism) to the mixed quasimonotone reaction functions (e.g., activator-inhibitor mechanism). By Schauder fixed point theorem, it is shown that the system admits at least one positive solution if there exist a coupled of upper and lower solutions. This result is applied to a Lotka-Volterra predator-prey model.

1. Introduction

We consider a quasilinear reaction diffusion system in a bounded domain under coupled nonlinear boundary conditions. The system of equations is given in the form𝜕𝑢_𝑖𝑎𝜕𝑡−∇⋅_𝑖𝐷_𝑖𝑢_𝑖∇𝑢_𝑖+𝐛_𝑖⋅𝐷_𝑖𝑢_𝑖∇𝑢_𝑖=𝑓_𝑖(),𝐷𝑡,𝑥,𝐮)(𝑡>0,𝑥∈Ω_𝑖𝑢_𝑖𝜕𝑢_𝑖𝜕𝜈=𝑔_𝑖(),𝑢𝑡,𝑥,𝐮)(𝑡>0,𝑥∈𝜕Ω_𝑖(0,𝑥)=𝜓_𝑖(𝑥)(𝑥∈Ω),𝑖=1,…,𝑁, (1.1) where 𝐮≡(𝑢₁,…,𝑢_𝑁), Ω is a bounded domain in ℝ^𝑛 with boundary 𝜕Ω, 𝜕/𝜕𝜈 denotes the outward normal derivative on 𝜕Ω. It is assumed that the boundary 𝜕Ω is of class 𝐶^1+𝛼. It is also assumed that, for each 𝑖=1,…,𝑁, the functions 𝑎_𝑖≡𝑎_𝑖(𝑡,𝑥), 𝐛_𝑖≡𝐛_𝑖(𝑡,𝑥)≡(𝑏_𝑖⁽¹⁾,…,𝑏_𝑖^(𝑛)), 𝑓_𝑖(𝑡,𝑥,⋅) and 𝑔_𝑖(𝑡,𝑥,⋅) are Hölder continuous in [0,∞)×Ω. The density-dependent diffusion coefficient 𝐷_𝑖(𝑢_𝑖) may have the property 𝐷_𝑖(0)=0, which means that the elliptic operators are degenerate.

The quasilinear reaction diffusion system has been investigated extensively in the literature [1–3]. Recently by use of upper and lower solutions and its associated monotone iterations, [4, 5] deal with the scalar equation and the system endowed with the nonlinear Neumann-Robin boundary conditions, respectively. The paper in [6] is concerned with the existence, uniqueness, and asymptotic behavior for the quasilinear parabolic systems with the Dirichlet boundary condition. However, the requirement of the reaction functions in [4–6] are monotone nondecreasing. This paper relaxed the condition to mixed quasimonotone reaction functions, which leads to the difficult point that the ordered upper and lower solutions do not exist. To overcome it, we construct the coupled upper and lower solutions.

The purpose of this paper is to study the existence for the system (1.1) by the Schauder fixed point theorem. The rest of this paper is organized as follows. In Section 2 we show the existence by the method of upper and lower solutions and the Schauder fixed point theorem. An application is given in Section 3 to the Lotka-Volterra predator-prey model. The paper ends with Section 4 for some discussions.

2. Existence of Solutions

To the simplicity, throughout this paper, we denote𝑄=(0,𝑇]×Ω,𝑆=(0,𝑇]×𝜕Ω,𝑄=[0,𝑇]×Ω, (2.1) and let 𝐶^𝑚(𝑄) and 𝐶^𝛼(𝑄) be the respective space of 𝑚-times differentiable and Hölder continuous functions in 𝑄, where 𝑄 represents a domain or a section between two functions. For vector functions with 𝑁-components we denote the above function space by 𝒞^𝑚(𝑄) and 𝒞^𝛼(𝑄), respectively.

In this paper, we make the following hypothesis.

(𝐻) For each 𝑖=1,…,𝑁, the following conditions hold:(i)𝑎_𝑖(𝑡,𝑥), 𝑏_𝑖^(𝑙)(𝑡,𝑥)(𝑙=1,…,𝑛) and 𝑓_𝑖(𝑡,𝑥,⋅) are in 𝐶^𝛼/2,𝛼(𝑄) with 𝑎_𝑖≥𝑎^∗_𝑖>0, 𝑔_𝑖(𝑡,𝑥,⋅)∈𝐶^{1+𝛼/2,2+𝛼}(𝑄);(ii)𝐷_𝑖(𝑢_𝑖)∈𝐶^{1+𝛼/2,1+𝛼}(Λ_𝑖) and 𝐷_𝑖(𝑢_𝑖)>0 for 𝑢_𝑖>0 and 𝐷_𝑖(0)≥0;(iii)𝐟(⋅,𝐮),𝐠(⋅,𝐮) are mixed quasimonotone 𝒞¹-functions in Λ.

In the above hypothesis, Λ and Λ_𝑖 are the sectors between a pair of coupled upper and lower solutions given by (2.8) below. It is allowed that 𝐷_𝑖(0)=0 for some 𝑖 and 𝐷_𝑖(0)>0 for a different 𝑖. Particularly, if 𝐷_𝑖(𝑢) is a positive constant for all 𝑖 then system (1.1) becomes the standard coupled system of semilinear parabolic equations. Recall that a vector function 𝐟(⋅,𝐮) is said to be mixed quasimonotone in Λ if for each 𝑖=1,…,𝑁, there exist nonnegative integers 𝑎_𝑖 and 𝑏_𝑖 with 𝑎_𝑖+𝑏_𝑖=𝑁−1 such that the function 𝑓_𝑖(⋅,𝐮)≡𝑓_𝑖(⋅,𝑢_𝑖,[𝐮]_𝑎𝑖,[𝐮]_𝑏𝑖) is nondecreasing with respect to all component [𝐮]_𝑎𝑖 and is nonincreasing with respect to all component [𝐮]_𝑏𝑖, where 𝐮≡(𝑢_𝑖,[𝐮]_𝑎𝑖,[𝐮]_𝑏𝑖)∈Λ. Similarly, 𝑔_𝑖(⋅,𝐮)≡𝑔_𝑖(⋅,𝑢_𝑖,[𝐮]_𝑐𝑖,[𝐮]_𝑑𝑖). Our approach to the existence problem is by the method of coupled upper and lower solutions which are defined as follows.

Definition 2.1. A pair of functions 𝐮=(𝑢₁,…,𝑢_𝑁),𝐮=(𝑢₁,…,𝑢_𝑁)∈𝒞(𝑄)∩𝒞^1,2(𝑄) are called coupled upper and lower solutions of (1.1) if 𝐮≥𝐮 and if 𝜕𝑢_𝑖𝑎𝜕𝑡−∇⋅_𝑖𝐷_𝑖𝑢_𝑖∇𝑢_𝑖+𝐛_𝐢⋅𝐷_𝑖𝑢_𝑖∇𝑢_𝑖≥𝑓_𝑖⋅,𝑢_𝑖,[𝐮]_𝑎𝑖,[𝐮]_𝑏𝑖in𝑄,𝜕𝑢_𝑖𝑎𝜕𝑡−∇⋅_𝑖𝐷_𝑖𝑢_𝑖∇𝑢_𝑖+𝐛_𝐢⋅𝐷_𝑖𝑢_𝑖∇𝑢_𝑖≤𝑓_𝑖⋅,𝑢_𝑖,[𝐮]_𝑎𝑖,[𝐮]_𝑏𝑖𝐷in𝑄,_𝑖𝑢_𝑖𝜕𝑢_𝑖𝜕𝜈≥𝑔_𝑖⋅,𝑢_𝑖,[𝐮]_𝑐𝑖,[𝐮]_𝑑𝑖𝐷on𝑆,_𝑖𝑢_𝑖𝜕𝑢_𝑖𝜕𝜈≤𝑔_𝑖⋅,𝑢_𝑖,[𝐮]_𝑐𝑖,[𝐮]_𝑑𝑖on𝑆,𝑢_𝑖(0,𝑥)≥𝜓_𝑖(𝑥),𝑢_𝑖(0,𝑥)≤𝜓_𝑖(𝑥)inΩ,𝑖=1,…,𝑁. (2.2)

Define𝑤_𝑖=𝐼_𝑖𝑢_𝑖=^𝑢𝑖₀𝐷_𝑖(𝑠)𝑑𝑠for𝑢_𝑖≥0,𝑖=1,…,𝑁, (2.3) it follows from the following Hypothesis (𝐻) that𝐼^′_𝑖𝑢_𝑖=𝑑𝐼_𝑖𝑑𝑢_𝑖=𝐷_𝑖𝑢_𝑖>0, (2.4) then the inverse 𝑢_𝑖≡𝑞_𝑖(𝑤_𝑖) exists and is an increasing function of 𝑤_𝑖>0. In view of𝜕𝑤_𝑖𝜕𝑡=𝐷_𝑖𝑢_𝑖𝜕𝑢_𝑖𝜕𝑡,∇𝑤_𝑖=𝐷_𝑖𝑢_𝑖∇𝑢_𝑖,𝜕𝑤_𝑖𝜕𝜈=𝐷_𝑖𝑢_𝑖𝜕𝑢_𝑖𝜕𝜈 (2.5) we may write (1.1) in the equivalent form𝐷_𝑖𝑢_𝑖⁻¹𝜕𝑤_𝑖𝑎𝜕𝑡−∇⋅_𝑖∇𝑤_𝑖+𝐛_𝐢⋅∇𝑤_𝑖=𝑓_𝑖𝑡,𝑥,𝑢_𝑖,[𝐮]_𝑎𝑖,[𝐮]_𝑏𝑖in𝑄,𝜕𝑤_𝑖𝜕𝜈=𝑔_𝑖𝑡,𝑥,𝑢_𝑖,[𝐮]_𝑐𝑖,[𝐮]_𝑑𝑖𝑤on𝑆,_𝑖(0,𝑥)=𝜂_𝑖(𝑥)𝑢inΩ,_𝑖=𝑞_𝑖𝑤_𝑖𝑖=1,…,𝑁inΩ, (2.6) where 𝜂_𝑖(𝑥)=𝐼_𝑖(𝜓_𝑖(𝑥)). Thus the pair (𝐮,𝐰) and (𝐮,𝐰), where 𝑤_𝑖=𝐼_𝑖(𝑢_𝑖) and 𝑤_𝑖=𝐼(𝑢_𝑖), satisfy the inequalities𝐷_𝑖𝑢_𝑖⁻¹𝜕𝑤_𝑖𝑎𝜕𝑡−∇⋅_𝑖∇𝑤_𝑖+𝐛_𝐢⋅∇𝑤_𝑖≥𝑓_𝑖⋅,𝑢_𝑖,[𝐮]_𝑎𝑖,[𝐮]_𝑏𝑖𝐷in𝑄,_𝑖𝑢_𝑖⁻¹𝜕𝑤_𝑖𝑎𝜕𝑡−∇⋅_𝑖∇𝑤_𝑖+𝐛_𝐢⋅∇𝑤_𝑖≤𝑓_𝑖⋅,𝑢_𝑖,[𝐮]_𝑎𝑖,[𝐮]_𝑏𝑖in𝑄,𝜕𝑤_𝑖𝜕𝜈≥𝑔_𝑖⋅,𝑢_𝑖,[𝐮]_𝑐𝑖,[𝐮]_𝑑𝑖on𝑆,𝜕𝑤_𝑖𝜕𝜈≤𝑔_𝑖⋅,𝑢_𝑖,[𝐮]_𝑐𝑖,[𝐮]_𝑑𝑖on𝑆,𝑤_𝑖(0,𝑥)≥𝜂_𝑖(𝑥)𝑤_𝑖(0,𝑥)≤𝜂_𝑖(𝑥)inΩ,𝑖=1,…,𝑁, (2.7) is referred to as coupled upper and lower solutions of (2.6). For a given pair of coupled upper and lower solutions 𝐮, 𝐮 we setΛ_𝑖=𝑢_𝑖∈𝐶𝑄∶𝑢_𝑖≤𝑢≤𝑢_𝑖,Λ=𝐮∈𝒞𝑄∶𝐮≤𝐮≤𝐮,Λ×Λ=(𝐮,𝐰)∈𝒞𝑄×𝒞𝑄∶(𝐮,𝐰)≤(𝐮,𝐰)≤(𝐮,𝐰). (2.8) In Hypothesis (𝐻)-(ii) we allow 𝐷_𝑖(0)=0 which leads to a degenerate diffusion coefficient. If 𝐷_𝑖(0)=0, we set 𝑢_𝑖≥𝛿_𝑖>0, which ensures that 𝐷_𝑖(𝑢_𝑖) has a positive lower bound. Since (𝐻)-(iii), there exist smooth nonnegative functions 𝑐_𝑖^(𝑙)≡𝑐_𝑖^(𝑙)(𝑡,𝑥),𝑙=1,2, such that𝑐_𝑖⁽¹⁾𝐷_𝑖𝑢_𝑖+𝜕𝑓_𝑖𝜕𝑢_i(⋅,𝐮)≥0,𝑐_𝑖⁽²⁾𝐷_𝑖𝑢_𝑖+𝜕𝑔_𝑖𝜕𝑢_𝑖(⋅,𝐮)≥0for𝐮∈Λ. (2.9) In fact, it suffices to choose any 𝐜^(𝟏)≡(𝑐₁⁽¹⁾,…,𝑐_𝑁⁽¹⁾), 𝐜^(𝟐)≡(𝑐₁⁽²⁾,…,𝑐_𝑁⁽²⁾) satisfying𝑐_𝑖⁽¹⁾(𝑡,𝑥)≥max−𝜕𝑓_𝑖/𝜕𝑢_𝑖(𝑡,𝑥,𝐮)𝐷_𝑖𝑢_𝑖,𝑐∶𝐮∈Λ_𝑖(2)(𝑡,𝑥)≥max−𝜕𝑔_𝑖/𝜕𝑢_𝑖(𝑡,𝑥,𝐮)𝐷_𝑖𝑢_𝑖∶𝐮∈Λ. (2.10) Define for each 𝑖=1,…,𝑁,𝐹_𝑖(𝑡,𝑥,𝐮)=𝑐_𝑖⁽¹⁾(𝑡,𝑥)𝐼_𝑖𝑢_𝑖+𝑓_𝑖(𝑡,𝑥,𝐮),𝐺_𝑖(𝑡,𝑥,𝐮)=𝑐_𝑖⁽²⁾(𝑡,𝑥)𝐼_𝑖𝑢_𝑖+𝑔_𝑖(),𝐿𝑡,𝑥,𝐮_𝑖𝑤_𝑖=∇⋅𝑎_𝑖∇𝑤_𝑖−𝐛_𝐢∇⋅𝑤_𝑖−𝑐_𝑖⁽¹⁾(𝑡,𝑥)𝑤_𝑖,𝐵_𝑖𝑤_𝑖=𝜕w_𝑖𝜕𝜈+𝑐_𝑖⁽²⁾(𝑡,𝑥)𝑤_𝑖. (2.11) Since (2.9), (𝐻) and 𝐼^′_𝑖(𝑢_𝑖)=𝐷_𝑖(𝑢_𝑖), 𝐹_𝑖(⋅,𝐮) and 𝐺_𝑖(⋅,𝐮) possess the property𝐹_𝑖⋅,𝑣_𝑖,[𝐯]_𝑎𝑖,[𝐮]_𝑏𝑖≤𝐹_𝑖⋅,𝑢_𝑖,[𝐮]_𝑎𝑖,[𝐯]_𝑏𝑖𝐺_𝑖⋅,𝑣_𝑖,[𝐯]_𝑐𝑖,[𝐮]_𝑑𝑖≤𝐺_𝑖⋅,𝑢_𝑖,[𝐮]_𝑐𝑖,[𝐯]_𝑑𝑖,whenever𝐮≤𝐯≤𝐮≤𝐮. (2.12) Moreover, (2.6) is equivalent to𝐷_𝑖𝑢_𝑖⁻¹𝜕𝑤_𝑖𝜕𝑡−𝐿_𝑖𝑤_𝑖=𝐹_𝑖⋅,𝑢_𝑖,[𝐮]_𝑎𝑖,[𝐮]_𝑏𝑖𝐵in𝑄,_𝑖𝑤_𝑖=𝐺_𝑖⋅,𝑢_𝑖,[𝐮]_𝑐𝑖,[𝐮]_𝑑𝑖𝑤on𝑆,_𝑖(0,𝑥)=𝜂_𝑖(𝑥)𝑢inΩ,_𝑖=𝑞_𝑖𝑤_𝑖,𝑖=1,…,𝑁,inΩ. (2.13) Thus the pair (𝐮,𝐰) and (𝐮,𝐰), where 𝑤_𝑖=𝐼_𝑖(𝑢_𝑖) and 𝑤_𝑖=𝐼_𝑖(𝑢_𝑖), satisfies the inequalities𝐷_𝑖𝑢_𝑖⁻¹𝜕𝑤_𝑖𝜕𝑡−𝐿_𝑖𝑤_𝑖≥𝐹_𝑖⋅,𝑢_𝑖,[𝐮]_𝑎𝑖,[𝐮]_𝑏𝑖𝐷in𝑄,_𝑖𝑢_𝑖⁻¹𝜕𝑤_𝑖𝜕𝑡−𝐿_𝑖𝑤_𝑖≤𝐹_𝑖⋅,𝑢_𝑖,[𝐮]_𝑎𝑖,[𝐮]_𝑏𝑖𝐵in𝑄,_𝑖𝑤_𝑖≥𝐺_𝑖⋅,𝑢_𝑖,[𝐮]_𝑐𝑖,[𝐮]_𝑑𝑖𝐵on𝑆,_𝑖𝑤_𝑖≤𝐺_𝑖⋅,𝑢_𝑖,[𝐮]_𝑐𝑖,[𝐮]_𝑑𝑖on𝑆,𝑤_𝑖(0,𝑥)≥𝜂_𝑖(𝑥)𝑤_𝑖(0,𝑥)≤𝜂_𝑖(𝑥)inΩ, (2.14) are referred to coupled upper and lower solutions of (2.13).

The property (2.12) is quite useful for the construction of monotone convergent sequences. To ensure the existence of the sequence to be constructed in the iteration process (2.16) below we assume that either 𝐷_𝑖(0)>0 or 𝐷_𝑖(0)=0 for 𝑢_𝑖≥𝛿_𝑖>0. Define a modified function 𝐷_𝑖(𝑢_𝑖) by𝐷_𝑖𝑢_𝑖=𝐷_𝑖𝑢_𝑖+𝑢_𝑖−𝑢_𝑖if𝑢_𝑖>𝑢_𝑖,𝐷_𝑖𝑢_𝑖if𝑢_𝑖≤𝑢_𝑖≤𝑢_𝑖,𝐷_𝑖𝑢_𝑖+𝑢_𝑖−𝑢_𝑖if𝑢_𝑖<𝑢_𝑖. (2.15) Then by the above assumption, there exists 𝑑₀>0 such that 𝐷_𝑖(𝑢)≥𝑑₀ for all 𝑢∈ℝ.

By using 𝐮⁽⁰⁾=𝐮 and 𝐮⁽⁰⁾=𝐮 as the initial iteration we can construct sequences {𝐮^(𝑚),𝐰^(𝑚)} and {𝐮^(𝑚),𝐰^(𝑚)} from the nonlinear iteration process𝐷_𝑖𝑢_𝑖^(𝑚)−1𝜕𝑤_𝑖^(𝑚)𝜕𝑡−𝐿_𝑖𝑤_𝑖^(𝑚)=𝐹_𝑖⋅,𝑢_𝑖^(𝑚−1),𝐮^(𝑚−1)_𝑎𝑖,𝐮^(𝑚−1)_𝑏𝑖in𝑄,𝐷_𝑖𝑢_𝑖^(𝑚)−1𝜕𝑤_𝑖^(𝑚)𝜕𝑡−𝐿_𝑖𝑤_𝑖^(𝑚)=𝐹_𝑖⋅,𝑢_𝑖^(𝑚−1),𝐮^(𝑚−1)_𝑎𝑖,𝐮^(𝑚−1)_𝑏𝑖𝐵in𝑄,_𝑖𝑤_𝑖^(𝑚)=𝐺_𝑖⋅,𝑢_𝑖^(𝑚−1),𝐮^(𝑚−1)_𝑐𝑖,𝐮^(𝑚−1)_𝑑𝑖𝐵on𝑆,_𝑖𝑤_𝑖^(𝑚)=𝐺_𝑖⋅,𝑢_𝑖^(𝑚−1),𝐮^(𝑚−1)_𝑐𝑖,𝐮^(𝑚−1)_𝑑𝑖on𝑆,𝑤_𝑖^(𝑚)(0,𝑥)=𝜂_𝑖(𝑥),𝑤_𝑖^(𝑚)(0,𝑥)=𝜂_𝑖(𝑥)inΩ. (2.16) The sequences {𝐮^(𝑚),𝐰^(𝑚)} and {𝐮^(𝑚),𝐰^(𝑚)} are well defined by the existence theorem of [1]. The following lemma gives the monotone property of these sequences.

Lemma 2.2. The sequences {𝐮^(𝑚),𝐰^(𝑚)}, {𝐮^(𝑚),𝐰^(𝑚)} governed by (2.16) possess the monotone property (𝐮,𝐰)≤𝐮^(𝐦),𝐰^(𝐦)≤𝐮^(𝐦+𝟏),𝐰^(𝐦+𝟏)≤𝐮^(𝐦+𝟏),𝐰^(𝐦+𝟏)≤𝐮^(𝐦),𝐰^(𝐦)≤(𝐮,𝐰)form=1,2,…. (2.17) Moreover, for each 𝑚=1,2,…,𝐮^(𝐦) and 𝐮^(𝐦) are coupled upper and lower solutions of (1.1).

Proof. Let 𝑧_𝑖⁽¹⁾=𝑤_𝑖⁽¹⁾−𝑤_𝑖⁽⁰⁾, 𝑖=1,…,𝑁. Then by (2.14) and (2.16), 𝑧_𝑖⁽¹⁾ satisfies 𝐷_𝑖𝑢_𝑖⁽¹⁾⁻¹𝜕𝑧_𝑖⁽¹⁾𝜕𝑡−𝐿_𝑖𝑧_𝑖⁽¹⁾=𝐹_𝑖⋅,𝑢_𝑖⁽⁰⁾,𝐮⁽⁰⁾_𝑎𝑖,𝐮⁽⁰⁾_𝑏𝑖−𝐷_𝑖𝑢_𝑖⁽¹⁾⁻¹𝜕𝑤_𝑖⁽⁰⁾𝜕𝑡−𝐿_𝑖𝑤_𝑖⁽⁰⁾=𝐹_𝑖⋅,𝑢_𝑖⁽⁰⁾,𝐮⁽⁰⁾_𝑎𝑖,𝐮⁽⁰⁾_𝑏𝑖−𝐷_𝑖𝑢_𝑖⁽⁰⁾⁻¹𝜕𝑤_𝑖⁽⁰⁾𝜕𝑡−𝐿_𝑖𝑤_𝑖⁽⁰⁾−𝐷_𝑖𝑢_𝑖⁽¹⁾⁻¹−𝐷_𝑖𝑢_𝑖⁽⁰⁾⁻¹𝜕𝑤_𝑖⁽⁰⁾𝜕𝑡≥−𝐷_𝑖𝑢_𝑖⁽¹⁾⁻¹−𝐷_𝑖𝑢_𝑖⁽⁰⁾⁻¹𝜕𝑤_𝑖⁽⁰⁾.𝜕𝑡 (2.18) Since by the mean value theorem, 𝐷_𝑖𝑢_𝑖⁽¹⁾⁻¹−𝐷_𝑖𝑢_𝑖⁽⁰⁾⁻¹=−𝐷^′_𝑖𝜉⁽⁰⁾𝐷_𝑖𝜉⁽⁰⁾²𝑢_𝑖⁽¹⁾−𝑢_𝑖⁽⁰⁾=−𝐷^′_𝑖𝜉⁽⁰⁾𝐷_𝑖𝜉⁽⁰⁾³𝑤_𝑖⁽¹⁾−𝑤_𝑖⁽⁰⁾, (2.19) for some intermediate value 𝜉⁽⁰⁾≡𝜉⁽⁰⁾(𝑡,𝑥) between 𝑢_𝑖⁽⁰⁾ and 𝑢_𝑖⁽¹⁾, we have 𝐷_𝑖𝑢_𝑖⁽¹⁾⁻¹𝜕𝑧_𝑖⁽¹⁾𝜕𝑡−𝐿_𝑖𝑧_𝑖⁽¹⁾+𝛾⁽⁰⁾𝑧_𝑖⁽¹⁾≥0, (2.20) where 𝛾⁽⁰⁾=−𝐷^′_𝑖𝜉⁽⁰⁾𝐷_𝑖𝜉⁽⁰⁾³𝜕𝑤_𝑖⁽⁰⁾.𝜕𝑡 (2.21) Since (2.14), the boundary and initial inequalities 𝐵_𝑖𝑧_𝑖⁽¹⁾=𝐺_𝑖⋅,𝑢_𝑖⁽⁰⁾,𝐮⁽⁰⁾_𝑐𝑖,𝐮⁽⁰⁾_𝑑𝑖−𝐵_𝑖𝑤_𝑖𝑧≥0on𝑆,_𝑖(1)(0,𝑥)=𝜂_𝑖(𝑥)−𝜂_𝑖(𝑥)=0inΩ. (2.22) In view of the definition of 𝐷_𝑖 in (2.15), the function 𝐷_𝑖(𝑢_𝑖⁽¹⁾)𝛾⁽⁰⁾ of (2.20) is bounded. From the weak maximum principle, it follows 𝑧_𝑖⁽¹⁾≥0 on 𝑄. This gives 𝑤_𝑖⁽¹⁾≥𝑤_𝑖⁽⁰⁾ and thus 𝑢_𝑖⁽¹⁾≥𝑢_𝑖⁽⁰⁾. A similar argument yields 𝑤_𝑖⁽¹⁾≤𝑤_𝑖⁽⁰⁾ and 𝑢_𝑖⁽¹⁾≤𝑢_𝑖⁽⁰⁾.
Moreover, letting 𝑧_𝑖⁽¹⁾=𝑤_𝑖⁽¹⁾−𝑤_𝑖⁽¹⁾, by (2.12), (2.16), and after the similar above argument 𝐷_𝑖𝑢_𝑖⁽¹⁾⁻¹𝜕𝑧_𝑖⁽¹⁾𝜕𝑡−𝐿_𝑖𝑧_𝑖⁽¹⁾+𝛾_𝑖⁽⁰⁾𝑧_𝑖⁽¹⁾=𝐹_𝑖⋅,𝑢_𝑖⁽⁰⁾,𝐮⁽⁰⁾_𝑎𝑖,𝐮⁽⁰⁾_𝑏𝑖−𝐹_𝑖⋅,𝑢_𝑖⁽⁰⁾,𝐮⁽⁰⁾_𝑎𝑖,𝐮⁽⁰⁾_𝑏𝑖𝐵≥0in𝑄,_𝑖𝑧_𝑖⁽¹⁾=𝐺_𝑖⋅,𝑢_𝑖⁽⁰⁾,𝐮⁽⁰⁾_𝑐𝑖,𝐮⁽⁰⁾_𝑑𝑖−𝐺_𝑖⋅,𝑢_𝑖⁽⁰⁾,𝐮⁽⁰⁾_𝑐𝑖,𝐮⁽⁰⁾_𝑑𝑖𝑧≥0on𝑆,_𝑖(1)(0,𝑥)=𝜂_𝑖(𝑥)−𝜂_𝑖(𝑥)=0inΩ, (2.23) where 𝛾_𝑖⁽⁰⁾=−𝐷^′_𝑖𝜉_𝑖⁽⁰⁾𝐷_𝑖𝜉_𝑖⁽⁰⁾³𝜕𝑤_𝑖⁽⁰⁾,𝜕𝑡 (2.24) for some intermediate value 𝜉_𝑖⁽⁰⁾≡𝜉_𝑖⁽⁰⁾(𝑡,𝑥) between 𝑢_𝑖⁽⁰⁾ and   𝑢_𝑖⁽¹⁾. It follows again from the weak maximum principle that 𝐰⁽¹⁾≥𝐰⁽¹⁾ and thus 𝐮⁽¹⁾≥𝐮⁽¹⁾. The above conclusions show that 𝐮⁽⁰⁾,𝐰⁽⁰⁾≤𝐮⁽¹⁾,𝐰⁽¹⁾≤𝐮⁽¹⁾,𝐰⁽¹⁾≤𝐮⁽⁰⁾,𝐰⁽⁰⁾. (2.25)
Now we show that 𝐮⁽¹⁾ and 𝐮⁽¹⁾ are coupled upper and lower solutions of (1.1). Since (2.25), 𝐷_𝑖(𝑢_𝑖⁽¹⁾)=𝐷_𝑖(𝑢_𝑖⁽¹⁾) for 𝑖=1,…,𝑁. It suffices to show that 𝐮⁽¹⁾ and 𝐮⁽¹⁾ satisfy (2.14). Since (2.12) and (2.16), we have 𝐷_𝑖𝑢_𝑖⁽¹⁾⁻¹𝜕𝑤_𝑖⁽¹⁾𝜕𝑡−𝐿_𝑖𝑤_𝑖⁽¹⁾=𝐹_𝑖⋅,𝑢_𝑖⁽⁰⁾,𝐮⁽⁰⁾_𝑎𝑖,𝐮⁽⁰⁾_𝑏𝑖≥𝐹_𝑖⋅,𝑢_𝑖⁽¹⁾,𝐮⁽¹⁾_𝑎𝑖,𝐮⁽¹⁾_𝑏𝑖,𝐷_𝑖𝑢_𝑖⁽¹⁾⁻¹𝜕𝑤_𝑖⁽¹⁾𝜕𝑡−𝐿_𝑖𝑤_𝑖⁽¹⁾=𝐹_𝑖⋅,𝑢_𝑖⁽⁰⁾,𝐮⁽⁰⁾_𝑎𝑖,𝐮⁽⁰⁾_𝑏𝑖≤𝐹_𝑖⋅,𝑢_𝑖⁽¹⁾,𝐮⁽¹⁾_𝑎𝑖,𝐮⁽¹⁾_𝑏𝑖,𝐵_𝑖𝑢_𝑖⁽¹⁾=𝐺_𝑖⋅,𝑢_𝑖⁽⁰⁾,𝐮⁽⁰⁾_𝑐𝑖,𝐮⁽⁰⁾_𝑑𝑖≥𝐺_𝑖⋅,𝑢_𝑖⁽¹⁾,𝐮⁽¹⁾_𝑐𝑖,𝐮⁽¹⁾_𝑑𝑖,𝐵_𝑖𝑢_𝑖⁽¹⁾=𝐺_𝑖⋅,𝑢_𝑖⁽⁰⁾,𝐮⁽⁰⁾_𝑐𝑖,𝐮⁽⁰⁾_𝑑𝑖≤𝐺_𝑖⋅,𝑢_𝑖⁽¹⁾,𝐮⁽¹⁾_𝑐𝑖,𝐮⁽¹⁾_𝑑𝑖,𝑢_𝑖⁽¹⁾(0,𝑥)=𝜂_𝑖(𝑥)𝑢_𝑖⁽¹⁾(0,𝑥)=𝜂_𝑖(𝑥). (2.26)
Next we use an induction method. We assume that 𝐮^(𝑚) and 𝐮^(𝑚) are coupled upper and lower solutions of (1.1) and satisfying the following relation: (𝐮,𝐰)≤𝐮^(𝐦),𝐰^(𝐦)≤𝐮^(𝐦),𝐰^(𝐦)≤(𝐮,𝐰). (2.27) Then by choosing 𝐮^(𝑚) and 𝐮^(𝑚) as the coupled upper and lower solutions 𝐮 and 𝐮, after the similar above argument, we have 𝐮^(𝑚),𝐰^(𝑚)≤𝐮^(𝑚+1),𝐰^(𝑚+1)≤𝐮^(𝑚+1),𝐰^(𝑚+1)≤𝐮^(𝑚),𝐰^(𝑚), (2.28)𝐮^(𝑚+1) and 𝐮^(𝑚+1) are coupled upper and lower solutions of (1.1). The conclusion of the lemma follows from the induction principle.

Theorem 2.3. Let 𝐮, 𝐮 be a pair of coupled upper and lower solutions of (1.1), and let hypothesis (𝐻) hold. Assume that either 𝐷_𝑖(0)>0 for some 𝑖 or 𝑢_𝑖≥𝛿_𝑖>0. Then the problem (1.1) has at least one solution 𝐮∈Λ.

Proof. We first consider the problem (2.13), where 𝐷_𝑖 is replaced by 𝐷_𝑖. For each 𝑖=1,…,𝑁, we define operators ℒ_𝑖∶𝒟_𝑖×𝒟_𝑖→ℛ_𝑖 and ℒ∶𝒟×𝒟→ℛ by ℒ_𝑖𝑢_𝑖,𝑤_𝑖=𝐷_𝑖𝑢_𝑖⁻¹𝜕𝑤_𝑖𝜕𝑡−𝐿_𝑖𝑤_𝑖(),ℒ()=ℒ𝑖=1,…,𝑁𝐮,𝐰₁𝑢₁,𝑤₁,…,ℒ_𝑁𝑢_𝑁,𝑤_𝑁, (2.29) where 𝒟_𝑖=𝑢_𝑖∈𝐶^{1+𝛼/2,2+𝛼}(𝑄);𝑢_𝑖(0,𝑥)=𝜓_𝑖(𝑥),ℛinΩ_𝑖=𝑢_𝑖∈𝐶^𝛼/2,𝛼(𝑄),𝒟=𝒟₁×…×𝒟_𝑁,ℛ=ℛ₁×…×ℛ_𝑁. (2.30) Define also ℱ(𝐮)=𝐹₁⋅,𝑢₁,[𝐮]_𝑎𝑖,[𝐮]_𝑏𝑖,…,𝐹_𝑁⋅,𝑢_𝑁,[𝐮]_𝑎𝑖,[𝐮]_𝑏𝑖. (2.31) Then the system (2.13), in which 𝐷_𝑖 is replaced by 𝐷_𝑖, may be written in the form ℒ(𝐮,𝐰)=ℱ(𝐮),𝐁(𝐰)=𝐺(𝐮),(𝐮,𝐰)∈𝒟×𝒟, (2.32) where 𝐁=(𝐵₁,…,𝐵_𝑁) and 𝐆=(𝐺₁,…,𝐺_𝑁) are given in (2.11). Given any 𝐯∈Λ and any 𝑖=1,…,𝑁, we consider the scalar problem ℒ_𝑖𝑢_𝑖,𝑤_𝑖=ℱ_𝑖(𝐯)in𝑄,𝐵_𝑖𝑤_𝑖=𝐺_𝑖(𝐯)on𝑆,𝑢_𝑖(0,𝑥)=𝜓_𝑖(𝑥)inΩ. (2.33) It follows from the existence theorem of [1] (Chapter V, Section 7) that (2.33) has a unique solution (𝑢^∗_𝑖,𝑤^∗_𝑖)∈𝒟_𝑖×𝒟_𝑖. In fact, the inverse ℒ_𝑖⁻¹∶ℛ_𝑖→𝒟_𝑖×𝒟_𝑖 exists and is a positive compact operator on ℛ_𝑖. This implies that the equation ℒ(𝐮,𝐰)=ℱ(𝐯)𝐁(𝐰)=𝐺(𝐯), (2.34) has a unique solution (𝐮,𝐰)=ℒ⁻¹[ℱ(𝐯)],𝐰=𝐁⁻¹𝐺(𝐯). Let 𝜒 be the closed bounded convex sunset given by 𝜒=(𝐮,𝐰)∈ℛ×ℛ∶𝐮≤𝐮≤𝐮,𝐰≤𝐰≤𝐰. (2.35) By the compact property on ℒ⁻¹ and the hypothesis on 𝐟 the operator ℒ⁻¹ℱ is compact on 𝜒. We show that ℒ⁻¹ℱ maps 𝜒 to itself.
Let (𝐯,𝐳)∈𝜒 be given, and (𝐮,𝐰)=ℒ⁻¹[ℱ(𝐯)]. After the similar argument of the proof of Lemma 2.2, we conclude 𝐮≤𝐮≤𝐮, therefore ℒ⁻¹ℱ maps 𝜒 to itself. It follows from the Schauder fixed point theorem that (2.13) with 𝐷_𝑖 being replaced by 𝐷_𝑖 has at least one solution 𝐮∈𝜒. Since 𝐮≤𝐮≤𝐮, it follows from (2.15) that 𝐷_𝑖(𝑢_𝑖)=𝐷_𝑖(𝑢_𝑖) for 𝑖=1,…,𝑁. Thus 𝐮 is also the solution of (2.13). Therefore the existence of the solution to (1.1) is proved.

3. Applications

As an application of the results obtained in the previous section we consider a Lotka-Volterra predator model. This model involves two species 𝑢₁ and 𝑢₂ that are governed by the system𝜕𝑢₁𝐷𝜕𝑡−∇⋅₁𝑢₁∇𝑢₁=𝑢₁𝑎₁−𝑏₁₁𝑢₁−𝑏₁₂𝑢₂(𝑡>0,𝑥∈Ω)𝜕𝑢₂𝐷𝜕𝑡−∇⋅₂𝑢₂∇𝑢₂=𝑢₂𝑎₂+𝑏₂₁𝑢₁−𝑏₂₂𝑢₂(𝑡>0,𝑥∈Ω),𝜕𝑢₁𝜕𝜈+𝛽₁𝑢₁=0,𝜕𝑢₂𝜕𝜈+𝛽₂𝑢₂(),𝑢=0𝑡>0,𝑥∈𝜕Ω_𝑖(0,𝑥)=𝜓_𝑖(𝑥)𝑖=1,2,(𝑥∈Ω), (3.1) where 𝑎_𝑖,𝑏_𝑖𝑗 are positive constants 𝛽_𝑖≡𝛽_𝑖(𝑥)≥0∈𝜕Ω, the initial functions 𝜓_𝑖(𝑥) for 𝑖=1,2 have a positive lower bound. The density-dependent diffusion coefficients 𝐷₁(0)=𝐷₂(0)=0.

It is easy to verify that if (𝑢₁,𝑢₂) and (𝑢₁,𝑢₂) satisfy (𝑢₁,𝑢₂)≥(𝑢₁,𝑢₂) and the following inequalities:𝜕𝑢₁𝐷𝜕𝑡−∇⋅₁𝑢₁∇𝑢₁≥𝑢₁𝑎₁−𝑏₁₁𝑢₁−𝑏₁₂𝑢₂,𝜕𝑢₂𝐷𝜕t−∇⋅₂𝑢₂∇𝑢₂≥𝑢₂𝑎₂+𝑏₂₁𝑢₁−𝑏₂₂𝑢₂,𝜕𝑢₁𝐷𝜕𝑡−∇⋅₁𝑢₁∇𝑢₁≤𝑢₁𝑎₁−𝑏₁₁𝑢₁−𝑏₁₂𝑢₂,𝜕𝑢₂𝐷𝜕𝑡−∇⋅₂𝑢₂∇𝑢₂≤𝑢₂𝑎₂+𝑏₂₁𝑢₁−𝑏₂₂𝑢₂,𝜕𝑢₁𝜕𝜈+𝛽₁𝑢₁≥0,𝜕𝑢₂𝜕𝜈+𝛽₂𝑢₂≥0,𝜕𝑢₁𝜕𝜈+𝛽₁𝑢₁≤0,𝜕𝑢₂𝜕𝜈+𝛽₂𝑢₂≤0,𝑢_𝑖(0,𝑥)≤𝜓_𝑖(𝑥)≤𝑢_𝑖(0,𝑥),𝑖=1,2, (3.2) then the pair (𝑢₁,𝑢₂),(𝑢₁,𝑢₂) are coupled upper and lower solutions of (3.1).

To guarantee (3.2), we seek such a pair in the form𝑢₁,𝑢₂=𝑀₁,𝑀₂,𝑢₁,𝑢₂=𝑞₁𝛿₁𝜙₁,𝑞₂𝛿₂𝜙₂, (3.3) where for each 𝑖=1,2,  𝑀_𝑖 and 𝛿_𝑖 are positive constants to be chosen, 𝑞_𝑖 is the inverse of (2.3), and 𝜙_𝑖 is the (normalized) positive eigenfunction corresponding to the smallest eigenvalue of the eigenvalue problem∇²𝜙_𝑖+𝜆_𝑖𝜙=0inΩ,𝜕𝜙_𝑖𝜕𝜈+𝛾_𝑖𝜙_𝑖=0on𝜕Ω(𝑖=1,2). (3.4) The constant 𝛾_𝑖>0 will be determined in the following discussion. If we set𝑀₁=𝑎₁𝑏₁₁𝑀₂=𝑎₂+𝑏₂₁𝑀₁𝑏₂₂, (3.5) then the first and second inequalities of (3.2) are satisfied. The third and fourth inequalities become−∇²𝛿₁𝜙₁≤𝑞₁𝛿₁𝜙₁𝑎₁−𝑏₁₁𝑞₁𝛿₁𝜙₁−𝑏₁₂𝑀₂,−∇²𝛿₂𝜙₂≤𝑞₂𝛿₂𝜙₂𝑎₂+𝑏₂₁𝑞₁𝛿₁𝜙₁−𝑏₂₂𝑞₂𝛿₂𝜙₂. (3.6) By (3.4) and 𝑞_𝑖(𝛿_𝑖𝜙_𝑖)>0, the above inequalities are satisfied by some sufficiently small 𝛿_𝑖>0 if𝜆₁<𝑞₁𝛿₁𝜙₁𝛿₁𝜙₁𝑎₁−𝑏₁₂𝑀₂,𝜆₂<𝑞₂𝛿₂𝜙₂𝛿₂𝜙₂𝑎₂. (3.7) Since 𝐷_𝑖(0)=0, by L’Hopital’s rule,lim_𝑤→0+𝑞_𝑖(𝑤)𝑤=lim_𝑤→0+𝑞^′_𝑖(𝑤)=lim_𝑧→0+1𝐷_𝑖(𝑧)=∞, (3.8) we see that there exists 𝛿^∗_𝑖>0 such that the inequalities in (3.7) are satisfied by every 𝛿_𝑖≤𝛿^∗_𝑖 if we impose the condition𝑀₂<𝑎₁𝑏₁₂. (3.9) By (3.3), the fifth inequalities of (3.2) are trivially satisfied, and the sixth inequalities of (3.2) become𝐷₁𝑞₁𝛿₁𝜙₁𝜕𝑞₁𝛿₁𝜙₁𝜕𝜈+𝛽₁𝐷₁𝑞₁𝛿₁𝜙₁𝑞₁𝛿₁𝜙₁𝐷≤0,₂𝑞₂𝛿₂𝜙₂𝜕𝑞₂𝛿₂𝜙₂𝜕𝜈+𝛽₂𝐷₂𝑞₂𝛿₂𝜙₂𝑞₂𝛿₂𝜙₂≤0, (3.10) Substituting (3.4) into (3.10) yields−𝛾₁𝛿₁𝜙₁≤−𝛽₁𝐷₁𝑞₁𝛿₁𝜙₁𝑞₁𝛿₁𝜙₁,−𝛾₂𝛿₂𝜙₂≤−𝛽₂𝐷₂𝑞₂𝛿₂𝜙₂𝑞₂𝛿₂𝜙₂, (3.11) It is obvious that the above relations hold for any 𝛾_𝑖≥0 if 𝛽_𝑖(𝑥)≡0. In the general case 𝛽_𝑖(𝑥)≢0 the relations 𝛿₁𝜙₁=𝐼₁(𝑢₁) and 𝛿₂𝜙₂=𝐼₂(𝑢₂), where 𝐼_𝑖 is defined in (2.3), implies that (3.11) is satisfied if𝛾₁≥𝛽₁𝑢₁𝐷₁𝑢₁𝐼₁𝑢₁,𝛾₂≥𝛽₂𝑢₂𝐷₂𝑢₂𝐼₂𝑢₂. (3.12) Since 𝐼^′_𝑖(𝑧)=𝐷_𝑖(𝑧),lim_𝑧→0+𝑧𝐷_𝑖(𝑧)𝐼_𝑖(𝑧)=lim_𝑧→0+𝐷_𝑖(𝑧)+𝑧𝐷^′_𝑖(𝑧)𝐷_𝑖(𝑧)=1+lim_𝑧→0+𝑧𝐷^′_𝑖(𝑧)𝐷_𝑖(𝑧). (3.13) If we impose the conditionlim_𝑧→0+𝑧𝐷^′_𝑖(𝑧)𝐷_𝑖(𝑧)=𝜌_𝑖, (3.14) then by setting𝛾_𝑖>𝛽_𝑖1+𝜌_𝑖, (3.15)

(3.12) is satisfied. If the below (3.16) holds, then (3.5) and (3.9) are satisfied. Thus all inequalities of (3.2) are satisfied. Directly applying Theorem 2.3, we have the following theorem.

Theorem 3.1. Suppose the initial functions 𝜓_𝑖(𝑥)≤𝑀_𝑖 in (2.8) for 𝑖=1,2. Let 𝑎₁𝑎₂>𝑏₁₁𝑏₁₂𝑏₁₁𝑏₂₂−𝑏₁₂𝑏₂₁, (3.16) and let 𝛽_𝑖(𝑥)≥0 and 𝐷₁(𝑢₁),𝐷₂(𝑢₂) satisfy (𝐻)-(𝑖𝑖) with 𝐷₁(0)=𝐷₂(0)=0. Assume that either 𝛽_𝑖(𝑥)≡0 or lim_𝑧→0+𝑧𝐷^′_𝑖(𝑧)𝐷_𝑖(𝑧)=𝜌_𝑖,𝑖=1,2, (3.17) for some constants 𝜌_𝑖. Then the system (3.1) admits at least one positive solution.

Remark 3.2. Pao and Ruan [5] have considered a Lotka-Volterra competition model with density-dependent diffusion, where the coefficient 𝑏₂₁ of the system (3.1) is negative. The difference between them is that our method does not require that the reaction functions possess the monotone nondecreasing property. The condition for the existence for the solutions of the competition model is 𝑏₁₂/𝑏₂₂<𝑎₁/𝑎₂<𝑏₁₁/𝑏₂₁, while the condition for the existence for the solutions of the predator model is 𝑎₁/𝑎₂>𝑏₁₁𝑏₁₂/(𝑏₁₁𝑏₂₂−𝑏₁₂𝑏₂₁).

Remark 3.3. In a special case 𝐷₁(𝑢₁)=(𝑚−1)𝑢₁^𝑚−1, 𝐷₂(𝑢₂)=(𝑚−1)𝑢₂^𝑚−1 for 𝑚>1, (3.1) becomes 𝜕𝑢₁𝜕𝑡−∇²𝑢^𝑚₁=𝑢₁𝑎₁−𝑏₁₁𝑢₁−𝑏₁₂𝑢₂(𝑡>0,𝑥∈Ω),𝜕𝑢₂𝜕𝑡−∇²𝑢^𝑚₂=𝑣𝑎₂+𝑏₂₁𝑢₁−𝑏₂₂𝑢₂(𝑡>0,𝑥∈Ω),𝜕𝑢₁𝜕𝜈+𝛽₁𝑢₁=0,𝜕𝑢₂𝜕𝜈+𝛽₂𝑢₂=0(𝑡>0,𝑥∈𝜕Ω). (3.18) Then the condition (3.17) is trivially satisfied. The conclusions in Theorem 3.1 hold true for (3.18). In fact, if 𝐷₁(𝑢₁)=𝐷₂(𝑢₂)=𝑑, the condition (3.17) is also trivial true, hence Theorem 2.3 is also valid. After the similar proof as Theorem 3.1, we conclude that Theorem 3.1 holds true for semilinear parabolic system.

4. Discussions

The intension of the present paper is to demonstrate the existence of solutions for the degenerate diffusion reaction system with nonlinear boundary condition. Our method is to look for the positive solution by constructing the coupled upper and lower solutions. The virtue of the technique is that it helps to extend the results for the scalar equation to the coupled system. Our existence theorem of Theorem 2.3 in this paper is applicable to various Lotka-Volterra models, such as competition, predator-prey, or mutualism model, while the method in [6] is not applicable to predator-prey model.

Since Levin and Segel illuminated the important role of the diffusion on the patterns in [7], a number of Lotka-Volterra models with constant diffusion have been investigated in the past three decades. In fact the concern of the density-dependent diffusion is also reasonable in animal disperse model (see [8] for a review). Our study is a starting attempt to consider the role of the density-dependent diffusion on Lotka-Volterra model. In biological terms, the results of Theorems 3.1 imply that if the rate of intraspecific competition of the predator is large, the two species are coexistent. The results also have applicability to 3 species model. Note that for Lotka-Volterra predator-prey model with constant diffusion, when the rate of intraspecific competition of the prey is large, the two species are both extinct. When the density-dependent diffusion is taken into account, it is an open problem whether there exist the extinct phenomena.

Acknowledgment

The work is partially supported by PRC Grant NSFC 10801115.