Analysis of Coupled Reaction-Diffusion Equations for RNA Interactions

Abstract

We consider a system of coupled reaction-diffusion equations that models the interaction between multiple types of chemical species, particularly the interaction between one messenger RNA and different types of non-coding microRNAs in biological cells. We construct various modeling systems with different levels of complexity for the reaction, nonlinear diffusion, and coupled reaction and diffusion of the RNA interactions, respectively, with the most complex one being the full coupled reaction-diffusion equations. The simplest system consists of ordinary differential equations (ODE) modeling the chemical reaction. We present a derivation of this system using the chemical master equation and the mean-field approximation, and prove the existence, uniqueness, and linear stability of equilibrium solution of the ODE system. Next, we consider a single, nonlinear diffusion equation for one species that results from the slow diffusion of the others. Using variational techniques, we prove the existence and uniqueness of solution to a boundary-value problem of this nonlinear diffusion equation. Finally, we consider the full system of reaction-diffusion equations, both steady-state and time-dependent. We use the monotone method to construct iteratively upper and lower solutions and show that their respective limits are solutions to the reaction-diffusion system. For the time-dependent system of reaction-diffusion equations, we obtain the existence and uniqueness of global solutions. We also obtain some asymptotic properties of such solutions.

1 Introduction

Let Ω be a bounded domain in ℝ³ with a smooth boundary ∂Ω. Let N ≥ 1 be an integer. Let D_i (i = 1, …, N), D, β_i (i = 1, …, N), β, and k_i (i = 1, …, N) be positive numbers. Let α_i (i = 1, …, N) and α be nonnegative functions on Ω × (0, ∞). We consider the following system of coupled reaction-diffusion equations:

$$\frac{\partial u_i}{\partial t}=D_i\mathrm{\Delta }{u}_i-{\beta }_i{u}_i-k_i{u}_i\upsilon +{\alpha }_i\text{in}\mathrm{\Omega }\times (0,\infty ),i=1,\dots ,N,$$ (1.1)

$$\frac{\partial \upsilon }{\partial t}=D\mathrm{\Delta }\upsilon -\beta \upsilon -\sum _{i=1}^N{k}_i{u}_i\upsilon +\alpha \text{in}\mathrm{\Omega }\times (0,\infty ),$$ (1.2)

together with the boundary and initial conditions

$$\frac{\partial u_i}{\partial \mathrm{n}}=\frac{\partial \upsilon }{\partial \mathrm{n}}=0\text{on}\partial \mathrm{\Omega }\times (0,\infty ),i=1,\dots ,N,$$ (1.3)

$$u_i(·,0)=u_{i0}\text{and}\upsilon (·,0)={\upsilon }_0\text{in}\mathrm{\Omega },i=1,\dots ,N,$$ (1.4)

where ∂/∂n denotes the normal derivative along the exterior unit normal n at the boundary ∂Ω, and all u_i0 (i = 1, …, N) and υ₀ are nonnegative functions on Ω.

The reaction-diffusion system (1.1)–(1.4) is a biophysical model of the interaction between different types of Ribonucleic acid (RNA) molecules, a class of biological molecules that are crucial in the coding and decoding, regulation, and expression of genes [23]. Small, non-coding RNAs (sRNA) regulate developmental events such as cell growth and tissue differentiation through binding and reacting with messenger RNA (mRNA) in a cell. Different sRNA species may competitively bind to different mRNA targets to regulate genes [4, 6, 12, 13, 16, 21]. Recent experiments suggest that the concentration of mRNA and different sRNA in cells and across tissue is linked to the expression of a gene [22]. One of the main and long-term goals of our study of the reaction-diffusion system (1.1)–(1.4) is therefore to possibly provide some insight into how different RNA concentrations can contribute to turning genes “on” or “off” across various length scales, and eventually to the gene expression.

In Eqs. (1.1) and (1.2), the function u_i = u_i(x, t) for each i (1 ≤ i ≤ N) represents the local concentration of the ith sRNA species at x ∈ Ω and time t. We assume a total of N sRNA species. The function υ = υ(x, t) represents the local concentration of the mRNA species at x ∈ Ω and time t. For each i (1 ≤ i ≤ N), D_i is the diffusion coefficient and β_i is the self-degradation rate of the ith sRNA species. Similarly, D is the diffusion coefficient and β is the self-degradation rate of mRNA. For each i (1 ≤ i ≤ N), k_i is the rate of reaction between the ith sRNA and mRNA. We neglect the interactions among different sRNA species as they can be effectively described through their diffusion and self-degradation coefficients. The reaction terms u_iυ (i = 1, …, N) result from the mean-field approximation. The nonnegative functions α_i = α_i(x, t) (i = 1, …, N) and α = α(x, t) (x ∈ Ω, t > 0) are the production rates of the corresponding RNA species, and are termed transcription profiles. Notice that we set the linear size of the region Ω to be of tissue length to account for the RNA interaction across different cells [22].

The reaction-diffusion system model (1.1)–(1.4) was first proposed for the special case N = 1 and one space dimension in [14]; cf. also [12, 15, 19]. The full model with N(≥ 2) sRNA species was proposed in [7].

An interesting feature of the reaction-diffusion system (1.1)–(1.4), first discovered in [14], is that the increase in the diffusivity (within certain range) of an sRNA species sharpens the concentration profile of mRNA. Figure 1 depicts numerically computed steady-state solutions to the system (1.1)–(1.3) in one space dimension with N = 1, Ω = (0, 1), D = 0, a few selected values of D₁, β₁ = β = 0.01, k₁ = 1, and

$${\alpha }_1=0.1+0.1\text{tan}\mathrm{h}(5x-2.5),$$ (1.5)

$$\alpha =0.1+0.1\text{tan}\mathrm{h}(2.5-5x).$$ (1.6)

One can see that as the diffusion constant D₁ of the sRNA increases, the profile of the steady-state concentration υ = υ(x) of the mRNA sharpens.

Figure 1.

Numerical solutions to the steady-state equations with the boundary conditions (1.1)–(1.3) in one space dimension with N = 1, Ω = (0, 1), D = 0, β₁ = β = 0.01, k₁ = 1, and α₁ and α given in (1.5) and (1.6), respectively. The numerically computed, steady-state concentration of mRNA υ = υ(x) (0 < x < 1) sharpens as the the diffusion constant D₁ of the sRNA increases from 0.00001 to 0.0005, 0.0001, and 0.001.

As one of a series of studies on the reaction-diffusion system modeling, analysis, and computation of the the RNA interactions, the present work focuses on: (1) the construction of various modeling systems with different levels of complexity for the reaction, nonlinear diffusion, and coupled reaction and diffusion, respectively, with the most complex one being the full reaction-diffusion system (1.1)–(1.4); and (2) the mathematical justification for each of the models, proving the well-posedness of the corresponding differential equations. To understand how the reaction terms (i.e., the product terms u_iυ in (1.1) and (1.2)) come from, we shall first, however, present a brief derivation of the corresponding reaction system (i.e., no diffusion) for the case N = 1 using a chemical master equation and the mean-field approximation [19].

We shall consider our different modeling systems in four cases.

Case 1. We consider the following system of ordinary different equations (ODE) for the concentrations u_i = u_i(t) ≥ 0 (i = 1, …, N) and υ = υ(t) ≥ 0:

$$\frac{d{u}_i}{dt}=-{\beta }_i{u}_i-k_i{u}_i\upsilon +{\alpha }_{i}i=1,\dots ,N,$$ (1.7)

$$\frac{d\upsilon }{dt}=-\beta \upsilon -\sum _{i=1}^N{k}_i{u}_i\upsilon +\alpha ,$$ (1.8)

where all α_i (i = 1, …, N) and α are nonnegative numbers. We shall prove the existence, uniqueness, and linear stability of the steady-state solution to this ODE system; cf. Theorem 3.1.

Case 2. We consider the situation where all the diffusion coefficients D_i (i = 1, …, N) are much smaller than the diffusion coefficient D. That is, we consider the approximation D_i = 0 (i = 1, …, N). Assume all α and α_i (i = 1, …, N) are independent of time t. The steady-state solution of u_i in Eq. (1.1) with D_i = 0 leads to u_i = α_i/(β_i+k_iυ) (i = 1, …, N). These expressions, coupled with the υ-equation (1.2), imply that the steady-state solution υ ≥ 0 should satisfy the following nonlinear diffusion equation and boundary condition:

$$D\mathrm{\Delta }\upsilon -\beta \upsilon -\sum _{i=1}^N\frac{k_i{\alpha }_i\upsilon }{{\beta }_i+k_i\upsilon }+\alpha =0\text{in}\mathrm{\Omega },$$ (1.9)

$$\frac{\partial \upsilon }{\partial n}=0\text{on}\partial \mathrm{\Omega }.$$ (1.10)

The single, nonlinear equation (1.9) is the Euler–Lagrange equation of some energy functional. We shall use the direct method in the calculus of variations to prove the existence and uniqueness of the nonnegative solution to the boundary-value problem (1.9) and (1.10); cf. Theorem 4.1.

Case 3. We consider the following steady-state system corresponding to (1.1)–(1.4) for the concentrations u_i ≥ 0 (i = 1, …, N) and υ ≥ 0:

$$D_i\mathrm{\Delta }{u}_i-{\beta }_i{u}_i-k_i{u}_i\upsilon +{\alpha }_i=0\text{in}\mathrm{\Omega },i=1,\dots ,N,$$ (1.11)

$$D\mathrm{\Delta }\upsilon -\beta \upsilon -\sum _{i=1}^N{k}_i{u}_i\upsilon +\alpha =0\text{in}\mathrm{\Omega },$$ (1.12)

$$\frac{\partial u_i}{\partial n}=\frac{\partial \upsilon }{\partial n}=0\text{on}\partial \mathrm{\Omega },i=1,\dots ,N.$$ (1.13)

Here again we assume that α and α_i (i = 1, …, N) are independent of time t. We shall use the monotone method [20] to prove the existence of a solution to this system of reaction-diffusion equations; cf. Theorem 5.1. The monotone method amounts to constructing sequences of upper and lower solutions, extracting convergent subsequences, and proving that the limits are desired solutions.

Case 4. This is the full reaction-diffusion system (1.1)–(1.4). We shall prove the existence and uniqueness of global solution to this system; cf. Theorem 6.1. To do so, we first consider local solutions, i.e., solutions defined on a finite time interval. Again, we use the monotone method to construct iteratively upper and lower solutions and show their limits are the desired solutions. Unlike in the case of steady-state solutions, we are not able to use high solution regularity, as that would require compatibility conditions. Rather, we use an integral representation of solution to our initial-boundary-value problem. We then use the Maximum Principle for systems of linear parabolic equations to obtain the existence and uniqueness of global solution. We also study some additional properties such as the asymptotic behavior of solutions to the full system.

While our underlying reaction-diffusion system has been proposed to model RNA interactions in molecular biology, its basic mathematical properties are similar to some of those reaction-diffusion systems modeling other physical and biological processes. Our preliminary analysis presented here therefore shares some common features in the study of reaction-diffusion systems; cf. e.g., [10, 17] and the references therein. Our continuing mathematical effort in understanding the reaction and diffusion of RNA is to analyze the qualitative properties of solutions to the corresponding equations, in particular, the asymptotic behavior of such solutions as certain parameters become very small or large.

The rest of this paper is organized as follows: In Section 2, we present a brief derivation of the reaction system (1.7) and (1.8) for the case N = 1 using a chemical master equation and the mean-field approximation. In Section 3, we consider the system of ODE (1.7) and (1.8) and prove the existence, uniqueness, and linear stability of steady-state solution. In Section 4, we prove the existence and uniqueness of the boundary-value problem of the single nonlinear diffusion equation (1.9) and (1.10) for the concentration υ of mRNA. In Section 5 we prove the existence of a steady-state solution to the system of reaction-diffusion equations (1.11)–(1.13). In Section 6, we prove the existence and uniqueness of global solution to the full system of time-dependent, reaction-diffusion equations (1.1)–(1.4). Finally, in Section 7, we prove some asymptotic properties of solutions to the full system of time-dependent reacation-diffusion equations.

2 Derivation of the Reaction System

We give a brief derivation of the reaction system (1.7) and (1.8), and make a remark on how the full reaction-diffusion system (1.1)–(1.4) is formulated.

For simplicity, we shall consider two chemical species: mRNA and one sRNA. Figure 2 describes sRNA-mediated gene silencing within the cell and depicts the different rates in which mRNA and sRNA populations may change at time t. In the figure, α_s and α_m describe the sRNA and mRNA production rates, β_s and β_m describe the sRNA and mRNA independent degradation rates, and γ describes the coupled degradation rate at time t. Notice in the rate diagram that the mRNA and sRNA binding process is irreversible. The numerical value of each of these rates can be determined via experimental data [15].

Figure 2.

Kinetic scheme of the interaction of sRNA and mRNA in a cell. Here, α_s and α_m represent the production rates of sRNA and mRNA, respectively; β_s and β_m represent the independent degradation rates of sRNA and mRNA, respectively; and γ represents the coupled degradation rate of sRNA and mRNA.

We denote by M_t and S_t the numbers of mRNA and sRNA, respectively, in a given cell at time t, and consider the two continuous-time processes (M_t)_t≥0 and (S_t)_t≥0. We assume that (M_t, S_t)_t≥0 is a stationary continuous-time Markov chain with state space S with the following ordering:

$$S=\{(0,0),\dots ,(m-1,s-1),(m-1,s),(m,s-1),(m,s),(m,s+1),\dots \}.$$

We assume that the total numbers of mRNA and sRNA are finite, and hence the state space is finite. For any given state (m, s) ∈ S, we denote by P_m,s(t) the probability that the system is in this state, i.e., P_m,s(t) = P(M_t = m, S_t = s). For convenience, we extend S to include integer pairs (m, s) for m < 0 or s < 0 and set P_m,s(t) = 0 if m < 0 or s < 0. Note that

$$\sum _{(m,s)\in S}{P}_{m,s}(t)=1$$ (2.1)

for any t ≥ 0. Note also that the averages 〈M_t〉, 〈S_t〉, and 〈M_tS_t〉 are defined by

$$\langle M_t\rangle =\sum _{(m,s)\in S}m{P}_{m,s}(t),\langle S_t\rangle =\sum _{(m,s)\in S}s{P}_{m,s}(t),\text{and}\langle M_t{S}_t\rangle =\sum _{(m,s)\in S}ms{P}_{m,s}(t).$$ (2.2)

The following master equation describes the reactions defined in Figure 2:

$${Ṗ}_{m,s}(t)={\alpha }_m{P}_{m-1,s}(t)+{\alpha }_s{P}_{m,s-1}(t)+{\beta }_s(s+1)P_{m,s+1}(t)+{\beta }_m(m+1)P_{m+1,s}(t)+\gamma (m+1)(s+1)P_{m+1,s+1}(t)-({\alpha }_m+{\alpha }_s+{\beta }_{m}m+{\beta }_{s}s+\gamma ms)P_{m,s}(t),$$

where a dot denotes the time derivative. Using this and (2.2), we obtain by a series of calculations that

$$\frac{d}{dt}\langle M_t\rangle =\sum _{(m,s)\in S}m{Ṗ}_{m,s}(t)={\alpha }_m{A}_m+{\alpha }_s{A}_s+{\beta }_m{B}_m+{\beta }_s{B}_s+\gamma C,$$ (2.3)

where

$$A_m=\sum _{(m,s)\in S}m{P}_{m-1,s}(t)-\sum _{(m,s)\in S}m{P}_{m,s}(t),A_s=\sum _{(m,s)\in S}m{P}_{m,s-1}(t)-\sum _{(m,s)\in S}m{P}_{m,s}(t),B_m=\sum _{(m,s)\in S}m(m+1)P_{m+1,s}(t)-\sum _{(m,s)\in S}{m}^2{P}_{m,s}(t),B_s=\sum _{(m,s)\in S}m(s+1)P_{m,s+1}(t)-\sum _{(m,s)\in S}ms{P}_{m,s}(t),C=\sum _{(m,s)\in S}m(m+1)(s+1)P_{m+1,s+1}(t)-\sum _{(m,s)\in S}{m}^{2}s{P}_{m,s}(t).$$

By our convention that P_m,s(t) = 0 if m < 0 or s < 0, we have by the change of index m − 1 → m and (2.1) that

$$A_m=\sum _{(m+1,s)\in S}(m+1)P_{m,s}(t)-\sum _{(m,s)\in S}m{P}_{m,s}(t)=\sum _{(m,s)\in S}(m+1)P_{m,s}(t)-\sum _{(m,s)\in S}m{P}_{m,s}(t)=\sum _{(m,s)\in S}{P}_{m,s}(t)=1.$$

Similarly, by changing the index s − 1 → s, we have

$$A_s=\sum _{(m,s+1)\in S}m{P}_{m,s}(t)-\sum _{(m,s)\in S}m{P}_{m,s}(t)=\sum _{(m,s)\in S}m{P}_{m,s}(t)-\sum _{(m,s)\in S}m{P}_{m,s}(t)=0.$$

Changing m + 1 → m, we obtain by (2.2) that

$$B_m=\sum _{(m-1,s)\in S}(m-1)m{P}_{m,s}(t)-\sum _{(m,s)\in S}{m}^2{P}_{m,s}(t)=\sum _{(m,s)\in S}(m-1)m{P}_{m,s}(t)-\sum _{(m,s)\in S}{m}^2{P}_{m,s}(t)=-\sum _{(m,s)\in S}m{P}_{m,s}(t)=-\langle M_t\rangle .$$

Changing s + 1 → s and noting ms = 0 when s = 0, we have

$$B_s=\sum _{(m,s-1)\in S}ms{P}_{m,s}(t)-\sum _{(m,s)\in S}ms{P}_{m,s}(t)=\sum _{(m,s)\in S}ms{P}_{m,s}(t)-\sum _{(m,s)\in S}ms{P}_{m,s}(t),=0.$$

Finally, changing m + 1 → m and s + 1 → s, we obtain by (2.2) that

$$C=\sum _{(m-1,s-1)\in S}(m-1)ms{P}_{m,s}(t)-\sum _{(m,s)\in S}{m}^{2}s{P}_{m,s}(t)=\sum _{(m,s)\in S}(m-1)ms{P}_{m,s}(t)-\sum _{(m,s)\in S}{m}^{2}s{P}_{m,s}(t)=-\sum _{(m,s)\in S}ms{P}_{m,s}(t)=-\langle M_t{S}_t\rangle .$$

Inserting all A_m, A_s, B_m, B_s and C into (2.3), we obtain

$$\frac{d}{dt}\langle M_t\rangle ={\alpha }_m-{\beta }_m\langle M_t\rangle -\gamma \langle M_t{S}_t\rangle .$$ (2.4)

Similarly, we have

$$\frac{d}{dt}\langle S_t\rangle ={\alpha }_s-{\beta }_s\langle S_t\rangle -\gamma \langle M_t{S}_t\rangle .$$ (2.5)

We now make the mean-field assumption: 〈M_tS_t〉 = 〈M_t〉 〈S_t〉. If we denote by υ(t) = 〈M_t〉 and u₁(t) = 〈S_t〉, the spatially homogeneous concentrations of mRNA and sRNA, respectively, then we obtain (1.7) (N = 1) from (2.4) and (1.8) from (2.5) (N = 1), respectively, with α₁ = α_m, β₁ = β_m, k₁ = γ, α = α_s, and β = β_s.

We remark that based on Fick’s law the spatial diffution of the underlying sRNA and mRNA molecules can be described by D_iΔu_i (i = 1, …, N) and DΔu with all D_i and D the diffusion constants, respectively [1, 11, 17]. Here we have neglected any possible and more complicated processes such as cross diffusion and anomalous diffusion. Combining these terms with the reaction system (1.7) and (1.8), we obtain the full reaction-diffusion system (1.1)–(1.4) as our mathematical model for the RNA interaction.

3 Reaction System: Steady-State Solution and Its Linear Stability

Theorem 3.1

Assume all β_i, k_i, and α_i (i = 1, …, N), and β, k, and α are positive numbers. The system of ODE

$$\frac{d{u}_i}{dt}=-{\beta }_i{u}_i-k_i{u}_i\upsilon +{\alpha }_i,i=1,\dots ,N,$$ (3.1)

$$\frac{d\upsilon }{dt}=-\beta \upsilon -\sum _{i=1}^N{k}_i{u}_i\upsilon +\alpha$$ (3.2)

has a unique equilibrium solution (u₁₀, …, u_N0, υ₀) ∈ ℝ^N+1 with all u_i0 > 0 (i = 1, …, N) and υ₀ > 0. Moreover, it is linearly stable.

Proof

If (u₁, …, u_N, υ) ∈ ℝ^N+1 is an equilibrium solution to (3.1) and (3.2) with all u_i > 0 (i = 1, …, N) and υ > 0, then $S≔{\sum }_{i=1}^N{k}_i{u}_i$ should satisfy

$$S=\sum _{i=1}^N\frac{k_i{\alpha }_i}{{\beta }_i+k_i\alpha {(\beta +S)}^{-1}},$$ (3.3)

and the solution should be given by

$$u_i=\frac{{\alpha }_i}{{\beta }_i+k_i\alpha {(\beta +S)}^{-1}}(i=1,\dots ,N)\text{and}\upsilon =\frac{\alpha }{\beta +S}.$$ (3.4)

Thus the key here is to prove that there is a unique solution S > 0 to (3.3).

Define g : [0, ∞) → ℝ by

$$g(s)=s-\sum _{i=1}^N\frac{k_i{\alpha }_i}{{\beta }_i+k_i\alpha {(\beta +s)}^{-1}}.$$ (3.5)

Clearly, g is smooth in [0, ∞), g(0) < 0, and g(+∞) = +∞. Thus F := {s ≥ 0 : g(s) ≥ 0} is nonempty, closed, and bounded below. Let s₀ = min F. Then s₀ > 0 and g(s₀) = 0. Moreover, g′(s₀) ≥ 0. By direct calculations, we have

$$g″(s)=2\alpha \sum _{i=1}^N\frac{k_i^2{\alpha }_i{\beta }_i}{{[{\beta }_i(\beta +s)+k_i\alpha ]}^3}>0\text{for all}s>0.$$

Thus g′(s) > g′(s₀) = 0 for s > s₀. Hence g(s) > g(s₀) = 0 for s > s₀. Therefore s₀ is the unique solution to g = 0 on [0, ∞).

Set now

$$u_{i0}=\frac{{\alpha }_i}{{\beta }_i+k_i\alpha {(\beta +s_0)}^{-1}},i=1,\dots ,N,$$ (3.6)

$${\upsilon }_0=\frac{\alpha }{\beta +s_0}.$$ (3.7)

Clearly, all u_i0 > 0 (i = 1, …, N) and υ_i0 > 0. Note that $S_0={\sum }_{i=1}^N{k}_i{u}_{i0}$, since g(s₀) = 0. Thus (3.7) implies which together with (3.6) further imply β_iu_i0 + k_iu_i0υ₀ = α_i (i = 1, …, N). Therefore (u₁₀, …, u_N0, υ₀) is an equilibrium solution to (3.1) and (3.2).

Assume both (u₁₀, …, u_N0, υ₀) and (ū₁₀, …, ū_N0, ῡ₀) are equilibrium solutions to (3.1) and (3.2) with all u_i0 > 0 and ū_i0 > 0 (i = 1, …, N), υ₀ > 0 and ῡ₀ > 0. Then, by (3.3) and (3.5), $S_0≔{\sum }_{i=1}^N{k}_i{u}_{i0}>0$ and ${\mathrm{S̅}}_0≔{\sum }_{i=1}^N{k}_i{ū}_{i0}>0$ satisfy g(S₀) = 0 and g(S̅₀) = 0, respectively. By the uniqueness of solution s₀ of g = 0, we then have S̅₀ = S₀ = s₀. It then follows from (3.6) and (3.7) that u_i0 = ū_i0 (i = 1, …, N) and υ₀ = ῡ₀. Therefore the equilibrium solution is unique.

The linearized system for (U₁, …, U_N, V) around the equilibrium solution (u₁₀, …, u_N0, υ₀) is given by

$$\frac{d{U}_i}{dt}=-({\beta }_i+k_i{\upsilon }_0)U_i-k_i{u}_{i0}V,i=1,\dots ,N,\frac{dV}{dt}=-\sum _{i=1}^N{k}_i{\upsilon }_0{U}_i-(\beta +\sum _{i=1}^N{k}_i{u}_{i0})V.$$

Let w = (U₁, …, U_N, V)^T, where the superscript T denotes the transpose. This system is then dw/dt = Mw, where

$$M=\left(\begin{array}{ccccc}\hfill -({\beta }_1+k_1{\upsilon }_0)\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill -k_1{u}_{10}\hfill \\ \hfill 0\hfill & \hfill -({\beta }_2+k_2{\upsilon }_0)\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill -k_2{u}_{20}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill \hfill & \hfill ⋮\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \hfill & \hfill -({\beta }_N+k_N{\upsilon }_0)\hfill & \hfill -k_N{u}_{N0}\hfill \\ \hfill -k_1{\upsilon }_0\hfill & \hfill -k_2{\upsilon }_0\hfill & \hfill \dots \hfill & \hfill -k_N{\upsilon }_0\hfill & \hfill -(\beta +{\sum }_{i=1}^N{k}_i{u}_{i0})\hfill \end{array}\right).$$

It is easy to see that M is strictly column diagonally dominant with negative diagonal entries. Gers̆gorin’s Theorem (with columns replacing rows) [8] then implies that the real part of any eigenvalue of M is negative. This leads to the desired linear stability.

4 A Single Nonlinear Diffusion Equation: Existence and Uniqueness of Solution

Theorem 4.1

Assume Ω is a bounded domain in ℝ³ with a Lipschitz-continuous boundary ∂Ω. Assume D, β, and all β_i and k_i (i = 1, …, N) are positive numbers. Assume α_i ∈ L²(Ω) with α_i ≥ 0 a.e. in Ω (i = 1, …, N) and α ∈ L²(Ω) with α ≥ 0 a.e. in Ω. Then there exists a unique weak solution υ ∈ H¹(Ω) with υ ≥ 0 a.e. in Ω to the boundary-value problem

$$D\mathrm{\Delta }\upsilon -\beta \upsilon -\sum _{i=1}^N\frac{k_i{\alpha }_i\upsilon }{{\beta }_i+k_i\upsilon }+\alpha =0\mathit{\text{in}}\mathrm{\Omega },$$ (4.1)

$$\frac{\partial \upsilon }{\partial \mathrm{n}}=0\mathit{\text{on}}\partial \mathrm{\Omega }.$$ (4.2)

The same statement is true if the Neumann boundary condition (4.2) is replaced by the Dirichlet boundary condition υ = υ₀ on ∂Ω for some υ₀ ∈ H¹(Ω) with υ₀ ≥ 0 on ∂Ω.

Proof

We prove the case with the Neumann boundary condition (4.2) as the Dirichlet boundary condition can be treated similarly. We define J : H¹(Ω) → ℝ ∪ {±∞} by

$$J[u]={\int }_{\mathrm{\Omega }}\{\frac{D}{2}|\nabla u{|}^2+\frac{\beta }{2}{u}^2+\sum _{i=1}^N\frac{{\alpha }_i{\beta }_i}{k_i}[\frac{k_i}{{\beta }_i}u-\text{ln}(1+\frac{k_i}{{\beta }_i}u\left)\right]-\alpha u\}dx,$$

where ln s = −∞ for s ≤ 0. Define g(s) = s − ln(1 + s) for s ∈ ℝ. It is easy to see that g = +∞ on (−∞,−1], and g is strictly convex and attains its unique minimum at 0 with g(0) = 0 on (−1,∞). Thus, since the term in the summation in J is (α_iβ_i/k_i)g(k_iu/β_i) ≥ 0, there exist constants C₁, C₂ ∈ ℝ with C₁ > 0 such that

$$J[u]\ge C_1{\Vert u\Vert }_{H^1(\mathrm{\Omega })}^2+C_2\forall u\in H^1(\mathrm{\Omega }).$$ (4.3)

Denote θ = inf_u∈H¹(Ω) J[u]. Clearly, θ is finite. Standard arguments [2, 5, 9] with an energy-minimizing sequence, using Fatou’s lemma to treat the lower-order terms in J, lead to the existence of u ∈ H¹(Ω) such that J[u] = θ.

Now, we prove that |u| is also a minimizer of J on H¹(Ω). In fact, we prove more generally that if w ∈ H¹(Ω) then J[|w|] ≤ J[w]. (If u = υ₀ on ∂Ω, then |u| = |υ₀| = υ₀ on ∂Ω, since υ₀ ≥ 0 on ∂Ω.) First, |w|² = w² and |∇|w|| = |∇w| a.e. in Ω. Since α is nonnegative in Ω, we also have −α|w| ≤ −αw a.e. in Ω. Consider h(s) = g(|s|) − g(s) with again g(s) = s − ln(1 + s). If s ≤ −1 then h(s) = −∞. For s ∈ (−1, 0), we have h(s) = −2s + ln(1 + s) − ln(1 − s) and h′(s) = 2s²/(1 − s²) > 0. Thus h(s) < h(0) = 0. If s ≥ 0 then h(s) = 0. Hence h(s) ≤ 0 for all s ∈ ℝ. Consequently, by the definition of J and the fact that all α ≥ 0 and α_i ≥ 0 (i = 1, …, N) a.e. in Ω, we have J[|w|] ≤ J[w].

Denote now υ = |u|. Then υ is also a minimizer of J on H¹(Ω) and υ ≥ 0 in Ω. Let η ∈ H¹(Ω) ∩ L^∞(Ω) and fix i (1 ≤ i ≤ N). It follows from the Mean-Value Theorem and Lebesgue Dominated Convergence Theorem that

$${\frac{d}{dt}|}_{t=0}{\int }_{\mathrm{\Omega }}\frac{{\alpha }_i{\beta }_i}{k_i}\text{ln}(1+\frac{k_i}{{\beta }_i}(\upsilon +t\eta ))dx={\int }_{\mathrm{\Omega }}\frac{{\alpha }_i{\beta }_i\eta }{{\beta }_i+k_i\upsilon }dx\forall \eta \in H^1(\mathrm{\Omega })\cap L^{\infty }(\mathrm{\Omega }).$$

Since υ minimizes J over H¹(Ω), we have (d/dt)|_t=0J[υ+tη] = 0 for all η ∈ H¹(Ω)∩L^∞(Ω). Standard calculations then imply that

$${\int }_{\mathrm{\Omega }}[D\nabla \upsilon ·\nabla \eta +\eta (\beta \upsilon +\sum _{i=1}^N\frac{{\alpha }_i{\beta }_i}{{\beta }_i+k_i\upsilon }-\alpha \left)\right]dx=0\forall \eta \in H^1(\mathrm{\Omega })\cap L^{\infty }(\mathrm{\Omega }).$$

Notice that 0 ≤ α_iβ_i/(β_i + k_iυ) ≤ α_i a.e. in Ω for all i = 1, …, N. Therefore, since H¹(Ω) ∩ L^∞(Ω) is dense in H¹(Ω), we can replace η ∈ H¹(Ω) ∩ L^∞(Ω) by η ∈ H¹(Ω). Consequently, the minimizer υ ∈ H¹(Ω) is a weak solution to (4.1) and (4.2).

If υ̂ ∈ H¹(Ω) is also a nonnegative weak solution to (4.1) and (4.2), then w = υ − υ̂ is a weak solution to DΔw−bw = 0 in Ω and ∂_nw = 0 on ∂Ω, where $b=\beta +{\sum }_{i=1}^N{\beta }_i{k}_i{\alpha }_i/[({\beta }_i+k_i\upsilon )({\beta }_i+k_i\mathrm{\upsilon ̂})]$ is in H¹(Ω) and b ≥ β > 0 in Ω. Therefore w = 0 a.e. in Ω. Hence the solution is unique.

We remark that the regularity of the solution υ to the boundary-value problem (4.1) and (4.2) depends on the smoothness of the domain Ω and that of the variable coefficients α_i (i = 1, …, N) and the source function α. Since the solution υ is nonnegative and the nonlinear term of υ is bounded, the regularity of υ is in fact similar to that of the solution to a linear elliptic problem. For instance, if ∂Ω is of the class C^k and all α, α_i ∈ W^k,p(Ω) (i = 1, …, N) for some nonnegative integer k and p ∈ [2, ∞), then υ ∈ W^k+2,p(Ω). If ∂Ω is of the class C^2,γ and all α, α_i ∈ C^0,γ(Ω̅) (i = 1, …, N) for some γ ∈ (0, 1), then υ ∈ C^2,γ (Ω̅).

5 Reaction-Diffusion System: Existence of Steady-State Solution

Theorem 5.1

Let Ω be a bounded domain in ℝ³. Assume all D_i, D, β_i, β, and k_i (i = 1, …, N) are positive constants. Assume all α_i and α (i = 1, …, N) are nonnegative functions on Ω.

1. Assume the boundary ∂Ω of Ω is in the class C^2,μ for some μ ∈ (0, 1). Assume also α_i, α ∈ C^0,μ (Ω̅) (i = 1, …, N). There exist u₁, …, u_N, υ ∈ C^2,μ (Ω̅) with u_i ≥ 0 (i = 1, …, N) and u ≥ 0 in Ω such that (u₁, …, u_N, υ) is a solution to the boundary-value problem

   $$D_i\mathrm{\Delta }{u}_i-{\beta }_i{u}_i-k_i{u}_i\upsilon +{\alpha }_i=0\mathit{\text{in}}\mathrm{\Omega },i=1,\dots ,N,$$ (5.1)

   $$D\mathrm{\Delta }\upsilon -\beta \upsilon -\sum _{i=1}^N{k}_i{u}_i\upsilon +\alpha =0\mathit{\text{in}}\mathrm{\Omega },$$ (5.2)

   $$\frac{\partial u_i}{\partial n}=\frac{\partial \upsilon }{\partial n}=0\mathit{\text{on}}\partial \mathrm{\Omega },i=1,\dots ,N.$$ (5.3)

2. Assume the boundary ∂Ω of Ω is in the class C² and all α_i, α ∈ L²(Ω) (i = 1, …, N). There exist u₁, …, u_N, υ ∈ H²(Ω) with u_i ≥ 0 (i = 1, …, N) and u ≥ 0 in Ω such that (u₁, …, u_N, υ) is a solution to the system of boundary-value problems (5.1)–(5.3).

We remark that we do not know if the solution is unique. Our numerical calculations indicate that the solution may not be unique. If α_i (i = 1, …, N) satisfy some additional assumptions, then we may have the solution uniqueness; see [18] (Theorem 6.2, Chapter 8).

Proof

(1) We divide our proof in five steps.

Step 1. Construction of upper solutions and lower solutions. We define ${ū}_i^{(0)}$, ῡ⁽⁰⁾, ${\mathrm{u̲}}_i^{(0)}$, υ̲⁽⁰⁾ (i = 1, …, N) to be constant functions such that:

$${ū}_i^{(0)}\ge \frac{{\Vert {\alpha }_i\Vert }_{L^{\infty }(\mathrm{\Omega })}}{{\beta }_i},{ῡ}^{(0)}\ge \frac{{\Vert \alpha \Vert }_{L^{\infty }(\mathrm{\Omega })}}{\beta },{\mathrm{u̲}}_i^{(0)}=0,{\mathrm{\upsilon ̲}}^{(0)}=0\text{in}\mathrm{\Omega },i=1,\dots ,N.$$ (5.4)

It is clear that

$$D_i\mathrm{\Delta }{ū}_i^{(0)}-{\beta }_i{ū}_i^{(0)}-k_i{ū}_i^{(0)}{\mathrm{\upsilon ̲}}^{(0)}+{\alpha }_i\le 0\text{in}\mathrm{\Omega },i=1,\dots ,N,$$ (5.5)

$$D\mathrm{\Delta }{\mathrm{\upsilon ̲}}^{(0)}-\beta {\mathrm{\upsilon ̲}}^{(0)}-\sum _{i=1}^N{k}_i{ū}_i^{(0)}{\mathrm{\upsilon ̲}}^{(0)}+\alpha \ge 0\text{in}\mathrm{\Omega },$$ (5.6)

$$\frac{\partial {ū}_i^{(0)}}{\partial n}=\frac{\partial {\mathrm{\upsilon ̲}}^{(0)}}{\partial n}=0\text{on}\partial \mathrm{\Omega },i=1,\dots ,N,$$ (5.7)

$$D_i\mathrm{\Delta }{\mathrm{u̲}}_i^{(0)}-{\beta }_i{\mathrm{u̲}}_i^{(0)}-k_i{\mathrm{u̲}}_i^{(0)}{ῡ}^{(0)}+{\alpha }_i\ge 0\text{in}\mathrm{\Omega },i=1,\dots ,N,$$ (5.8)

$$D\mathrm{\Delta }{ῡ}^{(0)}-\beta {ῡ}^{(0)}-\sum _{i=1}^N{k}_i{\mathrm{u̲}}_i^{(0)}{ῡ}^{(0)}+\alpha \le 0\text{in}\mathrm{\Omega },$$ (5.9)

$$\frac{\partial {\mathrm{u̲}}_i^{(0)}}{\partial n}=\frac{\partial {ῡ}^{(0)}}{\partial n}=0\text{on}\partial \mathrm{\Omega },i=1,\dots ,N.$$ (5.10)

Step 2. Iteration. Let

$$c=\text{max}\{\beta +\sum _{i=1}^N{k}_i{ū}_i^{(0)},{\beta }_1+k_1{ῡ}^{(0)},\dots ,{\beta }_N+k_N{ῡ}^{(0)}\}.$$ (5.11)

Define iteratively the functions ${ū}_i^{(k)}$, υ̲^(k), ${\mathrm{u̲}}_i^{(k)}$, ῡ^(k) (i = 1, …, N) for k = 1, 2, … by

$$-D_i\mathrm{\Delta }{ū}_i^{(k)}+c{ū}_i^{(k)}=c{ū}_i^{(k-1)}-{\beta }_i{ū}_i^{(k-1)}-k_i{ū}_i^{(k-1)}{\mathrm{\upsilon ̲}}^{(k-1)}+{\alpha }_i\text{in}\mathrm{\Omega },i=1,\dots ,N,$$ (5.12)

$$-D\mathrm{\Delta }{\mathrm{\upsilon ̲}}^{(k)}+c{\mathrm{\upsilon ̲}}^{(k)}=c{\mathrm{\upsilon ̲}}^{(k-1)}-\beta {\mathrm{\upsilon ̲}}^{(k-1)}-\sum _{i=1}^N{k}_i{ū}_i^{(k-1)}{\mathrm{\upsilon ̲}}^{(k-1)}+\alpha \text{in}\mathrm{\Omega },$$ (5.13)

$$\frac{\partial {ū}_i^{(k)}}{\partial n}=\frac{\partial {\mathrm{\upsilon ̲}}^{(k)}}{\partial n}=0\text{on}\partial \mathrm{\Omega },i=1,\dots ,N,$$ (5.14)

$$-D_i\mathrm{\Delta }{\mathrm{u̲}}_i^{(k)}+c{\mathrm{u̲}}_i^{(k)}=c{\mathrm{u̲}}_i^{(k-1)}-{\beta }_i{\mathrm{u̲}}_i^{(k-1)}-k_i{\mathrm{u̲}}_i^{(k-1)}{ῡ}^{(k-1)}+{\alpha }_i\text{in}\mathrm{\Omega },i=1,\dots ,N,$$ (5.15)

$$-D\mathrm{\Delta }{ῡ}^{(k)}+c{ῡ}^{(k)}=c{ῡ}^{(k-1)}-\beta {ῡ}^{(k-1)}-\sum _{i=1}^N{k}_i{\mathrm{u̲}}_i^{(k-1)}{ῡ}^{(k-1)}+\alpha \text{in}\mathrm{\Omega },$$ (5.16)

$$\frac{\partial {\mathrm{u̲}}_i^{(k)}}{\partial n}=\frac{\partial {ῡ}^{(k)}}{\partial n}=0\text{on}\partial \mathrm{\Omega },i=1,\dots ,N.$$ (5.17)

We recall that, for any constants D̂ > 0 and ĉ > 0, and any q ∈ C^0,μ(Ω̅), the standard theory of elliptic boundary-value problems guarantees the existence and uniqueness of solution w ∈ C^2,μ(Ω̅) to the boundary-value problem [5]

$$-\mathrm{D̂}\mathrm{\Delta }w+ĉw=q\text{in}\mathrm{\Omega },\frac{\partial w}{\partial n}=0\text{on}\partial \mathrm{\Omega }.$$

Moreover, there exists a constant C > 0, independent of q, such that

$${\Vert w\Vert }_{C^{2,\mu }(\mathrm{\Omega ̅})}\le C{\Vert q\Vert }_{C^{0,\mu }(\mathrm{\Omega ̅})}.$$ (5.18)

It therefore follows from (5.4) and a simple induction argument that there are unique solutions to the above boundary-values problems (5.12)–(5.17), defining our functions ${ū}_i^{(k)}$, υ̲^(k), ${\mathrm{u̲}}_i^{(k)}$, ῡ^(k), i = 1, …, N and k = 1, 2, …, all in C^2,μ(Ω̅).

Step 3. Comparison. We now prove for any k ≥ 1 that

$$0\le {\mathrm{u̲}}_i^{(0)}\le {\mathrm{u̲}}_i^{(k)}\le {\mathrm{u̲}}_i^{(k+1)}\le {ū}_i^{(k+1)}\le {ū}_i^{(k)}\le {ū}_i^{(0)}\text{in}\mathrm{\Omega ̅},i=1,\dots ,N,$$ (5.19)

$$0\le {\mathrm{\upsilon ̲}}^{(0)}\le {\mathrm{\upsilon ̲}}^{(k)}\le {\mathrm{\upsilon ̲}}^{(k+1)}\le {ῡ}^{(k+1)}\le {ῡ}^{(k)}\le {ῡ}^{(0)}\text{in}\mathrm{\Omega ̅}.$$ (5.20)

It follows from (5.4), (5.5), (5.12) with k = 1, (5.11), (5.7), and (5.14) with k = 1 that

$$-D_i\mathrm{\Delta }({ū}_i^{(0)}-{ū}_i^{(1)})+c({ū}_i^{(0)}-{ū}_i^{(1)})\ge 0\text{in}\mathrm{\Omega },i=1,\dots ,N,\frac{\partial ({ū}_i^{(0)}-{ū}_i^{(1)})}{\partial n}=0\text{on}\partial \mathrm{\Omega },i=1,\dots ,N.$$

The Maximum Principle [5] implies then ${ū}_i^{(0)}\ge {ū}_i^{(1)}$ in Ω̅ (i = 1, …, N). Similarly, we have ${\mathrm{u̲}}_i^{(0)}\le {\mathrm{u̲}}_i^{(1)}$, ῡ⁽⁰⁾ ≥ ῡ⁽¹⁾, and υ̲⁽⁰⁾ ≤ υ̲⁽¹⁾ in Ω̅ for all i = 1, …, N.

By (5.4), ${\mathrm{u̲}}_i^{(0)}\ge 0$ (i = 1, …, N) and υ̲⁽⁰⁾ ≥ 0. Next, by (5.12) with k = 1, (5.15) with k = 1, (5.11), and (5.4), we obtain

$$-D_i\mathrm{\Delta }({ū}_i^{(1)}-{\mathrm{u̲}}_i^{(1)})+c({ū}_i^{(1)}-{\mathrm{u̲}}_i^{(1)})=c({ū}_i^{(0)}-{\mathrm{u̲}}_i^{(0)})-{\beta }_i({ū}_i^{(0)}-{\mathrm{u̲}}_i^{(0)})-k_i({ū}_i^{(0)}{\mathrm{\upsilon ̲}}^{(0)}-{\mathrm{u̲}}_i^{(0)}{ῡ}^{(0)})=(c-{\beta }_i-k_i{\mathrm{\upsilon ̲}}^{(0)})({ū}_i^{(0)}-{\mathrm{u̲}}_i^{(0)})+k_i{\mathrm{u̲}}_i^{(0)}({ῡ}^{(0)}-{\mathrm{\upsilon ̲}}^{(0)})\ge 0\text{in}\mathrm{\Omega ̅},i=1,\dots ,N.$$

We also have by (5.14) and (5.17) with k = 1 that ${\partial }_n({ū}_i^{(1)}-{\mathrm{u̲}}_i^{(1)})=0$ on ∂Ω (i = 1, …, N). The Maximum Principle then implies ${ū}_i^{(1)}\ge {\mathrm{u̲}}_i^{(1)}$ in Ω̅ for all i = 1, …, N. Similarly, we have ῡ⁽¹⁾ ≥ υ̲⁽¹⁾ in Ω̅. We thus have proved

$$0\le {\mathrm{u̲}}_i^{(0)}\le {\mathrm{u̲}}_i^{(1)}\le {ū}_i^{(1)}\le {ū}_i^{(0)}\text{in}\mathrm{\Omega ̅},i=1,\dots ,N,0\le {\mathrm{\upsilon ̲}}^{(0)}\le {\mathrm{\upsilon ̲}}^{(1)}\le {ῡ}^{(1)}\le {ῡ}^{(0)}\text{in}\mathrm{\Omega ̅}.$$

Assume now k ≥ 2 and

$$0\le {\mathrm{u̲}}_i^{(0)}\le \cdots \le {\mathrm{u̲}}_i^{(k-1)}\le {ū}_i^{(k-1)}\le \cdots \le {ū}_i^{(0)}\text{in}\mathrm{\Omega ̅},i=1,\dots ,N,$$ (5.21)

$$0\le {\mathrm{\upsilon ̲}}^{(0)}\le \cdots \le {\mathrm{\upsilon ̲}}^{(k-1)}\le {ῡ}^{(k-1)}\le \cdots \le {ῡ}^{(0)}\text{in}\mathrm{\Omega ̅}.$$ (5.22)

We prove

$${\mathrm{u̲}}_i^{(k-1)}\le {\mathrm{u̲}}_i^{(k)}\le {ū}_i^{(k)}\le {ū}_i^{(k-1)}\text{in}\mathrm{\Omega ̅},i=1,\dots ,N,$$ (5.23)

$${\mathrm{\upsilon ̲}}^{(k-1)}\le {\mathrm{\upsilon ̲}}^{(k)}\le {ῡ}^{(k)}\le {ῡ}^{(k-1)}\text{in}\mathrm{\Omega ̅}.$$ (5.24)

By (5.12) with k − 1 replacing k, (5.12), (5.21), (5.22), and (5.11), we obtain

$$-D_i\mathrm{\Delta }({ū}_i^{(k-1)}-{ū}_i^{(k)})+c({ū}_i^{(k-1)}-{ū}_i^{(k)})=(c-{\beta }_i-k_i{\mathrm{\upsilon ̲}}^{(k-2)})({ū}_i^{(k-2)}-{ū}_i^{(k-1)})+k_i{ū}_i^{(k-1)}({\mathrm{\upsilon ̲}}^{(k-1)}-{\mathrm{\upsilon ̲}}^{(k-2)})\ge (c-{\beta }_i-k_i{ῡ}^{(0)})({ū}_i^{(k-2)}-{ū}_i^{(k-1)})\ge 0\text{in}\mathrm{\Omega ̅},i=1,\dots ,N.$$

We also have by the boundary conditions (5.14) that ${\partial }_n({ū}_i^{(k-1)}-{ū}_i^{(k)})=0$ on ∂Ω (i = 1, …, N). The Maximum Principle now implies ${ū}_i^{(k-1)}\ge {ū}_i^{(k)}$ in Ω̅ for all i = 1, …, N. Similarly, we have ${\mathrm{u̲}}_i^{(k-1)}\le {\mathrm{u̲}}_i^{(k)}$ in Ω̅ (i = 1, …, N). By (5.16), (5.21), (5.22), and (5.11), we have

$$-D\mathrm{\Delta }({ῡ}^{(k-1)}-{ῡ}^{(k)})+c({ῡ}^{(k-1)}-{ῡ}^{(k)})=(c-\beta -\sum _{i=1}^N{k}_i{\mathrm{u̲}}_i^{(k-2)})({ῡ}^{(k-2)}-{ῡ}^{(k-1)})+\sum _{i=1}^N{k}_i{ῡ}^{(k-1)}({\mathrm{u̲}}_i^{(k-1)}-{\mathrm{u̲}}_i^{(k-2)})\ge (c-\beta -\sum _{i=1}^N{k}_i{ū}_i^{(0)})({ῡ}^{(k-2)}-{ῡ}^{(k-1)})\ge 0\text{in}\mathrm{\Omega ̅}.$$

Since ∂_nῡ^(k−1) = ∂_nῡ^(k) on ∂Ω, we thus have ῡ^(k−1) ≥ ῡ^(k) in Ω̅. Similarly, υ̲^(k−1) ≤ υ̲^(k) in Ω̅.

By (5.12), (5.15), (5.21), (5.22), and (5.11), we have

$$-D_i\mathrm{\Delta }({ū}_i^{(k)}-{\mathrm{u̲}}_i^{(k)})+c({ū}_i^{(k)}-{\mathrm{u̲}}_i^{(k)})=(c-{\beta }_i-k_i{\mathrm{\upsilon ̲}}^{(k-1)})({ū}_i^{(k-1)}-{\mathrm{u̲}}_i^{(k-1)})+k_i{\mathrm{u̲}}_i^{(k-1)}({ῡ}^{(k-1)}-{\mathrm{\upsilon ̲}}^{(k-1)})\ge (c-{\beta }_i-{ῡ}^{(0)})({ū}_i^{(k-1)}-{\mathrm{u̲}}_i^{(k-1)})\ge 0\text{in}\mathrm{\Omega ̅},i=1,\dots ,N.$$

These and the corresponding boundary conditions for ${ū}_i^{(k)}$ and ${\mathrm{u̲}}_i^{(k)}$ lead to ${\mathrm{u̲}}_i^{(k)}\le {ū}_i^{(k)}$ in Ω̅ for all i = 1, …, N. By (5.13), (5.16), (5.21), (5.22), and (5.11), we have

$$-D\mathrm{\Delta }({ῡ}^{(k)}-{\mathrm{\upsilon ̲}}^{(k)})+c({ῡ}^{(k)}-{\mathrm{\upsilon ̲}}^{(k)})=(c-\beta -\sum _{i=1}^N{k}_i{\mathrm{u̲}}_i^{(k-1)})({ῡ}^{(k-1)}-{\mathrm{\upsilon ̲}}^{(k-1)})+\sum _{i=1}^N{k}_i{\mathrm{\upsilon ̲}}^{(k-1)}({ū}_i^{(k-1)}-{\mathrm{u̲}}_i^{(k-1)})\ge (c-\beta -\sum _{i=1}^N{k}_i{ū}_i^{(0)})({ῡ}^{(k-1)}-{\mathrm{\upsilon ̲}}^{(k-1)})\ge 0\text{in}\mathrm{\Omega ̅}.$$

This together with the fact that ∂_nῡ^(k−1) = ∂_nυ̲^(k−1) imply υ̲^(k) ≤ ῡ^(k) in Ω̅. We have proved (5.23) and (5.24). By induction, we have proved (5.19) and (5.20).

Step 4. Regularity and boundedness. From the above iteration (5.5)–(5.10), we obtain by (5.19) and (5.20) uniformly bounded sequences of nonnegative C^2,μ(Ω̅)-functions $\{{ū}_i^{(k)}\},\{{\mathrm{u̲}}_i^{(k)}\}$, {ῡ^(k)}, and {υ̲^(k)}. By standard Hölder estimates for elliptic problems, we conclude that all the sequences $\{{\Vert {ū}_i^{(k)}\Vert }_{C^{2,\mu }(\mathrm{\Omega ̅})}\}$ (1 ≤ i ≤ N), $\{{\Vert {\mathrm{u̲}}_i^{(k)}\Vert }_{C^{2,\mu }(\mathrm{\Omega ̅})}\}$ (1 ≤ i ≤ N), {‖ῡ^(k)‖_C^2,μ(Ω̅)}, and {‖υ̲^(k)‖_C^2,μ(Ω̅)} are bounded.

Step 5. Convergence to solution. From Step 4, the sequences $\{{ū}_i^{(k)}\}$ and $\{{\mathrm{u̲}}_i^{(k)}\}$ (1 ≤ i ≤ N), {ῡ^(k)}, and {υ̲^(k)} are bounded in C²(Ω̅) and pointwise monotonic on Ω̅. Therefore, they converge pointwise to some functions ū_i and u̲_i (1 ≤ i ≤ N), ῡ, and υ̲ on Ω̅, respectively. By the Arzela–Ascoli Theorem, there exist C²(Ω̅)-convergent subsequences ${\{{ū}_i^{(k_j)}\}}_{j=1}^{\infty }$ and ${\{{\mathrm{u̲}}_i^{(k_j)}\}}_{j=1}^{\infty }$ (1 ≤ i ≤ N), ${\{{ῡ}^{(k_j)}\}}_{j=1}^{\infty }$, and ${\{{\mathrm{\upsilon ̲}}^{(k_j)}\}}_{j=1}^{\infty }$, of $\{{ū}_i^{(k)}\}$ and $\{{\mathrm{u̲}}_i^{(k)}\}$ (1 ≤ i ≤ N), {υ̲^(k)}, and {ῡ^(k)}, respectively. Clearly, these subsequences converge in C²(Ω̅) to ū_i (1 ≤ i ≤ N), u̲_i (1 ≤ i ≤ N), ῡ, and υ̲, respectively. Note that each of the subsequences $\{{ū}_i^{(k_j-1)}\}$ and $\{{\mathrm{u̲}}_i^{(k_j-1)}\}$ (1 ≤ i ≤ N), {ῡ^(k_j−1)}, and {υ̲^(k_j−1)} also converges pointwise to its respective limit. Now replace k in (5.12)–(5.17) by k_j and sending j → ∞, we conclude that (ū₁, ⋯, ū_N, υ̲) and (u̲₁, ⋯, u̲_N, ῡ) are C²(Ω̅)-solutions of the system (5.1)–(5.3).

(2) This part can be proved similarly. The sequences of functions $\{{ū}_i^{(k)}\}$ and $\{{\mathrm{u̲}}_i^{(k)}\}$ (1 ≤ i ≤ N), {ῡ^(k)}, and {υ̲^(k)} can be defined as weak solutions of the corresponding boundary-value problems. Moreover, By using the estimate ‖w‖_H²(Ω) ≤ C‖q‖_L²(Ω), replacing (5.18), we have that all the sequences are bounded in H²(Ω). The monotonicity and pointwise boundedness imply that these sequences converge, respectively, to some functions ū_i and u̲_i (1 ≤ i ≤ N), ῡ, and υ̲ on Ω̅, all being nonnegative. Instead of using the Arzela–Ascoli Theorem, we use the fact that H²(Ω) is a Hilbert space and use also Sobolev compact embedding theorem to extract the subsequences ${\{{ū}_i^{(k_j)}\}}_{j=1}^{\infty }$ and ${\{{\mathrm{u̲}}_i^{(k_j)}\}}_{j=1}^{\infty }$ (1 ≤ i ≤ N), ${\{{ῡ}^{(k_j)}\}}_{j=1}^{\infty }$, and ${\{{\mathrm{\upsilon ̲}}^{(k_j)}\}}_{j=1}^{\infty }$ that converge weakly in H²(Ω) and strongly in H¹(Ω) to the pointwise limiting functions ū_i and u̲_i (1 ≤ i ≤ N), ῡ, and υ̲, respectively. Finally, we use the weak forms of the equations to pass to the limit. For instance, we have for any ϕ ∈ H¹(Ω) and all j = 1, 2, … that

$${\int }_{\mathrm{\Omega }}[D_i\nabla {ū}_i^{(k_j)}·\nabla \varphi +c{ū}_i^{(k_j)}\varphi ]dx={\int }_{\mathrm{\Omega }}[c{ū}_i^{(k_j-1)}-{\beta }_i{ū}_i^{(k_j-1)}-k_i{ū}_i^{(k_j-1)}{\mathrm{\upsilon ̲}}^{(k_j-1)}+{\alpha }_i]\varphi dx,i=1,\dots ,N.$$

Sending j → ∞, we have by the H¹(Ω)-convergence of ${\{{ū}_i^{(k_j)}\}}_{j=1}^{\infty }$ and the pointwise convergence of ${\{{ū}_i^{(k)}\}}_{k=1}^{\infty }$ and ${\{{\mathrm{\upsilon ̲}}^{(k)}\}}_{k=1}^{\infty }$ that

$${\int }_{\mathrm{\Omega }}[D_i\nabla {ū}_i·\nabla \varphi +c{ū}_i\varphi ]dx={\int }_{\mathrm{\Omega }}[c{ū}_i-{\beta }_i{ū}_i-k_i{ū}_i\mathrm{\upsilon ̲}+{\alpha }_i]\varphi dx,i=1,\dots ,N.$$

Therefore, (ū₁, …, ū_N, υ̲) and (u̲₁, ⋯, u̲_N, ῡ) are the weak (and nonnegative) solutions in H²(Ω) of the system (5.1)–(5.3).

6 Reaction-Diffusion System: Existence and Uniqueness of Global Solution to Time-Dependent Problem

Our goal in this section is to prove the existence and uniqueness of global solution for the reaction-diffusion system (1.1)–(1.4). We shall first prove the existence and uniqueness of local solution to this system. Our proof of the existence of a lcoal solution is similar to that of Theorem 5.1 on the existence of steady-state solution but involves the representation formula and regularity of solutions to linear parabolic equations. The uniqueness of a local solution is obtained by the Maximum Principle for systems of linear parabolic equations.

For any set Ω ⊆ ℝ³ and any T > 0, we denote Ω_T = Ω × (0, T]. If Ω ⊆ ℝ³ is open, we also denote by $C_1^2({\mathrm{\Omega }}_T)$ the class of functions u : Ω_T → ℝ of (x, t) that are continuously differentiable in t and twice continuously differentiable in x = (x₁, x₂, x₃) on Ω.

Theorem 6.1

(Existence and uniqueness of local solution). Let Ω be a bounded domain in ℝ³ with a C²-boundary ∂Ω. Let D_i, β_i, and k_i (i = 1, …, N), D and β, and T be all positive numbers. Let ${\alpha }_i\in C^1(\overline{{\mathrm{\Omega }}_T})$ (i = 1, …, N) and $\alpha \in C^1(\overline{{\mathrm{\Omega }}_T})$ be all nonnegative functions on $\overline{{\mathrm{\Omega }}_T}$. Assume u_i0 ∈ C¹(Ω̅) (i = 1, …, N) and υ₀ ∈ C¹(Ω̅) are all nonnegative functions on Ω̅. Then there exist unique $u_i\in C(\overline{{\mathrm{\Omega }}_T})\cap C_1^2({\mathrm{\Omega }}_T)$ (i = 1, …, N) and $\upsilon \in C(\overline{{\mathrm{\Omega }}_T})\cap C_1^2({\mathrm{\Omega }}_T)$ such that all u_i ≥ 0 (i = 1, …, N) and υ ≥ 0 in $\overline{{\mathrm{\Omega }}_T}$ and

$$\frac{\partial u_i}{\partial t}=D_i\mathrm{\Delta }{u}_i-{\beta }_i{u}_i-k_i{u}_i\upsilon +{\alpha }_i\mathit{\text{in}}{\mathrm{\Omega }}_T,i=1,\dots ,N,$$ (6.1)

$$\frac{\partial \upsilon }{\partial t}=D\mathrm{\Delta }\upsilon -\beta \upsilon -\sum _{i=1}^N{k}_i{u}_i\upsilon +\alpha \mathit{\text{in}}{\mathrm{\Omega }}_T,$$ (6.2)

$$\frac{\partial u_i}{\partial n}=\frac{\partial \upsilon }{\partial n}=0\mathit{\text{on}}\partial \mathrm{\Omega }\times (0,T],i=1,\dots ,N,$$ (6.3)

$$u_i(·,0)=u_{i0}\mathit{\text{and}}\upsilon (·,0)={\upsilon }_0\mathit{\text{in}}\mathrm{\Omega ̅},i=1,\dots ,N.$$ (6.4)

Proof

We first prove the existence of solution in four steps.

Step 1. Construction of upper solutions and lower solutions. We choose the constant functions ${ū}_i^{(0)}$, ῡ⁽⁰⁾, ${\mathrm{u̲}}_i^{(0)}$, and υ̲⁽⁰⁾ (i = 1, …, N) on $\overline{{\mathrm{\Omega }}_T}$ by

$${ū}_i^{(0)}\ge \frac{{\Vert {\alpha }_i\Vert }_{L^{\infty }({\mathrm{\Omega }}_T)}}{{\beta }_i}+{\Vert u_{i0}\Vert }_{L^{\infty }(\mathrm{\Omega })},{ῡ}^{(0)}\ge \frac{{\Vert \alpha \Vert }_{L^{\infty }({\mathrm{\Omega }}_T)}}{\beta }+{\Vert {\upsilon }_0\Vert }_{L^{\infty }(\mathrm{\Omega })},{\mathrm{u̲}}_i^{(0)}=0,{\mathrm{\upsilon ̲}}^{(0)}=0.$$

It is clear that

$$\frac{\partial {ū}_i^{(0)}}{\partial t}-D_i\mathrm{\Delta }{ū}_i^{(0)}+{\beta }_i{ū}_i^{(0)}+k_i{ū}_i^{(0)}{\mathrm{\upsilon ̲}}^{(0)}-{\alpha }_i\ge 0\text{in}{\mathrm{\Omega }}_T,i=1,\dots ,N,\frac{\partial {\mathrm{\upsilon ̲}}^{(0)}}{\partial t}-D\mathrm{\Delta }{\mathrm{\upsilon ̲}}^{(0)}+\beta {\mathrm{\upsilon ̲}}^{(0)}+\sum _{i=1}^N{k}_i{ū}_i^{(0)}{\mathrm{\upsilon ̲}}^{(0)}-\alpha \le 0\text{in}{\mathrm{\Omega }}_T,\frac{\partial {ū}_i^{(0)}}{\partial n}=0\text{and}\frac{\partial {\mathrm{\upsilon ̲}}^{(0)}}{\partial n}=0\text{on}\partial \mathrm{\Omega }\times (0,T],i=1,\dots ,N,{ū}_i^{(0)}(·,0)\ge u_{i0}\text{and}{\mathrm{\upsilon ̲}}^{(0)}(·,0)\le {\upsilon }_0\text{in}\mathrm{\Omega ̅},i=1,\dots ,N,\frac{\partial {\mathrm{u̲}}_i^{(0)}}{\partial t}-D_i\mathrm{\Delta }{\mathrm{u̲}}_i^{(0)}+{\beta }_i{\mathrm{u̲}}_i^{(0)}+k_i{\mathrm{u̲}}_i^{(0)}{ῡ}^{(0)}-{\alpha }_i\le 0\text{in}{\mathrm{\Omega }}_T,i=1,\dots ,N,\frac{\partial {ῡ}^{(0)}}{\partial t}-D\mathrm{\Delta }{ῡ}^{(0)}+\beta {ῡ}^{(0)}+\sum _{i=1}^N{k}_i{\mathrm{u̲}}_i^{(0)}{ῡ}^{(0)}-\alpha \ge 0\text{in}{\mathrm{\Omega }}_T,\frac{\partial {\mathrm{u̲}}_i^{(0)}}{\partial n}=0\text{and}\frac{\partial {ῡ}^{(0)}}{\partial n}=0\text{on}\partial \mathrm{\Omega }\times (0,T],{\mathrm{u̲}}_i^{(0)}(·,0)\le u_{i0}\text{and}{ῡ}^{(0)}(·,0)\ge {\upsilon }_0\text{in}\mathrm{\Omega ̅},i=1,\dots ,N.$$

Step 2. Iteration. Let

$$c=\text{max}\{\beta +\sum _{i=1}^N{k}_i{ū}_i^{(0)},{\beta }_1+k_1{ῡ}^{(0)},\dots ,{\beta }_N+k_N{ῡ}^{(0)}\}.$$ (6.5)

Define iteratively the functions ${ū}_i^{(k)}$, υ̲^(k), ${\mathrm{u̲}}_i^{(k)}$, and ῡ^(k) (i = 1, …, N) on Ω_T for k = 1, 2, … by

$$\frac{\partial {ū}_i^{(k)}}{\partial t}-D_i\mathrm{\Delta }{ū}_i^{(k)}+c{ū}_i^{(k)}=c{ū}_i^{(k-1)}-{\beta }_i{ū}_i^{(k-1)}-k_i{ū}_i^{(k-1)}{\mathrm{\upsilon ̲}}^{(k-1)}+{\alpha }_i\text{in}{\mathrm{\Omega }}_T,i=1,\dots ,N,$$ (6.6)

$$\frac{\partial {\mathrm{\upsilon ̲}}^{(k)}}{\partial t}-D\mathrm{\Delta }{\mathrm{\upsilon ̲}}^{(k)}+c{\mathrm{\upsilon ̲}}^{(k)}=c{\mathrm{\upsilon ̲}}^{(k-1)}-\beta {\mathrm{\upsilon ̲}}^{(k-1)}-\sum _{i=1}^N{k}_i{ū}_i^{(k-1)}{\mathrm{\upsilon ̲}}^{(k-1)}+\alpha \text{in}{\mathrm{\Omega }}_T,$$ (6.7)

$$\frac{\partial {ū}_i^{(k)}}{\partial n}=\frac{\partial {\mathrm{\upsilon ̲}}^{(k)}}{\partial n}=0\text{on}\partial \mathrm{\Omega }\times (0,T],i=1,\dots ,N,$$ (6.8)

$${ū}_i^{(k)}(·,0)=u_{i,0}\text{and}{\mathrm{\upsilon ̲}}^{(k)}(·,0)={\upsilon }_0\text{in}\mathrm{\Omega ̅},i=1,\dots ,N,$$ (6.9)

$$\frac{\partial {\mathrm{u̲}}_i^{(k)}}{\partial t}-D_i\mathrm{\Delta }{\mathrm{u̲}}_i^{(k)}+c{\mathrm{u̲}}_i^{(k)}=c{\mathrm{u̲}}_i^{(k-1)}-{\beta }_i{\mathrm{u̲}}_i^{(k-1}-k_i{\mathrm{u̲}}_i^{(k-1)}{ῡ}^{(k-1)}+{\alpha }_i\text{in}{\mathrm{\Omega }}_T,i=1,\dots ,N,$$ (6.10)

$$\frac{\partial {ῡ}^{(k)}}{\partial t}-D\mathrm{\Delta }{ῡ}^{(k)}+c{ῡ}^{(k)}=c{ῡ}^{(k-1)}-\beta {ῡ}^{(k-1)}-\sum _{i=1}^N{k}_i{\mathrm{u̲}}_i^{(k-1)}{ῡ}^{(k-1)}+\alpha \text{in}{\mathrm{\Omega }}_T,$$ (6.11)

$$\frac{\partial {\mathrm{u̲}}_i^{(k)}}{\partial n}=\frac{\partial {ῡ}^{(k)}}{\partial n}=0\text{on}\partial \mathrm{\Omega }\times (0,T],i=1,\dots ,N,$$ (6.12)

$${\mathrm{u̲}}_i^{(k)}(·,0)=u_{i0}\text{and}{ῡ}^{(k)}(·,0)={\upsilon }_0\text{in}\mathrm{\Omega ̅}.$$ (6.13)

The theory for initial-boundary-value problems of linear parabolic equations (cf. Theorem 2 in Chapter 5 of [3], or Theorem 1.2 in Chapter 2 of [18]) guarantees the existence of solutions ${ū}_i^{(1)}$, υ̲⁽¹⁾, ${\mathrm{u̲}}_i^{(1)}$, and ῡ⁽¹⁾ (i = 1, …, N) for k = 1 that are all in $C(\mathrm{\Omega ̅})\cap C_1^2({\mathrm{\Omega }}_T)$ and are all Hölder continuous in x uniformly in $\overline{{\mathrm{\Omega }}_T}$. Suppose ${ū}_i^{(k)}$, υ̲^(k), ${\mathrm{u̲}}_i^{(k)}$, and ῡ^(k) (i = 1, …, N) for k ≥ 1 exist, and are all in $C(\mathrm{\Omega ̅})\cap C_1^2({\mathrm{\Omega }}_T)$ and Hölder continuous in x uniformly in $\overline{{\mathrm{\Omega }}_T}$. Then the theory for initial-boundary-value problems of linear parabolic equations then implies that the solutions ${ū}_i^{(k+1)}$, υ̲^(k+1), ${\mathrm{u̲}}_i^{(k+1)}$, and ῡ^(k+1) (i = 1, …, N) all exist, are in $C(\mathrm{\Omega ̅})\cap C_1^2({\mathrm{\Omega }}_T)$, and are Hölder continuous in x uniformly in $\overline{{\mathrm{\Omega }}_T}$. By induction, we have for all k = 1, 2, … the existence of the solutions ${ū}_i^{(k)}$, υ̲^(k), ${\mathrm{u̲}}_i^{(k)}$, and ῡ^(k) (i = 1, …, N) in $C(\mathrm{\Omega ̅})\cap C_1^2({\mathrm{\Omega }}_T)$ that are Hölder continuous in x uniformly in $\overline{{\mathrm{\Omega }}_T}$.

In fact, there is a representation formula for our solutions. Suppose $q\in C(\overline{{\mathrm{\Omega }}_T})$ is Hölder continuous in x uniformly on $\overline{{\mathrm{\Omega }}_T}$ and suppose g ∈ C¹(Ω̅). Let $u\in C(\overline{{\mathrm{\Omega }}_T})\cap C_1^2({\mathrm{\Omega }}_T)$ satisfy

$$\frac{\partial u}{\partial t}-D\mathrm{\Delta }u+cu=q\text{in}{\mathrm{\Omega }}_T,$$ (6.14)

$$u(·,0)=g(·)\text{in}\mathrm{\Omega ̅},$$ (6.15)

$$\frac{\partial u}{\partial n}=0\text{on}\partial \mathrm{\Omega }\times (0,T].$$ (6.16)

Then we can extend g to a C¹-function on a neighborhood of Ω̅ as the boundary ∂Ω is of the class C². Hence we have the following representation of the solution to the initial-boundary-value problem (6.14)–(6.16) (cf. Section 3 of Chapter 5 of [3] and Theorem 8.3.2 of [18]):

$$u(x,t)={\int }_{\mathrm{\Omega }}\mathrm{\Gamma }(x,t;\xi ,0)g(\xi )d\xi +{\int }_0^t{\int }_{\mathrm{\Omega }}\mathrm{\Gamma }(x,t;\xi ,\tau )q(\xi ,\tau )d\xi d\tau +{\int }_0^t{\int }_{\partial \mathrm{\Omega }}\mathrm{\Gamma }(x,t;\xi ,\tau )\mathrm{\Psi }(\xi ,\tau )d{S}_{\xi }d\tau \forall (x,t)\in \mathrm{\Omega }\times (0,T],$$ (6.17)

where

$$\mathrm{\Gamma }(x,t;\xi ,\tau )={[4\pi D(t-\tau )]}^{-3/2}{e}^{-c(t-\tau )-|x-\xi {|}^2/4D(t-\tau )},$$ (6.18)

$$\mathrm{\Psi }(x,t)=2F(x,t)+2\sum _{j=1}^{\infty }{\int }_0^t{\int }_{\partial \mathrm{\Omega }}{M}_j(x,t;\xi ,\tau )F(\xi ,\tau )d{S}_{\xi }d\tau ,$$ (6.19)

$$F(x,t)={\int }_{\mathrm{\Omega }}\frac{\partial \mathrm{\Gamma }(x,t;\xi ,0)}{\partial n(x,t)}g(\xi )d\xi +{\int }_0^t{\int }_{\mathrm{\Omega }}\frac{\partial \mathrm{\Gamma }(x,t;\xi ,\tau )}{\partial n(x,t)}q(\xi ,\tau )d\xi d\tau ,$$

$$M_1(x,t;\xi ,\tau )=2\frac{\partial \mathrm{\Gamma }(x,t;\xi ,\tau )}{\partial n(x,t)},$$

$$M_{j+1}(x,t;\xi ,\tau )={\int }_0^t{\int }_{\partial \mathrm{\Omega }}{M}_1(x,t;y,\sigma )M_j(y,\sigma ;\xi ,\tau )d{S}_{y}d\sigma .$$

Here the infinite series converges and the function F(x, t) is bounded.

Step 3. Comparison. Notice by (6.5) that

$$c-\beta -\sum _{i=1}^N{k}_i{ū}_i^{(0)}\ge 0\text{and}c-{\beta }_i-k_i{ῡ}^{(0)}\ge 0\text{in}{\mathrm{\Omega }}_T.$$

By the Maximum Principle for parabolic equations [2, 3, 9, 18] and using arguments similar to those for the steady-state solutions (cf. Step 3 in the proof of Theorem 5.1 in Section 5), we then have from the iteration (6.6)–(6.13) in Step 2 that

$$0={\mathrm{u̲}}_i^{(0)}\le {\mathrm{u̲}}_i^{(k)}\le {\mathrm{u̲}}_i^{(k+1)}\le {ū}_i^{(k+1)}\le {ū}_i^{(k)}\le {ū}_i^{(0)}\text{in}\overline{{\mathrm{\Omega }}_T},$$ (6.20)

$$0={\mathrm{\upsilon ̲}}^{(0)}\le {\mathrm{\upsilon ̲}}^{(k)}\le {\mathrm{\upsilon ̲}}^{(k+1)}\le {ῡ}^{(k+1)}\le {ῡ}^{(k)}\le {ῡ}^{(0)}\text{in}\overline{{\mathrm{\Omega }}_T}.$$ (6.21)

Step 4. Convergence to solution. By the monotonicity (6.20) and (6.21), we have the pointwise limits

$${ū}_i=\underset{k\to \infty }{\text{lim}}{ū}_i^{(k)},ῡ=\underset{k\to \infty }{\text{lim}}{ῡ}^{(k)},{\mathrm{u̲}}_i=\underset{k\to \infty }{\text{lim}}{\mathrm{u̲}}_i^{(k)},\mathrm{\upsilon ̲}=\underset{k\to \infty }{\text{lim}}{\mathrm{\upsilon ̲}}^{(k)}\text{in}\overline{{\mathrm{\Omega }}_T},i=1,\dots ,N.$$

These limits are nonnegative bounded measurable functions. In particular,

$${ū}_i(x,0)={\mathrm{u̲}}_i(x,0)=u_{i0}(x)(i=1,\dots ,N)\text{and}ῡ(x,0)=\mathrm{\upsilon ̲}(x,0)={\upsilon }_0(x)\forall x\in \mathrm{\Omega ̅}.$$

Let us now fix i (1 ≤ i ≤ N) and set for each integer k ≥ 1

$$q_i^{(k)}=c{ū}_i^{(k-1)}-{\beta }_i{ū}_i^{(k-1)}-k_i{ū}_i^{(k-1)}{\mathrm{\upsilon ̲}}^{(k-1)}+{\alpha }_i\text{in}{\mathrm{\Omega }}_T,q_i=c{ū}_i-{\beta }_i{ū}_i-k_i{ū}_i\mathrm{\upsilon ̲}+{\alpha }_i\text{in}{\mathrm{\Omega }}_T.$$ (6.22)

Note that each $q^{(k)}\in C(\overline{{\mathrm{\Omega }}_T})$ is Hölder continuous in x uniformly in $\overline{{\mathrm{\Omega }}_T}$. Moreover,

$$\underset{k\to \infty }{\text{lim}}{q}_i^{(k)}=q_i\text{in}{\mathrm{\Omega }}_T.$$ (6.23)

It follows from (6.6), (6.8), and (6.9) that

$$\frac{\partial {ū}_i^{(k)}}{\partial t}-D_i\mathrm{\Delta }{ū}_i^{(k)}+c{ū}_i^{(k)}=q_i^{(k)}\text{in}{\mathrm{\Omega }}_T,$$ (6.24)

$${ū}_i^{(k)}(·,0)=u_{i0}(·)\text{in}\mathrm{\Omega ̅},$$ (6.25)

$$\frac{\partial {ū}_i^{(k)}}{\partial n}=0\text{on}\partial \mathrm{\Omega }\times (0,T].$$ (6.26)

Therefore, by the representation (6.17) and (6.19) for the solution to (6.14)–(6.16) that are now replaced by (6.24)–(6.26), we have

$${ū}_i^{(k)}(x,t)={\int }_{\mathrm{\Omega }}\mathrm{\Gamma }(x,t;\xi ,0)u_{i0}(\xi )d\xi +{\int }_0^t{\int }_{\mathrm{\Omega }}\mathrm{\Gamma }(x,t;\xi ,\tau )q_i^{(k)}(\xi ,\tau )d\xi d\tau +{\int }_0^t{\int }_{\partial \mathrm{\Omega }}\mathrm{\Gamma }(x,t;\xi ,\tau ){\mathrm{\Psi }}_i^{(k)}(\xi ,\tau )d{S}_{\xi }d\tau \forall (x,t)\in \mathrm{\Omega }\times (0,T],$$

$${\mathrm{\Psi }}_i^{(k)}(x,t)=2F_i^{(k)}(x,t)+2\sum _{j=1}^{\infty }{\int }_0^t{\int }_{\partial \mathrm{\Omega }}{M}_j(x,t;\xi ,\tau )F_i^{(k)}(\xi ,\tau )d{S}_{\xi }d\tau ,$$

$$F_i^{(k)}(x,t)={\int }_{\mathrm{\Omega }}\frac{\partial \mathrm{\Gamma }(x,t;\xi ,0)}{\partial n(x,t)}{u}_{i0}(\xi )d\xi +{\int }_0^t{\int }_{\mathrm{\Omega }}\frac{\partial \mathrm{\Gamma }(x,t;\xi ,\tau )}{\partial n(x,t)}{q}_i^{(k)}(\xi ,\tau )d\xi d\tau ,$$

where Γ is given in (6.18) with D replaced by D_i.

Since the sequence ${\{q_i^{(k)}\}}_{k=1}^{\infty }$ is uniformly bounded in $\overline{{\mathrm{\Omega }}_T}$ and converges (cf. (6.23)), the sequence ${\{F_i^{(k)}\}}_{k=1}^{\infty }$ is uniformly bounded in ∂Ω × (0, T] and converges. Further, the series in the expression of ${\mathrm{\Psi }}_i^{(k)}$ converges absolutely. Therefore, the sequence ${\{{\mathrm{\Psi }}_i^{(k)}\}}_{k=1}^{\infty }$ is also uniformly bounded on ∂Ω × (0, T] and converges. Let the limit be Ψ_i = Ψ_i(x, t). Taking the limit as k → ∞ and using the Lebesgue Dominated Convergence Theorem, we obtain by (6.23) that

$${ū}_i(x,t)={\int }_{\mathrm{\Omega }}\mathrm{\Gamma }(x,t;\xi ,0)u_{i0}(\xi )d\xi +{\int }_0^t{\int }_{\mathrm{\Omega }}\mathrm{\Gamma }(x,t;\xi ,\tau )q_i(\xi ,\tau )d\xi d\tau +{\int }_0^t{\int }_{\partial \mathrm{\Omega }}\mathrm{\Gamma }(x,t;\xi ,\tau ){\mathrm{\Psi }}_i(\xi ,\tau )d{S}_{\xi }d\tau ,\forall (x,t)\in \mathrm{\Omega }\times (0,T],$$ (6.27)

$${\mathrm{\Psi }}_i(x,t)=2F_i(x,t)+2\sum _{j=1}^{\infty }{\int }_0^t{\int }_{\partial \mathrm{\Omega }}{M}_j(x,t;\xi ,\tau )F_i(\xi ,\tau )d{S}_{\xi }d\tau ,$$ (6.28)

$$F_i(x,t)={\int }_{\mathrm{\Omega }}\frac{\partial \mathrm{\Gamma }(x,t;\xi ,0)}{\partial n(x,t)}{u}_{i0}(\xi )d\xi +{\int }_0^t{\int }_{\mathrm{\Omega }}\frac{\partial \mathrm{\Gamma }(x,t;\xi ,\tau )}{\partial n(x,t)}{q}_i(\xi ,\tau )d\xi d\tau ,$$

where Γ is given in (6.18) with D replaced by D_i.

Since u_i0 ∈ C¹(Ω̅), the first term in (6.27) is a function of (x. t) in C²(Ω_T). Since q_i is bounded, the second term in (6.27) is also a continuous function of (x, t) on $\overline{{\mathrm{\Omega }}_T}$. By (6.28), the function Ψ_i(x, t) is continuous on ∂Ω × [0, T]. Thus the third term in (6.27) and hence ū_i = ū_i(x, t) is a continuous function in $\overline{{\mathrm{\Omega }}_T}$. Similarly, υ̲ = υ̲(x, t) is a continuous function in $\overline{{\mathrm{\Omega }}_T}$. Therefore, q_i = q_i(x, t) as defined in (6.22) is continuous in $\overline{{\mathrm{\Omega }}_T}$. Repeat the same argument using (6.27) and (6.28), we have that ū_i is in fact Hölder continuous in x uniformly in $\overline{{\mathrm{\Omega }}_T}$. Similarly, υ̲ is Hölder continuous in x uniformly in $\overline{{\mathrm{\Omega }}_T}$. Finally, q_i is Hölder continuous in x uniformly in $\overline{{\mathrm{\Omega }}_T}$. Therefore, we have the interior regularity of all ū_i (i = 1, …, N) and υ̲; cf. [3] (Theorem 2 in Section 5.3). Now (6.22), (6.27), and (6.28) imply that ū_i, and υ̲, solve (6.1) with u and υ replaced by ū_i and υ̲, respectively. The existence of solutions to other equations can be obtained similarly.

We now prove the uniqueness in three steps.

Step 1. We prove that solutions (ū₁, …, ū_N, υ̲) and (u̲₁, …, u̲_N, ῡ) obtained above satisfy u̲_i = ū_i (i = 1, …, N) and υ̲ = ῡ in $\overline{{\mathrm{\Omega }}_T}$.

In fact, setting w_i = ū_i − u̲_i (i = 1, …, N) and w = ῡ − υ̲, we have

$$\frac{\partial w_i}{\partial t}=D_i\mathrm{\Delta }{w}_i-{\beta }_i{w}_i-k_i\mathrm{\upsilon ̲}{w}_i+k_i{\mathrm{u̲}}_{i}w\text{in}{\mathrm{\Omega }}_T,i=1,\dots ,N,\frac{\partial w}{\partial t}=D\mathrm{\Delta }w-\beta w-\sum _{i=1}^N{k}_i{\mathrm{u̲}}_{i}w+\sum _{i=1}^N{k}_i{w}_i\mathrm{\upsilon ̲}\text{in}{\mathrm{\Omega }}_T,\frac{\partial w_i}{\partial n}=\frac{\partial w}{\partial n}=0\text{on}\partial \mathrm{\Omega }\times (0,T],i=1,\dots ,N,w_i(·,0)=0\text{and}w(·,0)=0\text{in}\mathrm{\Omega ̅},i=1,\dots ,N.$$

The uniqueness of solution to linear systems of parabolic equations then lead to w_i = 0 (i = 1, …, N) and w = 0.

Step 2. We prove the following: If $(u_1^{*},\dots ,u_N^{*},{\upsilon }^{*})$ is any other solution to (6.1)–(6.4) such that ${\mathrm{u̲}}_i^{(0)}\le u_i^{*}\le {ū}_i^{(0)}$ (i = 1, …, N) and υ̲⁽⁰⁾ ≤ υ* ≤ ῡ⁽⁰⁾ in $\overline{{\mathrm{\Omega }}_T}$, then ${\mathrm{u̲}}_i=u_i^{*}={ū}_i$ (i = 1, …, N) and υ̲ = υ* = ῡ in $\overline{{\mathrm{\Omega }}_T}$.

In fact, let us replace $({ū}_1^{(0)},\dots ,{ū}_N^{(0)},{\mathrm{\upsilon ̲}}^{(0)})$ by $(u_1^{*},\dots ,u_N^{*},{\upsilon }^{*})$ and keep $({\mathrm{u̲}}_1^{(0)},\dots ,{\mathrm{u̲}}_N^{(0)},{ῡ}^{(0)})$ unchanged in the iteration in Step 2. Then, $({ū}_1^{(k)},\dots ,{ū}_N^{(k)},{\mathrm{\upsilon ̲}}^{(k)})=(u_1^{*},\dots ,u_N^{*},{\upsilon }^{*})$ (k = 0, 1, …), and the sequence $({\mathrm{u̲}}_1^{(k)},\dots ,{\mathrm{u̲}}_N^{(k)},{ῡ}^{(k)})$ (k = 0, 1, …) remains unchanged and it converges to the solution (u̲₁, …, u̲_N, ῡ). Therefore, by the iteration we get ${\mathrm{u̲}}_i\le u_i^{*}$ (i = 1, …, N) and υ* ≤ ῡ in $\overline{{\mathrm{\Omega }}_T}$. A similar argument leads to $u_i^{*}\le {ū}_i$ (i = 1, …, N) and υ̲ ≤ υ* in $\overline{{\mathrm{\Omega }}_T}$. These, together with the result proved in Step 1, imply ${\mathrm{u̲}}_i=u_i^{*}={ū}_i$ (i = 1, …, N) and υ̲ = υ* = ῡ in $\overline{{\mathrm{\Omega }}_T}$.

Step 3. If we have two nonnegative solutions defined on $\overline{{\mathrm{\Omega }}_T}$, then we can choose all ${ū}_i^{(0)}$ (i = 1, …, N) and ῡ⁽⁰⁾ (in the iteration in Step 2 of proving existence) large enough to bound from above these solutions in $\overline{{\mathrm{\Omega }}_T}$. Then, both of these solutions must be the same as those constructed by iterative upper and lower solutions. The uniqueness is therefore proved.

Theorem 6.2

(Existence and uniqueness of global solution). Let Ω be a bounded domain in ℝ³ with a C²-boundary ∂Ω. Let D_i (i = 1, …, N), D, β_i (i = 1, …, N), β, and k_i (i = 1, …, N) be all positive numbers. Let α_i ∈ C¹(Ω̅ × [0, ∞)) (i = 1, …, N) and α ∈ C¹(Ω̅ × [0, ∞)) be all nonnegative functions on Ω̅ × [0, ∞). Assume u_i0 ∈ C¹(Ω̅) (i = 1, …, N) and υ₀ ∈ C¹(Ω̅) are all nonnegative functions on Ω̅. Then there exists a unique nonnegative solution (u₁, …, u_N, υ) to the system (1.1)–(1.4) with all u_i (i = 1, …, N) and υ being continuous on Ω̅ × [0, ∞) and continuously differentiable in t ∈ (0, ∞) and twice continuously differentiable in x ∈ Ω.

Proof

Let T_m = m (m = 1, 2, …). Then for each T_m, the system has a unique solution defined on $\overline{{\mathrm{\Omega }}_{T_m}}$. By the uniqueness of local solution, the solution corresponding to T_m and that to T_n are identical on [0, T_m] if m ≤ n. Therefore, on each finite interval of time, all the solutions are the same as long as they are defined on that interval. Hence we have the existence of a global solution. It is unique since the local solution is unique.

7 Reaction-Diffusion System: Asymptotic Behavior

We now assume that all α_i (i = 1, …, N) and α are independent of time t and consider the initial-boundary-value problem of the full, time-dependent system of reaction-diffusion equations (1.1)–(1.4). Given the initial data u_i0 (i = 1, …, N) and υ₀, the system has a unique global solution u_i = u_i (x, t) (i = 1, …, N) and υ = υ(x, t) by Theorem 6.2. We ask if the limit of the solution as t → ∞ exists, and if so, if the limit is a steady-state solution.

We first state the following result and omit its proof as it is similar to that for the special case for two equations; cf. Corollary 8.3.1 in [18]:

Proposition 7.1

Let Ω, T, and D_i (i = 1, …, N) be all the same as in Theorem 6.1. Let $a_{i,j}\in C^1(\overline{{\mathrm{\Omega }}_T})$ (i, j = 1, …, N) be such that a_i,j ≥ 0 in Ω_T if i ≠ j. Suppose $w_i\in C(\overline{{\mathrm{\Omega }}_T})\cap C_1^2({\mathrm{\Omega }}_T)$ (i = 1, …, N) satisfy

$$\frac{\partial w_i}{\partial t}-D_i\mathrm{\Delta }{w}_i\ge \sum _{i,j=1}^N{a}_{i,j}(x,t)w_j\mathit{\text{in}}{\mathrm{\Omega }}_T,i=1,\dots ,N,\frac{\partial w_i}{\partial n}=0\mathit{\text{on}}\partial \mathrm{\Omega }\times (0,T],i=1,\dots ,N,w_i(·,0)\ge 0\mathit{\text{in}}\mathrm{\Omega ̅},i=1,\dots ,N.$$

Then w_i ≥ 0 in $\overline{{\mathrm{\Omega }}_T}$ (i = 1, …, N).

The following theorem states indicates particularly that if the initial values are large constant functions then the global solutions to the time-dependent problem are monotonic in time t and the limits as t → ∞ are steady-state solutions.

Theorem 7.1

Let Ω be a bounded domain in ℝ³ with its boundary ∂Ω in the class C^2,μ for some μ ∈ (0, 1). Let D_i, β_i, and k_i (i = 1, …, N), and D and β be all the same as in Theorem 6.1. Let α_i ∈ C¹(Ω̅) (i = 1, …, N) and α ∈ C¹(Ω̅) be all nonnegative on Ω̅. Let (Ū₁, …, Ū_N, V̲) be the nonnegative solution to the system (1.1)–(1.4) with the initial values ${Ū}_i(·,0)={ū}_i^{(0)}$ (i = 1, …, N) and V̲ (·, 0) = υ̲⁽⁰⁾ all being constant functions. Let (U̲₁, …, U̲_N, V̅) be the nonnegative solution to the system (1.1)–(1.4) with the initial values ${\mathrm{U̲}}_i(·,0)={\mathrm{u̲}}_i^{(0)}$ (i = 1, …, N) and V̅ (·, 0) = ῡ⁽⁰⁾ all being constant functions. Assume ${ū}_i^{(0)}\ge {\Vert {\alpha }_i\Vert }_{L^{\infty }(\mathrm{\Omega })}/{\beta }_i$, ῡ⁽⁰⁾ ≥ ‖α‖_L^∞(Ω)/β, and ${\mathrm{u̲}}_i^{(0)}={\mathrm{\upsilon ̲}}^{(0)}=0$ (i = 1, …, N). Then the following hold true:

1. Ū_i ≥ U̲_i (i = 1, …, N) and V̅ ≥ V̲ in Ω̅ × [0, ∞).

2. Ū_i (i = 1, …, N) and V̅ are monotonically nonincreasing in t and U̲_i (i = 1, …, N) and V̲ are monotonically nondecreasing in t.

3. Let

   $$({Ū}_{1,s},\dots ,{Ū}_{N,s},{\mathrm{V̲}}_s)=\underset{t\to \infty }{\text{lim}}({Ū}_1,\dots ,{Ū}_N,\mathrm{V̲}),({\mathrm{U̲}}_{1,s},\dots ,{\mathrm{U̲}}_{N,s},{\mathrm{V̅}}_s)=\underset{t\to \infty }{\text{lim}}({\mathrm{U̲}}_1,\dots ,{\mathrm{U̲}}_N,\mathrm{V̅}),$$

   Then Ū_i,s ≥ U̲_i,s (i = 1, …, N) and V̅_s ≥ V̲_s on Ω̅. Moreover, all Ū_i,s, U̲_i,s, V̅_s, and V̲_s are in C^2,μ (Ω̅), and (Ū_1,s, …, Ū_N,s, V̲_s) and (U̲_1,s, …, U̲_N,s, V̅_s) are solutions of the time-independent system (5.1)–(5.3).

4. If $(u_{1,s}^{*},\dots ,u_{N,s}^{*},{\upsilon }_s^{*})$ is any solution to (5.1)–(5.3) such that ${\mathrm{u̲}}_i^{(0)}\le u_{i,s}^{*}\le {ū}_i^{(0)}$ (i = 1, …, N) and ${\mathrm{\upsilon ̲}}^{(0)}\le {\upsilon }_s^{*}\le {ῡ}^{(0)}$ on Ω̅, then ${\mathrm{U̲}}_{i,s}\le u_{i,s}^{*}\le {Ū}_{i,s}$ (i = 1, …, N) and ${\mathrm{V̲}}_s\le {\upsilon }_s^{*}\le {\mathrm{V̅}}_s$ on Ω̅.

Proof

(1) Let T > 0. Let W_i = Ū_i − U̲_i (i = 1, …, N) and W = V̅ − V̲. We then have

$$\frac{\partial W_i}{\partial t}=D_i\mathrm{\Delta }{W}_i-{\beta }_i{W}_i-k_i\mathrm{V̲}{W}_i+k_i{\mathrm{U̲}}_{i}W\text{in}{\mathrm{\Omega }}_T,i=1,\dots ,N,\frac{\partial W}{\partial t}=D\mathrm{\Delta }W-\beta W-\sum _{i=1}^N{k}_i{\mathrm{U̲}}_{i}W+\sum _{i=1}^N{k}_i{W}_i\mathrm{V̲}\text{in}{\mathrm{\Omega }}_T,\frac{\partial W_i}{\partial n}=\frac{\partial W}{\partial n}=0\text{on}\partial \mathrm{\Omega }\times (0,T],i=1,\dots ,N,W_i(·,0)={ū}_i^{(0)}-{\mathrm{u̲}}_i^{(0)}\ge 0\text{and}W(·,0)={ῡ}^{(0)}-{\mathrm{\upsilon ̲}}^{(0)}\ge 0\text{in}\mathrm{\Omega ̅},i=1,\dots ,N.$$

Therefore, we get by Proposition 7.1 that W_i ≥ 0 (i = 1, …, N) and W ≥ 0 in $\overline{{\mathrm{\Omega }}_T}$. Since T > 0 is arbitrary, we obtain the desired inequality on Ω × [0, ∞).

(2) Let T > 0 and δ > 0. Set W̅_i(x, t) = Ū_i(x, t) − Ū_i(x, t + δ) (i = 1, …, N) and W̲(x, t) = V̲ (x, t + δ) − V̲ (x, t). Then

$$\frac{\partial {\mathrm{W̅}}_i(x,t)}{\partial t}=D_i\mathrm{\Delta }{\mathrm{W̅}}_i(x,t)-{\beta }_i{\mathrm{W̅}}_i(x,t)-k_i\mathrm{V̲}(x,t+\delta ){\mathrm{W̅}}_i(x,t)+k_i{\mathrm{U̲}}_i(x,t)\mathrm{W̲}(x,t)\forall (x,t)\in {\mathrm{\Omega }}_T,i=1,\dots ,N,\frac{\partial \mathrm{W̲}(x,t)}{\partial t}=D\mathrm{\Delta }\mathrm{W̲}(x,t)-\beta \mathrm{W̲}(x,t)-\sum _{i=1}^N{k}_i{\mathrm{U̲}}_i(x,t+\delta )\mathrm{W̲}(x,t)+\sum _{i=1}^N{k}_i{\mathrm{W̅}}_i(x,t)\mathrm{V̲}(x,t)\forall (x,t)\in {\mathrm{\Omega }}_T,\frac{\partial {\mathrm{W̅}}_i}{\partial n}=\frac{\partial \mathrm{W̲}}{\partial n}=0\text{on}\partial \mathrm{\Omega }\times (0,T],i=1,\dots ,N,{\mathrm{W̅}}_i(·,0)={ū}_i^{(0)}-{Ū}_i(\delta )\ge 0\text{and}\mathrm{W̲}(·,0)=\mathrm{V̲}(\delta )-{\mathrm{\upsilon ̲}}^{(0)}\ge 0\text{in}\mathrm{\Omega ̅},i=1,\dots ,N.$$

Again by Proposition 7.1 we get W̅_i ≥ 0 (i = 1, …, N) and W̲ ≥ 0. Hence Ū_i (i = 1, …, N) are monotonically nonincreasing in t and V̲ is monotonically nondecreasing in t. Similarly, U̲_i (i = 1, …, N) are monotonically nondecreasing in t and V̅ is monotonically nonincreasing in t.

(3) By Part (1) we have Ū_i,s ≥ U̲_i,s (i = 1, …, N) and V̅_s ≥ V̲_s on Ω̅. The claim that (Ū_1,s, …, Ū_N,s, V̲_s) and (U̲_1,s, …, U̲_N,s, V̅_s) are solutions of the time-independent system (5.1)–(5.3) can be proved similarly as the proof of Theorem 10.4.3 in [18] and that of Theorem 3.6 in [20].

(4) This part can proved by the same argument in Step 2 in the proof of uniqueness of solution of Theorem 6.1.

Theorem 7.2

Let Ω, D_i, D, β_i, β, k_i, α_i, α, ${ū}_i^{(0)},{\mathrm{u̲}}_i^{(0)}$, ῡ⁽⁰⁾, υ̲⁽⁰⁾, Ū_i, U̲_i, V̅, and V̲ (i = 1, …, N) be all the same as in Theorem 7.1. Let u_i0 ∈ C¹(Ω̅) (i = 1, …, N) and υ₀ ∈ C¹(Ω̅) be such that ${\mathrm{\upsilon ̲}}^{(0)}\le u_{i0}\le {ū}_i^{(0)}$ and υ̲⁽⁰⁾ ≤ υ₀ ≤ ῡ⁽⁰⁾ (i = 1, …, N) in Ω̅. Let (u₁, …, u_N, υ) be the unique nonnegative global solution to the time-dependent problem (1.1)–(1.4) with the initial data (u₁₀, …, u_N0, υ₀). Then Ū_i ≥ u_i ≥ U̲_i (i = 1, …, N) and V̅ ≥ υ ≥ V̲ in Ω̅ × [0, ∞).

Proof

This is similar to the proof of Part (1) of Theorem 7.1.

The following two corollaries relate the uniqueness of steady-state solution to the asymptotic behavior of solution to the time-dependent problem.

Corollary 7.1

With the assumption of Theorem 7.2, the following hold true:

1. That Ū_i,s = U̲_i,s (i = 1, …, N) and V̅_s = V̲_s in Ω̅ if and only if the steady-state solution (u_1,s, …, u_N,s) ∈ (C²(Ω̅))^N+1 satisfying ${\mathrm{u̲}}_i^{(0)}\le u_{i,s}\le {ū}_i^{(0)}$ (i = 1, …, N) and υ̲⁽⁰⁾ ≤ υ_s ≤ ῡ⁽⁰⁾ in Ω̅ is unique.

2. If the steady-state solution (u_1,s, …, u_N,s) ∈ (C²(Ω̅))^N+1 that satisfies ${\mathrm{u̲}}_i^{(0)}\le u_{i,s}\le {ū}_i^{(0)}$ (i = 1, …, N) and υ̲⁽⁰⁾ ≤ υ_s ≤ ῡ⁽⁰⁾ in Ω̅ is unique, then for any initial data (u₁₀, …, u_N0, υ₀) ∈ (C¹(Ω̅))^N+1 with ${\mathrm{\upsilon ̲}}^{(0)}\le u_{i0}\le {ū}_i^{(0)}$ (i = 1, …, N) and υ̲⁽⁰⁾ ≤ υ₀ ≤ ῡ⁽⁰⁾ in Ω̅, the corresponding nonnegative solution (u₁, …, u_N, υ) of the time-dependent problem (1.1)–(1.4) converges to (u_1,s, …, u_N,s, υ_s) as t → ∞.

3. If the nonnegative steady-state solution in (C²(Ω̅))^N+1 is unique, then the nonnegative global solution to the time-dependent problem with any nonnegative initial data in (C¹(Ω̅))^N+1 converges to this steady-state solution as t → ∞.

Proof

(1) This follows immediately from Theorem 7.2.

(2) This follows from Theorem 7.1 and Theorem 7.2.

(3) Choose ${ū}_i^{(0)}$ (i = 1, …, N) and ῡ⁽⁰⁾ all large enough and apply Part (2).

Corollary 7.2

With the same assumption as in Corollary 7.1, if Ū_i,s ≠ U̲_i,s for some i with 1 ≤ i ≤ N or V̅_s ≠ V̲_s, then: (1) For any initial data (u₁₀, …, u_N0, υ₀) ∈ (C¹(Ω̅))^N+1 with ${\mathrm{u̲}}_i^{(0)}\le u_{i0}\le {\mathrm{U̲}}_{i,s}$ (i = 1, …, N) and V̅_s ≤ υ₀ ≤ ῡ⁽⁰⁾ in Ω̅, the corresponding nonnegative global solution (u₁, …, u_N, υ) of the time-dependent problem (1.1)–(1.4) converges to (U̲_1,s, …, U̲_N,s, V̅_s) as t → ∞; and (2) For any initial data (u₁₀, …, u_N0, υ₀) ∈ (C¹(Ω̅))^N+1 with ${Ū}_{i,s}\le u_{i0}\le {ū}_i^{(0)}$ (i = 1, …, N) and υ̲⁽⁰⁾ ≤ υ₀ ≤ V̲_s in Ω̅, the corresponding nonnegative global solution (u₁, …, u_N, υ) of the time-dependent problem (1.1)–(1.4) converges to (Ū_1,s, …, Ū_N,s, V̲_s) as t → ∞.

Proof

This is similar to the proof of Theorem 7.1.