Analysis of Coupled Reaction-Diffusion Equations for

RNA Interactions

Maryann E. Hohn∗ Bo Li† Weihua Yang‡

November 11, 2014

Abstract

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

Key words: RNA, gene expression, reaction-diffusion systems, well-posedness, vari-ational methods, monotone methods, maximum principle.

AMS subject classifications: 34D20, 35J20, 35J25, 35J57, 35J60, 35K57, 92D25.

∗Department of Mathematics, University of Connecticut, Storrs, 196 Auditorium Road, Unit 3009, Storrs,CT 06269-3009, USA. Email: maryann.hohn@uconn.edu.

†Department of Mathematics and Center for Theoretical Biological Physics, University of California, SanDiego, 9500 Gilman Drive, Mail code: 0112, La Jolla, CA 92093-0112, USA. Email: bli@math.ucsd.edu.

‡Department of Mathematics and Institute of Mathematics and Physics, Beijing University of Technology,No. 100, Pingleyuan, Chaoyang District, Beijing, P. R. China, 100124, and Department of Mathematics,University of California, San Diego, 9500 Gilman Drive, Mail code: 0112, La Jolla, CA 92093-0112, USA.whyang@bjut.edu.cn.

1

1 Introduction

Let Ω be a bounded domain in R3 with a smooth boundary ∂Ω. Let N ≥ 1 be an integer. Let

Di (i = 1, . . . , N), D, βi (i = 1, . . . , N), β, and ki (i = 1, . . . , N) be positive numbers. Letαi (i = 1, . . . , N) and α be nonnegative functions on Ω× (0,∞). We consider the followingsystem of coupled reaction-diffusion equations:

∂ui

∂t= Di∆ui − βiui − kiuiv + αi in Ω× (0,∞), i = 1, . . . , N, (1.1)

∂v

∂t= D∆v − βv −

N∑

i=1

kiuiv + α in Ω× (0,∞), (1.2)

together with the boundary and initial conditions

∂ui

∂n=

∂v

∂n= 0 on ∂Ω× (0,∞), i = 1, . . . , N, (1.3)

ui(·, 0) = ui 0 and v(·, 0) = v0 in Ω, i = 1, . . . , N, (1.4)

where ∂/∂n denotes the normal derivative along the exterior unit normal n at the boundary∂Ω, and all ui 0 (i = 1, . . . , N) and v0 are nonnegative functions on Ω.

The reaction-diffusion system (1.1)–(1.4) is a biophysical model of the interaction be-tween different types of Ribonucleic acid (RNA) molecules, a class of biological moleculesthat 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 andtissue differentiation through binding and reacting with messenger RNA (mRNA) in acell. Different sRNA species may competitively bind to different mRNA targets to reg-ulate genes [4,6,12,13,16,21]. Recent experiments suggest that the concentration of mRNAand different sRNA in cells and across tissue is linked to the expression of a gene [22]. Oneof 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 cancontribute to turning genes “on” or “off” across various length scales, and eventually to thegene expression.

In Eqs. (1.1) and (1.2), the function ui = ui(x, t) for each i (1 ≤ i ≤ N) representsthe local concentration of the ith sRNA species at x ∈ Ω and time t. We assume a totalof N sRNA species. The function v = v(x, t) represents the local concentration of themRNA species at x ∈ Ω and time t. For each i (1 ≤ i ≤ N), Di is the diffusion coefficientand βi is the self-degradation rate of the ith sRNA species. Similarly, D is the diffusioncoefficient and β is the self-degradation rate of mRNA. For each i (1 ≤ i ≤ N), ki is therate of reaction between the ith sRNA and mRNA. We neglect the interactions amongdifferent sRNA species as they can be effectively described through their diffusion and self-degradation coefficients. The reaction terms uiv (i = 1, . . . , N) result from the mean-fieldapproximation. The nonnegative functions αi = αi(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 termedtranscription profiles. Notice that we set the linear size of the region Ω to be of tissue lengthto account for the RNA interaction across different cells [22].

2

The reaction-diffusion system model (1.1)–(1.4) was first proposed for the special caseN = 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 sharpensthe concentration profile of mRNA. Figure 1 depicts numerically computed steady-statesolutions 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

α1 = 0.1 + 0.1 tanh(5x− 2.5), (1.5)

α = 0.1 + 0.1 tanh(2.5− 5x). (1.6)

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

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

16

18

20

x

Co

nce

ntr

atio

ns

of

mR

NA

0.00001

0.0005

0.0001

0.001

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-stateconcentration of mRNA v = v(x) (0 < x < 1) sharpens as the the diffusion constant D1 ofthe 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, andcomputation of the the RNA interactions, the present work focuses on: (1) the constructionof various modeling systems with different levels of complexity for the reaction, nonlineardiffusion, and coupled reaction and diffusion, respectively, with the most complex one beingthe full reaction-diffusion system (1.1)–(1.4); and (2) the mathematical justification for eachof the models, proving the well-posedness of the corresponding differential equations. Tounderstand how the reaction terms (i.e., the product terms uiv in (1.1) and (1.2)) comefrom, we shall first, however, present a brief derivation of the corresponding reaction system

3

(i.e., no diffusion) for the case N = 1 using a chemical master equation and the mean-fieldapproximation [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 ui = ui(t) ≥ 0 (i = 1, . . . , N) and v = v(t) ≥ 0:

dui

dt= −βiui − kiuiv + αi i = 1, . . . , N, (1.7)

dv

dt= −βv −

N∑

i=1

kiuiv + α, (1.8)

where all αi (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. Theo-rem 3.1.

Case 2. We consider the situation where all the diffusion coefficients Di (i = 1, . . . , N)are much smaller than the diffusion coefficient D. That is, we consider the approximationDi = 0 (i = 1, . . . , N). Assume all α and αi (i = 1, . . . , N) are independent of time t. Thesteady-state solution of ui in Eq. (1.1) with Di = 0 leads to ui = αi/(βi+kiv) (i = 1, . . . , N).These expressions, coupled with the v-equation (1.2), imply that the steady-state solutionv ≥ 0 should satisfy the following nonlinear diffusion equation and boundary condition:

D∆v − βv −

N∑

i=1

kiαiv

βi + kiv+ α = 0 in Ω, (1.9)

∂v

∂n= 0 on ∂Ω. (1.10)

The single, nonlinear equation (1.9) is the Euler–Lagrange equation of some energy func-tional. We shall use the direct method in the calculus of variations to prove the existenceand 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) forthe concentrations ui ≥ 0 (i = 1, . . . , N) and v ≥ 0:

Di∆ui − βiui − kiuiv + αi = 0 in Ω, i = 1, . . . , N, (1.11)

D∆v − βv −N∑

i=1

kiuiv + α = 0 in Ω, (1.12)

∂ui

∂n=

∂v

∂n= 0 on ∂Ω, i = 1, . . . , N. (1.13)

Here again we assume that α and αi (i = 1, . . . , N) are independent of time t. We shall usethe 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 se-quences of upper and lower solutions, extracting convergent subsequences, and proving thatthe limits are desired solutions.

4

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

While our underlying reaction-diffusion system has been proposed to model RNA inter-actions in molecular biology, its basic mathematical properties are similar to some of thosereaction-diffusion systems modeling other physical and biological processes. Our prelimi-nary 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 mathematicaleffort in understanding the reaction and diffusion of RNA is to analyze the qualitative prop-erties of solutions to the corresponding equations, in particular, the asymptotic behavior ofsuch 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 derivationof the reaction system (1.7) and (1.8) for the case N = 1 using a chemical master equationand 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 ofthe single nonlinear diffusion equation (1.9) and (1.10) for the concentration v 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 ofglobal 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 oftime-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 onhow the full reaction-diffusion system (1.1)–(1.4) is formulated.

For simplicity, we shall consider two chemical species: mRNA and one sRNA. Figure 2describes sRNA-mediated gene silencing within the cell and depicts the different rates inwhich mRNA and sRNA populations may change at time t. In the figure, αs and αm

describe the sRNA and mRNA production rates, βs and βm describe the sRNA and mRNAindependent 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. Thenumerical value of each of these rates can be determined via experimental data [15].

We denote by Mt and St the numbers of mRNA and sRNA, respectively, in a given cellat time t, and consider the two continuous-time processes (Mt)t≥0 and (St)t≥0 . We assume

5

sRNA

mRNAαm

αs

βs

γ

βm

ø

ø

ø

Figure 2: Kinetic scheme of the interaction of sRNA and mRNA in a cell. Here, αs andαm represent the production rates of sRNA and mRNA, respectively; βs and βm representthe independent degradation rates of sRNA and mRNA, respectively; and γ represents thecoupled degradation rate of sRNA and mRNA.

that (Mt, St)t≥0 is a stationary continuous-time Markov chain with state space S with thefollowing ordering:

S = (0, 0), . . . , (m− 1, s− 1), (m− 1, s), (m, s− 1), (m, s), (m, s+ 1), . . . .

We assume that the total numbers of mRNA and sRNA are finite, and hence the statespace is finite. For any given state (m, s) ∈ S, we denote by Pm,s(t) the probability that thesystem is in this state, i.e., Pm,s(t) = P (Mt = m,St = s). For convenience, we extend S toinclude integer pairs (m, s) for m < 0 or s < 0 and set Pm,s(t) = 0 if m < 0 or s < 0. Notethat

∑

(m,s)∈S

Pm,s(t) = 1 (2.1)

for any t ≥ 0. Note also that the averages 〈Mt〉 , 〈St〉, and 〈MtSt〉 are defined by

〈Mt〉 =∑

(m,s)∈S

mPm,s(t), 〈St〉 =∑

(m,s)∈S

sPm,s(t), and 〈MtSt〉 =∑

(m,s)∈S

msPm,s(t).

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

Pm,s(t) = αmPm−1,s(t) + αsPm,s−1(t) + βs(s+ 1)Pm,s+1(t) + βm(m+ 1)Pm+1,s(t)

+ γ(m+ 1)(s+ 1)Pm+1,s+1(t)− (αm + αs + βmm+ βss+ γms)Pm,s(t),

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

d

dt〈Mt〉 =

∑

(m,s)∈S

mPm,s(t) = αmAm + αsAs + βmBm + βsBs + γC, (2.3)

6

where

Am =∑

(m,s)∈S

mPm−1,s(t)−∑

(m,s)∈S

mPm,s(t),

As =∑

(m,s)∈S

mPm,s−1(t)−∑

(m,s)∈S

mPm,s(t),

Bm =∑

(m,s)∈S

m(m+ 1)Pm+1,s(t)−∑

(m,s)∈S

m2Pm,s(t),

Bs =∑

(m,s)∈S

m(s+ 1)Pm,s+1(t)−∑

(m,s)∈S

msPm,s(t),

C =∑

(m,s)∈S

m(m+ 1)(s+ 1)Pm+1,s+1(t)−∑

(m,s)∈S

m2sPm,s(t).

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

Am =∑

(m+1,s)∈S

(m+ 1)Pm,s(t)−∑

(m,s)∈S

mPm,s(t)

=∑

(m,s)∈S

(m+ 1)Pm,s(t)−∑

(m,s)∈S

mPm,s(t)

=∑

(m,s)∈S

Pm,s(t)

= 1.

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

As =∑

(m,s+1)∈S

mPm,s(t)−∑

(m,s)∈S

mPm,s(t)

=∑

(m,s)∈S

mPm,s(t)−∑

(m,s)∈S

mPm,s(t)

= 0.

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

Bm =∑

(m−1,s)∈S

(m− 1)mPm,s(t)−∑

(m,s)∈S

m2Pm,s(t)

=∑

(m,s)∈S

(m− 1)mPm,s(t)−∑

(m,s)∈S

m2Pm,s(t)

= −∑

(m,s)∈S

mPm,s(t)

= −〈Mt〉 .

7

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

Bs =∑

(m,s−1)∈S

msPm,s(t)−∑

(m,s)∈S

msPm,s(t)

=∑

(m,s)∈S

msPm,s(t)−∑

(m,s)∈S

msPm,s(t),

= 0.

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

C =∑

(m−1,s−1)∈S

(m− 1)msPm,s(t)−∑

(m,s)∈S

m2sPm,s(t)

=∑

(m,s)∈S

(m− 1)msPm,s(t)−∑

(m,s)∈S

m2sPm,s(t)

= −∑

(m,s)∈S

msPm,s(t)

= −〈MtSt〉 .

Inserting all Am, As, Bm, Bs and C into (2.3), we obtain

d

dt〈Mt〉 = αm − βm 〈Mt〉 − γ 〈MtSt〉 . (2.4)

Similarly, we haved

dt〈St〉 = αs − βs 〈St〉 − γ 〈MtSt〉 . (2.5)

We now make the mean-field assumption: 〈MtSt〉 = 〈Mt〉 〈St〉 . If we denote by v(t) =〈Mt〉 and u1(t) = 〈St〉, 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), respec-tively, with α1 = αm, β1 = βm, k1 = γ, α = αs, and β = βs.

We remark that based on Fick’s law the spatial diffution of the underlying sRNA andmRNA molecules can be described by Di∆ui (i = 1, . . . , N) and D∆u with all Di and D thediffusion constants, respectively [1, 11, 17]. Here we have neglected any possible and morecomplicated processes such as cross diffusion and anomalous diffusion. Combining theseterms 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 Lin-

ear Stability

Theorem 3.1. Assume all βi, ki, and αi (i = 1, . . . , N), and β, k, and α are positivenumbers. The system of ODE

dui

dt= −βiui − kiuiv + αi, i = 1, . . . , N, (3.1)

8

dv

dt= −βv −

N∑

i=1

kiuiv + α (3.2)

has a unique equilibrium solution (u10, . . . , uN0, v0) ∈ RN+1 with all ui0 > 0 (i = 1, . . . , N)

and v0 > 0. Moreover, it is linearly stable.

Proof. If (u1, . . . , uN , v) ∈ RN+1 is an equilibrium solution to (3.1) and (3.2) with all ui > 0

(i = 1, . . . , N) and v > 0, then S :=∑N

i=1 kiui should satisfy

S =N∑

i=1

kiαi

βi + kiα (β + S)−1 , (3.3)

and the solution should be given by

ui =αi

βi + kiα (β + S)−1 (i = 1, . . . , N) and v =α

β + 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−

N∑

i=1

kiαi

βi + kiα (β + s)−1 . (3.5)

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

g′′(s) = 2αN∑

i=1

k2i αiβi

[βi (β + s) + kiα]3 > 0 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 theunique solution to g = 0 on [0,∞).

Set now

ui0 =αi

βi + kiα (β + s0)−1 , i = 1, . . . , N, (3.6)

v0 =α

β + s0. (3.7)

Clearly, all ui0 > 0 (i = 1, . . . , N) and vi0 > 0. Note that s0 =∑N

i=1 kiui0, since g(s0) = 0.Thus (3.7) implies which together with (3.6) further imply βiui0 + kiui0v0 = αi (i =1, . . . , N). Therefore (u10, . . . , uN0, v0) is an equilibrium solution to (3.1) and (3.2).

Assume both (u10, . . . , uN0, v0) and (u10, . . . , uN0, v0) are equilibrium solutions to (3.1)and (3.2) with all ui0 > 0 and ui0 > 0 (i = 1, . . . , N), v0 > 0 and v0 > 0. Then, by (3.3)and (3.5), S0 :=

∑Ni=1 kiui0 > 0 and S0 :=

∑Ni=1 kiui0 > 0 satisfy g(S0) = 0 and g(S0) = 0,

9

respectively. By the uniqueness of solution s0 of g = 0, we then have S0 = S0 = s0. Itthen follows from (3.6) and (3.7) that ui0 = ui0 (i = 1, . . . , N) and v0 = v0. Therefore theequilibrium solution is unique.

The linearized system for (U1, . . . , UN , V ) around the equilibrium solution (u10, . . . , uN0, v0)is given by

dUi

dt= −(βi + kiv0)Ui − kiui0V, i = 1, . . . , N,

dV

dt= −

N∑

i=1

kiv0Ui −

(

β +N∑

i=1

kiui0

)

V.

Let w =(

U1, . . . , UN , V)T

, where the superscript T denotes the transpose. This system isthen dw/dt = Mw, where

M =

−(β1 + k1v0) 0 . . . 0 −k1u10

0 −(β2 + k2v0) . . . 0 −k2u20...

.... . .

...0 0 −(βN + kNv0) −kNuN0

−k1v0 −k2v0 . . . −kNv0 −(

β +∑N

i=1 kiui0

)

.

It is easy to see that M is strictly column diagonally dominant with negative diagonalentries. Gersgorin’s Theorem (with columns replacing rows) [8] then implies that the realpart 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 βi and ki (i = 1, . . . , N) are positive numbers. Assume αi ∈ L2(Ω)with αi ≥ 0 a.e. in Ω (i = 1, . . . , N) and α ∈ L2(Ω) with α ≥ 0 a.e. in Ω. Then there existsa unique weak solution v ∈ H1(Ω) with v ≥ 0 a.e. in Ω to the boundary-value problem

D∆v − βv −N∑

i=1

kiαiv

βi + kiv+ α = 0 in Ω, (4.1)

∂v

∂n= 0 on ∂Ω. (4.2)

The same statement is true if the Neumann boundary condition (4.2) is replaced by theDirichlet boundary condition v = v0 on ∂Ω for some v0 ∈ H1(Ω) with v0 ≥ 0 on ∂Ω.

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

J [u] =

∫

Ω

D

2|∇u|2 +

β

2u2 +

N∑

i=1

αiβi

ki

[

kiβi

u− ln

(

1 +kiβi

u

)]

− αu

dx,

10

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

J [u] ≥ C1‖u‖2H1(Ω) + C2 ∀u ∈ H1(Ω). (4.3)

Denote θ = infu∈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 theexistence 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 moregenerally that if w ∈ H1(Ω) then J [|w|] ≤ J [w]. (If u = v0 on ∂Ω, then |u| = |v0| = v0on ∂Ω, since v0 ≥ 0 on ∂Ω.) First, |w|2 = w2 and |∇|w| | = |∇w| a.e. in Ω. Since α isnonnegative 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 haveh(s) = −2s + ln(1 + s) − ln(1 − s) and h′(s) = 2s2/(1 − s2) > 0. Thus h(s) < h(0) = 0. Ifs ≥ 0 then h(s) = 0. Hence h(s) ≤ 0 for all s ∈ R. Consequently, by the definition of J andthe fact that all α ≥ 0 and αi ≥ 0 (i = 1, . . . , N) a.e. in Ω, we have J [|w|] ≤ J [w].

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

d

dt

∣

∣

∣

∣

t=0

∫

Ω

αiβi

kiln

(

1 +kiβi

(v + tη)

)

dx =

∫

Ω

αiβiη

βi + kivdx ∀η ∈ H1(Ω) ∩ L∞(Ω).

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

∫

Ω

[

D∇v · ∇η + η

(

βv +N∑

i=1

αiβi

βi + kiv− α

)]

dx = 0 ∀η ∈ H1(Ω) ∩ L∞(Ω).

Notice that 0 ≤ αiβi/(βi + kiv) ≤ αi a.e. in Ω for all i = 1, . . . , N. Therefore, sinceH1(Ω) ∩ L∞(Ω) is dense in H1(Ω), we can replace η ∈ H1(Ω) ∩ L∞(Ω) by η ∈ H1(Ω).Consequently, the minimizer v ∈ H1(Ω) is a weak solution to (4.1) and (4.2).

If v ∈ H1(Ω) is also a nonnegative weak solution to (4.1) and (4.2), then w = v − v is aweak solution to D∆w− bw = 0 in Ω and ∂nw = 0 on ∂Ω, where b = β+

∑Ni=1 βikiαi/[(βi+

kiv)(βi+kiv)] is in H1(Ω) and b ≥ β > 0 in Ω. Therefore w = 0 a.e. in Ω. Hence the solutionis unique.

We remark that the regularity of the solution v to the boundary-value problem (4.1)and (4.2) depends on the smoothness of the domain Ω and that of the variable coefficientsαi (i = 1, . . . , N) and the source function α. Since the solution v is nonnegative and thenonlinear term of v is bounded, the regularity of v is in fact similar to that of the solutionto a linear elliptic problem. For instance, if ∂Ω is of the class Ck and all α, αi ∈ W k,p(Ω)(i = 1, . . . , N) for some nonnegative integer k and p ∈ [2,∞), then v ∈ W k+2,p(Ω). If ∂Ω isof the class C2,γ and all α, αi ∈ C0,γ(Ω) (i = 1, . . . , N) for some γ ∈ (0, 1), then v ∈ C2,γ(Ω).

11

5 Reaction-Diffusion System: Existence of Steady-State

Solution

Theorem 5.1. Let Ω be a bounded domain in R3. Assume all Di, D, βi, β, and ki (i =

1, . . . , N) are positive constants. Assume all αi and α (i = 1, . . . , N) are nonnegativefunctions on Ω.(1) Assume the boundary ∂Ω of Ω is in the class C2,µ for some µ ∈ (0, 1). Assume

also αi, α ∈ C0,µ(Ω) (i = 1, . . . , N). There exist u1, . . . , uN , v ∈ C2,µ(Ω) with ui ≥ 0(i = 1, . . . , N) and u ≥ 0 in Ω such that (u1, . . . , uN , v) is a solution to the boundary-value problem

Di∆ui − βiui − kiuiv + αi = 0 in Ω, i = 1, . . . , N, (5.1)

D∆v − βv −

N∑

i=1

kiuiv + α = 0 in Ω, (5.2)

∂ui

∂n=

∂v

∂n= 0 on ∂Ω, i = 1, . . . , N. (5.3)

(2) Assume the boundary ∂Ω of Ω is in the class C2 and all αi, α ∈ L2(Ω) (i = 1, . . . , N).There exist u1, . . . , uN , v ∈ H2(Ω) with ui ≥ 0 (i = 1, . . . , N) and u ≥ 0 in Ω suchthat (u1, . . . , uN , v) 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 calculationsindicate that the solution may not be unique. If αi (i = 1, . . . , N) satisfy some additionalassumptions, 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 u(0)i , v(0), u

(0)i , v(0)

(i = 1, . . . , N) to be constant functions such that:

u(0)i ≥

‖αi‖L∞(Ω)

βi

, v(0) ≥‖α‖L∞(Ω)

β, u

(0)i = 0, v(0) = 0 in Ω, i = 1, . . . , N. (5.4)

It is clear that

Di∆u(0)i − βiu

(0)i − kiu

(0)i v(0) + αi ≤ 0 in Ω, i = 1, . . . N, (5.5)

D∆v(0) − βv(0) −N∑

i=1

kiu(0)i v(0) + α ≥ 0 in Ω, (5.6)

∂u(0)i

∂n=

∂v(0)

∂n= 0 on ∂Ω, i = 1, . . . , N, (5.7)

Di∆u(0)i − βiu

(0)i − kiu

(0)i v(0) + αi ≥ 0 in Ω, i = 1, . . . , N, (5.8)

D∆v(0) − βv(0) −

N∑

i=1

kiu(0)i v(0) + α ≤ 0 in Ω, (5.9)

12

∂u(0)i

∂n=

∂v(0)

∂n= 0 on ∂Ω, i = 1, . . . , N. (5.10)

Step 2. Iteration. Let

c = max

β +N∑

i=1

kiu(0)i , β1 + k1v

(0), . . . , βN + kN v(0)

. (5.11)

Define iteratively the functions u(k)i , v(k), u

(k)i , v(k) (i = 1, . . . , N) for k = 1, 2, . . . by

−Di∆u(k)i + cu

(k)i = cu

(k−1)i − βiu

(k−1)i − kiu

(k−1)i v(k−1) + αi in Ω, i = 1, . . . , N, (5.12)

−D∆v(k) + cv(k) = cv(k−1) − βv(k−1) −N∑

i=1

kiu(k−1)i v(k−1) + α in Ω, (5.13)

∂u(k)i

∂n=

∂v(k)

∂n= 0 on ∂Ω, i = 1, . . . , N, (5.14)

−Di∆u(k)i + cu

(k)i = cu

(k−1)i − βiu

(k−1)i − kiu

(k−1)i v(k−1) + αi in Ω, i = 1, . . . , N, (5.15)

−D∆v(k) + cv(k) = cv(k−1) − βv(k−1) −

N∑

i=1

kiu(k−1)i v(k−1) + α in Ω, (5.16)

∂u(k)i

∂n=

∂v(k)

∂n= 0 on ∂Ω, i = 1, . . . , N. (5.17)

We recall that, for any constants D > 0 and c > 0, and any q ∈ C0,µ(Ω), the stan-dard theory of elliptic boundary-value problems guarantees the existence and uniqueness ofsolution w ∈ C2,µ(Ω) to the boundary-value problem [5]

− D∆w + cw = q in Ω,

∂w

∂n= 0 on ∂Ω.

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

‖w‖C2,µ(Ω) ≤ C‖q‖C0,µ(Ω). (5.18)

It therefore follows from (5.4) and a simple induction argument that there are unique solu-

tions to the above boundary-values problems (5.12)–(5.17), defining our functions u(k)i , v(k),

u(k)i , v(k), i = 1, . . . , N and k = 1, 2, . . . , all in C2,µ(Ω).Step 3. Comparison. We now prove for any k ≥ 1 that

0 ≤ u(0)i ≤ u

(k)i ≤ u

(k+1)i ≤ u

(k+1)i ≤ u

(k)i ≤ u

(0)i in Ω, i = 1, . . . , N, (5.19)

0 ≤ v(0) ≤ v(k) ≤ v(k+1) ≤ v(k+1) ≤ v(k) ≤ v(0) in Ω. (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

−Di∆(

u(0)i − u

(1)i

)

+ c(

u(0)i − u

(1)i

)

≥ 0 in Ω, i = 1, . . . , N,

13

∂(

u(0)i − u

(1)i

)

∂n= 0 on ∂Ω, i = 1, . . . , N.

The Maximum Principle [5] implies then u(0)i ≥ u

(1)i in Ω (i = 1, . . . , N). Similarly, we have

u(0)i ≤ u

(1)i , v(0) ≥ v(1), and v(0) ≤ v(1) in Ω for all i = 1, . . . , N.

By (5.4), u(0)i ≥ 0 (i = 1, . . . , N) and v(0) ≥ 0. Next, by (5.12) with k = 1, (5.15) with

k = 1, (5.11), and (5.4), we obtain

−Di∆(

u(1)i − u

(1)i

)

+ c(

u(1)i − u

(1)i

)

= c(

u(0)i − u

(0)i

)

− βi

(

u(0)i − u

(0)i

)

− ki

(

u(0)i v(0) − u

(0)i v(0)

)

=(

c− βi − kiv(0))

(

u(0)i − u

(0)i

)

+ kiu(0)i

(

v(0) − v(0))

≥ 0 in Ω, i = 1, . . . , N.

We also have by (5.14) and (5.17) with k = 1 that ∂n(u(1)i − u

(1)i ) = 0 on ∂Ω (i = 1, . . . , N).

The Maximum Principle then implies u(1)i ≥ u

(1)i in Ω for all i = 1, . . . , N. Similarly, we

have v(1) ≥ v(1) in Ω. We thus have proved

0 ≤ u(0)i ≤ u

(1)i ≤ u

(1)i ≤ u

(0)i in Ω, i = 1, . . . , N,

0 ≤ v(0) ≤ v(1) ≤ v(1) ≤ v(0) in Ω.

Assume now k ≥ 2 and

0 ≤ u(0)i ≤ · · · ≤ u

(k−1)i ≤ u

(k−1)i ≤ · · · ≤ u

(0)i in Ω, i = 1, . . . , N, (5.21)

0 ≤ v(0) ≤ · · · ≤ v(k−1) ≤ v(k−1) ≤ · · · ≤ v(0) in Ω. (5.22)

We prove

u(k−1)i ≤ u

(k)i ≤ u

(k)i ≤ u

(k−1)i in Ω, i = 1, . . . , N, (5.23)

v(k−1) ≤ v(k) ≤ v(k) ≤ v(k−1) in Ω. (5.24)

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

−Di∆(

u(k−1)i − u

(k)i

)

+ c(

u(k−1)i − u

(k)i

)

=(

c− βi − kiv(k−2)

)

(

u(k−2)i − u

(k−1)i

)

+ kiu(k−1)i

(

v(k−1) − v(k−2))

≥(

c− βi − kiv(0))

(

u(k−2)i − u

(k−1)i

)

≥ 0 in Ω, i = 1, . . . , N.

We also have by the boundary conditions (5.14) that ∂n(u(k−1)i − u

(k)i ) = 0 on ∂Ω (i =

1, . . . , N). The Maximum Principle now implies u(k−1)i ≥ u

(k)i in Ω for all i = 1, . . . , N.

14

Similarly, we have u(k−1)i ≤ u

(k)i in Ω (i = 1, . . . , N). By (5.16), (5.21), (5.22), and (5.11),

we have

−D∆(

v(k−1) − v(k))

+ c(

v(k−1) − v(k))

=

(

c− β −

N∑

i=1

kiu(k−2)i

)

(

v(k−2) − v(k−1))

+N∑

i=1

kiv(k−1)

(

u(k−1)i − u

(k−2)i

)

≥

(

c− β −N∑

i=1

kiu(0)i

)

(

v(k−2) − v(k−1))

≥ 0 in Ω.

Since ∂nv(k−1) = ∂nv

(k) on ∂Ω, we thus have v(k−1) ≥ v(k) in Ω. Similarly, v(k−1) ≤ v(k) in Ω.By (5.12), (5.15), (5.21), (5.22), and (5.11), we have

−Di∆(

u(k)i − u

(k)i

)

+ c(

u(k)i − u

(k)i

)

=(

c− βi − kiv(k−1)

)

(

u(k−1)i − u

(k−1)i

)

+ kiu(k−1)i

(

v(k−1) − v(k−1))

≥(

c− βi − v(0))

(

u(k−1)i − u

(k−1)i

)

≥ 0 in Ω, i = 1, . . . , N.

These and the corresponding boundary conditions for u(k)i and u

(k)i lead to u

(k)i ≤ u

(k)i in Ω

for all i = 1, . . . , N. By (5.13), (5.16), (5.21), (5.22), and (5.11), we have

−D∆(

v(k) − v(k))

+ c(

v(k) − v(k))

=

(

c− β −

N∑

i=1

kiu(k−1)i

)

(

v(k−1) − v(k−1))

+N∑

i=1

kiv(k−1)

(

u(k−1)i − u

(k−1)i

)

≥

(

c− β −N∑

i=1

kiu(0)i

)

(

v(k−1) − v(k−1))

≥ 0 in Ω.

This together with the fact that ∂nv(k−1) = ∂nv

(k−1) imply v(k) ≤ v(k) in Ω. 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 ob-tain by (5.19) and (5.20) uniformly bounded sequences of nonnegative C2,µ(Ω)-functions

u(k)i , u

(k)i , v(k), and v(k). By standard Holder estimates for elliptic problems, we

conclude that all the sequences ‖u(k)i ‖C2,µ(Ω) (1 ≤ i ≤ N), ‖u

(k)i ‖C2,µ(Ω) (1 ≤ i ≤ N),

‖v(k)‖C2,µ(Ω), and ‖v(k)‖C2,µ(Ω) are bounded.

Step 5. Convergence to solution. From Step 4, the sequences u(k)i and u

(k)i (1 ≤ i ≤

N), v(k), and v(k) are bounded in C2(Ω) and pointwise monotonic on Ω. Therefore, theyconverge pointwise to some functions ui and ui (1 ≤ i ≤ N), v, and v on Ω, respectively.

15

By the Arzela–Ascoli Theorem, there exist C2(Ω)-convergent subsequences u(kj)i ∞j=1 and

u(kj)i ∞j=1 (1 ≤ i ≤ N), v(kj)∞j=1, and v(kj)∞j=1, of u

(k)i and u

(k)i (1 ≤ i ≤ N),

v(k), and v(k), respectively. Clearly, these subsequences converge in C2(Ω) to ui (1 ≤i ≤ N), ui (1 ≤ i ≤ N), v, and v, respectively. Note that each of the subsequences

u(kj−1)i and u

(kj−1)i (1 ≤ i ≤ N), v(kj−1), and v(kj−1) also converges pointwise to

its respective limit. Now replace k in (5.12)–(5.17) by kj and sending j → ∞, we concludethat (u1, · · · , uN , v) and (u1, · · · , uN , v) are C2(Ω)-solutions of the system (5.1)–(5.3).

(2) This part can be proved similarly. The sequences of functions u(k)i and u

(k)i (1 ≤

i ≤ N), v(k), and v(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 pointwiseboundedness imply that these sequences converge, respectively, to some functions ui andui (1 ≤ i ≤ N), v, and v on Ω, all being nonnegative. Instead of using the Arzela–AscoliTheorem, we use the fact that H2(Ω) is a Hilbert space and use also Sobolev compact

embedding theorem to extract the subsequences u(kj)i ∞j=1 and u

(kj)i ∞j=1 (1 ≤ i ≤ N),

v(kj)∞j=1, and v(kj)∞j=1 that converge weakly in H2(Ω) and strongly in H1(Ω) to thepointwise limiting functions ui and ui (1 ≤ i ≤ N), v, and v, respectively. Finally, weuse the weak forms of the equations to pass to the limit. For instance, we have for anyφ ∈ H1(Ω) and all j = 1, 2, . . . that

∫

Ω

[

Di∇u(kj)i · ∇φ+ cu

(kj)i φ

]

dx

=

∫

Ω

[

cu(kj−1)i − βiu

(kj−1)i − kiu

(kj−1)i v(kj−1) + αi

]

φ dx, i = 1, . . . , N.

Sending j → ∞, we have by the H1(Ω)-convergence of u(kj)i ∞j=1 and the pointwise conver-

gence of u(k)i ∞k=1 and v(k)∞k=1 that

∫

Ω

[Di∇ui · ∇φ+ cuiφ]dx =

∫

Ω

[cui − βiui − kiuiv + αi]φ dx, i = 1, . . . , N.

Therefore, (u1, . . . , uN , v) and (u1, · · · , uN , v) are the weak (and nonnegative) solutions inH2(Ω) of the system (5.1)–(5.3).

6 Reaction-Diffusion System: Existence and Unique-

ness of Global Solution to Time-Dependent Problem

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

16

For any set Ω ⊆ R3 and any T > 0, we denote ΩT = Ω × (0, T ]. If Ω ⊆ R

3 is open,we also denote by C2

1(ΩT ) the class of functions u : ΩT → R of (x, t) that are continuouslydifferentiable 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 inR

3 with a C2-boundary ∂Ω. Let Di, βi, and ki (i = 1, . . . , N), D and β, and T be all positivenumbers. Let αi ∈ C1(ΩT ) (i = 1, . . . , N) and α ∈ C1(ΩT ) be all nonnegative functions onΩT . Assume ui 0 ∈ C1(Ω) (i = 1, . . . , N) and v0 ∈ C1(Ω) are all nonnegative functions onΩ. Then there exist unique ui ∈ C(ΩT ) ∩ C2

1(ΩT ) (i = 1, . . . , N) and v ∈ C(ΩT ) ∩ C21(ΩT )

such that all ui ≥ 0 (i = 1, . . . , N) and v ≥ 0 in ΩT and

∂ui

∂t= Di∆ui − βiui − kiuiv + αi in ΩT , i = 1, . . . , N, (6.1)

∂v

∂t= D∆v − βv −

N∑

i=1

kiuiv + α in ΩT , (6.2)

∂ui

∂n=

∂v

∂n= 0 on ∂Ω× (0, T ], i = 1, . . . , N, (6.3)

ui(·, 0) = ui 0 and v(·, 0) = v0 in Ω, i = 1, . . . , 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 u(0)i , v(0), u

(0)i , and v(0) (i = 1, . . . , N) on ΩT by

u(0)i ≥

‖αi‖L∞(ΩT )

βi

+ ‖ui 0‖L∞(Ω), v(0) ≥‖α‖L∞(ΩT )

β+ ‖v0‖L∞(Ω), u

(0)i = 0, v(0) = 0.

It is clear that

∂u(0)i

∂t−Di∆u

(0)i + βiu

(0)i + kiu

(0)i v(0) − αi ≥ 0 in ΩT , i = 1, . . . , N,

∂v(0)

∂t−D∆v(0) + βv(0) +

N∑

i=1

kiu(0)i v(0) − α ≤ 0 in ΩT ,

∂u(0)i

∂n= 0 and

∂v(0)

∂n= 0 on ∂Ω× (0, T ], i = 1, . . . , N,

u(0)i (·, 0) ≥ ui 0 and v(0)(·, 0) ≤ v0 in Ω, i = 1, . . . , N,

∂u(0)i

∂t−Di∆u

(0)i + βiu

(0)i + kiu

(0)i v(0) − αi ≤ 0 in ΩT , i = 1, . . . , N,

∂v(0)

∂t−D∆v(0) + βv(0) +

N∑

i=1

kiu(0)i v(0) − α ≥ 0 in ΩT ,

∂u(0)i

∂n= 0 and

∂v(0)

∂n= 0 on ∂Ω× (0, T ],

u(0)i (·, 0) ≤ ui 0 and v(0)(·, 0) ≥ v0 in Ω, i = 1, . . . , N.

17

Step 2. Iteration. Let

c = max

β +N∑

i=1

kiu(0)i , β1 + k1v

(0), . . . , βN + kN v(0)

. (6.5)

Define iteratively the functions u(k)i , v(k), u

(k)i , and v(k) (i = 1, . . . , N) on ΩT for k = 1, 2, . . .

by

∂u(k)i

∂t−Di∆u

(k)i + cu

(k)i = cu

(k−1)i − βiu

(k−1)i − kiu

(k−1)i v(k−1) + αi in ΩT , i = 1, . . . , N,

(6.6)

∂v(k)

∂t−D∆v(k) + cv(k) = cv(k−1) − βv(k−1) −

N∑

i=1

kiu(k−1)i v(k−1) + α in ΩT , (6.7)

∂u(k)i

∂n=

∂v(k)

∂n= 0 on ∂Ω× (0, T ], i = 1, . . . , N, (6.8)

u(k)i (·, 0) = ui 0 and v(k)(·, 0) = v0 in Ω, i = 1, . . . , N, (6.9)

∂u(k)i

∂t−Di∆u

(k)i + cu

(k)i = cu

(k−1)i − βiu

(k−1i − kiu

(k−1)i v(k−1) + αi in ΩT , i = 1, . . . , N,

(6.10)

∂v(k)

∂t−D∆v(k) + cv(k) = cv(k−1) − βv(k−1) −

N∑

i=1

kiu(k−1)i v(k−1) + α in ΩT , (6.11)

∂u(k)i

∂n=

∂v(k)

∂n= 0 on ∂Ω× (0, T ], i = 1, . . . , N, (6.12)

u(k)i (·, 0) = ui 0 and v(k)(·, 0) = v0 in Ω. (6.13)

The theory for initial-boundary-value problems of linear parabolic equations (cf. Theo-rem 2 in Chapter 5 of [3], or Theorem 1.2 in Chapter 2 of [18]) guarantees the existence of

solutions u(1)i , v(1), u

(1)i , and v(1) (i = 1, . . . , N) for k = 1 that are all in C(Ω)∩C2

1(ΩT ) and are

all Holder continuous in x uniformly in ΩT . Suppose u(k)i , v(k), u

(k)i , and v(k) (i = 1, . . . , N)

for k ≥ 1 exist, and are all in C(Ω) ∩ C21(ΩT ) and Holder continuous in x uniformly in

ΩT . Then the theory for initial-boundary-value problems of linear parabolic equations thenimplies that the solutions u

(k+1)i , v(k+1), u

(k+1)i , and v(k+1) (i = 1, . . . , N) all exist, are in

C(Ω) ∩ C21(ΩT ), and are Holder continuous in x uniformly in ΩT . By induction, we have

for all k = 1, 2, . . . the existence of the solutions u(k)i , v(k), u

(k)i , and v(k) (i = 1, . . . , N) in

C(Ω) ∩ C21(ΩT ) that are Holder continuous in x uniformly in ΩT .

In fact, there is a representation formula for our solutions. Suppose q ∈ C(ΩT ) is Holdercontinuous in x uniformly on ΩT and suppose g ∈ C1(Ω). Let u ∈ C(ΩT ) ∩ C2

1(ΩT ) satisfy

∂u

∂t−D∆u+ cu = q in ΩT , (6.14)

18

u(·, 0) = g(·) in Ω, (6.15)

∂u

∂n= 0 on ∂Ω× (0, T ]. (6.16)

Then we can extend g to a C1-function on a neighborhood of Ω as the boundary ∂Ω isof 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 8.3.2of [18]):

u(x, t) =

∫

Ω

Γ(x, t; ξ, 0)g(ξ) dξ +

∫ t

0

∫

Ω

Γ(x, t; ξ, τ)q(ξ, τ) dξ dτ

+

∫ t

0

∫

∂Ω

Γ(x, t; ξ, τ)Ψ(ξ, τ) dSξ dτ ∀(x, t) ∈ Ω× (0, T ], (6.17)

where

Γ(x, t; ξ, τ) = [4πD(t− τ)]−3/2e−c(t−τ)−|x−ξ|2/4D(t−τ), (6.18)

Ψ(x, t) = 2F (x, t) + 2∞∑

j=1

∫ t

0

∫

∂Ω

Mj(x, t; ξ, τ)F (ξ, τ) dSξdτ, (6.19)

F (x, t) =

∫

Ω

∂Γ(x, t; ξ, 0)

∂n(x, t)g(ξ) dξ +

∫ t

0

∫

Ω

∂Γ(x, t; ξ, τ)

∂n(x, t)q(ξ, τ) dξdτ,

M1(x, t; ξ, τ) = 2∂Γ(x, t; ξ, τ)

∂n(x, t),

Mj+1(x, t; ξ, τ) =

∫ t

0

∫

∂Ω

M1(x, t; y, σ)Mj(y, σ; ξ, τ) dSydσ.

Here the infinite series converges and the function F (x, t) is bounded.Step 3. Comparison. Notice by (6.5) that

c− β −N∑

i=1

kiu(0)i ≥ 0 and c− βi − kiv

(0) ≥ 0 in ΩT .

By the Maximum Principle for parabolic equations [2, 3, 9, 18] and using arguments similarto 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 = u(0)i ≤ u

(k)i ≤ u

(k+1)i ≤ u

(k+1)i ≤ u

(k)i ≤ u

(0)i in ΩT , (6.20)

0 = v(0) ≤ v(k) ≤ v(k+1) ≤ v(k+1) ≤ v(k) ≤ v(0) in ΩT . (6.21)

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

ui = limk→∞

u(k)i , v = lim

k→∞v(k), ui = lim

k→∞u(k)i , v = lim

k→∞v(k) in ΩT , i = 1, . . . , N.

19

These limits are nonnegative bounded measurable functions. In particular,

ui(x, 0) = ui(x, 0) = ui 0(x) (i = 1, . . . , N) and v(x, 0) = v(x, 0) = v0(x) ∀x ∈ Ω.

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

q(k)i = cu

(k−1)i − βiu

(k−1)i − kiu

(k−1)i v(k−1) + αi in ΩT ,

qi = cui − βiui − kiuiv + αi in ΩT . (6.22)

Note that each q(k) ∈ C(ΩT ) is Holder continuous in x uniformly in ΩT . Moreover,

limk→∞

q(k)i = qi in ΩT . (6.23)

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

∂u(k)i

∂t−Di∆u

(k)i + cu

(k)i = q

(k)i in ΩT , (6.24)

u(k)i (·, 0) = ui 0(·) in Ω, (6.25)

∂u(k)i

∂n= 0 on ∂Ω× (0, T ]. (6.26)

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

u(k)i (x, t) =

∫

Ω

Γ(x, t; ξ, 0)ui 0(ξ) dξ +

∫ t

0

∫

Ω

Γ(x, t; ξ, τ)q(k)i (ξ, τ) dξ dτ

+

∫ t

0

∫

∂Ω

Γ(x, t; ξ, τ)Ψ(k)i (ξ, τ) dSξ dτ ∀(x, t) ∈ Ω× (0, T ],

Ψ(k)i (x, t) = 2F

(k)i (x, t) + 2

∞∑

j=1

∫ t

0

∫

∂Ω

Mj(x, t; ξ, τ)F(k)i (ξ, τ) dSξdτ,

F(k)i (x, t) =

∫

Ω

∂Γ(x, t; ξ, 0)

∂n(x, t)ui0(ξ) dξ +

∫ t

0

∫

Ω

∂Γ(x, t; ξ, τ)

∂n(x, t)q(k)i (ξ, τ) dξdτ,

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

Since the sequence q(k)i ∞k=1 is uniformly bounded in ΩT and converges (cf. (6.23)), the

sequence F(k)i ∞k=1 is uniformly bounded in ∂Ω× (0, T ] and converges. Further, the series

in the expression of Ψ(k)i converges absolutely. Therefore, the sequence Ψ

(k)i ∞k=1 is also

uniformly bounded on ∂Ω × (0, T ] and converges. Let the limit be Ψi = Ψi(x, t). Takingthe limit as k → ∞ and using the Lebesgue Dominated Convergence Theorem, we obtainby (6.23) that

ui(x, t) =

∫

Ω

Γ(x, t; ξ, 0)ui 0(ξ) dξ +

∫ t

0

∫

Ω

Γ(x, t; ξ, τ)qi(ξ, τ) dξ dτ

20

+

∫ t

0

∫

∂Ω

Γ(x, t; ξ, τ)Ψi(ξ, τ) dSξ dτ, ∀(x, t) ∈ Ω× (0, T ], (6.27)

Ψi(x, t) = 2Fi(x, t) + 2∞∑

j=1

∫ t

0

∫

∂Ω

Mj(x, t; ξ, τ)Fi(ξ, τ) dSξdτ, (6.28)

Fi(x, t) =

∫

Ω

∂Γ(x, t; ξ, 0)

∂n(x, t)ui0(ξ) dξ +

∫ t

0

∫

Ω

∂Γ(x, t; ξ, τ)

∂n(x, t)qi(ξ, τ) dξdτ,

where Γ is given in (6.18) with D replaced by Di.Since ui 0 ∈ C1(Ω), the first term in (6.27) is a function of (x, t) in C2(ΩT ). Since qi is

bounded, the second term in (6.27) is also a continuous function of (x, t) on ΩT . By (6.28),the function Ψi(x, t) is continuous on ∂Ω× [0, T ]. Thus the third term in (6.27) and henceui = ui(x, t) is a continuous function in ΩT . Similarly, v = v(x, t) is a continuous functionin ΩT . Therefore, qi = qi(x, t) as defined in (6.22) is continuous in ΩT . Repeat the sameargument using (6.27) and (6.28), we have that ui is in fact Holder continuous in x uniformlyin ΩT . Similarly, v is Holder continuous in x uniformly in ΩT . Finally, qi is Holder continuousin x uniformly in ΩT . Therefore, we have the interior regularity of all ui (i = 1, . . . , N) andv; cf. [3] (Theorem 2 in Section 5.3). Now (6.22), (6.27), and (6.28) imply that ui, and v,solve (6.1) with u and v replaced by ui and v, respectively. The existence of solutions toother equations can be obtained similarly.

We now prove the uniqueness in three steps.Step 1. We prove that solutions (u1, . . . , uN , v) and (u1, . . . , uN , v) obtained above satisfy

ui = ui (i = 1, . . . , N) and v = v in ΩT .In fact, setting wi = ui − ui (i = 1, . . . , N) and w = v − v, we have

∂wi

∂t= Di∆wi − βiwi − kivwi + kiuiw in ΩT , i = 1, . . . , N,

∂w

∂t= D∆w − βw −

N∑

i=1

kiuiw +N∑

i=1

kiwiv in ΩT ,

∂wi

∂n=

∂w

∂n= 0 on ∂Ω× (0, T ], i = 1, . . . , N,

wi(·, 0) = 0 and w(·, 0) = 0 in Ω, i = 1, . . . , N.

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

Step 2. We prove the following: If (u∗1, . . . , u

∗N , v

∗) is any other solution to (6.1)–(6.4)

such that u(0)i ≤ u∗

i ≤ u(0)i (i = 1, . . . , N) and v(0) ≤ v∗ ≤ v(0) in ΩT , then ui = u∗

i = ui

(i = 1, . . . , N) and v = v∗ = v in ΩT .

In fact, let us replace (u(0)1 , . . . , u

(0)N , v(0)) by (u∗

1, . . . , u∗N , v

∗) and keep (u(0)1 , . . . , u

(0)N , v(0))

unchanged in the iteration in Step 2. Then, (u(k)1 , . . . , u

(k)N , v(k)) = (u∗

1, . . . , u∗N , v

∗) (k =

0, 1, . . . ), and the sequence (u(k)1 , . . . , u

(k)N , v(k)) (k = 0, 1, . . . ) remains unchanged and it

converges to the solution (u1, . . . , uN , v). Therefore, by the iteration we get ui ≤ u∗i (i =

1, . . . , N) and v∗ ≤ v in ΩT . A similar argument leads to u∗i ≤ ui (i = 1, . . . , N) and v ≤ v∗

21

in ΩT . These, together with the result proved in Step 1, imply ui = u∗i = ui (i = 1, . . . , N)

and v = v∗ = v in ΩT .

Step 3. If we have two nonnegative solutions defined on ΩT , then we can choose all u(0)i

(i = 1, . . . , N) and v(0) (in the iteration in Step 2 of proving existence) large enough tobound from above these solutions in ΩT . Then, both of these solutions must be the sameas those constructed by iterative upper and lower solutions. The uniqueness is thereforeproved.

Theorem 6.2 (Existence and uniqueness of global solution). Let Ω be a bounded domainin R

3 with a C2-boundary ∂Ω. Let Di (i = 1, . . . , N), D, βi (i = 1, . . . , N), β, and ki(i = 1, . . . , N) be all positive numbers. Let αi ∈ C1(Ω × [0,∞)) (i = 1, . . . , N) and α ∈C1(Ω × [0,∞)) be all nonnegative functions on Ω × [0,∞). Assume ui 0 ∈ C1(Ω) (i =1, . . . , N) and v0 ∈ C1(Ω) are all nonnegative functions on Ω. Then there exists a uniquenonnegative solution (u1, . . . , uN , v) to the system (1.1)–(1.4) with all ui (i = 1, . . . , N) andv being continuous on Ω × [0,∞) and continuously differentiable in t ∈ (0,∞) and twicecontinuously differentiable in x ∈ Ω.

Proof. Let Tm = m (m = 1, 2, . . . ). Then for each Tm, the system has a unique solutiondefined on ΩTm

. By the uniqueness of local solution, the solution corresponding to Tm andthat to Tn are identical on [0, Tm] if m ≤ n. Therefore, on each finite interval of time, allthe solutions are the same as long as they are defined on that interval. Hence we have theexistence of a global solution. It is unique since the local solution is unique.

7 Reaction-Diffusion System: Asymptotic Behavior

We now assume that all αi (i = 1, . . . , N) and α are independent of time t and considerthe initial-boundary-value problem of the full, time-dependent system of reaction-diffusionequations (1.1)–(1.4). Given the initial data ui0 (i = 1, . . . , N) and v0, the system has aunique global solution ui = ui(x, t) (i = 1, . . . , N) and v = v(x, t) by Theorem 6.2. We askif 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 specialcase for two equations; cf. Corollary 8.3.1 in [18]:

Proposition 7.1. Let Ω, T , and Di (i = 1, . . . , N) be all the same as in Theorem 6.1.Let ai,j ∈ C1(ΩT ) (i, j = 1, . . . , N) be such that ai,j ≥ 0 in ΩT if i 6= j. Suppose wi ∈C(ΩT ) ∩ C2

1(ΩT ) (i = 1, . . . , N) satisfy

∂wi

∂t−Di∆wi ≥

N∑

i,j=1

ai,j(x, t)wj in ΩT , i = 1, . . . , N,

∂wi

∂n= 0 on ∂Ω× (0, T ], i = 1, . . . , N,

wi(·, 0) ≥ 0 in Ω, i = 1, . . . , N.

Then wi ≥ 0 in ΩT (i = 1, . . . , N).

22

The following theorem states indicates particularly that if the initial values are largeconstant functions then the global solutions to the time-dependent problem are monotonicin 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 C2,µ

for some µ ∈ (0, 1). Let Di, βi, and ki (i = 1, . . . , N), and D and β be all the same asin Theorem 6.1. Let αi ∈ C1(Ω) (i = 1, . . . , N) and α ∈ C1(Ω) be all nonnegative on Ω.Let (U1, . . . , UN , V ) be the nonnegative solution to the system (1.1)–(1.4) with the initial

values U i(·, 0) = u(0)i (i = 1, . . . , N) and V (·, 0) = v(0) all being constant functions. Let

(U1, . . . , UN , V ) be the nonnegative solution to the system (1.1)–(1.4) with the initial values

U i(·, 0) = u(0)i (i = 1, . . . , N) and V (·, 0) = v(0) all being constant functions. Assume

u(0)i ≥ ‖αi‖L∞(Ω)/βi, v

(0) ≥ ‖α‖L∞(Ω)/β, and u(0)i = v(0) = 0 (i = 1, . . . , N). Then the

following hold true:(1) U i ≥ U i (i = 1, . . . , N) and V ≥ V in Ω× [0,∞).(2) U i (i = 1, . . . , N) and V are monotonically nonincreasing in t and U i (i = 1, . . . , N)

and V are monotonically nondecreasing in t.(3) Let

(U1,s, . . . , UN,s, V s) = limt→∞

(U1, . . . , UN , V ),

(U1,s, . . . , UN,s, V s) = limt→∞

(U1, . . . , UN , V ),

Then U i,s ≥ U i,s (i = 1, . . . , N) and V s ≥ V s on Ω. Moreover, all U i,s, U i,s, V s, and

V s are in C2,µ(Ω), and (U1,s, . . . , UN,s, V s) and (U1,s, . . . , UN,s, V s) are solutions ofthe time-independent system (5.1)–(5.3).

(4) If (u∗1,s, . . . , u

∗N,s, v

∗s) is any solution to (5.1)–(5.3) such that u

(0)i ≤ u∗

i,s ≤ u(0)i (i =

1, . . . , N) and v(0) ≤ v∗s ≤ v(0) on Ω, then U i,s ≤ u∗i,s ≤ U i,s (i = 1, . . . , N) and

V s ≤ v∗s ≤ V s on Ω.

Proof. (1) Let T > 0. Let Wi = U i − U i (i = 1, . . . , N) and W = V − V . We then have

∂Wi

∂t= Di∆Wi − βiWi − kiVWi + kiU iW in ΩT , i = 1, . . . , N,

∂W

∂t= D∆W − βW −

N∑

i=1

kiU iW +N∑

i=1

kiWiV in ΩT ,

∂Wi

∂n=

∂W

∂n= 0 on ∂Ω× (0, T ], i = 1, . . . , N,

Wi(·, 0) = u(0)i − u

(0)i ≥ 0 and W (·, 0) = v(0) − v(0) ≥ 0 in Ω, i = 1, . . . , N.

Therefore, we get by Proposition 7.1 that Wi ≥ 0 (i = 1, . . . , N) and W ≥ 0 in ΩT . SinceT > 0 is arbitrary, we obtain the desired inequality on Ω× [0,∞).

(2) Let T > 0 and δ > 0. Set W i(x, t) = U i(x, t) − U i(x, t + δ) (i = 1, . . . , N) andW (x, t) = V (x, t+ δ)− V (x, t). Then

∂W i(x, t)

∂t= Di∆W i(x, t)− βiW i(x, t)− kiV (x, t+ δ)W i(x, t) + kiU i(x, t)W (x, t)

23

∀(x, t) ∈ ΩT , i = 1, . . . , N,

∂W (x, t)

∂t= D∆W (x, t)− βW (x, t)−

N∑

i=1

kiU i(x, t+ δ)W (x, t) +N∑

i=1

kiW i(x, t)V (x, t)

∀(x, t) ∈ ΩT ,

∂W i

∂n=

∂W

∂n= 0 on ∂Ω× (0, T ], i = 1, . . . , N,

W i(·, 0) = u(0)i − U i(δ) ≥ 0 and W (·, 0) = V (δ)− v(0) ≥ 0 in Ω, i = 1, . . . , N.

Again by Proposition 7.1 we get W i ≥ 0 (i = 1, . . . , N) and W ≥ 0. Hence U i (i = 1, . . . , N)are monotonically nonincreasing in t and V is monotonically nondecreasing in t. Similarly, U i

(i = 1, . . . , N) are monotonically nondecreasing in t and V is monotonically nonincreasingin t.

(3) By Part (1) we have U i,s ≥ U i,s (i = 1, . . . , N) and V s ≥ V s on Ω. The claim that

(U1,s, . . . , UN,s, V s) and (U1,s, . . . , UN,s, V 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 ofTheorem 3.6 in [20].

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

Theorem 7.2. Let Ω, Di, D, βi, β, ki, αi, α, u(0)i , u

(0)i , v(0), v(0), U i, U i, V , and V

(i = 1, . . . , N) be all the same as in Theorem 7.1. Let ui0 ∈ C1(Ω) (i = 1, . . . , N) and

v0 ∈ C1(Ω) be such that v(0) ≤ ui 0 ≤ u(0)i and v(0) ≤ v0 ≤ v(0) (i = 1, . . . , N) in Ω.

Let (u1, . . . , uN , v) be the unique nonnegative global solution to the time-dependent problem(1.1)–(1.4) with the initial data (u10, . . . , uN0, v0). Then U i ≥ ui ≥ U i (i = 1, . . . , N) andV ≥ v ≥ V in Ω× [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 asymp-totic behavior of solution to the time-dependent problem.

Corollary 7.1. With the assumption of Theorem 7.2, the following hold true:(1) That U i,s = U i,s (i = 1, . . . , N) and V s = V s in Ω if and only if the steady-state

solution (u1,s, . . . , uN,s) ∈ (C2(Ω))N+1 satisfying u(0)i ≤ ui,s ≤ u

(0)i (i = 1, . . . , N) and

v(0) ≤ vs ≤ v(0) in Ω is unique.

(2) If the steady-state solution (u1,s, . . . , uN,s) ∈ (C2(Ω))N+1 that satisfies u(0)i ≤ ui,s ≤

u(0)i (i = 1, . . . , N) and v(0) ≤ vs ≤ v(0) in Ω is unique, then for any initial data

(u1 0, . . . , uN 0, v0) ∈ (C1(Ω))N+1 with v(0) ≤ ui 0 ≤ u(0)i (i = 1, . . . , N) and v(0) ≤ v0 ≤

v(0) in Ω, the corresponding nonnegative solution (u1, . . . , uN , v) of the time-dependentproblem (1.1)–(1.4) converges to (u1,s, . . . , uN,s, vs) as t → ∞.

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

24

Proof. (1) This follows immediately from Theorem 7.2.(2) This follows from Theorem 7.1 and Theorem 7.2.

(3) Choose u(0)i (i = 1, . . . , N) and v(0) all large enough and apply Part (2).

Corollary 7.2. With the same assumption as in Corollary 7.1, if U i,s 6= U i,s for some i

with 1 ≤ i ≤ N or V s 6= V s, then: (1) For any initial data (u1 0, . . . , uN 0, v0) ∈ (C1(Ω))N+1

with u(0)i ≤ ui 0 ≤ U i,s (i = 1, . . . , N) and V s ≤ v0 ≤ v(0) in Ω, the corresponding nonneg-

ative global solution (u1, . . . , uN , v) of the time-dependent problem (1.1)–(1.4) converges to(U1,s, . . . , UN,s, V s) as t → ∞; and (2) For any initial data (u1 0, . . . , uN 0, v0) ∈ (C1(Ω))N+1

with U i,s ≤ ui 0 ≤ u(0)i (i = 1, . . . , N) and v(0) ≤ v0 ≤ V s in Ω, the corresponding nonneg-

ative global solution (u1, . . . , uN , v) of the time-dependent problem (1.1)–(1.4) converges to(U1,s, . . . , UN,s, V s) as t → ∞.

Proof. This is similar to the proof of Theorem 7.1.

Acknowledgment. This work was supported by the San Diego Fellowship (M.E.H.) the USNational Science Foundation (NSF) through grant DMS-0811259 (B.L.) and DMS-1319731(B.L.), Center for Theoretical Biological Physics through NSF grant PHY-0822283 (B.L.),the National Institutes of Health (NIH) through grant R01GM096188 (B.L.), and Bei-jing Teachers Training Center for Higher Education (W.Y.). The authors thank ProfessorHerbert Levine for introducing to them the mean-field model for RNA interactions, andProfessor Daomin Cao and Professor Yuan Lou for helpful discussions. They also thank theanonymous referee for many helpful suggestions.

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