# Global Solutions for anm-Component System of Activator-Inhibitor Type

Authors: S. Abdelmalek; A. Gouadria; A. Youkana
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper deals with a reaction-diffusion system with fractional reactions modelingm-substances into interaction following activator-inhibitor's scheme. The existence of global solutions is obtained via a judicious Lyapunov functional that generalizes the one introduced by Masuda and Takahashi.


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## Body

## 1. Introduction

In this paper, we are concerned with the existence of global solutions to a reaction-diffusion system withm components generalizing the activator-inhibitor system:
(1)∂tu1-a1Δu1=f1(u)=σ-b1u1+u1p11∏‍j=2mujp1j,∂tui-aiΔui=fi(u)=-biui+u1pi1∏‍j=2mujpij,i=2,…,m,x∈Ω,t>0,
supplemented with Neumann boundary conditions
(2)∂ui∂η=0,on∂Ω×{t>0},i=1,…,m,
and the positive initial data
(3)ui(x,0)=φi(x)onΩ,i=1,…,m.Hereu=(u1,u2,…,um), Ω is an open bounded domain of class C1 in ℝN, with boundary ∂Ω, and ∂/∂η denotes the outward normal derivative on ∂Ω.Throughout the paper, we make the following hypotheses:The indexesai, pij are nonnegative for all i,j=1,…,m, with σ>0:
(4)0<p11-1<maxk=2,3,…,m{pk1min{1,p1kpkk+1,p1jpkj,p1kpkk+1,p1jpkjj=2,…,m,j≠k}};
we set Aij=(ai+aj)/(2aiaj) for all i,j=1,…,m. Let αi,i=1,…,m, be positive constants such that
(5)α1>2max{1,∑‍i=1mbib1},(6)Sll>0,l=2,…,m,
where
(7)Slr=Sr-1r-1·Slr-1-[Hlr-1]2,r=3,…,l,Hlr=det1≤i,j≤l((ai,j)i≠l,…r+1j≠l-1,..r)∏‍k=1k=r-2(det[k])2(r-k-2),r=3,…,l-1,Sl2=α12αl2a1al[12αl-A1l2],Hl2=α12α2αla1a2al[α1-1α1A2l-A12A1l],
where det1≤i,j≤l((ai,j)i≠l,…,r+1,j≠l-1,…,r) stands for the determinant of the r-square symmetric matrix obtained from the matrix (ai,j)1≤i,j≤m by removing the (r+1)th,(r+2)th,…,lth rows and the rth,(r+1)th,…,(l-1)th columns, where det[1],…,det[m] are the minors of the matrix (aij)1≤i,j≤m. The elements of the matrix are as follows:
(8)a11=a1α1(α1-1),a1i=-α1αi(a1+ai)2,i=2,…,m,aii=aiαi(αi+1),i=2,…,m,aij=αiαj(ai+aj)2,i,j=2,…,m,i≠j.The main result of the paper is the following.Theorem 1.
Assume that condition (4) is satisfied. Let u be a solution of (1)–(3) with positive and bounded initial data, and let
(9)L(t)=∫Ω‍u1α1(t,x)∏‍j=2mujαj(t,x)dx.
Then the functional L is uniformly bounded on the interval [0,T*], T*<Tmax, where Tmax(∥φ1∥∞,∥φ2∥∞,…,∥φm∥∞) denotes the eventual blow-up time.Corollary 2.
Under the assumptions of Theorem1 all solutions of (1)–(3) with positive initial data in C(Ω¯) are global. If in addition b1,…,bm,  σ>0, then u is uniformly bounded in Ω¯×[0,∞).Before we prove our results, let us dwell a while on the existing literature concerning Gierer-Meinhardt's type systems.In 1972, following an ingenious idea of Turing [1], Gierer and Meinhardt [2] proposed a mathematical model for pattern formations of spatial tissue structures of hydra in morphogenesis. It is a system of reaction-diffusion equations of the form:
(10)∂tu-a1Δu=σ-μu+upvq,∂tv-a2Δv=-νv+urvs,x∈Ω,t>0,
with Neumann boundary conditions
(11)∂u∂η=∂v∂η=0,x∈∂Ω,t>0,
and initial conditions
(12)u(x,0)=φ1(x)>0,v(x,0)=φ2(x)>0,x∈Ω,
where Ω⊂ℝN (N=1,2,3 in practice) is a bounded domain with smooth boundary ∂Ω,a1,a2,μ,ν,σ>0,  and  p, q, r and s are non negative windexes with p>1. Here u is the activator, and v is the inhibitor.Global existence of solutions in(0,∞) was proved by Rothe [3], more than ten years after Gierer and Meinhardt's original paper with special choice of the parameters: p=2, q=1, r=2, s=0, and N=3. Masuda and Takahashi [4] were able to prove global estimates and bounds of the solution for Gierer and Meinhardt's system in its general form. They proceeded by first proving lower bounds, then Lp bounds (for any p>1), then uniform estimates and bounds in appropriate Sobolev spaces. The key point is represented by the Lp bounds, which are derived using in a subtle way the specific structure of the equations.Li et al. [5] also studied the activator-inhibitor model.Very recently, Bernasconi [6] considered the larger system:
(13)∂ta(x,t)=daaxx+a2(x,t)h(x,t)-μa(x,t)+ρ,∂th(x,t)=dhhxx(x,t)+a2(x,t)-νh(x,t)+εs(x,t),∂ts(x,t)=dssxx(x,t)+a(x,t)-κs(x,t),
and Meinhardt et al. [7] proposed activator-inhibitor models to describe a theory of biological pattern:
(14)∂ta(x,t)=daaxx+a2(x,t)h(x,t)s(x,t)-μa(x,t)+ρ,∂th(x,t)=dhhxx(x,t)+a2(x,t)-νh(x,t),∂ts(x,t)=dssxx(x,t)+a(x,t)-κs(x,t),
which is Gierer and Meinhardt's system supplemented with a third equation, where a(x,t) is the activator, h(x,t) is the inhibitor, and s(x,t) is a source that acts as an inhomogeneous inhibitor.Our paper generalizes the system in [5] to m-components.


## 2. Preliminary Observations and Notations

The usual norms in the spacesLp(Ω), L∞(Ω), and C(Ω¯) are denoted, respectively, by the following:
(15)∥u∥pp=1|Ω|∫Ω‍|u(x)|pdx,∥u∥∞=esssupx∈Ω|u(x)|,∥u∥C(Ω¯)=maxx∈Ω¯|u(x)|.It is well known that to prove global existence of solutions to (1)–(3), it suffices to derive a uniform estimate of ∥fi(u1,u2,…,um)∥p,  i=1,…,m on [0;Tmax) in the space Lp(Ω) for some p>n/2 (see Henry [8]).Since the functionsfi are continuously differentiable on ℝ+m for all i=1,…,m, then for any initial data in C(Ω¯), the system (1)–(3) admits a unique, classical solution (u1,u2,…,um) on (0,Tmax)×Ω with the alternative(i)
eitherTmax=∞;(ii)
orTmax<∞, and limt↗Tmax∑i=1m∥ui(t,·)∥∞=∞.Using the maximum principle, one derives the lower bounds of the components of the solutionu of (1)–(3):
(16)ui(t,x)≥e-bitmin(φi(x))>0,i=1,…,m.Our aim is to construct a Lyapunov functional that allows us to obtainLp-bounds on ui leading to global existence.


## 3. Preparatory Lemmas

For the proof of Theorem1, we need some preparatory lemmas whose proofs will be in the appendix.Lemma 3.
Assume that the constantsqij satisfy
(17)q11-1qk1<min{1,q1kqkk+1,q1jqkj,j=2,…,m,j≠k}.
Then for all hi-1,αi>0, j,i=1,…,m, there exist C=C(hi-1,αi)>0 and θ=θ(α1)∈(0,1), such that
(18)α1u1q11-1+α1∏‍j=2mujq1j+αj≤αku1qk1+α1ukqkk+1+αk∏‍j=2,j≠kmujqkj+αj+C(u1α1∏‍j=2mujαj)θ,u1≥0,  ui≥hi-1,  i=1,…,m,  k∈{2,…,m}.Lemma 4 (see [9]).
LetA=(aij)1≤i,j≤m. Then one has:
(19)Kmm=det[m]·∏‍k=1k=m-2(det[k])2(m-k-2),m>2,K22=det[2],
where
(20)Kml=Kl-1l-1·Kml-1-(Hml-1)2,l=3,…,m,(21)Hml=det1≤i,j≤m((ai,j)i≠(l+1),…,m,j≠l,..,(m-1))·∏‍k=1k=l-2(det[k])2(l-k-2),l=3,…,m-1,(22)Km2=a11amm-(a1m)2,(23)Hm2=a11a2m-a12a1m.Lemma 5.
Letα1>2max{1,∑i=1m‍bi/b1}. One has
(24)Kll>Sll,l=2,…,m,
where
(25)Klr=Kr-1r-1·Klr-1-[Hlr-1]2,r=3,…,l,Hlr=det1≤i,j≤l((ai,j)i≠l,…r+1j≠l-1,..r)·∏‍k=1k=r-2(det[k])2(r-k-2),r=3,…,l-1,Kl2=α12αl2a1al[α1-1α1αl+1αl-A1l2],Hl2=α12α2αla1a2al[α1-1α1A2l-A12A1l].Lemma 6 (see Masuda and Takahashi [4]).
Letμ,T>0 and let fj=fj(t) be a nonnegative integrable function on [0,T) and 0<θj<1  (j=1,…,J). Let W=W(t) be a positive function on [0,T) satisfying the differential inequality
(26)dW(t)dt≤-μW(t)+∑‍Jj=1fj(t)Wθj(t),0≤t<T.
Then, one has
(27)W(t)≤κ,0≤t<T,
where κ is the maximal root of the algebraic equation:
(28)x-∑‍Jj=1(sup0<t<T∫0t‍e-μ(t-ξ)fj(ξ)dξ)xθj=W(0).


## 4. Proofs

Proof of Theorem1.
Sinceu1 satisfies ∂tu1-a1Δu1>0 on {u1<σ/b1}, the maximum principle implies u1≥δ:=min(σ/b1,minu0(x))>0.
DifferentiatingL(t) with respect to t yields
(29)L′(t)=∫Ω‍ddt(u1α1∏‍j=2mujαj)dx=∫Ω‍(α1u1α1-1∏‍j=2mujαj∂tu1∑‍i=2m-∑‍i=2mαiu1α1uiαi+1∏‍j=2,j≠imujαj∂tui)dx.
Replacing ∂tui, i=1,…,m, by its expression from (1), we get
(30)L′(t)=∫Ω‍(a1α1u1α1-1∏‍j=2mujαjΔu1u1pi1+α1ukpii+1+αi∏‍j=2j≠imujpi,j+αj-∑‍i=2mαiaiu1α1uiαi+1∏‍j=2,j≠imujαjΔui-b1αu1α1∏‍j=2mujαj+∑‍i=2mbiαiu1α1∏‍j=2mujαj+α1u1p11-1+α1∏‍j=2mujp1j+αj-∑‍i=2mαiu1pi1+α1ukpii+1+αi∏‍j=2,j≠imujpi,j+αju1pi1+α1ukpii+1+αi∏‍j=2j≠imujpi,j+αj+σα1u1α1-1∏‍j=2mujαj)dx∶=I+J,
where we have set
(31)I=a1α1∫Ω‍u1α1-1∏‍j=2mujαjΔu1dx-∑‍i=2mαiai∫Ω‍u1α1uiαi+1∏‍j=2,j≠imujαjΔuidx,(32)J=(-b1α1+∑‍i=2mbiαi)L(t)+α1∫Ω‍u1p11-1+α1∏‍j=2mujp1j+αjdx-∑‍i=2mαi∫Ω‍u1pi1+α1ukpii+1+αi∏‍j=2,j≠imujpij+αjdx+σα1∫Ω‍u1α1-1∏‍j=2mujαjdx.Estimation ofI.We are going to show that I≤0.
Using Green's formula, we obtain(33)I=∫Ω‍(a1α1[-(α1-1)u1α1-2∏‍j=2mujαj|∇u1|2u1α1-1uiαi+1∏‍jmujαj+∑‍i=2mαiu1α1-1uiαi+1∏‍j=2,j≠imujαj∇u1∇ui]+∑‍mi=2aiαi·[α1u1α1-1uiαi+1∏‍j=2,j≠imujαj∇u1∇uiu1α1ukαk+1uiαi+1∏‍jmujαj-(αi+1)u1α1uiαi+2∏‍j=2,j≠imujαj|∇ui|2-∑‍k=2k≠imαk×u1α1ukαk+1uiαi+1∏‍j=2,j≠i,j≠kmujαju1α1ukαk+1uiαi+1∏‍jmujαj×∇uk∇ui])dx,=-∫Ω‍(u1α1-2∏‍j=2mujαj+2(QT)·T)dx,
where Q=(ai,j)1≤i,j≤m is defined in (8) and
(34)T=(∏‍mj=2uj∇u1,…,∏‍mj=1j≠iuj∇ui,…,∏‍m-1j=1uj∇um)t.
The matrixQ is positive definite if and only if all its associated minor matrices Δ1,Δ2,…,Δm are positive. To see this, we have the following.(1)
Δ1=a1α1(α1-1)>0. Using (5), we get det[1]>0.(2)
According to Lemma4, we have
(35)det[2]=K22=α12α22a1a2[α1-1α1α2+1α2-A122].Using (6) and (24)  for l=2, we get det[2]>0.(3)
Again according to Lemma4, we have
(36)K33=det[3]det[1].Butdet[1]>0, thus sign(K33)=sign(det[3]).Using (6) and (24) for l=3, we get det[3]>0.(4)
We suppose thatdet[k]>0, k=1,2,…,l-1 and prove that det[l]>0; thus
(37)det[k]>0,k=1,…,(l-1)⟹∏‍k=1k=l-2(det[k])2(l-k-2)>0.
From Lemma4,
(38)Kll=det[l]·∏‍k=1k=l-2(det[k])2(l-k-2).
This along with (37) yields
(39)sign(Kll)=sign(det[l]).
But from (6) and (24) Kll>0; thus det[l]>0.
Consequently, we haveI≤0.
Estimation of
J. We are going to estimate J by a function of L(t).
According to the maximum principle, there existsC0 depending on φi(x),  i=1,…,m, such that ui≥C0>0,  i=2,…,m. We then have
(40)u1α1-1∏‍j=2mujαj=(u1α1∏‍j=2mujαj)(α1-1)/α1∏‍mj=2(1uj)αj/α1≤(u1α1∏‍j=2mujαj)(α1-1)/α1(1C0)∑‍j=2mαj/α1,
whereupon
(41)u1α1-1∏‍j=2mujαj≤C2(u1α1∏‍j=2mujαj)(α1-1)/α1,whereC2=(1C0)∑‍j=2mαj/α1.
We have
(42)J≤(-b1α1+∑‍i=2mbiαi)L(t)+α1∫Ω‍u1q11-1+α1∏‍j=2mujq1j+αjdx-∑‍i=2mαi∫Ω‍u1qi1+α1ukqii+1+αi∏‍j=2,j≠imujqij+αjdx+σα1∫Ω‍C2(u1α1∏‍j=2mujαj)(α1-1)/α1dx.
Using Lemma 3, we obtain
(43)J≤(-b1α1+∑‍i=2mbiαi)L(t)+∫Ω‍C(u1α1∏‍j=2mujαj)θdx+σα1∫Ω‍C2(u1α1∏‍j=2mujαj)(α1-1)/α1dx.
Applying Hölder's inequality, we obtain
(44)∫Ω‍C(u1α1∏‍j=2mujαj)θdx≤(∫Ω‍u1α1∏‍j=2mujαjdx)θC(meas(Ω))1-θ.
So
(45)∫Ω‍C(u1α1∏‍j=2mujαj)θdx≤C3Lθ(t),C3=C(meas(Ω))1-θ.
Also, we have
(46)∫Ω‍C2(u1α1∏‍j=2mujαj)(α1-1)/α1dx≤(∫Ω‍(u1α1∏‍j=2mujαj)dx)(α1-1)/α1·(∫Ω‍(C2)α1dx)1/α1.
So
(47)∫Ω‍C2(u1α1∏‍j=2mujαj)(α1-1)/α1dx≤C4L(α1-1)α1(t),whereC4=C2(meas(Ω))1/α1.
We then get
(48)J≤(-b1α1+∑‍i=2mbiαi)L(t)+C3Lθ(t)+α1σC4L(α1-1)/α1(t),
which implies
(49)J≤(-b1α1+∑‍i=2mbiαi)L(t)+C5(Lθ(t)+α1σL(α1-1)/α1(t)).
This yields the differential inequality:
(50)L′(t)≤(-b1α1+∑‍i=2mbiαi)L(t)+C5(Lθ(t)+α1σL(α1-1)/α1(t)).
Thus under conditions (5), (6), and (8), we obtain -b1α1+∑i=2mbiαi<0; using Lemma 6 we deduce that L(t) is bounded on (0,Tmax); that is, L(t)≤γ1, where γ1 depends on φi(x), i=1,…,m.Proof of Corollary2 (L∞-bounds).
By Theorem1, we have u1pi1/∏‍j=2mujpij∈L∞((0,Tmax),Lr(Ω)),  i=2,…,m for all r>N/2. By a simple argument relying on the variation-of-constants formula and the Lp-Lq-estimate (Proposition 48.4 see [10]), we deduce that u is uniformly bounded. Consequently, Tmax=∞.



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