Abstract and Applied Analysis

Volume 2013 (2013), Article ID 939405, 9 pages

http://dx.doi.org/10.1155/2013/939405

Research Article

Global Solutions for an -Component System of Activator-Inhibitor Type

S. Abdelmalek,1,2 A. Gouadria,³ and A. Youkana³

¹Department of Mathematics, College of Sciences, Taibah University, Yanbu, Saudi Arabia

²Department of Mathematics, University of Tebessa, 12002 Tebessa, Algeria

³Department of Mathematics, University of Batna, 05000 Batna, Algeria

Received 15 May 2013; Accepted 17 July 2013

Academic Editor: Khalil Ezzinbi

Copyright © 2013 S. Abdelmalek et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper deals with a reaction-diffusion system with fractional reactions modeling -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.

1. Introduction

In this paper, we are concerned with the existence of global solutions to a reaction-diffusion system with components generalizing the activator-inhibitor system:

supplemented with Neumann boundary conditions

and the positive initial data

Here , is an open bounded domain of class in , with boundary , and denotes the outward normal derivative on .

Throughout the paper, we make the following hypotheses:

The indexes , are nonnegative for all , with :

we set for all . Let , be positive constants such that

where

where stands for the determinant of the -square symmetric matrix obtained from the matrix by removing the rows and the columns, where are the minors of the matrix . The elements of the matrix are as follows:

The main result of the paper is the following.

Theorem 1. Assume that condition (4) is satisfied. Let be a solution of (1)–(3) with positive and bounded initial data, and let

Then the functional is uniformly bounded on the interval , , where denotes the eventual blow-up time.

Corollary 2. Under the assumptions of Theorem 1 all solutions of (1)–(3) with positive initial data in are global. If in addition ,  , then is uniformly bounded in .

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:

with Neumann boundary conditions

and initial conditions

where ( in practice) is a bounded domain with smooth boundary ,  and  , , and are non negative windexes with . Here is the activator, and is the inhibitor.

Global existence of solutions in was proved by Rothe [3], more than ten years after Gierer and Meinhardt's original paper with special choice of the parameters: , , , , and . 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 bounds (for any ), then uniform estimates and bounds in appropriate Sobolev spaces. The key point is represented by the 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:

and Meinhardt et al. [7] proposed activator-inhibitor models to describe a theory of biological pattern:

which is Gierer and Meinhardt's system supplemented with a third equation, where is the activator, is the inhibitor, and is a source that acts as an inhomogeneous inhibitor.

Our paper generalizes the system in [5] to -components.

2. Preliminary Observations and Notations

The usual norms in the spaces , , and are denoted, respectively, by the following:

It is well known that to prove global existence of solutions to (1)–(3), it suffices to derive a uniform estimate of ,   on in the space for some (see Henry [8]).

Since the functions are continuously differentiable on for all , then for any initial data in , the system (1)–(3) admits a unique, classical solution on with the alternative

(i)either ;(ii)or , and .

Using the maximum principle, one derives the lower bounds of the components of the solution of (1)–(3):

Our aim is to construct a Lyapunov functional that allows us to obtain -bounds on leading to global existence.

3. Preparatory Lemmas

For the proof of Theorem 1, we need some preparatory lemmas whose proofs will be in the appendix.

Lemma 3. Assume that the constants satisfy

Then for all , , there exist and , such that

,  ,  ,  .

Lemma 4 (see [9]). Let . Then one has:

where

Lemma 5. Let . One has

where

Lemma 6 (see Masuda and Takahashi [4]). Let and let be a nonnegative integrable function on and (). Let be a positive function on satisfying the differential inequality

Then, one has

where is the maximal root of the algebraic equation:

4. Proofs

Proof of Theorem 1. Since satisfies on , the maximum principle implies .

Differentiating with respect to yields

Replacing , , by its expression from (1), we get

where we have set

Estimation of . We are going to show that .

Using Green's formula, we obtain

where is defined in (8) and

The matrix is positive definite if and only if all its associated minor matrices are positive. To see this, we have the following.

(1). Using (5), we get .(2) According to Lemma 4, we have

 Using (6) and (24) for , we get .(3) Again according to Lemma 4, we have

 But , thus . Using (6) and (24) for , we get .(4) We suppose that , and prove that ; thus

From Lemma 4,

This along with (37) yields

But from (6) and (24) ; thus .

Consequently, we have .

Estimation of . We are going to estimate by a function of .

According to the maximum principle, there exists depending on ,  , such that ,  . We then have

whereupon

We have

Using Lemma 3, we obtain

Applying Hölder's inequality, we obtain

So

Also, we have

So

We then get

which implies

This yields the differential inequality:

Thus under conditions (5), (6), and (8), we obtain ; using Lemma 6 we deduce that is bounded on ; that is, , where depends on , .

Proof of Corollary 2 (-bounds). By Theorem 1, we have ,   for all . By a simple argument relying on the variation-of-constants formula and the --estimate (Proposition 48.4 see [10]), we deduce that is uniformly bounded. Consequently, .

Appendix

The purpose of this appendix is to prove the lemmas of Section 3 which have been used in the proof of Theorem 1.

Proof of Lemma 3. Inequality (18) is equivalent to

Let us set .

Now, we write

For each such that ,

where

Using Young's inequality for (A.3) with

where is sufficiently small, we get inequality (18).

Proof of Lemma 4. We prove this lemma by induction.

For , we have .

We consider the case .

By using the well-known Dodgson condensation [11] for the symmetric -square matrix:

But

So

Hence by using formula (20), formula (19) is correct for .

When , we suppose that formula (19) is correct for , and we prove it for .

It is sufficient to prove that

By putting in formula (21), we get

From the mathematical induction proof, we have

By putting in formula (20), we get

By replacing (A.9), (A.10), and (A.11) in (A.12), we obtain

and thus formula (19) is correct for .

Now, we prove formula (A.9); we may generalize formula (A.9) as follows:

Also, we prove formula (A.14) by induction. It is a second inductive proof included in the first one.

It is evident for .

For , formula (20) will be:

Since we already know that

simple substitution of these three formulas in the formula (A.15) followed by the application of the modified well-known Dodgson condensation which has been modified in [11] will lead to formula (A.14) for . directly.

When , we suppose that formula (A.14) is correct for , and we prove it for .

Formula (20) for reads

According to the first induction, we have

According to the second induction, we have

According to formula (21), we have:

By replacing (A.18), (A.19), and (A.20) in (A.17) and by using the well-known Dodgson condensation, we obtain formula (A.14) for . Therefore, the second inductive proof is finished and consequently the first one.

Proof of Lemma 5. We prove this lemma by induction:

For , we have

Because

Assuming , we suppose (24) is true for , and we prove it for . Hence, we aim to prove

Recall that

It is then sufficient to prove

which will satisfy the inequality

In order to prove (A.26), we first generalize it in the form

This can be proven by mathematical induction. It is a secondary inductive proof inside the primary one. For , it is evident that

For , the formula

is evident too.

When , we suppose formula (A.28) true for :

and we prove it for :

We have

Then

Accordingly, we have

This finishes the proof.

Acknowledgment

The authors would like to express their deepest gratitude for Professor KIRANE (Université de la Rochelle, France) for all his valuable input and guidance throughout the research and authoring of this paper.

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