## Abstract

We prove in this work the existence of a unique global nonnegative classical solution to the class of reaction–diffusion systems uttx=aΔutx−guvm,vttx=dΔvtx+λtxguvm, where a>0, d>0, t>0,x∈Rn, n,m∈N∗, λ is a nonnegative bounded function with λt.∈BUCRn for all t∈R+, the initial data u0, v0∈BUCRn, g:BUCRn→BUCRn is a of class C1,dgudu∈L∞R, g0=0 and gu≥0 for all u≥0. The ideas of the proof is inspired from the work of Collet and Xin who proved the same result in the particular case d>a=1, λ=1,gu=u. Moreover, they showed that the L∞-norm of v can not grow faster than Olnlnt for any space dimension.

### Keywords

- reaction–diffusion systems
- local existence
- positivity
- comparison principle
- global existence

## 1. Introduction

In the sequel, we use the notations.

For

For

For

Let

For

For

Reaction-Diffuison equations are nonlinear parabolic partial differential equations arises in many fields of sciences like chemistry, physics, biology, ecology and even medicine. It appears usually as coupled systems.

The somewhat general form of these systems of two equations is

where

In this chapter, we are concerned with the existence of global solutions to the reaction–diffusion system

with initial data

Whe assume that.

** (H1)**The constants

** i**.

** ii**.

** (H5)**The initial data

One of the essential questions for (1)–(3) is the existence of global solutions and possibly bounds uniform in time. Recently, Collet and Xin in their paper [3] published in 1996 have studied the system (1)–(3) but with

If we replace

It is worth mentioning here the result of S. Badraoui [6] who studied the system

where

If

If

Our work here is a continuation of the work of Collet and Xin [3]. We treat the same question in a slightly general case. Inspired by the same ideas in [3] we prove that the system (1)–(3) under the assumptions (H1) to (H5) has a unique global nonnegative classical solution.

The chapter is organized as follows: In section 2, we treat the existence of local solution and reveal its positivity using the maximum principle.

In section 3, firstly, we prove by a simple comparison argument that if

We emphazise here that I have engaged to calculate the constants encountered in all equations and inequalities exactly.

## 2. Existence of a local solution and its positivity

We convert the system (1)–(3) to an abstract first order system in the Banach space

Here

where

It is known that for

Hence, the operator

where

Since the map

For the positivity, let

We can write the first equation as

for some

The second equation can be written as

By the same theorem we get

For the existence of a global solution, we use the contraposed of the characterization of the maximal existence time

## 3. Existence of a global solution

For this task we will use the fact that the solution is nonnegative.

** Theorem 3.1.**Let

Then, the solution is global and uniformly bounded on

** Proof.**By the comparison principle we get (13).

The solution

Here

Since

From (17) and (18) into (16) we get

This last inequality leads to the veracity of (14).

Thus, from (13) and (14), we deduce that the solution

In the case where

** Theorem 3.2.**Let

the solution

** Proof.**In this case, it is not easy to prove global existence. But can derive estimates of solutions independent of

We need some lemmas.

** Lemma 3.3.**Let

Then for any

where

** Proof.**Note that

Using integration by parts, we get

In fact, let

where

We have

From (27) we obtain

We pass to the limit for

By the same way we get

From (30) we find that

From (25), (29) and (31) into (24) we get our basic identity (22).

** Lemma 3.4.**There exist two positive real constants

where

** Proof.**We seek

and

for

The inequality (33) is satisfied if

From (23); (35), then (33) is satisfied if

Also, (34) is satisfied if

Whence, if

As a consequence of (33) we have

** Lemma 3.5.**With the functional

We have

where

** Proof.**Calulate

whence

Calulate

whence

Using the Cauchy-Schwarz inequality

We pove that

To do this, it sufficies to compute the discriminant of the trinoma in

From (45) into (42) we find the desired result (40).

** Lemma 3.6.**For

we have

where

and

P** roof.**We seek a constant

The inequality (50) is equivalent to

Whence, from (50) into (40) we obtain

As

then, from (51) and (52) we get

where

Now, let us estimate

As

then

Thus, from (54) in (53) we get the estimate (47) with

In the following step we trie to control the second component

Let

** Lemma 3.7.**Let

** Proof.**It’s obvious that

and

Then

By induction we prove that

Let

Taking

From (60), (61) and (63) into (62) we get (56).

** Lemma 3.8.**Let

with

such that

** Proof.**By Pythagorean theorem we have

As

Also, it’s clear that

where

It’s easy to prove that

Then

As

We have obviously

From (71) and (73) we get (66).

Let

Firstly, using the fact that

By Hölder ineguality with

As

and by (56) we have

where

Then, from (76) and (77) into (75)

where

On the other hand, we deduce from the right-hand inequality in (66) that

Then

We have from (79) and (82) into (74)

Whence

and finally we have for all

As

** Remark.**We can extend the system to the case where instead of

** i**.

** ii.**There exist two constants

In this more general case, by examining the proof of the theorem 3.2; we see that under the same assumptions above, the system has also a global nonnegative classical solution.

## 4. Illustrative example

To illustrate the previous study about global existence, we give the following reaction–diffusion system

where

## 5. Conclusion and perspectives

We have prouved in the case where

As perspectives, we will replace the function