ON THE SOLVABILITY OF A CLASS OF REACTION-DIFFUSION SYSTEMS ABDELFATAH BOUZIANI

We deal with a class of parabolic reaction-diffusion systems. We use an iterative process based on results obtained for a linearized problem, then we derive some a priori estimates to establish the existence, uniqueness, and continuous dependence of the weak solution for a class of quasilinear systems.


Introduction
Reaction-diffusion systems of PDEs furnish valuable mathematical models for a great number of phenomena in engineering and biology.For instance, the following system describes the dynamics of a simple isothermal chemical reaction system [25]: where h is a positive parameter.Moreover, the next system is a model for the description of the patchy distributions of microscopic aquatic organisms known as plankton (see, [17]): and (a,a 1 ,a 2 ,b 1 ,b 2 ) are positive parameters.Likewise, if u(x,t) represents the population density, v(x,t) the concentration of the attractant, F(u) and H(u,v) describe the local kinetics of the population and the attractant respectively, t is the time, and x is the onedimensional spatial variable, then the system represents a model for a population with attractant and has growth-diffusion-chemotaxis type.Some versions of this model were investigated in [18,19,23].For other models, we refer the reader, for instance, to [1, 5, 7-9, 13, 20, 24].The purpose of this paper is to study the following quasilinear reaction-diffusion parabolic system: ) where d 1 , d 2 are positive constants, Ω is an open bounded subset of R N , with smooth boundary ∂Ω, Q T = Ω × I, T > 0, and For the semilinear case of (1.5)-(1.7)(when the functions f 1 and f 2 do not depend on the gradient), the existence of positive solutions has been established in [10][11][12]16], under the following assumptions.
Assumption 1.1.The total mass of the components u, v is controlled with time, which is ensured by (1.9) A. Bouziani and I. Mounir 3 Note that if Assumption 1.1 or 1.2 does not hold, the authors in [21] have been proved blowup in finite time of the solutions to some semilinear reaction-diffusion systems.
As for the quasilinear case, it is showed in [4] the existence of positive weak solutions when the initial data are in L 1 under Assumptions 1.1 and 1.2.
Assumption 1.3.The positivity of the solution is preserved with time, which is ensured by (1.10) Assumption 1.4.The nonlinear term with respect to the gradient is subquadratic, namely, where A more general result has been obtained later when the initial data are in L 2 (see [2]).The authors in [2] have investigated problems (1.5)-(1.7)under Assumptions 1.1, 1.3 together with the following assumptions.
Assumption 1.6.The function f 1 satisfies the "sign condition" u f 2 (x,t,u,v,∇u,∇v) ≤ 0 ∀u, v ≥ 0, a.e.(x,t) ∈ Q T . (1.12) Note that for a single equation (d 1 = d 2 and f 1 = f 2 ), existence results have been obtained by many authors; see for instance [1,3,6,15].Finally, we mention that in order to establish the existence, many authors have used some regularizations in time and some truncation based on the so-called natural truncation T k defined by where k is a positive real number.The present paper can be considered as a continuation of works cited above, especially [2,4].Our main goal is to extend those results, in a certain sense.Namely, we will establish the existence, uniqueness, and continuous dependence of a weak solution of problems (1.5)-(1.7)without supposing Assumptions 1.1-1.6.We will consider only the following.
Assumption 1.7.The functions f i (i = 1,2) are bounded in L 2 and satisfy where L is a positive constant.

Solvability of a class of reaction-diffusion systems
The paper is organized as follows.In Section 2 we transform problems (1.5)-(1.7) to an equivalent one which is easier to analyze, and we make precise in which sense we solve the reduced problem.Then, in Section 3, we formulate an approximate problem.In Section 4, we derive some useful a priori estimates.Section 5 is devoted to establish the existence of a weak solution, while the uniqueness and continuous dependence of the solution are given in Section 6.

An equivalent problem
In this section, we will consider the linearization of (1.5)-(1.7)obtained by assuming that The sum of the two components u and v satisfies the following linear parabolic equation: (2.1) Consequently, the function u of problems (1.5)-(1.7)fulfills ( It is well known that (2.1) has a unique solution in L 2 (I;H 1 0 (Ω)) ∩ C(I;L 2 (Ω)) that satisfies ∂w/∂t ∈ L 2 (I;L 2 (Ω)).Then, if we show that u, solution of problem (2.2), exists and sets then problems (1.5)-(1.7)will be solved.Consider now the following auxiliary problem: (2.4) A. Bouziani and I. Mounir 5 where Since problem (2.4) possesses a unique solution, our objective is to solve problem (2.5).
Let us now define the notion of the solution we are looking for.
Definition 2.1.Say that z(x,t) is a weak solution of problem (2.5) if the following properties are verified: holds for all θ ∈ H 1 0 (Ω), and all t ∈ I.As functions f i (i = 1,2), the function f verifies the following.
Assumption 2.2.The function f is bounded in L 2 and satisfies the Lipschitz condition: (2.8)

Formulation of an approximate problem
Let {z n } n be a sequence constructed as follows.
For n = 0, we set z 0 (x,t) = 0 for all (x,t) ∈ Q T , the other terms of the sequence are obtained iteratively as solutions of the linear parabolic equation where It is well known that for any fixed "n," problem (3.1) has a unique solution z n in L 2 (I; 6 Solvability of a class of reaction-diffusion systems Set one can easily check that y n verifies where

A priori estimates
In this section, we will establish useful estimates on y n in some suitable spaces in order to prove the convergence of the sequence {z n } n to the solution of problem (3.1).To this end, we consider the weak formulation of problem (3.4), in which we set θ = y n and integrate over (0,τ) to obtain According to the Cauchy inequality, it follows The application of a lemma of Gronwall's type leads to Therefore, by omitting the second term on the left-hand side of (4.4) and applying Assumption 2.2 to the right-hand side, we get In light of the Friedrichs inequality [22], we have A. Bouziani and I. Mounir 7 It follows that where On the other hand, by virtue of (4.1) in which we set θ = ∂y n /∂t, it yields In light of the Cauchy inequality and Assumption 2.2, the right-hand side of (4.9) is then dominated by where the integral over ∂y n /∂t will be absorbed in the left-hand side of (4.9).Thanks to the Friedrichs' inequality the second term on the left-hand side of (4.9) is controlled from below by Therefore, we have . (4.12) The right-hand side here is independent of τ, hence replacing the left-hand side by its upper bound with respect to τ from 0 to T, thus we obtain implying finally 8 Solvability of a class of reaction-diffusion systems where Hence we can present the following theorem.

Convergence and existence result
(5.4) hence by passing to the limit, we have Since H 1 0 (Ω) L 2 (Ω), we have also (5.7) We have to prove that ϕ equals ∂ z/∂t in L 2 (I;L 2 (Ω)).To this end, we consider the identity Then, by passing to the limit in (5.8)Consequently, for Lc 3 < 1 the limit relation (5.3) is satisfied.
Theorem 5.2.Suppose that assumption of Theorem 4.1 is satisfied, moreover assume that f (x,t,0,0) ∈ L 2 (I;L 2 (Ω)).Then the limit function z = z(x,t) is the weak solution of problem (2.5) in the sense of Definition 2.1.
Proof.According to Theorem 5.1, assertions (i) and (ii) of Definition 2.1 are fulfilled.Moreover, from (5.9) we conclude that z(•,0) = 0 holds in L 2 (Ω), and so assertion (iii) is verified.It remains to prove that z satisfies the integral identity (iv).Since z n is the solution of (3.1), we have for all θ ∈ L 2 (I;H 1 0 (Ω)), which can be written (5.13) 10 Solvability of a class of reaction-diffusion systems If we show that z will be a solution of (2.5) in the sense of Definition 2.1.
For I 1 , we have (5.15) Thanks to (5.3) we obtain lim n→∞ I 1 = 0.For the remaining term I 2 , we use the Schwarz inequality and Assumption 2.2 to get (5.16) Therefore by passing to the limit, we obtain lim n→∞ I 2 = 0.This completes the proof of Theorem 5.2.

Uniqueness and continuous dependence
Theorem 6.1.Suppose that assumptions of Theorem 5.2 are fulfilled.Let z 1 and z 2 be two weak solutions of (2.5) in L 2 (I;H 1 0 (Ω)).Then where c 1 is defined by (4.8).
A. Bouziani and I. Mounir 11 Considering the weak formulation of problem (6.2) and performing a similar calculation to that for the establishment of estimate (4.7), we derive the desired result.
As a consequence of Theorem 6.1, we obtain the following.
Theorem 6.3.Suppose that u = u(x,t) and u * = u * (x,t) are two solutions corresponding to (u 0 , f ) and (u * 0 , f * ).Moreover, assume that for some continuous nonnegative function K(t) and certain constant L. Then for all t ∈ I, where Proof.Considering the weak formulation of problem (2.2) written for u, subtracting from it the same integral identity written for u * and putting θ = u − u * , and performing an integration by parts, we get 12 Solvability of a class of reaction-diffusion systems Integrating (6.7) over (0,t), applying the Cauchy inequality, and using assumption (6.4), we obtain Choosing ε = 2/3L 2 and applying Gronwall's lemma to the obtained inequality, we get estimate (6.5), which completes the proof.
On the other hand, in estimate (4.14), if Lc 2 < 1, we deduce that the series ∞ n=0 y n and thus the sequence {z n } n converge in the space B. According to the definition of space B (see Theorem 4.1), we deduce that Δu = f 1 (x,t,u,v,∇u,∇v) + F(x) in Q T , Δv = f 2 (x,t,u,v,∇u,∇v) + G(x) in Q T , u x,t i,0 = Φ i x,t i , v x,t i,0 = Ψ i x,t i for (x,t) ∈ Q i 0 , u = v = 0, on Σ T ,(7.3)or more generally for a quasilinear pluriparabolic system with nonlocal initial conditions Δu = f 1 (x,t,u,v,∇u,∇v) + F(x) in Q T ,