Invariant Regions and Global Existence of Solutions for Reaction-Diffusion Systems with a Tridiagonal Matrix of Diffusion Coefficients and Nonhomogeneous Boundary Conditions

The purpose of this paper is the construction of invariant regions in which we establish the global existence of solutions for reaction-di ﬀ usion systems (three equations) with a tridiagonal matrix of di ﬀ usion coe ﬃ cients and with nonhomogeneous boundary conditions after the work of Kouachi (2004) on the system of reaction di ﬀ usion with a full 2-square matrix. Our techniques are based on invariant regions and Lyapunov functional methods. The nonlinear reaction term has been supposed to be of polynomial growth. and

The initial data are assumed to be in the following region: where One will treat the first case, the others will be discussed in the last section.

Abdelmalek Salem 3
We suppose that the reaction terms f , g, and h are continuously differentiable, polynomially bounded on Σ satisfying for all v,w ≥ 0. And for positive constants for all u, v, w in Σ, where C 1 is a positive constant.
In the trivial case where b = c = 0, nonnegative solutions exist globally in time.This class of systems motivated us to construct this type of functionals considered in this paper in the aim to prove global existence of solutions.

Existence
In this section, we prove that if ( f ,g,h) points into Σ on ∂Σ, then Σ is an invariant region for problem (1.1)-(1.5),that is, the solution remains in Σ for any intial data in Σ.Once the invariant regions are constructed, both problems of the local and global existensce become easier to be established.

Invariant regions.
The main result of this subsection is the following.Proposition 2.1.Suppose that the functions f , g, and h point into the region Σ on ∂Σ, then for any (u 0 ;v 0 ,w 0 ) in Σ, the solution (u(t;•);v(t;•),w(t;•)) of the problem (1.1)- (1.5) remains in Σ for any time [0;T * [.Proof.The proof follows the same way to that of Kouachi (see [4]).Then we multiply (1.1) by c, (1.2) by √ 2bc, and (1.3) by b.Adding the first result to the third one and subtracting the second result, we get (2.2).Subtracting the first result from the third one, we get (2.3).Adding the first, the second, and the third results to each other, we get (2.4): ) with the boundary conditions and the initial data where (2.9) First, let us notice that the condition of the parabolicity of the system (1.1)-(1. 3) implies the one of the (2.2)-(2.4)system; since Abdelmalek Salem 5 Now, it suffices to prove that the region Since, from (1.12)-(1.14),we have that F(0,V ,W)≥0 suffices to be f ( , for all V ,W ≥ 0 and all v,w ≥ 0, then U(t,x) ≥ 0, for all (t,x) in ]0,T * [×Ω, thanks to the invariant region method (see Smoller [5]), and Because G(U,0,W) ≥ 0 suffices to be − f (μw,v,w) + μh(μw,v,w) ≥ 0 for all U,W ≥ 0 and all v,w ≥ 0, then V (t,x)≥0 for all (t,x) in ]0,T * [×Ω, and H(U,V ,0) ≥ 0 suffices to be f (− ≥ 0 for all U,V ≥ 0 and all v,w ≥ 0, then W(t,x) ≥ 0 for all (t,x) in ]0,T * [×Ω, then Σ is an invariant region for the system (1.1)-(1.3).
As it has been mentioned at the beginning of this section and since ρ 1 , ρ 2 , and ρ 3 are positive, for any initial data in C(Ω) and L p (Ω), p ∈ (1,+∞).(The local existence and uniquness of the solutions to the initial value problem (2.2)-(2.6)gives us directly those of (1.1)-(1.5).) Once invariant regions are constructed, one can apply Lyapunov technique and establish global existence of unique solutions for (1.1)-(1.5).

Global existence.
As the determinant of the linear algebraic system (2.7) with regard to variables u,v, and w, is different from zero, then to prove global existence of solutions of problem (1.1)-(1.5),one needs to prove it for problem (2.2)-(2.6).To this subject; it is well-known that (see Henry [2]) it sufficies to derive anuniform estimate of F(U,V ,W) p , G(U,V ,W) p , and H(U,V ,W) p on [0, T * [ for some p > N/2.
Proof of Theorem 2.2.Differentiating L with respect to t yields (2.18) Using Green's formula and applying Lemma 2.5, we get where where (2.20) for q = 0, p, p = 0,n − 2 and T = (∇U,∇V ,∇W) t .We prove that there exists a positive constant C 2 independent of t ∈ [0,T max [ such that and that for several boundary conditions.
Substituting the expressions of the partial derivatives given by Lemma 2.5 in the second integralyields H σ (p+1) θ (q+1) dx. (2.26) Using the expressions (2.8), we obtain bh σ (p+1) θ (q+1) dx. (2.27) And so, we get the following inequality: (2.28) using condition (1.15) and relation (2.7) successively, we get q p U q V p−q W (n−1)−p (U + V + W + 1)dx. (2.29) Following the same reasoning as in Kouachi [6], a straightforward calculation shows that which for Z = L 1/n can be written as (2.31) A simple integration gives the uniform bound of the functional L on the interval [0,T * ]; this ends the proof of the Theorem 2.2.
Proof of corollary.The proof of this corollary is an immediate consequence of Theorem 2.2 and the inequality for some p ≥ 1, taking into consideration expressions (2.7).

Proof of proposition.
As it has been mentioned above; it suffices to derive a uniform estimate of F(U,V ,W) p , G(U,V ,W) p , and H(U,V ,W) p on [0, T * [ for some p > N/2.Since the functions f , g, and h are polynomially bounded on Σ, then using relations (2.5), (2.7), and (2.8), we get that F, G, and H are polynomially bounded, too, and the proof becomes an immediate consequence of Corollary 2.3.

Final remarks
The second, the third, and the fourth cases are to be studied in the same way as we have done with the first case.
The second case:.If 3) can be rewritten as follows: with the same boundary conditions (1.4) and initial data (1.5).
In this case, the diffusion matrix of the system becomes Then all the previous results remain valid in the region where And system (2.2)-(2.4)becomes The third case: 3) can be rewritten as follows: with the same boundary conditions (1.4) and initial data (1.5).
In this case, the diffusion matrix of the system becomes Then all the previous results remain valid in the region where And systems (2.2)-(2.4)become where For homogeneous Neumann boundary conditions, we useλ = β i = 0, i = 1, 2, 3.(1.7)(iii) For homogeneous Dirichlet boundary conditions, we use1 − λ = β i = 0, i = 1, 2,3.(1.8)Here, Ω is an open bounded domain of class C 1 in R N , with boundary ∂Ω and ∂/∂η denotes the outward normal derivative on ∂Ω.The a, b, and c are positive constants satisfying the condition √ 2a ≥ (b + c) which reflects the parabolicity of the system and implies at the same time that the matrix of diffusion
Corollary 2.3.Suppose that the functions f , g, and h are continuously differentiable on Σ, point into ∂Σ and satisfy condition(1.15).Then all solutions of (1.1)-(1.5)withinitialdata in Σ and uniformly bounded on Ω are in L ∞ (0,T * ;L p (Ω)) for all p ≥ 1.Proposition 2.4.Under the hypothesis of Corollary 2.3, if the reactions f , g, and h are polynomially bounded, then all solutions of (1.1)-(1.5)with the initial data in Σ and uniformly bounded on Ω are global time.