Int. J. Anal. Appl. (2023), 21:30 Generalized Result on Global Existence of Weak Solutions for Parabolic Reaction-Diffusion Systems Abdelhamid Bennoui1,∗, Nabila Barrouk2, Mounir Redjouh1 1Department of Mathematics and Computer Science, University Center of Barika, Algeria 2Faculty of Science and Technology, Department of Mathematics and Informatics, Mohamed Cherif Messaadia University, P.O. Box 1553, Souk Ahras 41000, Algeria Laboratory of Mathematics, Algeria ∗Corresponding author: abdelhamid.bennoui@cu-barika.dz Abstract. In this paper, we study global existence of weak solutions for 2 × 2 parabolic reaction- diffusion systems with a full matrix of diffusion coefficients on a bounded domain, such as, we treat the main properties related: the positivity of the solutions and the total mass of the components are preserved with time. Moreover, we suppose that the non-linearities have critical growth with respect to the gradient. The technique used is based on compact semigroup methods and some estimates. Our objective is to show, under appropriate hypotheses, that the proposed model has a global solution with a large choice of non-linearities. 1. Introduction The study of reaction-diffusion systems (or systems of parabolic partial differential equations) was extensively developed in the literature, see for example in [10,13,26,29,30]. The question on the existence of solution for reaction-diffusion systems have long been a subject of active research like their global existence, their positivity, and some other qualitative properties. In the present paper, we study a mathematical model of reaction-diffusion system  ∂u ∂t −a∆u −b∆v = f (t,x,u,v,∇u,∇v) , in QT , ∂v ∂t −c∆u −d∆v = −f (t,x,u,v,∇u,∇v) , in QT , u = v = 0, or ∂u ∂η = ∂v ∂η = 0, in ΣT , u (0,x) = u0 (x) , v (0,x) = v0 (x) , in Ω, (1.1) Received: Feb. 9, 2023. 2020 Mathematics Subject Classification. 35K57, 35K40, 35J20, 35J25, 35J57. Key words and phrases. semigroups; local solution; global solution; reaction-diffusion systems. https://doi.org/10.28924/2291-8639-21-2023-30 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-30 2 Int. J. Anal. Appl. (2023), 21:30 where Ω is an open bounded subset of RN, with smooth boundary ∂Ω, QT = ]0,T [×Ω, ΣT = ]0,T [× ∂Ω, T > 0, and ∆ denotes the Laplacian operator on L1 (Ω) with Dirichlet or Neumann boundary conditions, the constants a, b, c and d are supposed to be positives, a ≤ d, and (b + c)2 ≤ 4ad which reflects the parabolicity of the system and implies at the same time that the matrix of diffusion A = ( a b c d ) , is positive definite, that is the eigenvalues λ1 and λ2 (λ1 < λ2) of its transposed are positives. We consider the problem (1.1) where we suppose the following hypotheses  ( 1 + a−λ1 c )( u − a−λ1 c v ) f (t,x,u,v,∇u,∇v) ≤ 0, ∀a−λ2 c v ≤ u ≤ a−λ1 c v, a.e. (t,x) ∈ QT , (1.2)   ( 1 + a−λ1 c ) f ( t,x, a−λ1 c v,v, a−λ1 c s,s ) ≤ 0,( 1 + a−λ2 c ) f ( t,x, a−λ2 c v,v, a−λ2 c s,s ) ≥ 0, for all ∀v ≥ 0, ∀s ∈RN, a.e. (t,x) ∈ QT , (1.3)   λ1−λ2 c f (t,x,u,v,∇u,∇v) ≤ L1 ( λ2−λ1 c v + 1 ) ∀a−λ2 c v ≤ u ≤ a−λ1 c v, a.e. (t,x) ∈ QT , (1.4) where L1 is a positive constant. Now, we condider the following hypotheses:{ f : ]0,T [ × Ω ×R2 ×R2N →R is measurable, f : R2 ×R2N →R is function locally Lipschitz continuous. (1.5) − ( 1 + a−λ1 c ) f (t,x,u,v,∇u,∇v) ≤ C1 (∣∣∣−u (t,x) + a−λ1c v (t,x)∣∣∣) × ( F1 (t,x) + ∣∣∣−∇u (t,x) + a−λ1c ∇v (t,x)∣∣∣2 + ∣∣∣∇u (t,x) − a−λ2c ∇v (t,x)∣∣∣α ) , (1.6) ( 1 + a−λ2 c ) f (t,x,u,v,∇u,∇v) ≤ C2 (∣∣∣−u (t,x) + a−λ1c v (t,x)∣∣∣ ,∣∣∣u (t,x) − a−λ2c v (t,x)∣∣∣) × ( G1 (t,x) + ∣∣∣−∇u (t,x) + a−λ1c ∇v (t,x)∣∣∣2 + ∣∣∣∇u (t,x) − a−λ2c ∇v (t,x)∣∣∣α ) , (1.7) where C1 : [0,∞) → [0,∞) , C2 : [0,∞)2 → [0,∞) are non-decreasing, F1,G1 ∈ L1 (QT ) and 1 ≤ α < 2. In the diagonal case (i.e. when b = c = 0), Alikakos [5] established global existence and L∞−bounds of solutions for positive initial data for f (u,v) = −uvσ, and 1 < σ < n + 2 n . Masuda [23] showed that the solutions to this system exist globally for every σ > 1 and converge to a constant vector as t → +∞. Int. J. Anal. Appl. (2023), 21:30 3 Haraux and Youkana [14] have generalized the method of Masuda to non-linearities f (u,v) = −uΨ (v) satisfying lim v→+∞ [log (1 + Ψ (v))] v = 0. In [24, 25], Moumeni and Barrouk have obtained a global existence result of solutions for reaction- diffusion systems with a diagonal and triangular matrix of diffusion coefficents. By combining the compact semigroup methods and some L1 estimates, we show that global solutions exist for a large class of the function f . Recently, Kouachi and Youkana [18] have generalized the method of Haraux and Youkana to the triangular case, i.e. when b = 0. In the same direction, Kouachi [17] has proved the global existence of solutions for two-component reaction-diffusion systems with a general full matrix of diffusion coefficients, non-homogeneous bound- ary conditions and polynomial growth conditions on non-linear terms and he obtained in [18] the global existence of solutions for the same system with homogeneous Neumann boundary conditions. Rebiai and Benachour [28] have treated the case of a general full matrix of diffusion coefficients with homogeneous boundary conditions and non-linearities of exponential growth. This article is a continuation of [3] where c,b 6= 0. In that article the calculations were relatively simple since the system can be regarded as a perturbation of the simple and trivial case where b = c = 0; for which non-negative solutions exist globally in time. In the present paper, to show global existence result for reaction-diffusion system with critical growth with respect to the gradient (m = 2), we truncate the system (1.1) then we give suitable estimates. To that end, we show the convergence of the approximating problem by using a technique introduced by Boccardo et al. [7] and Dall’aglio and Orsina [11]. 2. Existence Multiplying second equation of (1.1) one time through by a−λ1 c and subtracting first equation of (1.1) and another time by −a−λ2 c and adding first equation of (1.1) we get,  ∂w ∂t −λ1∆w = F (t,x,w,z,∇w,∇z) , in QT , ∂z ∂t −λ2∆z = G (t,x,w,z,∇w,∇z) , in QT , w = z = 0 or ∂w ∂η = ∂z ∂η = 0, in ΣT , w (0,x) = w0 (x) ≥ 0, z (0,x) = z0 (x) ≥ 0, in Ω, (2.1) where w (t,x) = −u (t,x) + a−λ1 c v (t,x) , z (t,x) = u (t,x) − a−λ2 c v (t,x) , (2.2) and   F (t,x,w,z,∇w,∇z) = − ( 1 + a−λ1 c ) f (t,x,u,v,∇u,∇v) , G (t,x,w,z,∇w,∇z) = ( 1 + a−λ2 c ) f (t,x,u,v,∇u,∇v) . (2.3) 4 Int. J. Anal. Appl. (2023), 21:30 2.1. Assumptions. Suppose that the hypotheses (1.2)-(1.7) are satisfied, then the problem (2.1) true the following hypotheses: • The non-linearities F,G have critical growth with respect to |∇w| , |∇z|. With respect to w,z. We assume that the hypotheses (1.2) are satisfied and we obtain,  ( 1 + a−λ1 c )( u − a−λ1 c v ) f (t,x,u,v,∇u,∇v) ≤ 0, ∀a−λ2 c v ≤ u ≤ a−λ1 c v, a.e. (t,x) ∈ QT , i.e.   − ( 1 + a−λ1 c )( −u + a−λ1 c v ) f (t,x,u,v,∇u,∇v) ≤ 0, ∀−u + a−λ1 c v ≥ 0, ∀u − a−λ2 c v ≥ 0, a.e. (t,x) ∈ QT , by (2.2)-(2.3), then F satisfies the sign condition wF (t,x,w,z,∇w,∇z) ≤ 0, ∀w,z ≥ 0, a.e. (t,x) ∈ Q. (2.4) Moreover, the following properties hold: w (t,x) = −u (t,x) + a−λ1 c v (t,x) , z (t,x) = u (t,x) − a−λ2 c v (t,x) , (2.5) • By (2.2){ w (t,x) = 0, if u (t,x) = a−λ1 c v (t,x) , and in the case z (t,x) = λ2−λ1 c v (t,x) , z (t,x) = 0, if u (t,x) = a−λ2 c v (t,x) , and in the case w (t,x) = λ2−λ1 c v (t,x) , from (1.3), we get that  − ( 1 + a−λ1 c ) f ( t,x, a−λ1 c v,v, a−λ1 c s,s ) ≥ 0,( 1 + a−λ2 c ) f ( t,x, a−λ2 c v,v, a−λ2 c s,s ) ≥ 0. for all ∀v ≥ 0, ∀s ∈RN, a.e. (t,x) ∈ QT ,{ F (t,x, 0,z, 0,s) ≥ 0, G (t,x,w, 0, r, 0) ≥ 0, ∀w,z ≥ 0, ∀r,s ∈RN, a.e. (t,x) ∈ QT . (2.6) • From (2.3), we obtain that F + G = ( −1 − a−λ1 c + 1 + a−λ2 c ) f (t,x,u,v,∇u,∇v) = λ1 −λ2 c f (t,x,u,v,∇u,∇v) and (2.2) is given w (t,x) + z (t,x) = −u (t,x) + a−λ1 c v (t,x) + u (t,x) − a−λ2 c v (t,x) = λ2 −λ1 c v, Int. J. Anal. Appl. (2023), 21:30 5 then by the hypotheses (1.4), we obtain that F + G ≤ L1 (w + z + 1) , ∀w,z ≥ 0, a.e. (t,x) ∈ QT , (2.7) where L1 is a positive constant. • Let us, now by (2.3) and (1.5) introduce for F and G the hypotheses F,G : ]0,T [ × Ω ×R2 ×R2N →R are measurable. (2.8) F,G : R2 ×R2N →R are locally Lipschitz continuous, (2.9) namely |F (t,x,w,z,p,q) −F (t,x,ŵ, ẑ, p̂, q̂)| + |G (t,x,w,z,p,q) −G (t,x,ŵ, ẑ, p̂, q̂)| ≤ K (r) (|w − ŵ| + |z − ẑ| + ‖p− p̂‖ + ‖q − q̂‖) for a.e. (t,x) ∈ QT and for all 0 ≤ |w| , |ŵ| , |z| , |ẑ| ,‖p‖ ,‖p̂‖ ,‖q‖ ,‖q̂‖≤ r. • By (2.2)-(2.3) and the hypotheses (1.6)-(1.7), we obtain that F (t,x,w,z,∇w,∇z) ≤ C1 (|w|) ( F1 (t,x) + |∇w|2 + |∇z|α ) , (2.10) G (t,x,w,z,∇w,∇z) ≤ C2 (|w| , |z|) ( G1 (t,x) + |∇w|2 + |∇z|α ) , (2.11) where C1 : [0,∞) → [0,∞) , C2 : [0,∞)2 → [0,∞) are non-decreasing function, F1,G1 ∈ L1 (QT ) and 1 ≤ α < 2. Let us now point out that if the non-linearities F and G do not depend on the gradient (system (2.1) is semi-linear), the existence of global positive solutions has been obtained by Haraux and Youkana [14], Hollis et al. [15], Hollis and Morgan [16], and Martin and Pierre [22]. One can see that in all of these works, the triangular structure, namely hypotheses (2.7) and F ≤ L2 (w + z + 1) , ∀w,z ≥ 0, a.e. (t,x) ∈ QT , (2.12) plays an important role in the study of semi-linear systems (in our case, hypothesis (2.12) is satisfied since by (2.4), F ≤ 0 whenever w,z ≥ 0). Indeed, if (2.7) or (2.12) does not hold, Pierre and Schmitt [27] have proved blow up in finite time of the solutions to some semi-linear reaction-diffusion systems. When F and G are depend on the gradient, Boudiba [8] has solved the case where the triangular structure is satisfied and the growth of F and G with respect to |∇w|, |∇z| is sub-quadratic{ ∃ 1 ≤ m < 2, C : [0,∞)2 → [0,∞) non-decreasing such that |F (w,z,∇w,∇z)| + |G (w,z,∇w,∇z)| ≤ C (|w| , |z|) [ 1 + |∇w|m + |∇z|m ] . About the critical growth with respect to the gradient (m = 2), we recall that for the case of a single equation (λ1 = λ2 and F = G), existence results have been proved for the elliptic case in [4,6,20]. The corresponding parabolic equations have also been studied by many authors, see for instance [1,7,11,21]. 6 Int. J. Anal. Appl. (2023), 21:30 3. Statement of the result First, we have to clarify in which sense we want to solve problem (2.1). Definition 3.1. We say that (w,z) is a solution of (2.1) if  w,z ∈ C ( [0,T ] ; L1 (Ω) ) ∩L1 ( 0,T ; W 1,10 (Ω) ) F (t,x,w,z,∇w,∇z) and G (t,x,w,z,∇w,∇z) ∈ L1 (QT ) w (t) = Sλ1 (t) w0 + ∫ t 0 Sλ1 (t − s) F (s, .,w (s) ,z (s) ,∇w (s) ,∇z (s)) ds, ∀t ≥ 0 z (t) = Sλ2 (t) z0 + ∫ t 0 Sλ2 (t − s) G (s, .,w (s) ,z (s) ,∇w (s) ,∇z (s)) ds, ∀t ≥ 0 (3.1) where Sλ1 (t) and Sλ2 (t) denote the semigroups in L 1 (Ω) generated by−λ1∆ and−λ2∆ with Dirichlet or Neumann boundary conditions. Example 3.1. A typical example where the result of this paper can be applied is  ∂w ∂t −λ1∆w = −wϕ (z) |∇w|2 in QT ∂z ∂t −λ2∆z = wϕ (z) |∇w|2 in QT w = z = 0 or ∂w ∂η = ∂z ∂η = 0 on ΣT w (0,x) = w0 (x) , z (0,x) = z0 (x) in Ω, where ϕ is a bounded function. 3.1. Main Result. Theorem 3.1. Assume that (2.4)-(2.11) hold. If w0,z0 ∈ L2 (Ω), then there exists a positive global solution (w,z) of system (2.1). Moreover, w,z ∈ L2 ( 0,T ; H10 (Ω) ) . Before giving the proof of the above theorem, let us define the following functions. Given a real positive number k, we set Tk (s) = max{−k, min (k,s)} , and Gk (s) = s −Tk (s) . We note that { Tk (s) = s for 0 ≤ s ≤ k, Tk (s) = k for s > k. 0 ≤ s ≤ k, Tk (s) = s and Tk (s) = k s > k. 4. PROOF OF THEOREM 3.1 To prove Theorem 3.1, we will use the results which we will present in this section. Int. J. Anal. Appl. (2023), 21:30 7 4.1. Preliminaries. Theorem 4.1. Let Ω is an open bounded domain in Rn, and X = L1 (Ω) ∩H2 (Ω). The operator A defined by   D (A) = { u ∈ L1 (Ω) ∩H2 (Ω) , ∂u ∂η = 0 or u = 0 on ∂Ω } Au = ∆u , for all u ∈ D (A) is m−dissipative in L1 (Ω) ∩H2 (Ω). An important result of functional analysis which ensures the local existence of the solution is the following lemma: Lemma 4.1. Let A be a m−dissipative operator of the dense domain in the Banach space X and S (t) a semigroup engendered by A, F a function locally Lipchitz. Then for any w0 ∈ X it exists T (w0) = Tmax such that the problem  w ∈ C ([0,T ] ,D (A)) ∩C1 ([0,T ] ,X) , dw dt −Aw = F (s, .,w (s) ,∇w (s)) , w (0) = w0, admits a unique solution w verifying w (t) = S (t) w0 + ∫ t 0 S (t − s) F (s, .,w (s) ,∇w (s)) ds, ∀t ∈ [0,Tmax] . 4.1.1. Compactness result. In this subsection we will give a compactness result of the operator L defining the solution of the problem (2.1) where the initial value is equal to zero, i.e. L (F ) (t) = w (t) = ∫ t 0 S (t − s) F (s, .,w (s) ,∇w (s)) ds, ∀t ∈ [0,T ] . Theorem 4.2. For all t > 0, if the operators S (t) are compact, then L are compact of L1 ([0,T ] ,X) in L1 ([0,T ] ,X) . Proof. Step 1: We show that S (λ) L : F → S (λ) L (F ) is compact in L1 ([0,T ] ,X), i.e. show that the set {S (λ) L (F ) (t) , ‖F‖1 ≤ 1} is relatively compact in L 1 ([0,T ] ,X). Since S (t) is compact then, the application t 7→ S (t) is continuous of ]0, +∞[ in L(X), therefore ∀ε > 0, ∀δ > 0, ∃η > 0, ∀0 ≤ h ≤ η, ∀t ≥ δ, ‖S (t + h) −S (t)‖L(X) ≤ ε We choose λ = δ, we have for 0 ≤ t ≤ T −h S (λ) w (t + h) −S (λ) w (t) = ∫ t+h 0 S (λ + t + h− s) F (s, .,w (s) ,∇w (s)) ds 8 Int. J. Anal. Appl. (2023), 21:30 − ∫ t 0 S (λ + t − s) F (s, .,w (s) ,∇w (s)) ds = ∫ t+h t S (λ + t + h− s) F (s, .,w (s) ,∇w (s)) ds + ∫ t 0 (S (λ + t + h− s) −S (λ + t − s)) F (s, .,w (s) ,∇w (s)) ds, from where ‖S (λ) w (t + h) −S (λ) w (t)‖X ≤ ∫ t+h t ‖F (s, .,w (s) ,∇w (s))‖X ds + ε ∫ t 0 ‖F (s, .,w (s) ,∇w (s))‖X ds. We define z (t) by z (t) = { w (t) if 0 ≤ t ≤ T 0 if not Therefore ‖S (λ) z (t + h) −S (λ) z (t)‖1 ≤ (h + εT )‖F (s, .,w (s) ,∇w (s))‖1 , which implies that all {S (λ) z, ‖F‖1 ≤ 1} is equi-integrable, then it is conventional that all {S (λ) L (F ) (t) , ‖F‖1 ≤ 1} is relatively compact in L 1 ([0,T ] ,X), this way S (λ) L is compact. Step 2: We show that S (λ) L → L when λ → 0, in L1 ([0,T ] ,X). We have S (λ) w (t) −w (t) = ∫ t 0 S (λ + t − s) F (s, .,w (s) ,∇w (s)) ds − ∫ t 0 S (t − s) F (s, .,w (s) ,∇w (s)) ds. So for t ≥ δ, we have ‖S (λ) w (t) −w (t)‖ ≤ ∫ t δ ‖S (λ + s) −S (s)‖L(X) ‖F (s, .,w (s) ,∇w (s))‖ds +2 ∫ t t−δ ‖F (s, .,w (s) ,∇w (s))‖ds. We choose 0 < λ < η, then ‖S (λ) w (t) −w (t)‖≤ ε ∫ t δ ‖F (s, .,w (s) ,∇w (s))‖ds + 2 ∫ t t−δ ‖F (s, .,w (s) ,∇w (s))‖ds, and for 0 ≤ t < δ, we have ‖S (λ) w (t) −w (t)‖≤ 2 ∫ t 0 ‖F (s, .,w (s) ,∇w (s))‖ds. Since F ∈ L1 (0,T,X), from where ‖S (λ) w (t) −w (t)‖≤ (εT + 2δ)‖F (s, .,w (s) ,∇w (s))‖1 . Therefore, if λ → 0 then S (λ) w → w into L1 ([0,T ] ,X), where the operator L is a uniform limit with compact linear operator between two Banach spaces, then L is compact in L1 ([0,T ] ,X). � Int. J. Anal. Appl. (2023), 21:30 9 Remark 4.1. The semigroup S (t) generated by the operator ∆ is compact in L1 (Ω). 4.2. Approximating Scheme. For every function h defined from R+ × Ω ×R2 ×R2N into R, we associate ĥ such that ĥ (t,x,w,z,p,q) =   h (t,x,w,z,p,q) if w,z ≥ 0 h (t,x,w, 0,p,q) if w ≥ 0,z ≤ 0 h (t,x, 0,z,p,q) if u ≤ 0,z ≥ 0 h (t,x, 0, 0,p,q) if w,z ≤ 0, and consider the system   ∂w ∂t −λ1∆w = F̂ (t,x,w,z,∇w,∇z) in QT ∂z ∂t −λ2∆z = Ĝ (t,x,w,z,∇w,∇z) in QT w = z = 0 or ∂w ∂η = ∂z ∂η = 0, on ΣT w (0,x) = w0 (x) , z (0,x) = z0 (x) in Ω. (4.1) It is obviously seen, by the structure of F̂ and Ĝ, that systems (2.1) and (4.1) are equivalent on the set where w,z ≥ 0. Consequently, to prove theorem 3.1, we have to show that problem (4.1) has a weak solution which is positive. To this end, we consider the truncated function ψn in C ∞ c (R) such that 0 ≤ ψn ≤ 1 and ψn (r) = { 1 if |r| ≤ n 0 if |r| ≥ n + 1, and the mollification with respect to (t,x) is defined as follows. Let ρ ∈ C∞c ( R×RN ) such that suppρ ⊂ B (0, 1) , ∫ ρ = 1, ρ ≥ 0 on R × RN and ρn (y) = nNρ (ny) . One can see that ρn ∈ C∞c ( R×RN ) , suppρn ⊂ B ( 0, 1 n ) , ∫ ρn = 1 and ρn ≥ 0 on R×RN. For all n > 0, we define the functions wn0 and zn0 by wn0 = min{w0,n}∈ C ∞ c (Ω) and zn0 = min{z0,n}∈ C ∞ c (Ω) It is clear that wn0 and zn0 are non-negative sequences and wn0 → w0, zn0 → z0, in L 2 (Ω) , and define for all (t,x,w,z,p,q) in R+ × Ω ×R2 ×R2N; Fn (t,x,w,z,p,q) = [ψn (|w| + |z| + ‖p‖ + ‖q‖) F (., .,w,z,p,q)] ∗ρn (t,x) Gn (t,x,w,z,p,q) = [ψn (|w| + |z| + ‖p‖ + ‖q‖) G (., .,w,z,p,q)] ∗ρn (t,x) 10 Int. J. Anal. Appl. (2023), 21:30 Note that these functions enjoy the same properties as F and G, moreover they are Hölder continuous with respect to t,x and |Fn| , |Gn| ≤ Mn, where Mn is a constant depending only on n (these estimates can be derived from (2.9), the properties of the convolution product, and the fact that ∫ ρn = 1. Let us now consider the truncated system  ∂wn ∂t −λ1∆wn = Fn (t,x,wn,zn,∇wn,∇zn) in QT ∂zn ∂t −λ2∆zn = Gn (t,x,wn,zn,∇wn,∇zn) in QT wn = zn = 0 or ∂wn ∂η = ∂zn ∂η = 0, on ΣT wn (0,x) = wn0 (x) , zn (0,x) = zn0 (x) in Ω. (4.2) 4.2.1. Local existence of the solution of problem (4.2). We transform the system (4.2) into a first order system in the Banach space X = L1 (Ω) ×L1 (Ω), we obtain  ∂ωn ∂t = Aωn + Ψ (t,x,ωn,∇ωn) , t > 0 ωn (0) = ωn0 = (wn0,zn0 ) ∈ X. (4.3) Here ωn = col(wn,zn), the operator A is defined as follows A = ( λ1∆ 0 0 λ2∆ ) where D (A) := {ωn = col (wn,zn) ∈ X : col (∆wn, ∆zn) ∈ X} and the function Ψ is defined by Ψ (t,x,ωn,∇ωn) = col (Fn (t,x,ωn,∇ωn) ,Gn (t,x,ωn,∇ωn)) with Dirichlet (wn = zn = 0) or Neumann ( ∂wn ∂η = ∂zn ∂η = 0) boundary conditions. Theorem 4.3. There exist TM > 0 and (wn,zn) a local solution of (4.3) for all t ∈ [0,TM] . Proof. We know that Sλ1 (t) , Sλ2 (t) are contraction semigroups and that Ψ is locally Lipschitz in ωn, then there exists TM > 0 such that (wn,zn) is a local solution of (4.3) on [0,TM] . � It remains to show the positivity of the solutions 4.2.2. Positivity of the solution of problem (4.2). The positivity of the solution is preserved with time, which is ensured by 2.6. Lemma 4.2. Let (wn,zn) be a classical solution of (4.2) and suppose that wn0,zn0 ≥ 0. Then wn,zn ≥ 0. Int. J. Anal. Appl. (2023), 21:30 11 Proof. Let w̄n = e−σtwn and z̄n = e−σtzn σ > 0. then ∂wn ∂t = eσt ( ∂w̄n ∂t + σw̄n ) ∂zn ∂t = eσt ( ∂z̄n ∂t + σz̄n ) Consequently By the problem (4.2), we have (w̄n, z̄n) is a solution of the system  ∂w̄n ∂t + σw̄n −λ1∆w̄n = e−σtFn (t,x,w̄n, z̄n,∇w̄n,∇z̄n) in QT ∂z̄n ∂t + σz̄n −λ2∆z̄n = e−σtGn (t,x,w̄n, z̄n,∇w̄n,∇z̄n) in QT w̄n = z̄n = 0 or ∂w̄n ∂η = ∂z̄n ∂η = 0, on ΣT w̄n (0,x) = wn0 (x) , z̄n (0,x) = zn0 (x) in Ω, (4.4) Let U0 = (t0,x0) be the minimum of w̄n on QT . We will show that w̄n (U0) ≥ 0 which will imply that w̄n ≥ 0 on QT and then wn ≥ 0 on QT . Suppose the contrary, namely w̄n (U0) < 0. By the properties of the minimum, we can ensure that U0 ∈ ]0,T ] × Ω and ∂w̄n ∂t (U0) = 0, ∇w̄n (U0) = 0, ∆w̄n (U0) ≥ 0 if 0 < t0 < T ∂w̄n ∂t (U0) ≤ 0, ∇w̄n (U0) = 0, ∆w̄n (U0) ≥ 0 if t0 = T. Hence the first equation in (4.4) yields σw̄n (U0) = − ∂w̄n ∂t (U0) + λ1∆w̄n (U0) + e −σt0Fn (U0, w̄n (U0) , z̄n (U0) , 0,∇z̄n (U0)) ≥ e−σt0Fn (U0, w̄n (U0) , z̄n (U0) , 0,∇z̄n (U0)) . Now we use the structure of w̄n (U0) and hypothesis (2.6) to write Fn (U0, w̄n (U0) , z̄n (U0) , 0,∇z̄n (U0)) = Fn (U0, 0, z̄n (U0) , 0,∇z̄n (U0)) ≥ 0. This implies that w̄n (U0) ≥ 0 which is impossible by the hypotheses. Arguing in the same way for the second component z̄n, we obtain the positivity of (wn,zn). � 4.2.3. Global existence of the solution of problem (4.2). The total mass of the components w,z is controlled with time, which is ensured by the following lemma. Lemma 4.3. There exists a constant M depending on ‖w0‖L1(Ω) , ‖z0‖L1(Ω) , L1, T and |Ω| such that ‖wn (t) + zn (t)‖L1(Ω) ≤ M, ∀t ∈ [0,T ] . (4.5) Proof. Of the first and second equation of (4.2) with: ∂ ∂t (wn + zn) − ∆ (λ1wn + λ2zn) = Fn + Gn. 12 Int. J. Anal. Appl. (2023), 21:30 The hypothesis (2.7) allowed the following estimate ∂ ∂t (wn + zn) − ∆ (λ1wn + λ2zn) ≤ L1 (wn + zn + 1) . let us integrate on Ω and apply the formula of Green, then∫ Ω ∆wn = 0, and ∫ Ω ∆zn = 0, (4.6) we find ∫ Ω ∂ ∂t (wn + zn) ≤ L1 ∫ Ω (wn + zn + 1) . so ∂ ∂t ∫ Ω (wn + zn) dx∫ Ω (wn + zn + 1) dx ≤ L1, Integrating this inequality on [0,t] , ∀t ∈ ]0,T ] yields ln ∫ Ω (wn + zn + 1) dx ∣∣∣∣t 0 ≤ L1t, thus ln ∫ Ω (wn (t) + zn (t) + 1) dx∫ Ω (wn0 + zn0 + 1) dx ≤ L1t, which implies ∫ Ω (wn (t) + zn (t) + 1) dx∫ Ω (wn0 + zn0 + 1) dx ≤ exp (L1t) then we have ∫ Ω (wn (t) + zn (t) + 1) dx ≤ exp (L1t) ∫ Ω (wn0 + zn0 + 1) dx also ∫ Ω (wn + zn) (t) dx ≤ ∫ Ω (wn (t) + zn (t) + 1) dx ≤ exp (L1t) ∫ Ω (wn0 + zn0 + 1) dx = exp (L1t) [∫ Ω (wn0 + zn0 ) dx + |Ω| ] ≤ exp (L1T ) [∫ Ω (w0 + z0) dx + |Ω| ] as if wn0 ≤ w0, zn0 ≤ z0 ≤ exp (L1T ) [ ‖w0‖L1(Ω) + ‖z0‖L1(Ω) + |Ω| ] . This ends the proof of the lemma. � We can conclude from this estimate that the solution (wn,zn) given by the Theorem 4.3 is a global solution. Int. J. Anal. Appl. (2023), 21:30 13 Lemma 4.4. There exists a constant R1 depending on T, ‖w0‖L1(Ω) , ‖z0‖L1(Ω) , L1, L2 and |Ω| such that ∫ QT |Fn (t,x,wn,zn,∇wn,∇zn)| + |Gn (t,x,wn,zn,∇wn,∇zn)| ≤ R1 Proof. Considering the equations satisfied by wn and zn, we can write −Fn = − ∂wn ∂t + λ1∆wn, and −Gn = − ∂zn ∂t + λ2∆zn. Integrating on QT and using (4.6), the positivity of the solutions yield − ∫ QT Fn ≤ ∫ Ω wn0. Hence by hypothesis (2.4) ∫ QT |Fn| = − ∫ QT Fn ≤ ∫ Ω w0. (4.7) Similarly, we get − ∫ QT Gn ≤ ∫ Ω z0. (4.8) Integrating on QT and by hypothesis (2.7) we get∫ QT Gn ≤− ∫ QT Fn + ∫ QT L1 (wn + zn + 1) Moreover, by (4.5) and (4.7) we have∫ QT Gn ≤ L1T (M + |Ω|) + ∫ Ω w0. (4.9) By (4.8) and (4.9) we conclude that∫ QT |Gn| ≤ L1T (M + |Ω|) + ∫ Ω w0. (4.10) By (4.7) and (4.10) we get∫ QT |Fn (t,x,wn,zn,∇wn,∇zn)| + |Gn (t,x,wn,zn,∇wn,∇zn)| ≤ R1 Let us put: R1 = L1T (M + |Ω|) + 2‖w0‖L1(Ω) � Lemma 4.5. (i) There exists a constant R2 depending on λ1, ‖w0‖L2(Ω) such that∫ QT |∇wn|2 ≤ R2, ∫ QT |∇Tk (wn)| 2 ≤ R2. (ii) There exists a constant R3 depending on λ1, λ2, L1, ‖w0‖L2(Ω) , ‖z0‖L2(Ω) , |Ω| such that∫ QT |∇zn|2 ≤ R3, ∫ QT |∇Tk (zn)| 2 ≤ R3. 14 Int. J. Anal. Appl. (2023), 21:30 (iii) There exists a constant R4 depending on λ1, λ2, T, ‖w0‖L2(Ω) , ‖z0‖L2(Ω) , L1, |Ω| such that∫ QT (2wn + zn) (|Fn (t,x,wn,zn,∇wn,∇zn)| + |Gn (t,x,wn,zn,∇wn,∇zn)|) ≤ R4. Proof. (i) We multiply the first equation in the truncated problem by wn and we integrate on QT . We obtain ∫ QT wn ∂wn ∂t −λ1 ∫ QT wn∆wn = ∫ QT Fnwn. Since, by hypothesis (2.4), wnFn ≤ 0, we have∫ QT |∇wn|2 ≤ 1 λ1 ∫ Ω (wn0 ) 2 ≤ 1 λ1 ∫ Ω (w0) 2 . Then ∫ QT |∇wn|2 ≤ R2, where R2 ≥ 1 λ1 ‖w0‖2L2(Ω) . We have ∫ QT |∇wn|2 = ∫ [wn 0) which we define for a function w ∈ L2 ( 0,T ; H10 (Ω) ) such that w (0) = w0 ∈ L2 (Ω). In the following we will denote by ω (ε) a quantity that tends to zero as ε tends to zero, and ωσ (ε) a quantity that tends to zero for every fixed σ as ε tends to zero. Lemma 4.7. Let (wn)n be a sequence in L 2 ( 0,T ; H10 (Ω) ) ∩C([0,T ]) such that wn (0) = wn0 ∈ L 2 (Ω) and ∂wn ∂t = ρ1,n + ρ2,n with ρ1,n ∈ L2 ( 0,T ; H−1 (Ω) ) and ρ2,n ∈ L1 (QT ) . Moreover assume that wn converges to w in L2 (QT ) , and wn0 converges to w (0) in L 2 (Ω) . Let Ψ be a function in C1([0,T ]) such that Ψ ≥ 0, Ψ′ ≤ 0, Ψ(T ) = 0. Let ϕ be a Lipschitz increasing function in C0(R) such that ϕ (0) = 0. Then for all k,γ > 0, 〈 ρ1,n, Ψϕ ( Tk (wn) −Tk (wm)γ )〉 + ∫ QT ρ2,nΨϕ ( Tk (wn) −Tk (wm)γ ) ≥ ωγ,n ( 1 m ) + ωγ ( 1 n ) + ∫ Ω Ψ (0) Φ ( Tk (w) −Tk (w)γ ) (0) dx − ∫ Ω Gk (w) (0) Ψ (0) ϕ ( Tk (w) −Tk (w)γ ) (0) dx where Φ(t) = ∫ t 0 ϕ(s)ds and Gk(s) = s −Tk(s). Proof. See N. Alaa and I. Mounir [3]. � Int. J. Anal. Appl. (2023), 21:30 19 Proof of Lemma 4.6. Let K be a measurable subset of QT . We have∫ K |Fn (t,x,wn,zn,∇wn,∇zn)| = ∫ K∩[wn>k] |Fn| + ∫ K∩[wn≤k,zn>k] |Fn| + ∫ K∩[wn≤k,zn≤k] |Fn| = I1 + I2 + I3 Using Lemma 4.5, we obtain for k large enough I1≤ R4 k ≤ ε 3 I2≤ R4 k ≤ ε 3 Now, using hypothesis (2.10), we write I3 ≤ ∫ K∩[wn≤k,zn≤k] C1 (|wn|) [ F1 (t,x) + |∇wn|2 + |∇zn|α ] Then I3 ≤ C1(k) [∫ K F1 (t,x) + ∫ K∩[wn≤k,zn≤k] |∇wn|2 + ∫ K∩[wn≤k,zn≤k] |∇zn|α ] The third integral can be controlled by using Hölder’s inequality for α < 2∫ K∩[wn≤k,zn≤k] |∇zn|α ≤ [∫ K |∇zn|2 α 2 |K| 2−α 2 ≤ R α 2 3 |K| 2−α 2 ] , where in the last inequality we used Lemma 4.5. Therefore I3 ≤ C1(k) [∫ K F1 (t,x) + R α 2 3 |K| 2−α 2 + ∫ K |∇Tk (wn)| 2 ] Similarly by hypothesis (2.11), we get∫ K |Gn| ≤ 1 k ∫ QT wn |Gn| + 1 k ∫ QT zn |Gn| + ∫ K∩[wn≤k,zn≤k] |Gn| ≤ ε 3 + ε 3 + C2(k,k) [∫ K G1 (t,x) + R α 2 3 |K| 2−α 2 + ∫ K |∇Tk (wn)| 2 ] . For the remaining term, we must prove that ( |∇Tk (wn)| 2 ) n is equi-integrable in L1 (QT ) . To do this we will show that Tk (wn) converges to Tk (w) in L2 ( 0,T ; H10 (Ω) ) ; more precisely we will show that lim n→∞ ∫ QT |∇Tk (wn) −∇Tk (w)| 2 = 0 Let k and γ be positive real numbers, let m ∈ N, and choose Ψ a test function as in Lemma 4.7, define ϕ by ϕ(s) = s exp ( βs2 ) , with β to be fixed later. We will use a technique introduced by Boccardo et al. [7], we will multiply the first equation in the truncated problem (4.2) by the function test Ψϕ ( Tk (wn) −Tk (wm)γ ) , then we will integrate on QT . Finally we will use Lemma 4.7 to get the result. 20 Int. J. Anal. Appl. (2023), 21:30 Since ∂wn ∂t = ρ1,n +ρ2,n, where ρ1,n = λ1∆wn ∈ L2 ( 0,T ; H−1 (Ω) ) and ρ2,n = Fn ∈ L1 (QT ) , we have by Lemma 4.7 ∫ QT ∂wn ∂t Ψϕ ( Tk (wn) −Tk (wm)γ ) ≥ ωγ,n ( 1 m ) + ωγ ( 1 n ) − ∫ Ω Ψ (0) Φ ( Tk (w) −Tk (w)γ ) dx − ∫ Ω Gk (w) (0) Ψ (0) ϕ ( Tk (w) −Tk (w)γ ) (0) dx Hence λ1 ∫ QT ∇wnΨϕ′ ( Tk (wn) −Tk (wm)γ ) ∇ ( Tk (wn) −Tk (wm)γ ) − ∫ QT FnΨϕ ( Tk (wn) −Tk (wm)γ ) ≤ ωγ,n ( 1 m ) + ωγ ( 1 n ) + ∫ Ω Ψ (0) Φ ( Tk (w) −Tk (w)γ ) + ∫ Ω Gk (w) (0) Ψ (0) ϕ ( Tk (w) −Tk (w)γ ) (0) ≤ ωγ,n ( 1 m ) + ωγ ( 1 n ) + ω ( 1 γ ) , since Tk (w)γ → Tk (w) strongly in L 2 ( 0,T ; H10 (Ω) ) . We have I = λ1 ∫ QT ∇wnΨϕ′ ( Tk (wn) −Tk (wm)γ ) ∇ ( Tk (wn) −Tk (wm)γ ) J = − ∫ QT FnΨϕ ( Tk (wn) −Tk (wm)γ ) . The term I can be written as I = λ1 ∫ QT ∇Tk (wn) Ψϕ′ ( Tk (wn) −Tk (wm)γ ) ∇ ( Tk (wn) −Tk (wm)γ ) +λ1 ∫ [wn≥k] ∇wnΨϕ′ ( Tk (wn) −Tk (wm)γ ) ∇ ( Tk (wn) −Tk (wm)γ ) = I1 + I2 For I2, we have I2 = −λ1 ∫ QT ∇wnΨϕ′ ( Tk (wn) −Tk (wm)γ ) ∇ ( Tk (wm)γ ) χ[wn≥k] = ωγ,n ( 1 m ) −λ1 ∫ QT ∇wnΨϕ′ ( Tk (wn) −Tk (w)γ ) ∇ ( Tk (w)γ ) χ[wn≥k] = ωγ,n ( 1 m ) −λ1 ∫ QT ∇wnΨϕ′ ( Tk (wn) −Tk (w)γ ) ×∇ ( Tk (w)γ ) χ[wn≥k]X[χ≥k] −λ1 ∫ QT ∇wnΨϕ′ ( Tk (wn) −Tk (w)γ ) ∇ ( Tk (w)γ ) χ[wn≥k]χ[wk] FnΨϕ ( Tk (wn) −Tk (w)γ ) − ∫ [wn≤k] FnΨϕ ( Tk (wn) −Tk (w)γ ) Then J ≥ ωγ,n ( 1 m ) − ∫ [wn≤k] FnΨϕ ( Tk (wn) −Tk (w)γ ) since ϕ ( Tk (wn) −Tk (w)γ ) ≥ 0 on [wn > k] , Ψ ≥ 0 and −Fn ≥ 0 by hypotheses (2.4). On the other hand ∣∣∣∫[wn≤k] FnΨϕ(Tk (wn) −Tk (w)γ)∣∣∣ ≤ C1 (k) ∫ [wn≤k] F1 (t,x) Ψ ∣∣ϕ(Tk (wn) −Tk (w)γ)∣∣ +C1 (k) ∫ [wn≤k] |∇zn| α Ψ ∣∣ϕ(Tk (wn) −Tk (w)γ)∣∣ +C1 (k) ∫ [wn≤k] |∇Tk (wn)| 2 Ψ ∣∣ϕ(Tk (wn) −Tk (w)γ)∣∣ We set J1 = C1(k) ∫ [wn≤k] F1 (t,x) Ψ ∣∣ϕ(Tk (wn) −Tk (w)γ)∣∣ = ωγ ( 1 n ) + ω ( 1 γ ) since α < 2, we have J2 = C1(k) ∫ [wn≤k] |∇zn|α Ψ ∣∣ϕ(Tk (wn) −Tk (w)γ)∣∣ = ωγ ( 1 n ) + ω ( 1 γ ) and J3 = C1(k) ∫ [wn≤k] |∇Tk (wn)| 2 Ψ ∣∣ϕ(Tk (wn) −Tk (w)γ)∣∣ = C1(k) ∫ [wn≤k] |∇(Tk (wn) −Tk (w))| 2 Ψ ∣∣ϕ(Tk (wn) −Tk (w)γ)∣∣ +2C1(k) ∫ [wn≤k] ∇Tk (wn)∇Tk (w) Ψ ∣∣ϕ(Tk (wn) −Tk (w)γ)∣∣ −C1(k) ∫ [wn≤k] |∇Tk (w)| 2 Ψ ∣∣ϕ(Tk (wn) −Tk (w)γ)∣∣ = ωγ ( 1 n ) + ω ( 1 γ ) +C1(k) ∫ [wn≤k] |∇(Tk (wn) −Tk (w))| 2 Ψ ∣∣ϕ(Tk (wn) −Tk (w)γ)∣∣ . Int. J. Anal. Appl. (2023), 21:30 23 Thus − ∫ [wn≤k] FnΨϕ ( Tk (wn) −Tk (w)γ ) ≥ ωγ ( 1 n ) + ω ( 1 γ ) −C1(k) ∫ [wn≤k] |∇(Tk (wn) −Tk (w))| 2 Ψ ∣∣ϕ(Tk (wn) −Tk (w)γ)∣∣ hence J ≥ ωγ ( 1 n ) + ω ( 1 γ ) −C1(k) ∫ [wn≤k] |∇(Tk (wn) −Tk (w))| 2 Ψ ∣∣ϕ(Tk (wn) −Tk (w)γ)∣∣ Then I + J ≤ ωγ,n ( 1 m ) + ωγ ( 1 n ) + ω ( 1 γ ) We conclude that ∫ QT Ψ |∇(Tk (wn) −Tk (w))| 2 λ1ϕ ′(Tk (wn) −Tk (w)γ) −C1(k) ∣∣ϕ(Tk (wn) −Tk (w)γ)∣∣ ≤ ωγ,n ( 1 m ) + ωγ ( 1 n ) + ω ( 1 γ ) . Now, choose β such that β ≥ C21 (k)/4λ 2 1. Then we have λ1ϕ ′(s) −C1(k)|ϕ(s)| > λ1 2 , and this ends the proof. � Consequently by (2.2), we proved the existence of global solutions to a reaction-diffusion system (1.1). Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi- cation of this paper. References [1] N. Alaa, Solutions Faibles d’Équations Paraboliques Quasi-linéaires Avec Données Initiales Mesures, Ann. Math. Blaise Pascal. 3 (1996), 1-15. http://www.numdam.org/item?id=AMBP_1996__3_2_1_0. [2] N. Alaa, S. Mesbahi, Existence Result for Triangular Reaction Diffusion Systems With L1 Data and Critical Growth With Respect to the Gradient, Mediterr. J. Math. 10 (2013), 255-275. https://doi.org/10.1007/ s00009-012-0238-9. [3] N. Alaa, I. Mounir, Global Existence for Reaction-Diffusion Systems with Mass Control and Critical Growth with Respect to the Gradient, J. Math. Anal. Appl. 253 (2001), 532-557. https://doi.org/10.1006/jmaa.2000.7163. [4] N.E. Alaa, M. Pierre, Weak Solutions of Some Quasilinear Elliptic Equations with Data Measures, SIAM J. Math. Anal. 24 (1993), 23-35. https://doi.org/10.1137/0524002. [5] N.D. Alikakos, Lp Bounds of Solutions of Reaction-Diffusion Equations, Commun. Part. Differ. Equ. 4 (1979), 827-868. https://doi.org/10.1080/03605307908820113. [6] A. Bensoussan, L. Boccardo, F. Murat, On a Non-Linear P.D.E. Having Natural Growth Terms and Unbounded Solutions, Ann. Inst. H. Poincaré, Anal. Non Linéaire. 5 (1988), 347-364. http://www.numdam.org/item?id=AMBP_1996__3_2_1_0 https://doi.org/10.1007/s00009-012-0238-9 https://doi.org/10.1007/s00009-012-0238-9 https://doi.org/10.1006/jmaa.2000.7163 https://doi.org/10.1137/0524002 https://doi.org/10.1080/03605307908820113 24 Int. J. Anal. Appl. (2023), 21:30 [7] L. Boccardo, F. Murat, J.P. Puel, Existence Results for Some Quasilinear Parabolic Equations, Nonlinear Anal.: Theory Meth. Appl. 13 (1989), 373-392. https://doi.org/10.1016/0362-546x(89)90045-x. [8] N. Boudiba, Existence globale pour des systèmes de réaction-diffusion paraboliques quasilinéaires, Thèse de troisième cycle, Université des Sciences et de la Technologie Houari Boumediene d’Alger, (1995). [9] H. Brezis, W. Strauss, Semi-Linear Second Order Elliptic Equations in L1, J. Math. Soc. Japan. 25 (1973), 565-590. https://doi.org/10.2969/jmsj/02540565. [10] N.F. Britton, Reaction-Diffusion Equations and Their Applications to Biology, Academic Press, London, (1986). [11] A. Dall’aglio, L. Orsina, Nonlinear Parabolic Equations With Natural Growth Conditions and L1 Data, Nonlinear Anal.: Theory Meth. Appl. 27 (1996), 59-73. https://doi.org/10.1016/0362-546x(94)00363-m. [12] T. Diagana, Some Remarks on Some Strongly Coupled Reaction-Diffusion Equations, (2003). https://doi.org/ 10.48550/ARXIV.MATH/0305152. [13] P.C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer Berlin Heidelberg, Berlin, Heidelberg, 1979. https://doi.org/10.1007/978-3-642-93111-6. [14] A. Haraux, A. Youkana, On a Result of K. Masuda Concerning Reaction-Diffusion Equations, Tohoku Math. J. (2). 40 (1988), 159-163. https://doi.org/10.2748/tmj/1178228084. [15] S.L. Hollis, R.H. Martin, Jr., M. Pierre, Global Existence and Boundedness in Reaction-Diffusion Systems, SIAM J. Math. Anal. 18 (1987), 744-761. https://doi.org/10.1137/0518057. [16] S.L. Hollis, J. Morgan, Interior Estimates for a Class of Reaction-Diffusion Systems from L1 a Priori Estimates, J. Differ. Equ. 98 (1992), 260–276. https://doi.org/10.1016/0022-0396(92)90093-3. [17] S. Kouachi, Invariant Regions and Global Existence of Solutions for Reaction-Diffusion Systems With Full Matrix of Diffusion Coefficients and Nonhomogeneous Boundary Conditions, Georgian Math. J. 11 (2004), 349-359. https://doi.org/10.1515/gmj.2004.349. [18] S. Kouachi, A. Youkana, Global Existence for a Class of Reaction-Diffusion Systems, Bull. Polish. Acad. Sci. Math. 49 (2001), 1-6. [19] O.A. Ladyzenskaya, V.A. Solonnikov, N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Trans. Math. Monographs. vol. 23, American Mathematical Society, Providence, RI, (1968). [20] R. Landes, Solvability of Perturbed Elliptic Equations With Critical Growth Exponent for the Gradient, J. Math. Anal. Appl. 139 (1989), 63-77. https://doi.org/10.1016/0022-247x(89)90230-8. [21] R. Landes, V. Mustonen, On Parabolic Initial-Boundary Value Problems With Critical Growth for the Gradient, Ann. Inst. H. Poincaré Anal. Non Linéaire. 11 (1994), 135-158. https://doi.org/10.1016/S0294-1449(16)30189-5. [22] R.H. Martin, M. Pierre, Nonlinear Reaction-Diffusion Systems, in: Nonlinear Equations in the Applied Sciences, Math. Sci. Eng. Academic Press, New York (1991). [23] K. Masuda, On the Global Existence and Asymptotic Behavior of Solutions of Reaction-Diffusion Equations, Hokkaido Math. J. 12 (1983), 360-370. https://doi.org/10.14492/hokmj/1470081012. [24] A. Moumeni, N. Barrouk, Existence of Global Solutions for Systems of Reaction-Diffusion With Compact Result, Int. J. Pure Appl. Math. 102 (2015), 169-186. https://doi.org/10.12732/ijpam.v102i2.1. [25] A. Moumeni, N. Barrouk, Triangular Reaction-Diffusion Systems With Compact Result, Glob. J. Pure Appl. Math. 11 (2015), 4729-4747. [26] J.D. Murray, Mathematical Biology, Springer-Verlag, New York, (1993). [27] M. Pierre, D. Schmitt, Existence Globale ou Explosion Pour les Systèmes De réaction-Diffusion Avec Contrôle de Masse, Thèse de Doctorat, Université Henri Poincaré, Nancy I, (1995). [28] B. Rebiai, S. Benachour, Global Classical Solutions for Reaction–diffusion Systems With Nonlinearities of Exponen- tial Growth, J. Evol. Equ. 10 (2010), 511-527. https://doi.org/10.1007/s00028-010-0059-x. [29] J. Smoller, Shock Waves and Reaction-Difussion Systems, Springer-Verlag, New York, (1983). https://doi.org/10.1016/0362-546x(89)90045-x https://doi.org/10.2969/jmsj/02540565 https://doi.org/10.1016/0362-546x(94)00363-m https://doi.org/10.48550/ARXIV.MATH/0305152 https://doi.org/10.48550/ARXIV.MATH/0305152 https://doi.org/10.1007/978-3-642-93111-6 https://doi.org/10.2748/tmj/1178228084 https://doi.org/10.1137/0518057 https://doi.org/10.1515/gmj.2004.349 https://doi.org/10.1016/0022-247x(89)90230-8 https://doi.org/10.1016/S0294-1449(16)30189-5 https://doi.org/10.14492/hokmj/1470081012 https://doi.org/10.12732/ijpam.v102i2.1 https://doi.org/10.1007/s00028-010-0059-x Int. J. Anal. Appl. (2023), 21:30 25 [30] A.I. Volpert, V.A. Volpert, Traveling Wave Solutions of Parabolic Systems, American Mathematical Society, Prov- idence, RI, (1994). 1. Introduction 2. Existence 2.1. Assumptions 3. Statement of the result 3.1. Main Result 4. PROOF OF THEOREM 3.1 4.1. Preliminaries 4.2. Approximating Scheme 4.3. Convergence References