Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 110, pp. 1–28. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF GLOBAL SOLUTIONS AND BLOW-UP OF SOLUTIONS FOR COUPLED SYSTEMS OF FRACTIONAL DIFFUSION EQUATIONS BASHIR AHMAD, AHMED ALSAEDI, MOHAMED BERBICHE, MOKHTAR KIRANE Communicated by Jesus Idelfonso Diaz Abstract. We study the Cauchy problem for a system of semi-linear coupled fractional-diffusion equations with polynomial nonlinearities posed in R+×RN . Under appropriate conditions on the exponents and the orders of the fractional time derivatives, we present a critical value of the dimension N , for which global solutions with small data exist, otherwise solutions blow-up in finite time. Furthermore, the large time behavior of global solutions is discussed. 1. Introduction We consider the system CDγ1 0|tu−∆u = f(v), t > 0, x ∈ RN , CDγ2 0|tv −∆v = g(u), t > 0, x ∈ RN , (1.1) subject to the initial conditions u(0, x) = u0(x), v(0, x) = v0(x), x ∈ RN , (1.2) where 0 < γ1, γ2 < 1, for 0 < α < 1, CDα 0|tu denotes the Caputo time fractional derivative defined, for an absolutely continuous function u, by( CDα 0|tu ) (t) = 1 Γ(1− α) ∫ t 0 (t− s)−α∂tu(s, ·) ds, 0 < α < 1, where ∆ is the Laplace operator in RN . The functions f(v) and g(u) are the nonlinear source terms that will be determined later, and u0, v0 are given functions. Before we present our results and comment on them, let us dwell on existing results concerning the limiting case γ1 = γ2 = 1. Escobedo and Herrero [7] studied the existence of global solutions, and blowing-up of solutions for the system ut −∆u = vp, t > 0, x ∈ RN , v > 0, vt −∆v = uq, t > 0, x ∈ RN , u > 0. (1.3) 2010 Mathematics Subject Classification. 35A01, 35R09, 35K10, 45K05. Key words and phrases. Coupled fractional-diffusion equations; polynomial nonlinearities; global solution; blow-up. c©2020 Texas State University. Submitted May 1, 2019. Published November 2, 2020. 1 2 B. AHMAD, A. ALSAEDI, M. BERBICHE, M. KIRANE EJDE-2020/110 They showed, in particular, that for pq > 1, N 2 ≤ max{p, q}+ 1 pq − 1 , every nontrivial solution of (1.3) blows-up in a finite time T ∗ = T ∗(‖u‖∞, ‖v‖∞), in the sense that lim sup t→T∗ ‖u(t)‖∞ = lim sup t→T∗ ‖v(t)‖∞ = +∞. The work [7] has been followed by works of Escobedo and Herrero in a bounded domain, Escobedo and Levine [8] for more general nonlinear forcing terms, Uda [33], Fila, Levine and Uda [9] for differing diffusive coefficients, Lu [18], Lu and Sleeman [17], Mochizuki [22], Mochizuki and Huang [23], Takase and Sleeman [31, 32] , Samarskii et al. [29], and many other authors; see the review papers [4, 1, 24]. Time-fractional differential equations/systems for global or blowing-up solutions have been studied, for example, in [5, 6, 12, 14, 20, 21, 28, 34, 39]. Kirane, Laskri and Tatar [14] studied the more general system CDγ1 0|tu+ (−∆)β/2u = |v|p, t > 0, x ∈ RN , p > 1, CDγ2 0|tv + (−∆)γ/2v = |u|q, t > 0, x ∈ RN , q > 1, (1.4) (for the definition of (−∆)σ/2, 1 ≤ σ ≤ 2 see [14]) with nonnegative initial data, and proved the non-existence of global solutions under the condition pq > 1, N ≤ max { γ2 q + γ1 − (1− 1 pq ) γ2 γqp′ + γ1 βq′ , γ1 p + γ2 − (1− 1 pq ) γ1 βpq′ + γ2 γp′ } , where p+ p′ = pp′ and q + q′ = qq′. Here, we consider problem (1.1)-(1.2) and will give conditions relating the space dimension N with parameters γ1, γ2, p, and q for which the solution of (1.1)-(1.2) exists globally in time and satisfies L∞-decay estimates. We also discuss blowing- up in finite time solutions with initial data having positive average. Our study of the existence of global solutions relies on the semigroup theory, while for the blow-up of solutions result, we use the test function approach due to Zhang [41] and developed by Mitidieri and Pohozaev [24], and used by several authors (see for example [14, 10, 39]). Our result on blowing-up solutions improves the one obtained in [14]. We should mention that to the best of our knowledge there are no global existence and large time behavior results for the time-fractional diffusion system with two different fractional powers. The paper of Zhang et al. [40] does not treat the case of different time fractional operators. Also in [40], the authors do not obtain the decay rate of the solution in the space L∞(RN ). The rest of this article is organized as follows. In section 2, we present some preliminary lemmas. In section 3, we present the main results of this paper. Finally, section 4 and section 5 are devoted to the proofs of small data global existence and blow-up in finite time of the solutions of problem (1.1)-(1.2). Throughout this article, C will denote a positive constant. The space Lp(RN ) (1 ≤ p < ∞) will be equipped with the usual norm ‖u‖p Lp(RN ) = ∫ RN |u(x)|pdx. The space C0(RN ) denotes the set of all continuous functions decaying to zero at infinity, equipped with Chebychev’s norm ‖u‖∞. EJDE-2020/110 EXISTENCE OF GLOBAL SOLUTIONS 3 2. Preliminaries The left-sided and right-sided Riemann-Liouville integrals (see [30]), for Ψ ∈ L1(0, T ), 0 < α < 1, are defined as (Iα0|tΨ)(t) = 1 Γ(α) ∫ t 0 Ψ(σ) (t− σ)α−1 dσ, (Iαt|TΨ)(t) = 1 Γ(α) ∫ T t Ψ(σ) (σ − t)α−1 dσ, respectively, Γ stands for the Euler gamma function. The left-handed and right-handed Riemann-Liouville derivatives (see [30]), for Ψ ∈ AC1([0, T ]), 0 < α < 1, are defined as (Dα 0|tΨ)(t) = ( d dt ◦ I1−α 0|t Ψ)(t), (Dα t|TΨ)(t) = −( d dt ◦ I1−α t|T Ψ)(t), respectively. The Caputo fractional derivative for a function Ψ ∈ AC1([0, T ]) is defined by (CDα 0|tΨ)(t) = 1 Γ(1− α) ∫ t 0 Ψ′(σ) (t− σ)α dσ, (CDα t|TΨ)(t) = − 1 Γ(1− α) ∫ T t Ψ′(σ) (σ − t)α dσ. For 0 < α < 1 and Ψ ∈ AC1([0, T ]), we have( Dα 0|tΨ ) (t) = 1 Γ(1− α) [Ψ(0) tα + ∫ t 0 Ψ′(σ) (t− σ)α dσ ] , and ( Dα t|TΨ ) (t) = 1 Γ(1− α) [ Ψ(T ) (T − t)α − ∫ T t Ψ′(σ) (σ − t)α dσ ] . (2.1) The Caputo derivative is related to the Riemann-Liouville derivative by CDα 0|tΨ(t) = (Dα 0|t)(Ψ(t)−Ψ(0)), for Ψ ∈ AC1([0, T ]). Let 0 < α < 1, f ∈ AC1([0, T ]) and g ∈ AC1([0, T ]). Then∫ T 0 f(t)(Dα 0|tg)(t) dt = ∫ T 0 g(t)(CDα t|T f)(t) dt+ f(T )(I1−α 0|T g)(T ). If f(T ) = 0, then ∫ T 0 f(t)(Dα 0|tg)(t) dt = ∫ T 0 g(t)(CDα t|T f)(t) dt. For later use, let ϕ(t) = ( 1− t T )l + , for t ≥ 0, l ≥ 2. By a direct calculation, we obtain CDα t|Tϕ(t) = Γ(l + 1) Γ(l + 1− α) T−α ( 1− t T )l−α + , t ≥ 0. Now, we present some properties of two special functions. The two parameter Mittag-Leffler function [30] is defined for z ∈ C as Eα,β(z) = ∞∑ k=0 zk Γ(αk + β) , α, β ∈ C, <(α) > 0. 4 B. AHMAD, A. ALSAEDI, M. BERBICHE, M. KIRANE EJDE-2020/110 It satisfies I1−α 0|t (tα−1Eα,α(λtα)) = Eα,1(λtα) for λ ∈ C, 0 < α < 1. The Wright type function φα(z) = ∞∑ k=0 (−z)k k!Γ(−αk + 1− α) = 1 π ∞∑ k=0 (−z)kΓ(α(k + 1)) sin(π(k + 1)α) k! , for 0 < α < 1, is an entire function; it has the following properties: (a) φα(θ) ≥ 0 for θ ≥ 0 and ∫ +∞ 0 φα(θ)dθ = 1; (b) ∫ +∞ 0 φα(θ)θrdθ = Γ(1+r) Γ(1+αr) for r > −1; (c) ∫ +∞ 0 φα(θ)e−zθdθ = Eα,1(−z), z ∈ C; (d) α ∫ +∞ 0 θφα(θ)e−zθdθ = Eα,α(−z), z ∈ C. The operator A = −∆ with domain D(A) = {u ∈ C0(RN ) : ∆u ∈ C0(RN )}, generates, on C0(RN ), a semigroup {T (t)}t≥0, where T (t)u0(x) = ∫ RN G(t, x− y)u0(y)dy, G(t, x) = 1 (4πt)N/2 e−|x| 2/4t; it is analytic and contractive on Lq(RN ) [3] and, for t > 0, x ∈ RN , it satisfies ‖T (t)u0‖Lp(RN ) ≤ (4πt)− N 2 (1/q−1/p)‖u0‖Lq(RN ), (2.2) for 1 ≤ q ≤ p ≤ +∞. Let the operators Pα(t) and Sα(t) be defined by Pα(t)u0 = ∫ ∞ 0 φα(θ)T (tαθ)u0dθ, t ≥ 0, u0 ∈ C0(RN ), (2.3) Sα(t)u0 = α ∫ ∞ 0 θφα(θ)T (tαθ)u0dθ, t ≥ 0, u0 ∈ C0(RN ). (2.4) The operators Pα(t) and Sα(t) acting on the space C0(RN ) into itself, see [39, Lemma 2.3, Lemma 2.4] Consider the problem CDα 0|tu−∆u = f(t, x), t > 0, x ∈ RN , u(0, x) = u0(x), x ∈ RN , (2.5) where u0 ∈ C0(RN ) and f ∈ L1((0, T ), C0(RN )). If u is a solution of (2.5), then by [39], it satisfies u(t, x) = Pα(t)u0(x) + ∫ t 0 (t− s)α−1Sα(t− s)f(s, x) ds. The following lemmas play an important role in obtaining the results of this paper; their proofs are obtained by combining smoothing effect of the heat semigroup property (2.2) with formulas (2.3) and (2.4) (see [39]). Lemma 2.1. The operator {Pα(t)}t>0 has the following properties: (a) If u0 ≥ 0, u0 6≡ 0, then Pα(t)u0 > 0 and ‖Pα(t)u0‖L1(RN ) = ‖u0‖L1(RN ); EJDE-2020/110 EXISTENCE OF GLOBAL SOLUTIONS 5 (b) If p ≤ q ≤ +∞ and 1/r = 1/p− 1/q, 1/r < 2/N , then ‖Pα(t)u0‖Lq(RN ) ≤ (4πtα)− N 2r Γ(1−N/(2r)) Γ(1− αN/(2r)) ‖u0‖Lp(RN ). Lemma 2.2. For the operator family {Sα(t)}t>0, we have the following estimates: (a) If u0 ≥ 0 and u0 6≡ 0, then Sα(t)u0 > 0 and ‖Sα(t)u0‖L1(RN ) = 1 Γ(α) ‖u0‖L1(RN ); (b) If p ≤ q ≤ +∞ and 1/r = 1/p− 1/q, 1/r < 4/N , then ‖Sα(t)u0‖Lq(RN ) ≤ α(4πtα)− N 2r Γ(1−N/(2r)) Γ(1 + α− αN/(2r)) ‖u0‖Lp(RN ). Lemma 2.3. Let l ≥ 1, and let the function f(t, x) satisfy ‖f(t, ·)‖l ≤ { C1, 0 ≤ t ≤ 1, C2t −α, t > 1, for some positive constants C1, C2 and α. Then ‖f(t, ·)‖l ≤ max{C1, C2}(1 + t)−β , for all 0 < β ≤ α and t ≥ 0. Proof. For 0 ≤ t ≤ 1, we have ‖f(t, ·)‖l ≤ C1 ≤ C12α(1 + t)−α, so ‖f(t, ·)‖l ≤ K(1 + t)−β , for some positive constant K > 0, and for all 0 < β ≤ α. When t ≥ 1, it follows from ‖f(t, ·)‖l ≤ C2t −α that there is a constant K ′ > 0, such that ‖f(t, ·)‖l ≤ K ′(1 + t)−α, and so for all 0 < β ≤ α and any t ≥ 1, we have ‖f(t, ·)‖l ≤ K ′(1 + t)−β , for 0 < β ≤ α. Therefore ‖f(t, ·)‖l ≤ max{K,K ′}(1 + t)−β , for all 0 < β ≤ α and t ≥ 0. � 3. Main results In this section, we state our main result. First, we present the definition of a mild solution of problem (1.1)-(1.2). Definition 3.1. Let (u0, v0) ∈ C0(RN ) × C0(RN ), 0 < γ1, γ2 < 1, p, q ≥ 1 and T > 0. We say that (u, v) ∈ C([0, T ];C0(RN )×C0(RN )) a mild solution of system (1.1)-(1.2) if (u, v) satisfies the integral equations u(t, x) = Pγ1(t)u0 + ∫ t 0 (t− τ)γ1−1Sγ1(t− τ)f(v(τ, ·)) dτ, v(t, x) = Pγ2 (t)v0 + ∫ t 0 (t− τ)γ2−1Sγ2 (t− τ)g(u(τ, ·)) dτ. (3.1) Using the results in [39, Theorem 3.2] and [40, Theorem 3.2], the local solvability and uniqueness of (1.1)-(1.2) can be established. Proposition 3.2 (Existence of a local mild solution). Given u0 and v0 in C0(RN ), 0 < γ1, γ2 < 1, p, q ≥ 1, there exist a maximal time Tmax > 0 and a unique mild solution (u, v) ∈ C([0, Tmax];C0(RN ) × C0(RN )) to problem (1.1)-(1.2), such that either (i) Tmax =∞ (the solution is global), or 6 B. AHMAD, A. ALSAEDI, M. BERBICHE, M. KIRANE EJDE-2020/110 (ii) Tmax <∞ and limt→Tmax (‖u(t)‖∞ + ‖v(t)‖∞) =∞ (the solution blows up in a finite time). If, in addition, u0 ≥ 0, v0 ≥ 0, u0, v0 6≡ 0, then u(t) > 0, v(t) > 0 and u(t) ≥ Pγ1 (t)u0, v(t) ≥ Pγ2 (t)v0 for t ∈ (0, Tmax). Moreover, if (u0, v0) ∈ L1(RN ) × L1(RN ), then for all s1, s2 ∈ (1,+∞), (u, v) ∈ C([0, Tmax];Ls1(RN )× Ls2(RN )). Now, we state the first main result of this section concerning the existence of a global solution and large time behavior of solutions of (1.1)- (1.2). Theorem 3.3 (Existence of a global mild solution). Let N ≥ 1, let q ≥ p ≥ 1, be such that pq > 1, let (f(v), g(u)) = (±|v|p−1v,±|u|q−1u), or (±|v|p,±|u|q), and let 0 < γ1 ≤ γ2 < 1. If N 2 ≥ (γ2 − γ1)pq + qγ2 + γ1 γ1(pq − 1) , (3.2) then, for ‖u0‖1 + ‖u0‖∞ + ‖v0‖1 + ‖v0‖∞ ≤ ε0, with some ε0 > 0, there exist s1 > q, s2 > p such that problem (1.1)-(1.2) admits a global mild solution with u ∈ L∞([0,∞), L∞(RN )) ∩ L∞([0,∞), Ls1(RN )), v ∈ L∞([0,∞), L∞(RN )) ∩ L∞([0,∞), Ls2(RN )). Furthermore, for all δ > 0, max { 1− (pq − 1) γ2q(p+ 1) , 1− γ1(pq − 1) γ2(p+ 1) , 1− (pq − 1) q + 1 } < δ < min { 1, N(pq − 1) 2q(p+ 1) } , ‖u(t)‖s1 ≤ C(t+ 1)− (1−δ)(γ1+pγ2) pq−1 , ‖v(t)‖s2 ≤ C(t+ 1)− (1−δ)(γ2+qγ1) pq−1 , t ≥ 0. If, in addition, pN 2s2 < 1 and qN 2s1 < 1, or N > 2, pN/(2s2) < 1 and qN/(2s1) ≥ 1, or N > 2, qN/(2s1) ≥ 1, pN/(2s2) ≥ 1 and q ≥ p > 1 with max { q + 1 pq(p+ 1) , pq − 1 pq(p+ 1) , γ2/p, √ γ2 pq } < γ1 ≤ γ2 < 1, then u, v ∈ L∞([0,∞), L∞(RN )), ‖u(t)‖∞ ≤ C(t+ 1)−σ̃, ‖v(t)‖∞ ≤ C(t+ 1)−σ̂, t ≥ 0, for some constants σ̃ > 0 and σ̂ > 0. Definition 3.4 (Weak solution). Let u0, v0 ∈ L∞loc(RN ), T > 0. We say that (u, v) ∈ Lq((0, T ), L∞loc(RN ))× Lp((0, T ), L∞loc(RN )) is a weak solution of (1.1)-(1.2) if∫ T 0 ∫ RN (|v|pϕ+ u0D γ1 t|Tϕ) dx dt = ∫ T 0 ∫ RN u(−∆ϕ) dx dt+ ∫ T 0 ∫ RN uDγ1 t|Tϕdx dt, EJDE-2020/110 EXISTENCE OF GLOBAL SOLUTIONS 7∫ T 0 ∫ RN (|u|qϕ+ v0D γ2 t|Tϕ) dx dt = ∫ T 0 ∫ RN v(−∆ϕ) dx dt+ ∫ T 0 ∫ RN vDγ2 t|Tϕdx dt, for every ϕ ∈ C1,2 t,x ([0, T ]× RN ) such that suppx ϕ b RN and ϕ(T, ·) = 0. Similar to the proof in [39], we can easily obtain the following lemma asserting that the mild solution is the weak solution. Lemma 3.5. Assume u0, v0 ∈ C0(RN ), and let (u, v) ∈ C([0, T ], C0(RN )×C0(RN ) be a mild solution of (1.1)-(1.2), then (u, v) is a weak solution of (1.1)-(1.2). Our next result concerns the blowing-up of solutions of (1.1)-(1.2). Theorem 3.6 (Blow-up of mild solutions). Let N ≥ 1, p > 1, q > 1, 0 < γ1, γ2 < 1, let (f(v), g(u)) = (|v|p, |u|q), let u0, v0 ∈ C0(RN ), u0 ≥ 0, v0 ≥ 0, u0 6≡ 0 and v0 6≡ 0. If N 2 < min { (pγ2 + γ1) γ1(pq − 1) , (qγ1 + γ2) γ1(pq − 1) , (pq(γ1 − γ2) + qγ1 + γ2) γ1(pq − 1) , (p+ 1) pq − 1 } or N 2 < min { (qγ1 + γ2) γ2(pq − 1) , (pγ2 + γ1) γ2(pq − 1) , (pq(γ2 − γ1) + pγ2 + γ1) γ2(pq − 1) , (q + 1) pq − 1 } , then the mild solution (u, v) of (1.1)-(1.2) blows up in a finite time. Also if p = 1 and 1 < q < 1 + 2 N , or 1 < p < 1 + 2 N and q = 1, then the solution blows-up in a finite time. A result of blowing-up solutions can be obtained via differential inequalities. Let χ(x) = (∫ RN e− √ N2+|x|2 dx )−1 e− √ N2+|x|2 , x ∈ RN , which satisfies ∫ RN χ(x) dx = 1. In the next theorem, we take f(v) = |v|p and g(u) = |u|q. Theorem 3.7. Let γ1 = γ2 = γ ∈ (0, 1), u0, v0 ∈ C0(RN ) and u0, v0 ≥ 0. Let p > 1, q > 1 such that p ≤ q and let (f(v), g(u)) = (|v|p, |u|q). If Z0 := ∫ RN (u0(x) + v0(x))χ(x)dx > 2 p p−1 , then the solution of problem (1.1)-(1.2) blows-up in a finite time. Moreover, we have estimate of the time blowing up t̄∗∗ ≤ [ ln(1−2pZ1−p 0 ) 2(1−p) Γ(γ + 1) ]1/γ . The next lemma plays an important role in establishing lower solution for Caputo fractional differential equation. Lemma 3.8 ([35, Lemma 3.1]). Let u = u(t) is a solution of the ordinary differ- ential equation du dt = F (u), u(0) = u0, (3.3) where F is a function of u such that F (0) ≥ 0, F (u) > 0, Fu(u) ≥ 0 for u ≥ 0 then v(t) = u(t̄) is a lower solution of a Caputo fractional differential equation CDα 0|tu(t) = F (u), u(0) = u0, (3.4) 8 B. AHMAD, A. ALSAEDI, M. BERBICHE, M. KIRANE EJDE-2020/110 where t̄ = tα Γ(α+1) . That means CDα 0|tv(t) ≤ F (v), v(0) ≤ u0. 4. Proofs of main results Proof of Theorem 3.3. We proceed in three steps. Step 1: Global existence for (u, v) in Ls1(RN )×Ls2(RN ). Since q ≥ p ≥ 1, pq > 1, 0 < γ1 ≤ γ2 < 1 and N 2 ≥ (γ2 − γ1)pq + qγ2 + γ1 γ1(pq − 1) > γ2q(p+ 1)− γ1q(pq − 1) γ2(pq − 1) , we have N(pq − 1) 2q(p+ 1) > γ2(p+ 1)− γ1(pq − 1) γ2(p+ 1) = 1− γ1(pq − 1) γ2(p+ 1) . Note that, from N 2 ≥ (γ2 − γ1)pq + qγ2 + γ1 γ1(pq − 1) > q + 1 pq − 1 , we obtain N(pq − 1) 2q(p+ 1) > q + 1 pq − 1 × (pq − 1) q(p+ 1) = q + 1 q(p+ 1) > 1− (pq − 1) γ2q(p+ 1) , N(pq − 1) 2q(p+ 1) > q + 1 q(p+ 1) > 1− (pq − 1) q + 1 . From these facts, we can choose δ > 0 such that max { 1− (pq − 1) γ2q(p+ 1) , 1− γ1(pq − 1) γ2(p+ 1) , 1− (pq − 1) q + 1 } < δ < min { 1, N(pq − 1) 2q(p+ 1) } . (4.1) We set r1 = Nγ1(pq − 1) 2[γ1(1 + δp) + γ2p(1− δ)] , r2 = Nγ2(pq − 1) 2[γ2(1 + δq) + γ1q(1− δ)] , 1 s1 = 2δ N p+ 1 pq − 1 , 1 s2 = 2δ N q + 1 pq − 1 , σ1 = (1− δ)(γ1 + γ2p) pq − 1 , σ2 = (1− δ)(γ2 + γ1q) pq − 1 . (4.2) Clearly, we have 1 r1 = 2 Nγ1 (1− δ)(γ1 + γ2p) pq − 1 + 2δ N (p+ 1) pq − 1 , 1 r2 = 2 Nγ2 (1− δ)(γ2 + γ1q) pq − 1 + 2δ N (q + 1) pq − 1 . It is easy to check that s1 > q, s2 > p, ps1 > s2, qs2 > s1, s1 > r1 > 1, s2 > r2 > 1, N 2 γ1 ( 1 r1 − 1 s1 ) q < 1, N 2 γ2 ( 1 r2 − 1 s2 ) p < 1, N 2 ( p s2 − 1 s1 ) = δ = N 2 ( q s1 − 1 s2 ), pσ2 < 1, and qσ1 < 1. From pq(γ2 − 1) + 1 + qγ1 [pγ2 + γ1]q = q(pγ2 + γ1)− (pq − 1) [pγ2 + γ1]q < 1− (pq − 1) γ2q(p+ 1) < δ EJDE-2020/110 EXISTENCE OF GLOBAL SOLUTIONS 9 we obtain δ > pq(γ2−1)+qγ1+1 (γ1+pγ2)q which is equivalent to( γ1 − N 2 γ1( p s2 − 1 s1 )− pσ2 ) q > −1. In fact, since δ = N 2 ( ps2 − 1 s1 ), the above inequality gives( γ1 − δγ1 − pσ2 ) q > −1, using definition of σ2 = (1−δ)(γ2+γ1q) pq−1 , we obtain( γ1 − δγ1 − p (1− δ)(γ2 + γ1q) pq − 1 ) q > −1, so, we obtain (1− δ) ( γ1 − p (γ2 + γ1q) pq − 1 ) q > −1. Therefore (1− δ) ( (pq − 1)γ1 − p(γ2 + γ1q) pq − 1 ) q > −1. By simplification, we obtain (δ − 1) (γ1 + pγ2 pq − 1 ) q > −1, or δ (γ1 + pγ2 pq − 1 ) q > (γ1 + pγ2 pq − 1 ) q − 1. Thus δ > 1− pq − 1 (γ1 + pγ2)q = pq(γ2 − 1) + qγ1 + 1 (γ1 + pγ2)q . Similarly, we have ( γ2 − N 2 γ2( q s1 − 1 s2 )− qσ1 ) p > −1. equivalent to δ > pq(γ1−1)+pγ2+1 (γ2+qγ1)p . Let (u0, v0) ∈ C0(RN )× C0(RN ) ∩ Lr1(RN )× Lr2(RN ), and let (u, v) ∈ C([0, Tmax);C0(RN ) ∩ Ls1(RN ))× C([0, Tmax);C0(RN ) ∩ Ls2(RN )). For t ∈ [0, Tmax), from (3.1), we have ‖u(t)‖s1 ≤ ‖Pγ1 (t)u0‖s1 + ∥∥∫ t 0 (t− τ)γ1−1Sγ1 (t− τ)|v(τ)|pdτ ∥∥ s1 = ‖Pγ1 (t)u0‖s1 + [ ∫ RN ∣∣∣ ∫ t 0 (t− τ)γ1−1Sγ1 (t− τ)|v(τ)|pdτ ∣∣∣s1dx]1/s1 . We have [ ∫ RN ∣∣∣ ∫ t 0 (t− τ)γ1−1Sγ1 (t− τ)|v(τ)|pdτ ∣∣∣s1dx]1/s1 ≤ [ ∫ RN (∫ t 0 (t− τ)γ1−1|Sγ1 (t− τ)|v(τ)|pdτ | )s1 dx ]1/s1 . Using Minkowski’s integral inequality, we obtain[ ∫ RN (∫ t 0 (t− τ)γ1−1|Sγ1 (t− τ)|v(τ)|pdτ | )s1 dx ]1/s1 10 B. AHMAD, A. ALSAEDI, M. BERBICHE, M. KIRANE EJDE-2020/110 ≤ ∫ t 0 (∫ RN (t− τ)s1(γ1−1)|Sγ1(t− τ)|v(τ)|p|s1dx )1/s1 dτ = ∫ t 0 (t− τ)γ1−1 (∫ RN |Sγ1(t− τ)|v(τ)|p|s1dx )1/s1 dτ = ∫ t 0 (t− τ)γ1−1‖Sγ1 (t− τ)|v(τ)|p‖s1dτ. Hence, for t ∈ [0, Tmax), we obtain ‖u(t)‖s1 ≤ ‖Pγ1 (t)u0‖s1 + ∫ t 0 (t− τ)γ1−1‖Sγ1 (t− τ)|v(τ)|p‖s1dτ, (4.3) ‖v(t)‖s2 ≤ ‖Pγ2 (t)v0‖s2 + ∫ t 0 (t− τ)γ2−1‖Sγ2 (t− τ)|u(τ)|q‖s2dτ. (4.4) Applying lemmas 2.2 and 2.1, we obtain ‖u(t)‖s1 ≤ ‖u0‖r1t−σ1 + C ∫ t 0 (t− τ)γ1−1(t− τ)− N 2 γ1( ps2 − 1 s1 )‖v(τ)‖ps2dτ, (4.5) ‖v(t)‖s2 ≤ ‖v0‖r2t−σ2 + C ∫ t 0 (t− τ)γ2−1(t− τ)− N 2 γ2( qs1 − 1 s2 )‖u(τ)‖qs1dτ. (4.6) By using (4.6) in (4.5), we obtain ‖u(t)‖s1 ≤ ‖u0‖r1t−σ1 + C ∫ t 0 (t− τ)γ1−1(t− τ)− N 2 γ1( ps2 − 1 s1 )dτ × ( ‖v0‖r2t−σ2 + C ∫ t 0 (t− τ)γ2−1(t− τ)− N 2 γ2( qs1 − 1 s2 )‖u‖qs1dτ )p . Hence ‖u(t)‖s1 ≤ ‖u0‖r1t−σ1 + C ∫ t 0 (t− τ)γ1−1−N2 γ1( ps2 − 1 s1 )τ−pσ2dτ‖v0‖pr2 + C ∫ t 0 (t− τ)γ1−1−N2 γ1( ps2 − 1 s1 )τ (γ2−N2 γ2( qs1 − 1 s2 )−qσ1)p × ( τ σ1‖u(τ)‖s1 )pq dτ. (4.7) Multiplying both sides of (4.7) by tσ1 , where σ1 = (1−δ)(γ1+γ2p) pq−1 , we find that tσ1‖u‖s1 ≤ ‖u0‖r1 + Ctσ1 ∫ t 0 (t− τ)γ1−1−N2 γ1( ps2 − 1 s1 )τ−pσ2dτ‖v0‖pr2 + Ctσ1 ∫ t 0 (t− τ)γ1−1−N2 γ1( ps2 − 1 s1 )τ (γ2−N2 γ2( qs1 − 1 s2 )−qσ1)p × ( τ σ1 ‖u‖s1 )pq dτ. (4.8) Since γ1 − 1− N 2 γ1( ps2 − 1 s1 ) > −1, and ( γ2 − N 2 γ2( qs1 − 1 s2 )− qσ1 ) p > −1, we have tσ1‖u‖s1 ≤ ‖u0‖r1 + Ctσ1+γ1−N2 γ1( ps2 − 1 s1 )−pσ2‖v0‖pr2 + Ctσ1+γ1−N2 γ1( ps2 − 1 s1 )+(γ2−N2 γ2( qs1 − 1 s2 )−qσ1)p ( sup 0≤τ 2CA; by the intermediate value theorem, since h is continuous and h(0) = 0, there exists t1 ∈ (0, t0) such that h(t1) = 2CA. (4.12) Using (4.12) in (4.10), we obtain h(t1) ≤ C(A+ h(t1)pq) = (h(t1) 2 + Ch(t1)pq ) , from which, we infer h(t1) 2 ≤ Ch(t1)pq, (4.13) using (4.12) in (4.13), we obtain CA ≤ C(2CA)pq, so A ≤ (2CA)pq = (2C)pqApq, then it yields (2C)−pq ≤ Apq−1, which is equivalent to A ≥ (2C) pq 1−pq . This contradicts the choice of A. It then follows that h(t) remains bounded in all time t > 0 provided that ‖u0‖r1 and ‖v0‖r2 are small. Therefore tσ1‖u(t)‖s1 ≤ C, for all t > 0. (4.14) Similarly, we obtain tσ2‖v(t)‖s2 ≤ C, for all t > 0. (4.15) Step 2: L∞-global existence estimates of (u, v) in L∞(RN )×L∞(RN ). Let s1, s2 be the same as in (4.2). Since p ≤ q, we have Np 2s2 ≤ Nq 2s1 . We further assume for some ξ > q, w > p, k1 > 0, k2 > 0 that u(t) ∈ Lw(RN ), v(t) ∈ Lξ(RN ) and ‖u(t)‖w ≤ C(1 + tk1), ‖v(t)‖ξ ≤ C(1 + tk2) for every t ∈ [0, Tmax). (4.16) 12 B. AHMAD, A. ALSAEDI, M. BERBICHE, M. KIRANE EJDE-2020/110 Then, by applying Lemmas 2.1 and 2.2 again to (3.1), we obtain ‖u(t)‖∞ ≤ ‖Pγ1 (t)u0‖∞ + ∫ t 0 (t− τ)γ1−1−Nγ1p 2ξ ‖v(τ)‖pξdτ, (4.17) ‖v(t)‖∞ ≤ ‖Pγ2 (t)v0‖∞ + ∫ t 0 (t− τ)γ2−1−Nγ2q 2w ‖u(τ)‖qwdτ, (4.18) for all t ∈ [0, Tmax). Now, if one can find ξ and w such that Np 2ξ < 1 or Nq 2w < 1, (4.19) then the L∞-estimates of (u, v) can be obtained. In fact, if Np 2ξ < 1, in view of (4.16), from (4.17) we have ‖u(t)‖∞ ≤ ‖Pγ1 (t)u0‖∞ + C max τ∈[0,t] ‖v(τ)‖pξt (1−Np2ξ )γ1 ≤ C(1 + t(1− Np 2ξ )γ1+pk2), (4.20) and by taking w =∞ in (4.18), we obtain ‖v(t)‖∞ ≤ ‖Pγ2 (t)v0‖∞ + ∫ t 0 (t− τ)γ2−1‖u(τ)‖q∞dτ ≤ ‖Pγ2 (t)v0‖∞ + ∫ t 0 (t− τ)γ2−1 ( 1 + t(1− Np 2ξ )γ1+pk2 )q dτ ≤ C ( 1 + tγ2+[(1−Np2ξ )γ1+pk2]q ) . (4.21) These estimates show that Tmax =∞ and u, v ∈ L∞loc([0,∞);L∞(RN )). (4.22) In a similar way, we can deal with the case Nq 2w < 1. To find such appropriate ξ and w, we note that if Nq 2s1 < 1 or Np 2s2 < 1, then (4.20) and (4.21) hold by taking ξ = s1 or w = s2. This is certainly the case if N ≤ 2 as s1 > q and s2 > p. Thus it remains to deal with the case N > 2, Nq 2s1 ≥ 1 and Np 2s2 ≥ 1. We will do this via an iterative process. Define s′1 = s1, s′′1 = s2. Since s′1 > q and s′′1 > p, using the Hölder inequality and lemmas 2.1,2.2, we obtain from (4.3) and (4.4) that ‖u(t)‖s′2 ≤ ‖Pγ1 (t)u0‖s′2 + ∫ t 0 (t− τ) γ1−1−Nγ1 2 ( p s′′2 − 1 s′2 )‖v(τ)‖ps′′1 dτ, ‖v(t)‖s′′2 ≤ ‖Pγ2 (t)v0‖s′′2 + ∫ t 0 (t− τ) γ2−1−Nγ2 2 ( q s′1 − 1 s′′2 )‖u(τ)‖qs′1dτ, where s′2 and s′′2 are such that N 2 ( p s′′1 − 1 s′2 ) < 1, N 2 ( q s′1 − 1 s′′2 ) < 1. This can be verified by taking 1 s′2 = p s′′1 − 2 N + η, 1 s′′2 = q s′1 − 2 N + η, EJDE-2020/110 EXISTENCE OF GLOBAL SOLUTIONS 13 where 0 < η < 2(1− δ)/N . Observe that 1 s′1 − 1 s′2 = 2 N (1− δ)− η > 0, 1 s′′1 − 1 s′′2 = 2 N (1− δ)− η > 0, (4.23) and hence s′2 > s′1 > q and s′′2 > s′′1 > p. Next, we define the sequences {s′i}i≥1 and {s′′i }i≥1 iteratively as follows 1 s′i = p s′′i−1 − 2 N + η, 1 s′′i = q s′i−1 − 2 N + η, i ≥ 3. (4.24) Then 1 s′i − 1 s′i+1 = p ( 1 s′′i−1 − 1 s′′i ) = pq ( 1 s′i−2 − 1 s′i−1 ) , 1 s′′i − 1 s′′i+1 = q ( 1 s′i−1 − 1 s′i ) = pq ( 1 s′′i−2 − 1 s′′i−1 ) . Since pq > 1, in view of (4.23), we obtain 1 s′i > 1 s′i+1 , 1 s′′i > 1 s′′i+1 , i ≥ 1 (4.25) lim i→+∞ ( 1 s′i − 1 s′i+1 ) = lim i→+∞ ( 1 s′′i − 1 s′′i+1 ) = +∞. (4.26) Now, we ensure that there exists i0 such that p s′′i0 < 2 N or q s′i0 < 2 N . (4.27) In fact, if (4.27) is not true, that is p s′′i ≥ 2 N and q s′i ≥ 2 N for all i ≥ 1. Then, by (4.24), we see that s′i > 0, s′′i > 0 for all i ≥ 1 and hence, by (4.25), q < s′1 < · · · < s′i < . . . , p < s′′1 < · · · < s′′i < · · · . Therefore ∣∣ 1 s′i − 1 s′i+1 ∣∣ ≤ 1 s′i + 1 s′i+1 < 2 q < 2, for all i ≥ 1, which contradicts (4.26). Let i0 be the smallest number that satisfies (4.27). We note that i0 ≥ 2. Without loss of generality, we assume that p s′′i0 < 2 N , p s′′i ≥ 2 N for 1 ≤ i ≤ i0 − 1, q s′i ≥ 2 N for 1 ≤ i ≤ i0. (4.28) Thus (4.24) yields s′i > 0 for 1 ≤ i ≤ i0, s′′i > 0 for 1 ≤ i ≤ i0 + 1, which together with (4.25) leads to q < · · · < s′i0−1 < s′i0 p < · · · < s′′i0 < s′′i0+1. Now, from (4.24), for all i ≥ 2 we have N 2 ( p s′′i−1 − 1 s′i ) = 1− N 2 η = N 2 ( q s′i−1 − 1 s′′i ) . Now, let us deal with the boundedness of (u(t), v(t)) in Ls ′ i(RN )× Ls′′i (RN ). 14 B. AHMAD, A. ALSAEDI, M. BERBICHE, M. KIRANE EJDE-2020/110 By using Lemmas 2.1 and 2.2, it follows from (3.1) inductively that, for all 2 ≤ i ≤ i0 and for all t ∈ (0, Tmax), ‖u(t)‖s′i ≤ ‖Pγ1 (t)u0‖s′i + C ∫ t 0 (t− τ) γ1−1−Nγ1 2 ( p s′′ i−1 − 1 s′ i ) ‖v(τ)‖ps′′i−1 dτ ≤ C‖u0‖s′i + C ∫ t 0 (t− τ)γ1−1−γ1(1−Nη2 )‖v(τ)‖ps′′i−1 dτ, (4.29) and ‖v(t)‖s′′i ≤ ‖Pγ2 (t)v0‖s′′i + C ∫ t 0 (t− τ) γ2−1+ Nγ2 2 ( q s′ i−1 − 1 s′′ i ) ‖u(τ)‖qs′i−1 dτ ≤ C‖v0‖s′′i + C ∫ t 0 (t− τ)γ2−1−γ2(1−Nη2 )‖u(τ)‖qs′i−1 dτ, (4.30) for 2 ≤ i ≤ i0 + 1 and t ∈ (0, Tmax). From the Hölder inequality, we have ‖Pγ1 (t)u0‖s′i ≤ ‖u0‖s′i ≤ ‖u0‖ s1 s′ i s1 ‖u0‖ 1− s1 s′ i∞ <∞, ‖Pγ2(t)v0‖s′′i ≤ ‖v0‖s′′i ≤ ‖v0‖ s1 s′′ i s1 ‖v0‖ 1− s1 s′′ i∞ <∞, (4.31) since u0 ∈ Ls1 ∩ L∞ and v0 ∈ Ls2 ∩ L∞. From (4.29), (4.30) and (4.31) it follows that u(t) ∈ Ls ′ i(RN ), ‖u(t)‖s′i ≤ C(1 + tai), 1 ≤ ∀i ≤ i0, v(t) ∈ Ls ′′ i (RN ), ‖v(t)‖s′′i ≤ C(1 + tbi), 1 ≤ ∀i ≤ i0 + 1, (4.32) for all t ∈ (0, Tmax) and for some positive constants ai, bi. Since Np 2si′′0 < 1, taking s2 = s′′i0 , (4.19) holds; hence Tmax = +∞ and (4.22) holds. Step 3: First, we show the following decay estimates ‖u(t)‖s1 ≤ C(t+ 1)−σ1 , ‖v(t)‖s2 ≤ C(t+ 1)−σ2 , for t ≥ 0, where s1 and s2 are given by (4.2). According to Lemma 2.3, it suffices to prove that ‖u(t)‖s1 ≤ C, ‖v(t)‖s2 ≤ C, for all t ∈ [0, 1]. To do this, we need to show that ‖u(t)‖∞ ≤ C, ‖v(t)‖∞ ≤ C, for t ∈ [0, 1]. (4.33) In fact, by applying Lemmas 2.1 and 2.2 to (3.1) we see that ‖u(s)‖∞ ≤ ‖Pγ1 (s)u0‖∞ + ∫ s 0 (s− τ)γ1−1‖v(τ)‖p∞dτ ≤ ‖u0‖∞ + ∫ s 0 (s− τ)γ1−1‖v(τ)‖p∞dτ, ‖v(s)‖∞ ≤ ‖Pγ2(s)v0‖∞ + ∫ s 0 (s− τ)γ2−1‖u(τ)‖q∞dτ ≤ ‖v0‖∞ + ∫ s 0 (s− τ)γ2−1‖u(τ)‖q∞dτ, EJDE-2020/110 EXISTENCE OF GLOBAL SOLUTIONS 15 for 0 ≤ s ≤ t. For 0 ≤ s ≤ t ≤ 1, the two inequalities above give sup 0≤s≤t ‖u(s)‖∞ ≤ ‖u0‖∞ + 1 γ1 ( sup 0≤τ≤t ‖v(τ)‖∞ )p tγ1 ≤ ‖u0‖∞ + 1 γ1 ( sup 0≤τ≤t ‖v(τ)‖∞)p, sup 0≤s≤t ‖v(s)‖∞ ≤ ‖v0‖∞ + 1 γ2 ( sup 0≤τ≤t ‖u(τ)‖∞ )q tγ2 ≤ ‖v0‖∞ + 1 γ2 ( sup 0≤τ≤t ‖u(τ)‖∞)q. Using the second inequality into first inequality, it yields that sup 0≤s≤t ‖u(s)‖∞ ≤ ‖u0‖∞ + 1 γ1 ( ‖v0‖∞ + 1 γ2 ( sup 0≤τ≤t ‖u(τ)‖∞ )q)p ≤ C ( ‖u0‖∞ + ‖v0‖p∞ + ( sup 0≤τ≤t ‖u(τ)‖∞ )pq) So, arguing as in the first step by setting h(t) = sup0≤s≤t ‖u(s)‖∞ and A = ‖u0‖∞+ ‖v0‖p∞, we obtain h(t) ≤ A+ Chpq(t), for all t ≤ 1, which implies (4.33) for A small since pq > 1. We see from (4.3) and Lemmas 2.1, 2.2 that ‖u(t)‖s1 ≤ C‖u0‖s1 + C ∫ t 0 (t− τ)γ1−1‖|v(τ)|p‖s1dτ, where s1 given explicitly by (4.2). Therefore ‖u(t)‖s1 = C‖u0‖s1 + C ∫ t 0 (t− τ)γ1−1‖v(τ)‖pps1dτ. By the interpolation inequality ‖v(τ)‖pps1 ≤ ‖v(τ)‖ s2 s1 s2 ‖v(τ)‖ p(1− s2 ps1 ) ∞ , we obtain ‖u(t)‖s1 = C‖u0‖s1 + C sup τ∈(0,t) ‖v(τ)‖ p(1− s2 ps1 ) ∞ ∫ t 0 (t− τ)γ1−1‖v(τ)‖ s2 s1 s2 dτ. (4.34) Now, using (4.15) and (4.33) in (4.34), we obtain ‖u(t)‖s1 ≤ C‖u0‖s1 + C ∫ t 0 (t− τ)γ1−1τ− s2 s1 σ2dτ, provided that s2 s1 σ2 < 1. On the other hand, since s1and s2 satisfy s1 s2 σ1 = (1− δ)(pγ2 + γ1)s1 (pq − 1)s2 ≤ γ2, s2 s1 σ2 = (1− δ)(qγ1 + γ2)s2 (pq − 1)s1 ≤ γ1, we obtain γ1 − s2 s1 σ2 ≥ 0 and consequently ‖u(t)‖s1 ≤ C‖u0‖s1 + Ctγ1− s2s1 σ2 ≤ C, for all t ∈ [0, 1]. Analogously, ‖v(t)‖s2 ≤ C for all t ∈ [0, 1]. From (4.14), (4.15), (4.33) and Lemma 2.3, we conclude that ‖u(t)‖s1 ≤ C(t+ 1)− (1−δ)(γ1+pγ2) pq−1 , ‖v(t)‖s2 ≤ C(t+ 1)− (1−δ)(γ2+qγ1) pq−1 , (4.35) 16 B. AHMAD, A. ALSAEDI, M. BERBICHE, M. KIRANE EJDE-2020/110 for all t ≥ 0. Next, we derive L∞-decay estimates. Let σ1 = (1− δ)(pγ2 + γ1) (pq − 1) , σ2 = (1− δ)(qγ1 + γ2) (pq − 1) . If pN 2s2 < 1, by taking ξ = s2 in (4.18) and using (4.35), we obtain ‖u(t)‖∞ ≤ Ct− N 2 γ1‖u0‖1 + C ∫ t 0 (t− τ)γ1−1−Nγ1 2 p s2 τ−pσ2dτ (4.36) and pσ2 < 1, γ1 − Nγ1 2 p s2 − pσ2 = − [γ1 + γ1pδ + (1− δ)pγ2] pq − 1 . On the other hand, we have γ1 + γ1pδ + pγ2(1− δ) pq − 1 < γ1 + pγ2 pq − 1 ≤ N 2 γ1. (4.37) Then, it follows from (4.36), (4.37) and lemma 2.3 that ‖u(t)‖∞ ≤ Ct− N 2 γ1 + Ct− [γ1+γ1pδ+(1−δ)pγ2] pq−1 . Thus ‖u(t)‖∞ ≤ C(1 + t)− [γ1+γ1pδ+(1−δ)pγ2] pq−1 . (4.38) for all t ≥ 0. Similarly, if qN 2s1 < 1, then one can find that ‖v(t)‖∞ ≤ C(1 + t)− [γ2+γ2qδ+(1−δ)qγ1] pq−1 , for t ≥ 0. (4.39) At the same time, (4.38) holds as pN/(2s2) ≤ qN/(2s1). In particular, if pq > q + 2, and γ1q 2 > 2q + 1, we can choose δ > max { 1− (pq − 1) γ2q(p+ 1) , 1− γ1(pq − 1) γ2(p+ 1) } and δ ≈ max{1 − (pq−1) γ2q(p+1) , 1 − γ1(pq−1) γ2(p+1) } such that qN/(2s1) < 1. Therefore, estimates (4.38) and (4.39) hold. It is useful to note that N ≤ 2 implies pN/(2s2) < 1 and qN/(2s1) < 1 implies pq > 2 + q. It remains to consider the following two cases: (1) The case N > 2, Np 2s2 < 1 and Nq 2s1 ≥ 1. Let σ′ = γ1 + γ1pδ + (1− δ)pγ2 pq − 1 . For a positive µ such that µ < min{σ′, σ1} and qµ < 1, since N > 2 and q > 1, we can choose k > 0 such that k > qN 2 and qµ+ qNγ2 2k > γ2. Since s1 ≤ qN/2, we have k > s1. Using the interpolation inequality ‖u(t)‖k ≤ ‖u(t)‖(k−s1)/k ∞ ‖u(t)‖s1/ks1 ≤ Ct−σ ′(k−s1)/kt−σ1s1/k for all t > 0, it follows from (4.14) and (4.38) that ‖u(t)‖k ≤ Ct−µ for all t > 0. EJDE-2020/110 EXISTENCE OF GLOBAL SOLUTIONS 17 Whereupon, ‖v(t)‖∞ ≤ ‖Pγ2 (t)v0‖∞ + C ∫ t 0 (t− τ)γ2−1−Nq2k γ2‖u(τ)‖qkdτ ≤ Ct−N2 γ2‖v0‖1 + C ∫ t 0 (t− τ)γ2−1−Nγ2q 2k τ−qµdτ ≤ C(t− N 2 γ2 + tγ2−Nγ2q 2k −qµ) ≤ Ct−α, (4.40) for all t > 0, where α = min{N2 γ2,−γ2 + Nγ2q 2k + qµ} > 0. From (4.33) and (4.40), we infer that ‖v(t)‖∞ ≤ C(1 + t)−α for all t ≥ 0. We remark that, in the particular case p = 1, q > 3 and q2 > max{4γ2q+1, 4γ2+γ1 γ1 }, we can choose δ > max { 1− (pq − 1) γ2q(p+ 1) , 1− γ1(pq − 1) γ2(p+ 1) , 1− (pq − 1) q + 1 } = max { 1− (q − 1) 2γ2q , 1− γ1(q − 1) 2γ2 , 1− (q − 1) q + 1 } and δ ≈ max{1 − (q−1) 2γ2q , 1 − γ1(q−1) 2γ2 , 1 − (q−1) q+1 } such that N/(2s2) < 1. Therefore, we have the estimate (4.38). (2) The case: N > 2, qN/(2s1) ≥ 1, pN/(2s2) ≥ 1, q ≥ p > 1, and γ1 ≤ γ2. It needs a careful handling and we need to restrict further the choice of δ. From max{ q+1 pq(p+1) , pq−1 pq(p+1) , γ2/p, √ γ2 pq} < γ1 ≤ γ2 < 1 and pq > 1, we obtain max { 1− (pq − 1) γ2q(p+ 1) , 1− γ1(pq − 1) γ2(p+ 1) , 1− (pq − 1) q + 1 } < min { 1− (pq − 1) γ1pq(p+ 1) , N(pq − 1) 2q(p+ 1) } . So, we select δ > 0 such that max { 1− (pq − 1) γ2q(p+ 1) , 1− γ1(pq − 1) γ2(p+ 1) , 1− (pq − 1) q + 1 } < δ < min {N(pq − 1) 2q(p+ 1) , 1− (pq − 1) γ1pq(p+ 1) } . We get immediately pσ2 > 1/q and qσ1 > 1/p. Further, we notice that there exist ε ∈ (0, 1) and β < 1 close to 1 such that pσ2 − ε > 1/q, qσ1 − ε > 1/p, 1/p < β − ε, 1/q < β − ε. (4.41) By taking η = 2ε(1− δ)/N , we find the integer i0 as in the step 2, and, without loss of generality, we assume that (4.28) holds. We choose β in addition to (4.41) satisfying γ1 < γ1 pN 2s′′i0 + γ2β, since γ2 ≥ γ1. (4.42) As δ < N(pq − 1) 2(p+ 1)q ≤ N(pq − 1) 2(q + 1)p , 18 B. AHMAD, A. ALSAEDI, M. BERBICHE, M. KIRANE EJDE-2020/110 and β < 1, we have β + (p+ 1)qδ (pq − 1) < 1 + N 2 , β + (q + 1)pδ (pq − 1) < 1 + N 2 . (4.43) For 2 ≤ i ≤ i0 − 1, define r′i+1 and r′′i+1 inductively as follows: 1 r′2 = 1 s′2 + 2 N [pσ2 − ε(1− δ)], 1 r′′2 = 1 s′′2 + 2 N [qσ1 − ε(1− δ)], 1 r′i+1 = 1 s′i+1 + 2 N [β − ε(1− δ)], 1 r′′i+1 = 1 s′′i+1 + 2 N [β − ε(1− δ)]. It is clear that r′i, r ′′ i > 0 and r′i < s′i, r ′′ i < s′′i for all 2 ≤ i ≤ i0. A simple calculation shows that r′i, r ′′ i > 1. As s′i and s′′i are increasing in i for 1 ≤ i ≤ i0; we have 1 r′i+1 < 1 s′2 + 2 N [β − ε(1− δ)] = p s′′1 − 2 N + 2 N ε(1− δ) + 2 N [β − ε(1− δ)] = 2 N (p(q + 1)δ pq − 1 + β − 1 ) < 1, from (4.43); therefore r′i+1 > 1. Similarly, we can check that r′′i+1 > 1. From (4.22) and (4.32), we see that there exists a positive constant C such that ‖u(t)‖∞, ‖v(t)‖∞, ‖u(t)‖k1 , ‖v(t)‖k2 ≤ C for 0 ≤ t ≤ 1, (4.44) for all s′1 ≤ k1 ≤ s′i0 , s′′1 ≤ k2 ≤ s′′i0 . Furthermore, since 1 − ηN/2 = 1 − ε(1 − δ) and pσ2 < 1, using (4.29), (4.30) with the help of (4.14) and (4.15), we arrive at the estimate ‖u(t)‖s′2 ≤ ‖Pγ1(t)u0‖s′2 + C ∫ t 0 (t− τ)γ1−1−γ1(1−ε(1−δ))‖u(τ)‖ps′′2 dτ ≤ Ct− N 2 γ1( 1 r′2 − 1 s′2 )‖u0‖s′2 + C ∫ t 0 (t− τ)γ1−1−γ1(1−ε(1−δ))τ−pσ2dτ ≤ Ct−γ1(pσ2−ε(1−δ))‖u0‖s′2 + C ∫ t 0 (t− τ)γ1−1−γ1(1−ε(1−δ))τ−pσ2dτ ≤ Ct−γ1(pσ2−ε(1−δ)) for all t > 0. Similarly, ‖v(t)‖s′′2 ≤ Ct −γ2(pσ1−ε(1−δ)) for all t > 0. In view of (4.41) and β < 1, we conclude, thanks to lemma 2.3, that ‖u(t)‖s′2 ≤ Ct −γ1β/q, ‖v(t)‖s′′2 ≤ Ct −γ2β/p for all t > 0. An iterative argument gives ‖u(t)‖s′i0 ≤ Ct −γ1(β−ε(1−δ)) ≤ Ct−γ1β/q for all t > 0, ‖v(t)‖s′′i0 ≤ Ct −γ2(β−ε(1−δ)) ≤ Ct−γ2β/p for all t > 0. Therefore, by (4.17) and (4.18), we have ‖u(t)‖∞ ≤ Ct− N 2 γ1‖u0‖1 + C ∫ t 0 (t− τ) γ1−1−γ1 pN 2s′′ i0 ‖v(τ)‖ps′′i0 dτ EJDE-2020/110 EXISTENCE OF GLOBAL SOLUTIONS 19 ≤ Ct−N2 γ1‖u0‖1 + C ∫ t 0 (t− τ) γ1−1−γ1 pN 2s′′ i0 τ−γ2βdτ ≤ C ( t− N 2 γ1 + t γ1−γ1 pN 2s′′ i0 −γ2β ) ≤ Ct−σ̃, where σ′′ = min{N2 γ1, γ1 pN 2s′′i0 − γ1 + γ2β} > 0 from (4.42). Since Nq 2s1 ≥ 1, using similar arguments as for the case Np 2s2 < 1 and Nq 2s1 ≥ 1, we obtain ‖v(t)‖∞ ≤ Ct−σ ′′ 1 for some σ′′1 > 0 and for every t > 0. This completes the proof. � Remark 4.1. In the particular case N > 2, qN/(2s1) ≥ 1, pN/(2s2) ≥ 1, q > p = 1 and q2 ≤ 4γ1q + 1, using the above method, we obtain ‖u(t)‖∞ ≤ Ct−σ ′′ , t > 0, where σ′′ = min{N2 γ1, pN 2s′′i0 γ1−γ1+γ2(β−ε(1−δ))p}. Here, ε > 0 can be arbitrarily small, and β can be arbitrarily close to 1. However, since s′′i0 depends on ε and s′′i0 is decreasing in ε, it is not clear that σ′′ is positive. Proof of Theorem 3.6. Case: p > 1, q > 1. The proof proceeds by contradiction. Suppose that (u, v) is a nontrivial solution of (1.1) which exists globally in time. We make the judicious choice ϕ(t, x) = ϕ1(t)ϕ2(x) = ϕ1 ( t Tλ ) Φl ( |x| T 2 ) , where Φ ∈ C∞0 (R), 0 ≤ Φ(z) ≤ 1 is such that Φ(z) = { 1 if |z| ≤ 1, 0 if |z| > 2, ϕ1(t) = { (1− t Tλ )l if t ≤ Tλ, 0 if t > Tλ, where l > max{1, q q−1γ1 − 1, p p−1γ2 − 1}. We denote by QTλ := RN × [0, Tλ]. From Definition 3.4, of the weak solution, we have∫ Q Tλ |v|pϕ2(x)ϕ1(t)dx dt+ Tλ(1−γ1) ∫ RN u0(x)ϕ2(x)dx = ∫ Q Tλ ϕ2(x)uDγ1 t|Tλϕ1(t)dx dt− ∫ Q Tλ ∆ϕ2(x)ϕ1(t)u dx dt, (4.45) ∫ Q Tλ |u|qϕ2(x)ϕ1(t)dx dt+ Tλ(1−γ2) ∫ RN v0ϕ2(x)dx = ∫ Q Tλ ϕ2(x)uDγ2 t|Tλϕ1(t)dx dt− ∫ Q Tλ ϕ1(t)∆ϕ2(x)u dx dt. (4.46) Using Hölder’s inequality with exponents q and q′ (q + q′ = qq′), to the right-hand sides of (4.45) and (4.46), we obtain∫ Q Tλ uϕ2(x)Dγ1 t|Tλϕ1(t)dx dt = ∫ Q Tλ u|ϕ1(t)|1/q|ϕ2(x)|1− 1 q+ 1 q |ϕ1(t)|−1/qDγ1 t|Tλϕ1(t)dx dt 20 B. AHMAD, A. ALSAEDI, M. BERBICHE, M. KIRANE EJDE-2020/110 ≤ (∫ Q Tλ1 |Dγ1 t|Tλϕ1(t)|q ′ |ϕ1(t)|−q ′/q|ϕ2(x)|(1− 1 q )q′dx dt )1/q′ × (∫ Q Tλ |u|qϕ1ϕ2dx dt )1/q , and ∫ Q Tλ u|∆ϕ2(x)|ϕ1(t)dx dt ≤ (∫ RN |∆ϕ2(x)|q ′ |ϕ2(x)|−q ′/qdx ∫ Tλ 0 |ϕ1(t)|(1− 1 q )q′dt )1/q′ × (∫ Q Tλ |u|qϕ1ϕ2dx dt )1/q . Setting A(σ, κ, κ′) = (∫ Q Tλ1 |Dσ t|Tλϕ1(t)|κ ′ |ϕ1(t)|−κ ′ κ |ϕ2(x)|(1− 1 κ )κ′dx dt )1/κ′ , B(κ, κ′) = (∫ Q Tλ1 |∆ϕ2(x)|κ ′ |ϕ2(x)|−κ ′ κ |ϕ1(t)|(1− 1 κ )κ′dx dt )1/κ′ , and gathering the above estimates, we obtain∫ Q Tλ |v|pϕ1(t)ϕ2(x)dx dt+ Tλ(1−γ1) ∫ RN u0ϕ2(x)dx ≤ A(γ1, q, q ′) (∫ Q Tλ |u|qϕ1(t)ϕ2dx dt )1/q + B(q, q′) (∫ Q Tλ |u|qϕ1(t)ϕ2dx dt )1/q . (4.47) Similarly, we obtain∫ Q Tλ |u|qϕ2(x)ϕ1(t)dtdx+ Tλ(1−γ2) ∫ RN v0ϕ2(x)dx ≤ A(γ2, p, p ′) (∫ Q Tλ |v|pϕ1ϕ2dx dt )1/p + B(p, p′) (∫ Q Tλ |v|pϕ1ϕ2dx dt )1/p (4.48) Consequently, ∫ Q Tλ |v|pϕ1(t)ϕ2(x)dx dt+ CTλ(1−γ1) ∫ RN u0ϕ2(x)dx ≤ A (∫ Q Tλ |u|qϕ1ϕ2dx dt )1/q , and ∫ Q Tλ |u|qϕ1(t)ϕ2(x)dx dt+ CTλ(1−γ2) ∫ RN v0ϕ2(x)dx ≤ B (∫ Q Tλ |v|pϕ1ϕ2dx dt )1/p , EJDE-2020/110 EXISTENCE OF GLOBAL SOLUTIONS 21 where A = A(γ1, q, q ′) + B(q, q′), B = A(γ2, p, p ′) + B(p, p′). Using inequalities (4.47) and (4.48) in the last two inequalities, we obtain∫ Q Tλ |v|pϕ1(t)ϕ2(x) dx dt+ CTλ(1−γ1) ∫ RN u0ϕ2(x)dx ≤ AB1/q (∫ Q Tλ |v|pϕ1ϕ2dx dt ) 1 pq ,∫ Q Tλ |u|qϕ1(t)ϕ2(x) dx dt+ CTλ(1−γ2) ∫ RN v0ϕ2(x)dx ≤ BA1/p (∫ Q Tλ |u|qϕ1ϕ2dx dt ) 1 pq . Now, applying Young’s inequality, we obtain (pq − 1) ∫ Tλ 0 ∫ RN |v|pϕ2(x)ϕ1(t) dx dt+ CpqTλ(1−γ1) ∫ RN u0(x)ϕ2(x)dx ≤ (pq − 1) ( AB1/q ) pq pq−1 , (pq − 1) ∫ Tλ 0 ∫ RN |u|qϕ2(x)ϕ1(t) dx dt+ CpqTλ(1−γ2) ∫ RN v0(x)ϕ2(x)dx ≤ (pq − 1) ( BA1/p ) pq pq−1 . At this stage, using the change of variables, x = T 2y, t = Tλτ , with λ > 0 to be chosen later, we obtain∫ Tλ 0 ∫ RN |v|pϕ2(x)ϕ1(t)dx dt+ Tλ(1−γ1) ∫ RN u0ϕ2(x)dx ≤ C ( T −λγ1+(λ+2N) 1 q′ + T −4+(λ+2N) 1 q′ )(∫ Q Tλ |u|qϕ1(t)ϕ2(x)dx dt )1/q ≤ CT−λγ1+(λ+2N) 1 q′ (∫ Q Tλ |u|qϕ1ϕ2dx dt )1/q . Analogously, we have∫ Tλ 0 ∫ RN |u|qϕ2(x)ϕ1(t)dx dt+ CTλ(1−γ2) ∫ RN v0ϕ2(x)dx ≤ C ( T −λγ2+(λ+2N) 1 p′ + T −4+(λ+2N) 1 p′ )(∫ Q Tλ |v|pϕ1ϕ2dx dt )1/p = CT −λγ2+(λ+2N) 1 p′ (∫ Q Tλ |v|pϕ1ϕ2dx dt )1/p . Choosing γ1λ = 4, we have∫ Tλ 0 ∫ RN |v|pϕ2(x)ϕ1(t)dx dt+ Tλ(1−γ1) ∫ RN u0ϕ2(x)dx ≤ C ( T − 4 qγ1 γ2+(λ+2N) 1 p′ 1 q−4+( 4 γ1 +2N) 1 q′ + T −4 1 q+(λ+2N) 1 p′q−4+( 4 γ1 +2N) 1 q′ ) 22 B. AHMAD, A. ALSAEDI, M. BERBICHE, M. KIRANE EJDE-2020/110 × (∫ Q Tλ |v|pϕ1ϕ2dx dt ) 1 pq , and∫ Tλ 0 ∫ RN |u|qϕ2(x)ϕ1(t)dx dt+ CTλ(1−γ2) ∫ RN v0ϕ2(x)dx ≤ C ( T −λγ2+(λ+2N) 1 p′ + T −4+(λ+2N) 1 p′ )(∫ Q Tλ |v|pϕ1ϕ2dx dt )1/p ≤ C ( T −λγ2+(λ+2N) 1 p′ + T −4+(λ+2N) 1 p′ ) T −4 1 p+( 4 γ1 +2N) 1 pq′ × (∫ Q Tλ |u|qϕ1ϕ2dx dt ) 1 pq = C ( T − 4 γ1 γ2+( 4 γ1 +2N) 1 p′−4 1 p+( 4 γ1 +2N) 1 pq′ + T −4+( 4 γ1 +2N) 1 p′−4 1 p+( 4 γ1 +2N) 1 pq′ ) × (∫ Q Tλ |u|qϕ1(t)ϕ2dx dt ) 1 pq . Therefore, using the ε-Young inequality, we obtain∫ RN u0(x)ϕ2(x)dx ≤ CT δ1 , (4.49)∫ RN v0(x)ϕ2(x)dx ≤ CT δ2 , (4.50) where δ1 = max { (− 4 qγ1 γ2 + ( 4 γ1 + 2N) 1 p′ 1 q − 4 + ( 4 γ1 + 2N) 1 q′ ) pq pq − 1 + 4 γ1 (γ1 − 1), (−4 1 q + ( 4 γ1 + 2N) 1 p′q − 4 + ( 4 γ1 + 2N) 1 q′ ) pq pq − 1 + 4 γ1 (γ1 − 1) } , and δ2 = max { (− 4 γ1 γ2 + ( 4 γ1 + 2N) 1 p′ − 4 1 p + ( 4 γ1 + 2N) 1 pq′ ) pq pq − 1 + 4 γ1 (γ2 − 1), (−4 + ( 4 γ1 + 2N) 1 p′ − 4 1 p + ( 4 γ1 + 2N) 1 pq′ ) pq pq − 1 + 4 γ1 (γ2 − 1) } . The condition (3.2) leads to either δ1 < 0 or δ2 < 0. Then as T → ∞, the right- hand side of (4.49) (resp. (4.50)) tends to zero while the left-hand side tends to∫ RN u0(x)dx > 0 (resp. ∫ RN v0(x)dx > 0); a contradiction. We repeat the same argument for γ2λ = 4 to conclude the proof of Theorem 3.6. Case p = 1, q > 1 (the case p > 1, q = 1 is treated similarly). We still use the weak formulation of the solution and argue by contradiction. Let us set I = ∫ T 0 ∫ Ω vϕ dx dt, J = (∫ T 0 ∫ Ω uqϕdx dt )1/q . Then, applying Holder’s inequality as above, we obtain I + ∫ T 0 ∫ Ω u0D γ1 t|Tϕdx dt ≤ J (A+ B), (4.51) EJDE-2020/110 EXISTENCE OF GLOBAL SOLUTIONS 23 where A = (∫ T 0 ∫ Ω ϕ− q′ q |∆ϕ|q ′ dx dt )1/q′ , B = (∫ T 0 ∫ Ω ϕ− q′ q |Dγ1 t|Tϕ| q′ dx dt )1/q′ ; and J q + ∫ T 0 ∫ Ω v0D γ1 t|Tϕdx dt ≤ λ ∫ T 0 ∫ Ω vϕ dx dt+ ∫ T 0 ∫ Ω vDγ2 t|Tϕdx dt ≤ (λ+ ε)I, (4.52) thanks to the ε-Young inequality and where we have chosen ϕ as a the first eigen- function of the spectral problem −∆ϕ = λϕ, x ∈ BT (0), ϕ|∂Ω = 0 , where (Ω = BT (0) ⊂ RN is the ball centered in zero and of radius T and ∂Ω is the boundary of Ω). Adding equation (4.52) to (λ+ε) times equation (4.51), we obtain J q + (λ+ ε) ∫ T 0 ∫ Ω u0D γ1 t|Tϕdx dt+ ∫ T 0 ∫ Ω v0D γ2 t|Tϕdx dt ≤ (λ+ ε)J (A+ B), whereupon, J q−1 ≤ A+ B. Replacing ϕ(x) by ϕ( xT ) and passing to the new variables y = T−1x and τ = T−1t, and then letting T go to infinity, we obtain a contradiction whenever q < 1+ 2 N . � Proof of Theorem 3.7. Let u0, v0 ∈ C0(RN ) be nonnegative and (u, v) be the corre- sponding solution of (1.1)-(1.2). We proceed by contradiction. Assume that (u, v) exists globally in time, that is (u, v) exists in (0, t∗(u0, v0)), for all t∗(u0, v0) > 0. Let T ∈ (0, t∗(u0, v0)) be arbitrarily fixed. Taking χ as test-function and setting X(t) := ∫ RN u(t, x)χ(x)dx, Y (t) := ∫ RN v(t, x)χ(x)dx Z(t) = ∫ Ω χ(u(t, x) + v(t, x))dx, Z0 = ∫ RN (u0 + v0)χ(x) dx. It follows from (1.1)-(1.2) that CDγ 0|t ∫ RN u(t, x)χ(x)dx− ∫ RN u∆χ(x)dx = ∫ RN |v(t, x)|pχ(x)dx, t ∈ (0, T ), CDγ 0|t ∫ RN v(t, x)χ(x)dx− ∫ RN v∆χ(x)dx = ∫ RN |u(t, x)|qχ(x)dx, t ∈ (0, T ), (4.53) supplemented with the initial conditions X(0) = ∫ RN u0(x)χ(x)dx, Y (0) = ∫ RN v0(x)χ(x)dx. (4.54) From (4.53)-(4.54), we have Dγ 0|t([Z − Z0])(t)− ∫ RN (u(t, x) + v(t, x))∆χ(x)dx = ∫ RN (|v(t, x)|p + |u(t, x)|q)χ(x)dx, t ∈ (0, T ). (4.55) 24 B. AHMAD, A. ALSAEDI, M. BERBICHE, M. KIRANE EJDE-2020/110 We observe that∫ RN v(x, t)χ(x) dx = ∫ RN v(x, t)χ 1 p (x)χ1− 1 p (x) dx. Since the function χ satisfies ∫ RN χ(x) dx = 1, then it yields by Hölder’s inequality that ∫ RN v(x, t)χ(x) dx ≤ (∫ RN |v(x, t)|pχ(x) dx )1/p . So ∫ RN |v(x, t)|pχ(x) dx ≥ (∫ RN v(x, t)χ(x) dx )p = Y p(t). (4.56) Similarly, we obtain∫ RN |u(x, t)|qχ(x) dx ≥ (∫ RN u(x, t)χ(x) dx )q = Xq(t). (4.57) Using estimates (4.56), (4.57) in (4.55) and the fact that the funtion χ satisfies ∆χ ≥ −χ, it yields Dγ 0|t([Z − Z0]) + Z(t) ≥ Y p(t) +Xq(t), t ∈ (0, T ). (4.58) By adding Z(t) to the two members of (4.58), we obtain Dγ 0|t([Z(t)− Z0]) + 2Z(t) ≥ Y p(t) +Xq(t) +X(t) + Y (t) ≥ Y p(t) +Xq(t) +X(t). We assume that q ≥ p, by using the fact that Xq(t) +X(t) ≥ Xp(t) and (a+ b)r ≤ 2r−1(ar + br), a, b > 0, r ≥ 1, we obtain Dγ 0|t([Z(t)− Z0]) + 2Z(t) ≥ 21−pZ(t)p. (4.59) We put F (y) = 21−pyp − 2y, the function F is convex on (0,∞) (since F ∈ C2(0,+∞), F ′′ ≥ 0). Writing ∂t(k ∗ [Z − Z0])(t) instead of Dγ 0|t([Z(t) − Z0]) with k(t) = t−γ Γ(1−γ) in (4.59), we obtain ∂t(k ∗ [Z − Z0])(t) ≥ F (Z(t)), t ∈ (0, T ). (4.60) It is clear that F (y) > 0 and F ′(y) > 0 for all y > 2 p p−1 := α1. Suppose now that Z0 > α1. We claim that (4.60) implies that Z(t) > α1 for all t ∈ (0, T ). In fact, for Z(0) = Z0 > α1, we have by continuity of Z, there exists δ ∈ (0, T ] such that Z(t) > α1 for all t ∈ (0, δ). This implies that F (Z(t)) > 0 for all t ∈ (0, δ). By the comparison principle, it follows that Z(t) ≥ Z0 for all t ∈ (0, δ). Setting δ1 := sup{s ∈ (0, T ) : Z(t) ≥ Z0 t ∈ (0, s)}, then δ1 > 0. We want to show that δ1 = T . Indeed, if δ1 < T , then by setting s = t− δ1 for t ∈ (δ1, T ) and Z̃(s) = Z(s+ δ1), s ∈ (0, T − δ1), it follows from positivity of Z − Z0 on (0, δ1) and k being non- increasing that ∂s(k ∗ [Z̃ − Z0])(s) ≥ ∂t(k ∗ [Z − Z0])(s+ δ1), s ∈ (0, T − δ1). (4.61) From (4.60) and (4.61) we deduce that ∂s(k ∗ [Z̃ − Z0])(s) ≥ F (Z̃(s)), s ∈ (0, T − δ1). EJDE-2020/110 EXISTENCE OF GLOBAL SOLUTIONS 25 This time-shifting property can be already found in [34]. So we may repeat the argument from above to see that there exists δ̃ ∈ (0, T − δ1] such that Z̃(s) ≥ Z0 for all s ∈ (0, δ̃). This leads to a contradiction with the definition of δ1. Hence, the assumption δ1 < T was not true. This proves the claim. Knowing that Z(t) ≥ Z0 > α1 for all t ∈ (0, T ) it follows from (4.60) that CDγ 0|tZ(t) = ∂t(k ∗ [Z − Z0])(t) ≥ F (Z(t)) > 0, for all t ∈ (0, T ). (4.62) Therefore the function Z(t) satisfying (4.62) is an upper solution of the problem CDγ 0|ty = F (y) = 21−pyp − 2y, y(0) = Z0, (4.63) we have by comparison principle Z(t) ≥ y(t) (see [15, Theorem 2.3],[16, Theorem 4.10.]). On the other hand, since F (0) ≥ 0, F (y) > 0 and F ′(y) > 0, for all y ≥ Z0 > 2 p p−1 . It then follows from Lemma 3.8, that v(t) = w( tγ Γ(γ+1) ) is a lower solution for (4.63) (which means CDγ 0|tv ≤ F (v) = 21−pvp − 2v, v(0) = Z0 ≤ Z0), where w(t) solves the ordinary differential equation dw dt = F (w) = 21−pwp − 2w, w(0) = Z0. (4.64) By the comparison principle (see [15, Theorem 2.3],[16, Theorem 4.10.]), we obtain y(t) ≥ v(t). So, by solving the Cauchy problem (4.64), which is equivalent to d dt (e2tw) = 21−pe2(1−p)t(e2tw)p, w(0) = Z(0), in which the explicit blow-up solution is w(t) = ( e2(1−p)t − 1 2p + Z1−p 0 ) 1 1−p e−2t which blows up in finite time t∗∗ = ln(1−2pZ1−p 0 ) 2(1−p) . By the comparison principle (see [15, Theorem 2.3],[16, Theorem 4.10.]), we conclude that Z(t) ≥ y(t) ≥ v(t) = w ( tγ Γ(γ + 1) ) = (e2(1−p) tγ Γ(γ+1) − 1 2p + Z1−p 0 ) 1 1−p e−2 tγ Γ(γ+1) which in turn leads to Z(t) blows-up in finite time at t̄∗∗ ≤ [ ln(1−2pZ1−p 0 ) 2(1−p) Γ(γ+1)]1/γ . Thus the same holds for the solution (u, v) of (1.1)-(1.2), which in turn leads to a contradiction. � Remark 4.2. Similar results were obtained in [40, Theorem 3.5] using another method, while the authors did not address the estimation of the time blow up. Acknowledgements. This project was funded by the Deanship of Scientific Re- search (DSR) at King Abdulaziz University, Jeddah, Saudi Arabia, under grant no. RG-39-130-38. The authors, therefore, acknowledge with thanks DSR technical and financial support. 26 B. AHMAD, A. ALSAEDI, M. BERBICHE, M. KIRANE EJDE-2020/110 References [1] C. Bandle, H. A. Levine, Q. S. Zhang; Critical exponents of Fujita type for inhomogeneous parabolic equations and systems, J. Math. Anal. Appl., 251 (2000), 624–648. [2] E. Bajlekova; Fractional Evolution Equations in Banach Spaces, PhD thesis, 2001, Technische Universiteit Eindhoven, DOI:10.6100/IR549476. [3] T. Cazenave, A. Haraux; An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 1998. [4] K. Deng, H. A. Levine; The role of critical exponents in blow-up theorems, the sequel, J. Math. Anal. Appl., 243 (2000), 85–126. [5] J. I. Diaz, T. Pierantozzi, L. Vazquez; Finite time extinction for nonlinear fractional evolution equations and related properties, Electron. J. Differential Equations, Vol. 2016 (2016), No. 239, pp. 1–13. [6] S. E. Eidelman, A. N. Kochubei; Cauchy problem for fractional differential equations, Journal of differential equations, 199 (2004), 211–255. [7] M. Escobedo, M. A. Herrero; Boundedness and blowup for a semilinear reaction-diffusion systems, J. Differ. Equat., 89 (1991), 176–202. [8] M. Escobedo, H. A. Levine; Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations, Arch. Rational Mech. Anal. 129 (1995), 47–100. [9] M. Fila, H. A. Levine, Y. Uda; A Fujita-type global existence-global non-existence theorem for a system of reaction diffusion equations with differing diffusivities, Mathematical Methods in the Appl. Sciences, 17(10) (1994), 807–835. [10] A. Z. Fino, M. Kirane; Qualitative properties of solutions to a time-space fractional evolution equation, Quart. Appl. Math. 70 (2012), no. 1, 133–157. [11] K. M. Furati, M. Kirane; Necessary conditions for the existence of global solutions to systems of fractional differential equations . Fract. Calc. Appl. Anal. 11 (2008), no. 3, 281–298. [12] V. Gafiychuk, B. Datsko, V. Meleshko; Mathematical modeling of time fractional reaction- diffusion systems, Journal of Computational and Applied Mathematics, 220 (2008) 215–225. [13] M. Guedda, M. Kirane; Criticality for some evolution equations, Differ. Uravn. 37, No. 4 (2001), 574–575; translation in Differ. Equ. 37, No. 4 (2001), 540–550. [14] M. Kirane, Y. Laskri, N. E. Tatar; Critical exponents of Fujita type for certain evolution equations and systems with spatio-temporal fractional derivatives, Journal of Mathematical Analysis and Applications, vol. 312, no. 2, pp. 488–501, 2005. [15] V. Lakshmikantham, A. S. Vatsala; Theory of fractional differential inequalities and appli- cations, Commun. Appl. Anal. 11 (2007), no. 3-4, 395-402. [16] Lei Li, Jian-Guo Liu; A generalized definition of Caputo derivatives and its application to fractional ODEs, SIAM J. Math. Anal., 50(3):2867–2900, 2018. [17] G. Lu, B. D. Sleeman; Sub-solutions and super-solutions to systems of parabolic equations with applications to generalized Fujita-type systems, Math. Methods Appl. Sci., 17 (1994), 1005–1016. [18] G. Lu; Global existence and blow-up for a class of semi-linear parabolic systems: A Cauchy problem, Nonlinear Anal., 24, No. 8 (1995), 1193–1206. [19] Y. Ma, F. Zhang, C. Li; The asymptotics of the solutions to the anomalous diffusion equa- tions, Computers and Mathematics with Applications, 66 (2013), 682–692. [20] R. L. Magin; Fractional calculus models of complex dynamics in biological tissues, J. Comput. Math. Appl., 59, 1586–1593 (2010). [21] R. Metzler, J. Klafter; The random walk’s guide to anomalous diffusion: A fractional dy- namics approach, Phys. Rep., 339 (2000), pp. 1–77. [22] K. Mochizuki; Blow-up, lifespan and large time behavior of solutions of a weakly coupled system of reaction diffusion equations, Adv. Math. Appl. Sci., 48, World Scientific (1998), 175–198. [23] K. Mochizuki, Q. Huang; Existence and behavior of solutions for a weakly coupled system of reaction-diffusion equations, Methods Appl. Anal., 5 (1998), 109–124. [24] E. Mitidieri, S. I. Pohozaev; A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Proceedings of the Steklov Institute of Mathematics 2001; 234:1–383. [25] M. A. Pozio, A. Tesei; Global existence of solutions for a strongly coupled semilinear parabolic system, in Recent Advances in Nonlinear Elliptic and Parabolic Problems, Nancy, 1988, EJDE-2020/110 EXISTENCE OF GLOBAL SOLUTIONS 27 Pitman Research Notes in Mathematics Series, Vol. 208, pp. 172–183, Longman, Harlow, 1989. [26] Quittner, P.; Souplet, Ph.; Superlinear parabolic problems. Blow-up, global existence and steady states. Birkhäuser Advanced Texts. Birkhäuser, Basel, 2007. [27] R. Redlinger; Pointwise a priori bounds for strongly coupled semilinear parabolics ystems, Indiana Univ. Math. J., 36, No. 2, (1987), 441–454. [28] K. M. Saada, J. F. Gomez-Aguilar; Coupled reaction-diffusion waves in a chemical system via fractional derivatives in Liouville-Caputo sense, Revista Mexicana de Fisica, 64 (2018) 539–547. [29] A. Samarskii, V. Galaktionov, S. Kurdyumov, A. Mikhailov; Blow-up in Quasilinear Para- bolic Equations, de Gruyter Expositions in Mathematics, Vol. 19, de Gruyter, Berlin, 1995. [30] S. G. Samko, A. A. Kilbas, O. I. Marichev; Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Newark, New Jersey, 1993. [31] H. Takase, B. D. Sleeman; Nonexistence of global solutions to anisotropic Fujita-type systems of semi-linear parabolic equations, Proc. R. Soc. London Ser. A 456 (2000), 365–386. [32] H. Takase, B. D. Sleeman; Existence and Nonexistence of Fujita-Type Critical Exponents for Isotropic and Anisotropic Semi-Linear Parabolic Systems, Journal of Mathematical Analysis and Applications, 265, 395–413 (2002) . [33] Y. Uda; The critical exponent for a weakly coupled system of the generalized Fujita type reaction-diffusion equations, Z. Angew Math. Phys., 46 (1995), 366–383. [34] V. Vergara, R. Zacher; Stability, instability, and blow-up for time fractional and other non- local in time semi-linear sub-diffusion equations, J. Evol. Eq. 17 (2017), 599–626. [35] S. Subedi, A. S. Vatsala; Blow-up results for one dimensional Caputo fractional reaction diffusion equation, Mathematics in Engineering, Science & Aerospace, 10, (2019), 175–190. [36] R. N. Wang, D. H. Chen, T. J. Xiao; Abdtract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252 (2012), 202–235. [37] F. B. Weissler; Single point blow-up of semi-linear initial value problem, J. Differential Equa- tions, 55 (1984), 204–224. [38] S. Zhang; Monotone Method for Initial Value Problem for Fractional Diffusion Equations, Science in China Series A: Mathematics (2006), 1223–1230. [39] Q.-G. Zhang, H.-R. Sun; The blow-up and global existence of solutions of Cauchy problems for a time fractional diffusion equation, Topol. Methods Nonlinear Anal. 46 (2015), no. 1, 69–92. [40] Q. Zhang, H.R. Sun, Y. Li; Global existence and blow-up of solutions of the Cauchy problem for a time fractional diffusion system, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.03.013. [41] Qi S. Zhang; Blow-up results for nonlinear parabolic equations on manifolds, Duke Math. J., Vol. 97 (1999), 515–539., [42] Y. Zhou, X. H. Shen, L. Zhang; Cauchy problem for fractional evolution equations with Caputo derivative, Eur. Phys. J. Spec. Top. 222 (2013) 1747–1764. Bashir Ahmad NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdu- laziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Email address: bashirahmad qau@yahoo.com Ahmed Alsaedi NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdu- laziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Email address: aalsaedi@hotmail.com Mohamed Berbiche Laboratory of Mathematical Analysis, Probability and Optimizations, Mohamed Khider University, Biskra, PO. Box 145, Biskra (07000) Algeria Email address: berbichemed@yahoo.fr, mohamed.berbiche@univ-biskra.dz 28 B. AHMAD, A. ALSAEDI, M. BERBICHE, M. KIRANE EJDE-2020/110 Mokhtar Kirane Department of Mathematics and Statistics, College of Art and Sciences, Khalifa Uni- versity of Science and Technology, Abu Dhabi, United Arab Emirates. LaSIE, Université de La Rochelle, Pôle Sciences et Technologies, Avenue Michel Crépeau, 17000 La Rochelle, France. NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdu- laziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Email address: mokhtar.kirane@ku.ac.ae, mokhtar.kirane@univ-lr.fr 1. Introduction 2. Preliminaries 3. Main results 4. Proofs of main results Acknowledgements References