Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 67, pp. 1–19. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.67 GLOBAL SOLUTION FOR COUPLED PARABOLIC SYSTEMS WITH DEGENERATE COEFFICIENTS AND TIME-WEIGHTED SOURCES RICARDO CASTILLO, OMAR GUZMÁN-REA, MIGUEL LOAYZA, MARÍA ZEGARRA Abstract. In this article we obtained the so-called Fujita exponent for the degenerate parabolic coupled system ut − div(ω(x)∇u) = trvp vt − div(ω(x)∇v) = tsup in RN × (0, T ) with initial data belonging to [L∞(RN )]2, where p, q > 0 with pq > 1; r, s > −1, and either ω(x) = |x1|a or ω(x) = |x|b with a, b > 0. 1. Introduction Several authors have studied models associated with elliptic and parabolic partial differential equations, which presents a diffusion operator of the form div(ω(x)∇·), where div is the divergent, ∇ is the gradient, and the spatial function ω : RN → [0,∞) is a weight representing the part of thermal diffusion, which can degenerate. See for example the works of Kamin and Rosenau [21, 22, 23]; Kohn and Niren- berg [26]; Fabes, Kenig, and Serapioni [11]; Gutierrez and Nelson [15]; Fujishima, Kawakami, and Sire [12]; Dong and Phan [9]; Sire, Terracini, and Vita [28]; Zel- dovich [40]; Jleli, Kirane, and Samet [19]; and Jing, Nie, and Wang [20]. See also the works of Wang and Zhao [37, 38], where it is studied parabolic problems related to biological population models. We are interested in the degenerate coupled parabolic problem with time-weighted sources, ut − div(ω(x)∇u) = h1(t)v p in RN × (0, T ), vt − div(ω(x)∇v) = h2(t)u q in RN × (0, T ), u(0) = u0, v(0) = v0 in RN , (1.1) where (u0, v0) ∈ L∞(RN )×L∞(RN ) ≡ [L∞(RN )]2; u0, v0 ≥ 0; p, q > 0 with pq > 1; h1(t) = tr, h2(t) = ts with r, s > −1; and the weighted function ω : RN → [0,∞) satisfies one of the the following two conditions: either (A1) ω(x) = |x1|a with a ∈ [0, 1) for N = 1, 2, and a ∈ [0, 2/N) for N ≥ 3, or 2020 Mathematics Subject Classification. 35K05, 35A01, 35K58, 35K65, 35B33. Key words and phrases. Heat equation; global solution; degenerate coefficients; Fujita exponent; coupled system. ©2024. This work is licensed under a CC BY 4.0 license. Submitted April 11, 2024. Published October 31, 2024. 1 2 R. CASTILLO, O. GUZMÁN-REA, M. LOAYZA, M. ZEGARRA EJDE-2024/67 (A2) ω(x) = |x|b with b ∈ [0, 1). The function ω, with these characteristics, belongs to the Muckenhoupt class of functions A1+2/N . Moreover, the operator div(ω(x)∇·) is not self-adjoint, as noted in the observations made by Fujishima et al. [12]. In scenario (A1), the function ω exhibits a line of singularities. Consequently, problem (1.1) connects to the fractional Laplacian via the Caffarelli-Silvestre exten- sion, as referenced in [2, 12, 30, 5]. Additionally, the fractional Laplacian is linked to nonlocal diffusion and is present in the Levy diffusion process, as illustrated in [8, 24]. Fujishima et al. [12] studied the problem ut − div(ω(x)∇u) = up in RN × (0, T ), u(0) = u0 in RN , (1.2) and obtained the Fujita exponent p⋆(α) = 1 + 2− α N , where α = a in case (A1) and α = b in case (A2). When ω = 1, the problem defined in (1.2) has been studied by various re- searchers. Hirose Fujita [13] linked the critical exponent p⋆(0) to the global ex- istence of solutions for problem (1.2). He demonstrated that for 1 < p < p∗(0), problem (1.2) lacks any non-negative global solutions. When p > p∗(0), both global and non-global solutions may arise, contingent on the size of the initial conditions; for further information, refer to [27, 31]. In the critical scenario where p = p∗(0), Hayakawa [16] (for N = 1, 2), and subsequently Aronson and Weinberger [1] (for N ≥ 3), proved that problem (1.2) does not possess a global solution. Problem (1.1), with ω = 1 and h1 = h2 = 1, was studied firstly by Escobedo and Herrero [10]. They showed that (pq)∗ = 1 + 2 N (max{p, q}+ 1) is the Fujita exponent for problem (1.1), that is, if 1 < pq ≤ (pq)∗, then any non- trivial nonnegative solution blows up in finite time, and when pq > (pq)∗, there exist both global and nonglobal solutions. The case h1(t) = (1+ t)r and h2(t) = (1+ t)s was analyzed later in Cao et al. [6] who showed the existence of the Fujita exponent (pq)∗ = 1 + 2max{(r + 1)q + s+ 1, (s+ 1)p+ r + 1} N , for problem (1.1). See also [3, 4, 18] and the references therein for other related results. The primary aim of this study is to ascertain the Fujita exponent for problem (1.1). To achieve this, we employ the methods outlined in [12, 10], which are adapted to address the challenges specific to the degenerate coupled system and to handle the scenario where pq > 1 with 0 < p < 1 (or 0 < q < 1). Notably, we rely solely on the properties (A3)-(A7) that are confirmed by the fundamental solution Γ linked to the linear problem (2.1) (detailed in Section 2). Consequently, the conventional approaches for addressing problem (1.1) (where h1 = h2 = ω = 1) require refinement. EJDE-2024/67 COUPLED PARABOLIC SYSTEMS WITH DEGENERATE COEFFICIENTS 3 The approach that we use can also be applied to determine the critical Fujita exponent of the following problems: (ui)t − div(ω(x)∇ui) = triuqii+1, i = 1, . . . ,m− 1 in RN × (0, T ), (um)t − div(ω(x)∇um) = trmuqm1 in RN × (0, T ), (1.3) and ut − div(ω(x)∇u) = tr1up + tr2vq in RN × (0, T ), vt − div(ω(x)∇v) = tr3ur + tr4vs in RN × (0, T ). (1.4) When ω = 1, problem (1.3) was investigated in [32, 35, 4], while problem (1.4) was examined in [7, 34, 3]. Moreover, similar outcomes can be achieved by consid- ering the operator ω(x)−1 div(ω(x)∇ui) in place of the operator div(ω(x)∇ui), as demonstrated in the recent findings in [17, 25]. Solutions for problem (1.1) with initial data (u0, v0) ∈ [L∞(RN )]2 are understood in the following sense. Definition 1.1. Let u and v be a.e. finite, measurable functions defined on RN × (0.T ) for some T > 0. A pair (u, v) is called a solution of (1.1) with initial condition (u0, v0) ∈ [L∞(RN )]2, if (u, v) ∈ [L∞((0, T );L∞(RN ))]2 and satisfies u(x, t) = ∫ RN Γ(x, y, t)u0(y) dy + ∫ t 0 ∫ RN Γ(x, y, t− σ)h1(σ)v(y, σ) p dy dσ <∞, v(x, t) = ∫ RN Γ(x, y, t)v0(y) dy + ∫ t 0 ∫ RN Γ(x, y, t− σ)h2(σ)u(y, σ) q dy dσ <∞, (1.5) for almost all x ∈ RN and t ∈ (0, T ). If T = ∞, we say that (u, v) is a global-in-time solution of (1.1). Here S(t)ϕ(x) := [S(t)ϕ](x) := ∫ RN Γ(x, y, t)ϕ(y) dy where Γ(x, y, t) is the fundamental solution of the linear problem ut−div(ω∇u) = 0 in RN × (0,∞). Henceforth, we consider the following values: γ1 := (r + 1) + (s+ 1)p pq − 1 , (1.6) γ2 := (s+ 1) + (r + 1)q pq − 1 , (1.7) r1⋆ := N (2− α)γ1 , (1.8) r2⋆ := N (2− α)γ2 . (1.9) Our main result is the following. Theorem 1.2. Let r, s > −1, p, q > 0, with pq > 1. Suppose that α = a in the case that ω satisfies the condition (A1), and α = b in the case that ω satisfies the condition rm (A2). (i) If γ := max{γ1, γ2} ≥ N/(2 − α), then problem (1.1) has no nontrivial global- in-time solution. 4 R. CASTILLO, O. GUZMÁN-REA, M. LOAYZA, M. ZEGARRA EJDE-2024/67 (ii) If γ := max{γ1, γ2} < N/(2 − α), then there are nontrivial global-in-time solutions to (1.1). Moreover, there exists a constant δ > 0 such that for any (u0, v0) ∈ [L∞(RN ) ∩ Lr1⋆,∞(RN )]× [L∞(RN ) ∩ Lr2⋆,∞(RN )] with max{∥u0∥r1⋆,∞, ∥v0∥r2⋆,∞} < δ, then problem (1.1) has a global-in- time solution (u, v) satisfying: sup t>0 (1 + t) N 2−α ( 1 r1⋆ − 1 µ ) ∥u(t)∥µ,∞ <∞, sup t>0 (1 + t) N 2−α ( 1 r2⋆ − 1 µ ) ∥v(t)∥µ,∞ <∞ for max{r1⋆, r2⋆} < µ ≤ ∞. Remark 1.3. Here are some comments on Theorem 1.2. (i) When α = 0, Theorem 1.2 coincides with the result in [6, Theorem 1]. (ii) When α = 0 and r = s = 0, this theorem coincides with the results in [10]. Moreover, the values r1⋆ = N(pq−1)/2(p+1) and r2⋆ = N(pq−1)/2(q+1) are the same used in [10] to determine the global existence. (iii) The result is sharp and shows that the critical value of Fujita is given by (pq)∗(α) = 1 + (2− α)max{(s+ 1)p+ r + 1, (r + 1)q + s+ 1} N . This work is organized as follows. In section 2, we present the necessary prelim- inaries. Then in section 3, we prove the non-global existence. Finally, in section 4, we prove the global existence. 2. Preliminaries and technical results In that follows, C denotes a generic positive constant that may vary in different places, and its change is not essential to the analysis. The positive part of ϕ(x) is defined by ϕ+(x) = max{ϕ(x), 0}. The negative part of ϕ is defined analogously. For x = (x1, . . . , xN ) ∈ RN , |x| = (∑N i=1 x 2 i )1/2 is the Euclidean norm of RN . The spaces L∞(RN ) and Lζ(RN )(ζ ≥ 1) are defined as usual, and their norms are denoted by ∥ · ∥∞ and ∥ · ∥ζ , respectively. For 1 ≤ ζ ≤ ∞ and 1 ≤ σ ≤ ∞, the Lorentz space Lζ,σ(RN ) is defined as Lζ,σ := {ψ : RN → R;ψ is measurable and ∥ψ∥Lζ,σ(RN ) <∞}, where ∥ψ∥Lζ,σ(RN ) := ∥ψ∥Lζ,σ = {( ∫∞ 0 [s 1 ζ ψ⋆(s)]σ dss )1/σ if 1 ≤ σ <∞, sups>0 s 1/ζψ⋆(s) if σ = ∞, ψ⋆(s) := inf{λ > 0;µψ(λ) ≤ s}, µψ(λ) := {x : |ψ(x)| > λ}|, λ ≥ 0, is the distribution function of ψ. By definition, L∞,∞(RN ) = L∞(RN ). The Lorentz space Lζ,σ(RN ) is a Banach space; see [14, 41] for details. EJDE-2024/67 COUPLED PARABOLIC SYSTEMS WITH DEGENERATE COEFFICIENTS 5 Definition 2.1. The Muckenhoupt class Ap, with 1 < p <∞, is the set of locally integrable nonnegative functions w that satisfy( − ∫ Q w dx )( − ∫ Q w− 1 (p−1) dx )p−1 < K, for every cube Q and some constant K > 0. For p = 1, the function w belongs to the Muckenhoupt class A1 if there exists a constant K > 0 such that − ∫ Q w dx ≤ K inf Q w, for all cube Q. We will denote by Γ := Γ(x, y, t) the fundamental solution of the homogeneous problem ut − div(ω(x)∇u) = 0 (2.1) in RN × (0, T ), with a pole at point (y, 0), and ω verifying either (A1) or (A2) condition. Since ω belongs to the classes A1+2/N and A2 (see[29]), we have that the fundamental solution Γ = Γ(x, y, t) satisfies the following properties (see [15, 12] for more details): (A3) ∫ RN Γ(x, y, t)dx = ∫ RN Γ(x, y, t) dy = 1 for x, y ∈ RN and t > 0; (A4) Γ(x, y, t) = ∫ RN Γ(x, ξ, t− s)Γ(ξ, y, s)dξ for x, y ∈ RN and t > s > 0; (A5) Let c0 := supQ ( − ∫ Q ω(x)dx )( − ∫ Q ω(x)−1dx ) < ∞, where the supremum is taken over all cubes Q ∈ RN , and hx(r) = (∫ Br(x) ω(y)−N/2 dy )2/N . Then there exist constants C0⋆, c0⋆ > 0, depending only on N and c0, such that c−1 0⋆ ( 1 [h−1 x (t)]N + 1 [h−1 y (t)]N ) exp [ − c0⋆ (hx(| x− y |) t )1/(1−α)] ≤ Γ(x, y, t) ≤ C−1 0⋆ ( 1 [h−1 x (t)]N + 1 [h−1 y (t)]N ) exp [ − C0⋆ (hx(| x− y |) t )1/(1−α)] for x, y ∈ RN , t > 0, and α ∈ {a, b}, where h−1 x denotes the inverse function of hx. Also, by [12, estimates (2.11), (2.12)], we have (A6) ∫ |x|≤t1/(2−α) Γ(x, y, t)dx ≥ C, for all |y| ≤ t1/(2−α), and some constant C > 0. (A7) Γ(x, y, t) ≥ Ct−N/(2−α), for |x|, |y| ≤ t1/(2−α), t > 0, and some constant C > 0. Remark 2.2. From (A5), we deduce that the fundamental solution Γ is nonnega- tive. Moreover, if u0 ∈ L∞(RN ) is such that u0 ≥ 0 and u0 ̸= 0, then according to (A6) (or (A7), there exists a τ0 := τ(u0) > 0 for which S(t)u0(x) = ∫ RN Γ(x, y, t)u0(y) dy > 0, for almost every x ∈ RN and all t > τ0. 6 R. CASTILLO, O. GUZMÁN-REA, M. LOAYZA, M. ZEGARRA EJDE-2024/67 The subsequent results will be utilized to demonstrate the existence of solutions global-in-time for equation (1.1). Proposition 2.3 ([12]). (i) Let ϕ ∈ Lq1(RN ) and 1 ≤ q1 ≤ q2 ≤ ∞, then ∥S(t)ϕ∥q2 ≤ c1t − N 2−α ( 1 q1 − 1 q2 )∥ϕ∥q1 , (2.2) for t > 0. The constant c1 > 0 can be taken so that it depends only on N , α ∈ {a, b}. (ii) Let ϕ ∈ Lq1,∞(RN ) with 1 < q1 ≤ q2 ≤ ∞, then ∥S(t)ϕ∥q2,∞ ≤ c2t − N 2−α ( 1 q1 − 1 q2 )∥ϕ∥q1,∞, (2.3) for t > 0. The constant c2 > 0 can be taken so that it depends only on q1, N , and α ∈ {a, b}. In particular, c2 is bounded in q1 ∈ (1 + ε,∞) for any fixed ε > 0 and c2 → ∞ as q1 → 1. Another tool used is the following interpolation result in Lorentz space. Proposition 2.4 ([14]). Let 1 ≤ r0 ≤ r2 ≤ r1 ≤ ∞ be such that 1 r2 = θ r0 + 1−θ r1 , for θ ∈ [0, 1]. Then ∥f∥r2,∞ ≤ ∥f∥θr0,∞∥f∥1−θr1,∞, (2.4) for f ∈ Lr0,∞ ∩ Lr1,∞. The subsequent results will be utilized to demonstrate the existence of non-global solutions to equation (1.1). Lemma 2.5 ([12]). Assume that ω satisfies either (A1) or (A2). Let ϕ ∈ L∞(RN ), ϕ ≥ 0, and ϕ ̸= 0. Then there exists a positive constant C(α,N), depending only on α and N , such that S(t)ϕ(x) ≥ C(α,N)−1t− N 2−α ∫ |y|≤t 1 2−α ϕ(y)dy, for |x| ≤ t 1 2−α and t > 0, where α is defined by α = a in the case (A1) and α = b in the case (A2). Lemma 2.6. Assume that ω satisfies either (A1) or rm (A2). If u0 ∈ L∞(RN ) is a nonnegative function and q ≥ 1, then∫ RN Γ(x, y, t)[u0(y)] q dy ≥ (∫ RN Γ(x, y, t)u0(y) dy )q . If 0 < q < 1, then(∫ RN Γ(x, y, t)u0(y) dy )q ≥ ∫ RN Γ(x, y, t)[u0(y)] qdy. Proof. Since the fundamental solution Γ is nonnegative, by (A5), ∫ RN Γ(x, y, t) dy = 1, and by (A3), we can use Jensen’s inequality for q > 1 in the estimate∫ RN Γ(x, y, t)[u0(y)] q dy ≥ (∫ RN Γ(x, y, t)u0(y) dy )q . For 0 < q < 1, we observe that uq0 ∈ L∞(RN ). Thus, the conclusion follows as the anterior case replacing q by 1/q. □ EJDE-2024/67 COUPLED PARABOLIC SYSTEMS WITH DEGENERATE COEFFICIENTS 7 3. Nonglobal existence To demonstrate the non-global existence aspect of Theorem 1.2, we require the subsequent result. The method employed is traditional, albeit with necessary ad- justments (refer to [39]). Proposition 3.1. Assume that ω satisfies either (A1) or (A2), and u0, v0 ∈ L∞(RN ) with u0, v0 ≥ 0. Suppose that (u, v) ∈ [L∞((0, T ), L∞(RN ))]2 is a so- lution of problem (1.1) with 0 < T ≤ ∞, and p, q > 0 with pq > 1. Then there exists a constant C⋆ > 0 (which depends only on p, q, r, and s), such that tγ1∥S(t)u0∥∞ ≤ C⋆, if q > 1, tqγ1∥S(t)uq0∥∞ ≤ C⋆, if 0 < q < 1, tγ2∥S(t)v0∥∞ ≤ C⋆, if p > 1, tpγ2∥S(t)vp0∥∞ ≤ C⋆, if 0 < p < 1, (3.1) for all t ∈ [0, T ), where γ1, γ2 are given by (1.6) and (1.7). Proof. Since u0 ∈ L∞(RN ), from (2.2) we have S(t)u0(x) < ∞ for a.e. x ∈ RN . We will prove the first inequality of (3.1). To do this, we will show the estimate u(x, t) ≥ Ckt (βk−1)γ1 [S(t)u0(x)] βk (k ∈ N ∪ {0}), (3.2) for a.e. x ∈ RN and t ∈ (0, T ), where C0 = 1, β = pq and Ck = Cβk−1[(β k−1 − 1)qγ1 + s+ 1]−p[(βk−1 − 1)γ1β + p(s+ 1) + (r + 1)]−1, (3.3) for k ∈ N ∪ {0}. We proceed by induction on k. From (1.5) and property (A5), it follows that u(x, t) ≥ S(t)u0(x) for almost every x ∈ RN and all t > 0; thus, (3.2) is satisfied for k = 0. Now, assuming that estimate (3.2) is valid for k ≥ 1, we apply (1.5), properties (A3), (A4), (A5), and Lemma 2.6 to obtain v(x, t) ≥ ∫ t 0 ∫ RN Γ(x, y, t− σ)h1(σ)[u(y, σ)] q dydσ ≥ ∫ t 0 ∫ RN Γ(x, y, t− σ)σs[Ckσ (βk−1)γ1 [S(σ)u0(y)] βk ]qdσ ≥ Cqk ∫ t 0 σ(βk−1)γ1q+s [ ∫ RN (∫ RN Γ(x, y, t− σ)Γ(y, z, σ) dy ) u0(z)dz ]qβk dσ ≥ Cqk [S(t)u0(x)] qβk ∫ t 0 σ(βk−1)γ1q+sdσ = Ck,1t (βk−1)γ1q+s+1[S(t)u0(x)] qβk (3.4) for a.e. x ∈ RN and t > 0, where Ck,1 = Cqk/((β k − 1)γ1q + s+ 1). Similarly, from (3.4), we obtain u(x, t) ≥ ∫ t 0 ∫ RN Γ(x, y, t− σ)σs [ Ck,1σ (βk−1)γ1q+s+1[S(σ)u0(y) ]qβk ]pdydσ ≥ Cpk,1[S(t)u0(x)] βk+1 ∫ t 0 σ(βk−1)γ1β+(s+1)p+rdσ = Ck,2t (βk−1)γ1β+(s+1)p+(r+1)[S(t)u0(x)] βk+1 8 R. CASTILLO, O. GUZMÁN-REA, M. LOAYZA, M. ZEGARRA EJDE-2024/67 for a.e. x ∈ RN and t > 0, where Ck,2 = Cpk,1/[(β k − 1)γ1β + (s + 1)p + (r + 1)]. Since (βk − 1)γ1β + (s+ 1)p+ (r + 1) = (βk+1 − 1)γ1, we have u(x, t) ≥ Ck,2t (βk+1−1)γ1 [S(t)u0(x)] βk+1 , for a.e. x ∈ RN and t > 0. Setting Ck+1 = Ck,2 and inserting the value of Ck,1, we obtain (3.3). Thus, the induction process is complete. Now we show that there exists κ0 > 0 such that Ck ≥ κβ k 0 for all k ≥ 2. Defining θk = −β−k ln(Ck) it is sufficient to prove that the sequence {θk}k∈N is bounded from above. From relation (3.3), we have θi − θi−1 = β−i ln (Cβi−1 Ci ) = β−i ln ( [(βi−1 − 1)qγ1 + s+ 1]p[(βi−1 − 1)γ1β + p(s+ 1) + (r + 1)] ) ≤ { β−i ln[γ1(β i − 1)]p+1 if p > 1, β−i ln[q[γ1(β i − 1)]2 if 0 < p ≤ 1 ≤ Cβ−i(i+ 1). This implies that θk − θ1 = ∑k i=1(θi − θi−1) ≤ C ∑k i=1 β −i(i+ 1) <∞. From (3.2) and the estimate Ck ≥ κβ k 0 we have that u(x, t)1/β k ≥ κ0t γ1(1−1/βk)S(t)u0(x), for a.e x ∈ RN and t ∈ (0, T ). Since β > 1, letting k → ∞, we obtain the first inequality of (3.1). For the proof of the second inequality of (3.1), we argue similarly to the previous case. We use properties (A3)–(A5), and Lemma 2.6 iteratively, starting with v(x, t) ≥ ts+1S(t)[u0(x)] q, (3.5) until the inequality u(x, t) ≥ Dkt (βk−1)γ1 [S(t)[u0(x)] q]pβ k−1 , (3.6) for a.e. x ∈ RN , t ∈ (0, T ), and k ∈ N, where β = pq, and Dk ≥ ηβk 1 (η1 > 0). So, from (3.6), we obtain u(x, t)q/β k ≥ η1t qγ1(1−1/βk)S(t)[u0(x)] q, for a.e. x ∈ RN , t ∈ (0, T ) and some positive constant η1. Letting k tends to infinity, we obtain the desired estimate. By the symmetry the problem, the other inequalities can be proved analogously. □ The following result is a direct consequence of the above proposition. Corollary 3.2. Assume that ω satisfies either (A1) or (A2) condition, and u0, v0 ∈ L∞(RN ) with u0, v0 ≥ 0. If (u, v) ∈ [L∞((0,∞), L∞(RN ))]2 is a global-in-time solution of (1.1) then there exists a constant C⋆⋆ > 0 (which depends only on p, q, r, and s) such that tγ1∥S(t)u(t)∥∞ ≤ C⋆⋆, if q > 1, tqγ1∥S(t)[u(t)]q∥∞ ≤ C⋆⋆, if 0 < q < 1, EJDE-2024/67 COUPLED PARABOLIC SYSTEMS WITH DEGENERATE COEFFICIENTS 9 tγ2∥S(t)v(t)∥∞ ≤ C⋆⋆, if p > 1, tpγ2∥S(t)[v(t)]p∥∞ ≤ C⋆⋆, if 0 < p < 1, for all t ∈ (0,∞). Proof. Since (u, v) is a global-in-time solution to equation (1.1), the pair (u(· + σ), v(·+σ)) for σ > 0 also constitutes a global-in-time solution to the same problem with the initial condition (u(σ), v(σ)). Consequently, the estimate in equation (3.1) applies with (u(σ), v(σ)) replacing (u0, v0). Therefore, the result is obtained by setting σ = t in this estimate. □ Lemma 3.3. Under the assumptions of Proposition 3.1, let (u, v) be a global-in- time solution of (1.1) with initial condition (0, 0) ̸= (u0, v0) ∈ [L∞(RN )]2. Then there exist τ0 = τ0(u0, v0) > 0 such that u(x, t) > 0 and v(x, t) > 0 a.e. x ∈ RN and t > τ0. Proof. Assuming u0 ̸= 0, Remark 2.2 implies that [S(t)u0(x)] > 0 for almost every x ∈ RN and t > τ0 with some τ0 > 0. Following the reasoning used to derive (3.4), we obtain u(x, t) ≥ [S(t)u0](x) > 0, and v(x, t) ≥ (s+ 1)−1[(S(t)u0)(x)] qts+1 > 0, for almost every x ∈ RN and t > τ0. A similar approach applies when v0 ̸= 0. □ Proof of the nonglobal existence (Theorem 1.2(i)). Assuming without loss of generality that γ = γ1, we proceed by contradiction. Suppose there exists a global-in-time solution (u, v) to problem (1.1) with the initial condition (u0, v0) ̸= (0, 0). We will consider two cases: Case I: q > 1. Let us assume first that γ1 > N/(2 − α). By Lemma 3.3, there exists τ0 such that u(x, t) > 0 and v(x, t) > 0, (3.7) for a.e. x ∈ RN and t > τ0. Define w(t) := u(t+τ) and z(t) := v(t+τ) for all t ≥ 0 and some τ > min{1, τ0}. It follows from (3.7) that w0 := w(0) ̸= 0 and z0 := z(0) ̸= 0. Given that (w, z) forms a global-in-time solution to (1.1) with the initial condition (w0, z0) = (u(τ), v(τ)), Proposition 3.1 ensures that tγ1∥S(t)w0∥∞ ≤ C⋆ for all t ≥ 0. (3.8) On the other hand, since w0 > 0 there exists a non-trivial function 0 ≤ U1 ∈ L∞(RN ) such that suppU1 ⊂ B(t 1/(2−α) 0 ) (the ball of center 0 and radius t 1/(2−α) 0 ) for some t0 ≥ 1, and 0 ≤ U1 ≤ w0. By Lemma 2.5, we obtain S(t)U1(x) ≥ CMt− N 2−α , M := ∫ B(t 1/(2−α) 0 ) U1(y) dy, (3.9) for |x| ≤ t1/(2−α), t ≥ t0 and C > 0. Consequently, by property (A5), it follows that tγ1∥S(t)w0∥∞ ≥ tγ1∥S(t)U1∥∞ ≥ CMtγ1− N 2−α , for all t ≥ t2, which contradicts (3.8). Now, reconsider the previously mentioned global-in-time solution (w(t), z(t)) with γ1 = N 2−α . Following a computation similar to that in the derivation of (3.4), we obtain z(x, t) ≥ Cts+1[(S(t)w0)(x)] q, (3.10) 10 R. CASTILLO, O. GUZMÁN-REA, M. LOAYZA, M. ZEGARRA EJDE-2024/67 for almost every x ∈ RN and for all t > 0, and some constant C > 0. Conversely, from (3.9), it follows that S(t)w0(x) ≥ Ct− N 2−α = Ct−γ1 , (3.11) for all t ≥ t0, and for |x| ≤ t 1 2−α . Note that, t + 1 − σ ≤ t and σ ≤ t + 1 − σ for 1 ≤ σ ≤ t/2. Thus, from (1.5), (A5), (A6), (A7), (3.10), and (3.11), we obtain∫ |x|≤(t+1)1/(2−α) w(x, t+ 1)dx ≥ ∫ |x|≤t1/(2−α) w(x, t+ 1)dx ≥ ∫ |x|≤t1/(2−α) ∫ t/2 1 ∫ |y|≤(t+1−σ)1/(2−α) σrΓ(x, y, t+ 1− σ)z(y, σ)p dydσdx ≥ ∫ t/2 t1 ∫ |y|≤(t+1−σ)1/(2−α) σr (∫ |x|≤(t+1−σ) 1 2−α Γ(x, y, t+ 1− σ)dx ) z(y, σ)p dy dσ ≥ C ∫ t/2 t1 ∫ |y|≤(t+1−σ)1/(2−α) σr(σs+1[S(σ)w0(y)] q)p dy dσ ≥ C ∫ t/2 t1 ∫ |y|≤(t+1−σ)1/(2−α) σr+(s+1)p[S(σ)w0(y)] pq−1[S(σ)w0(y)] dy dσ ≥ C ∫ t/2 t1 σr+(s+1)p · σ−(pq−1)γ1 (∫ |y|≤σ1/(2−α) σ−γ1 dy ) dσ ≥ C ∫ t/2 t1 σ−1dσ = C ln (t/(2t1)) > 0, (3.12) for t/2 > t1 = max{t0, 2} and some constant C > 0. From (3.12), we deduce that for any R > 0, there exists t2 − 1 > 2t1 such that the function U2, defined by U2(x) := w(x, t2) ∈ L∞(RN ), satisfies∫ |x|≤t1/(2−α) 2 U2(x) dx ≥ C ln ( t2 − 1 2t1 ) > R. (3.13) Define (w1(t), z1(t)) = (w(t+ t2), z(t+ t2)). Note that (w1, z1) constitutes a global- in-time solution of (1.5) with the initial condition (w1(0), z1(0)) = (U2(x), z(t2)). Consequently, by Proposition 3.1, it follows that tγ1∥S(t)U2∥∞ ≤ C⋆ for all t ≥ 0. (3.14) However, from (3.13) and Lemma 2.5, it is established that S(t)U2(x) ≥ C(α,N)−1Rt− N 2−α , for |x| ≤ t1/(2−α) and t > t2. Consequently, tγ1∥S(t)U2∥∞ = t N 2−α ∥S(t)U2∥∞ ≥ C(α,N)−1R, for all t > t2. This contradicts (3.14) because of the arbitrariness of R > 0. EJDE-2024/67 COUPLED PARABOLIC SYSTEMS WITH DEGENERATE COEFFICIENTS 11 Case II: 0 < q < 1. From Lemma 3.3, we can assume without loss of generality, that u(t) > 0 and v(t) > 0 for all t ≥ 0. Thus, Corollary 3.2 implies tqγ1∥S(t)uq(t)∥∞ ≤ C⋆⋆, for all t > 0. (3.15) First, assume that γ1 > N 2−α . We can then find a non-trivial function 0 ≤ U3 ∈ L∞(RN ) such that suppU3 ⊂ B(t 1 2−α 0 ) for some t0 > 1 and 0 ≤ U3 ≤ u0. Following a similar argument to the derivation of (3.9), we obtain u(x, t) ≥ S(t)u0(x) ≥ Ct− N 2−αX t 1 2−α (x), (3.16) for t ≥ t0 and some constant C > 0, where Xt1/(2−α) is the characteristic function on the ball centered at 0 with radius t1/(2−α). Consequently, [u(x, t)]q ≥ Ct−q N 2−αXt1/(2−α)(x), for t ≥ t0 and some constant C > 0. This leads to tqγ1∥S(t)[u(t)]q∥∞ ≥ Ctq(γ1− N 2−α )S(t)Xt1/(2−α)(x), (3.17) for t ≥ t0. Moreover, by (A7), we have S(t)X t 1 2−α (x) ≥ ∫ |y| 0. Hence, estimate (3.17) contradicts (3.15). Now, let us assume that γ1 = N 2−α . Given that (u, v) is a global-in-time solution to equation (1.1), it follows that for any τ > 0: u(x, t+ τ) = ∫ RN Γ(x, y, t)u(y, τ) dy + ∫ t 0 ∫ RN Γ(x, y, t− σ)σrvp(y, σ + τ) dy dσ, v(x, t+ τ) = ∫ RN Γ(x, y, t)v(y, τ) dy + ∫ t 0 ∫ RN Γ(x, y, t− σ)σsuq(y, σ + τ) dy dσ, and (u(·+τ), v(·+τ)) is also a global-in-time solution of (1.1) with initial condition (u(τ), v(τ)). Then, recalling that 0 < q < 1 and proceeding similarly as in (3.5), we obtain v(x, t+ τ) ≥ Cts+1S(t)[u(τ)]q(x), (3.19) for a.e. x ∈ RN and t > 0. Thus, taking t = τ in (3.19) and arguing similarly as in (3.16)-(3.18), we have v(x, 2t) ≥ Cts+1S(t)[u(t)]q(x), ≥ Cts+1S(t)[t− N 2−αX t 1 2−α ]q(x) ≥ Cts+1 · t−qγ1 , (3.20) for |x| ≤ t1/(2−α) and t > t0 > 1. Let t > 4t0. Since γ1 = N/(2−α), from (3.20) and proceeding as in the derivation of (3.12), we have∫ |x|≤(t+1) 1 2−α u(x, t+ 1)dx ≥ C ∫ t/2 1 ∫ |y|≤(t+1−σ) 1 2−α σr(v(y, σ))p dy dσ 12 R. CASTILLO, O. GUZMÁN-REA, M. LOAYZA, M. ZEGARRA EJDE-2024/67 ≥ C ∫ t/2 2t0 ∫ |y|≤σ 1 2−α σr(v(y, 2 · 2−1σ))p dy dσ ≥ C ∫ t/2 2t0 ∫ |y|≤σ 1 2−α σr ( [ σ 2 ]s+1[ σ 2 ]−qγ1 )p dy dσ ≥ C ∫ t/2 2t0 ∫ |y|≤σ 1 2−α σr+(s+1)p[σ−γ1 ]pq−1[σ−γ1 ] dy dσ ≥ C ∫ t/2 2t0 σr+(s+1)pσ−(pq−1)γ1 (∫ |y|≤σ 1 2−α σ−γ1 dy ) dσ ≥ C ∫ t/2 2t0 σ−1 dσ = C ln ( t 4t0 ) . Thus, we can use the same argument given in the previous case, using Corollary 3.2 in place of Proposition 3.1, to obtain a contradiction. 4. Global Existence 4.1. Local existence. Lemma 4.1 (Comparison principle). Assume that either (A1) or (A2) is verified, and (u0,i, v0,i) ∈ [L∞(RN )]2, for i = 1, 2. Let f, g : [0,∞) → [0,∞) be nondecreas- ing and locally Lipschitz functions; r, s > −1; and (ui, vi) ∈ [L∞((0, T ), L∞(RN )]2, such that ui(x, t) = ∫ RN Γ(t, x, y)u0,i(y)dy + ∫ t 0 ∫ RN Γ(t− σ, x, y)σrf(vi(y, σ)) dydσ, vi(x, t) = ∫ RN Γ(t, x, y)v0,i(y)dy + ∫ t 0 ∫ RN Γ(t− σ, x, y)σsg(ui(y, σ)) dy dσ, (4.1) for a.e. x ∈ RN and t > 0. If u0,1 ≤ u0,2 and v0,1 ≤ v0,2, then u1(t) ≤ u2(t) and v1(t) ≤ v2(t) for all t ∈ (0, T ). Proof. Note that it is sufficient to show that [u1(t)− u2(t)] + = [v1(t)− v2(t)] + = 0 for t ∈ (0, T ). Let M0 = max{∥ui(t)∥∞, ∥vi(t)∥∞; t ∈ [0, T ], i = 1, 2}. Since u0,1 ≤ u0,2 and v0,1 ≤ v0,2, from property (A4) and (4.1) we have u1(t)− u2(t) ≤ ∫ t 0 S(t− σ)σr[f(v1(σ))− f(v2(σ))] dσ, v1(t)− v2(t) ≤ ∫ t 0 S(t− σ)σs[g(u1(σ))− g(u2(σ))] dσ. (4.2) Since f and g are nondecreasing and locally Lipschitz, we obtain g(u1(t))− g(u2(t)) ≤ [g(u1(t))− g(u2(t))] + ≤ LM1 [u1(t)− u2(t)] +, f(v1(t))− f(v2(t)) ≤ [f(v1(t))− f(v2(t))] + ≤ LM2 [v1(t)− v2(t)] +, (4.3) where LM1 and LM2 are the Lipschitz constants on the interval [0;M0]. EJDE-2024/67 COUPLED PARABOLIC SYSTEMS WITH DEGENERATE COEFFICIENTS 13 It follows from estimate (2.2), (4.2), and (4.3) that ∥[u1(t)− u2(t)] +∥∞ ≤ LM2 ∫ t 0 σr∥[v1(σ)− v2(σ)] +∥∞ dσ, ∥[v1(t)− v2(t)] +∥∞ ≤ LM1 ∫ t 0 σs∥[u1(σ)− u2(σ)] +∥∞ dσ, The results is now a direct consequence of Gronwall’s inequality (see for example [36]). □ Theorem 4.2. Suppose p, q > 0 with pq > 1, ω satisfies either (A1) or (A2), and (u0, v0) ∈ [L∞(RN )]2, u0, v0 ≥ 0. Then there exists T > 0 and a constant C0 > 0 such that problem (1.1) possesses a unique solution (u, v) on (0, T ) satisfying sup 0 0. The sequences {un}n≥1 and {vn}n≥1 satisfy 0 ≤ un(x, t) ≤ un+1(x, t) and 0 ≤ vn(x, t) ≤ vn+1(x, t) (4.4) for a.e. x ∈ RN , t > 0. This is clear since Γ, u0, and v0 are non-negative functions (Γ is nonnegative by (A5) property). Thus, we define u∞(x, t) = lim n→∞ un(x, t), v∞ = lim n→∞ vn(x, t). (4.5) Furthermore, we see that u∞(x, t), v∞(x, t) ∈ [0,∞]. Now, we show that the sequences {un}n≥1 and {vn}n≥1 are bounded in a small interval (0, T ), that is, sup 0 0 sufficiently small. The constant c1 > 0 is given by inequality (2.2). To show (4.6), we argue by induction on n. This is clear for n = 1 due to (2.2). Suppose that (4.6) holds for some n ∈ N. Then, by (2.2) we have ∥un+1(t)∥∞ ≤ ∥u1(t)∥∞ + ∫ t 0 σr∥S(t− σ)vn(σ) p∥∞ dσ ≤ c1∥u0∥∞ + c1 ∫ t 0 σr∥vn(σ)∥p∞ dσ ≤ c1∥u0∥∞ + c1[2c1(∥u0∥∞ + ∥v0∥∞)]p ∫ t 0 σr dσ, (4.7) 14 R. CASTILLO, O. GUZMÁN-REA, M. LOAYZA, M. ZEGARRA EJDE-2024/67 for t ∈ (0, T ). Similarly, we have ∥vn+1(t)∥∞ ≤ ∥v1(t)∥∞ + ∫ t 0 σs∥S(t− σ)un(σ) q∥∞ dσ ≤ c1∥v0∥∞ + c1 ∫ t 0 σs∥un(σ)∥q∞ dσ ≤ c1∥v0∥∞ + c1[2c1(∥u0∥∞ + ∥v0∥∞)]q ∫ t 0 σs dσ, (4.8) for t ∈ (0, T ). Thus, the inequality (4.6) holds by adding (4.7) and (4.8) and taking T > 0 small enough. Finally, by (4.4), (4.5), and (4.6), we have that the limits functions u∞ and v∞ satisfies (1.5) and sup 0 1. Let (u0, v0) ∈ [L∞(RN ) ∩ Lr1⋆,∞(RN )]× [L∞(RN ) ∩ Lr2⋆,∞(RN )] with max{∥u0∥r1⋆,∞, ∥v0∥r2⋆,∞} < δ, where δ > 0 will be chosen small enough later. From (1.6)-(1.9), we obtain the following estimates: pγ2 = γ1 + (r + 1), qγ1 = γ2 + (s+ 1), pr1⋆ > r2⋆, qr2⋆ > r1⋆. (4.9) Also, since γ < N/(2− α), we have r1⋆, r2⋆ > 1. Let {(un, vn)}n≥0 be the sequence defined by u0(t) = S(t)u0, v 0(t) = S(t)v0 and uN (t) = S(t)u0 + ∫ t 0 S(t− σ)h1(σ)[v n−1(σ)]p dσ, vN (t) = S(t)v0 + ∫ t 0 S(t− σ)h2(σ)[u n−1(σ)]q dσ, (4.10) for all t > 0. Note that the sequences {uN}n≥0 and {vN}n≥0 are non-decreasing. By induction, we prove that there exists a constant C̃ > 0 such that ∥uN (t)∥r1⋆,∞ ≤ c⋆⋆δ + C̃δp, ∥uN (t)∥∞ ≤ (c⋆⋆δ + C̃δp)t − N (2−α)r1⋆ , ∥vN (t)∥r2⋆,∞ ≤ 2c⋆⋆δ, ∥vN (t)∥∞ ≤ 2c⋆⋆δt − N (2−α)r2⋆ . (4.11) for all n ∈ N∪{0}, where c⋆⋆ = max{c1, c2} and the constants ci(i = 1, 2) are given in Proposition 2.3. EJDE-2024/67 COUPLED PARABOLIC SYSTEMS WITH DEGENERATE COEFFICIENTS 15 From estimate (2.3) we have ∥u0(t)∥r1⋆,∞ ≤ c∗∗∥u0∥r1⋆,∞, ∥u0(t)∥µ,∞ ≤ c∗∗t − N 2−α ( 1 r1⋆ − 1 µ )∥u0∥r1⋆,∞, ∥v0(t)∥r2⋆,∞ ≤ c∗∗∥v0∥r2⋆,∞, ∥v0(t)∥µ,∞ ≤ c∗∗t − N 2−α ( 1 r2⋆ − 1 µ )∥v0∥r2⋆,∞, (4.12) for all t > 0, µ ∈ [r1⋆,∞]. This implies that (4.11) holds for n = 0. Assume that (4.11) holds for some n ∈ N. By symmetry, we only prove that (4.11) holds for un+1. From estimates (2.4) and (4.11), we have ∥vn(t)∥µ,∞ ≤ ∥vn(t)∥ r2⋆ µ r2⋆,∞∥vn(t)∥1− r2⋆ µ ∞ ≤ 2c∗∗δt − N 2−α ( 1 r2⋆ − 1 µ ) (4.13) for all t > 0 and µ ∈ [r2⋆,∞]. Then, from (4.9) and (4.13), we have ∥vn(t)p∥η,∞ = ∥vn(t)∥pηp,∞ ≤ [ 2c∗∗δt − N 2−α ( 1 r2⋆ − 1 ηp ) ]p = [2c∗∗] pδpt N (2−α)η − N (2−α)r1⋆ −(r+1) (4.14) for any η > 1 with r2⋆ ≤ ηp. Similarly, from (4.9) and (4.11), we obtain ∥vN (t)p∥∞ = ∥vN (t)∥p∞ ≤ ( 2c⋆⋆δt − N (2−α)r2⋆ )p = (2c⋆⋆δ) pt−pγ2 = [2c∗∗] pδpt − N (2−α)r1⋆ −(r+1) (4.15) for all t > 0. Thus, by (2.2), (2.3), (4.9), (4.14) (with η = r1⋆), r2⋆ < r1⋆p, and (4.15) we have ∥ ∫ t t/2 S(t− σ)σrvN (σ)pdσ∥∞ ≤ c1c3[2c∗∗] pδpt − N (2−α)r1⋆ , (4.16) where c3 = (2−α)r1⋆ N (2N/(2−α)r1⋆ − 1), and ∥ ∫ t t 2 S(t− σ)σrvn(σ)p dσ∥r1⋆,∞ ≤ c2 ∫ t t 2 σr∥vn(σ)p∥r1⋆,∞ dσ = c2 ∫ t t 2 σr∥vn(σ)∥ppr1⋆,∞ dσ = c2[2c∗∗] pδp ∫ t t 2 σ−1 dσ ≤ c2[2c∗∗] pδp, (4.17) for all t > 0. 16 R. CASTILLO, O. GUZMÁN-REA, M. LOAYZA, M. ZEGARRA EJDE-2024/67 On the other hand, since t−σ ≥ t/2 for all σ ∈ [0, t/2], by (2.3) and (4.14) (with η = η1 > 1, which will be chosen later), we obtain ∥ ∫ t/2 0 S(t− σ)σrvN (σ)pdσ∥∞ ≤ ∫ t/2 0 ∥S(t− σ)σrvN (σ)p∥∞dσ ≤ c2 ∫ t/2 0 (t− σ) − N (2−α)η1 σr∥vN (σ)p∥η1,∞dσ ≤ 2 N (2−α)η1 c2t − N (2−α)η1 ∫ t/2 0 σr∥vN (σ)p∥η1,∞dσ ≤ 2 N (2−α)η1 c2[2c∗∗] pδpt − N (2−α)η1 ∫ t/2 0 σ N (2−α)η1 − N (2−α)r1⋆ −1 dσ ≤ c2c4[2c∗∗] pδpt − N (2−α)r1⋆ , (4.18) for some 1 < η1 < r1⋆ so that r2⋆ < η1p (this is possible since p r1⋆ > r2⋆ > 1), and c4 = 2 N (2−α)r1⋆ [ N (2−α)η1 − N (2−α)r1⋆ ] −1. Analogously (using the above η1 again), we have ∥ ∫ t/2 0 S(t− σ)σrvn(σ)pdσ∥r1⋆,∞ ≤ ∫ t/2 0 ∥S(t− σ)σrvn(σ)p∥r1⋆,∞ dσ ≤ c2 ∫ t/2 0 (t− σ)− N 2−α ( 1 η1 − 1 r1⋆ )σr∥vn(σ)p∥η1,∞ dσ ≤ c22 N 2−α ( 1 η1 − 1 r1⋆ )t− N 2−α ( 1 η1 − 1 r1⋆ ) ∫ t/2 0 σr∥vn(σ)p∥η1,∞ dσ ≤ c2[2c∗∗] pδp2 N 2−α ( 1 η1 − 1 r1⋆ )t− N 2−α ( 1 η1 − 1 r1⋆ ) ∫ t/2 0 σ N (2−α)η1 − N (2−α)r1⋆ −1 dσ ≤ c2c5[2c∗∗] pδp (4.19) for all t > 0, where c5 = [ N (2−α)η1 − N (2−α)r1⋆ ] −1. Then, from (4.10), (4.12), (4.16), (4.17), (4.18), (4.19), and taking δ > 0 suffi- ciently small, we obtain t N (2−α)r1⋆ ∥un+1(t)∥∞ ≤ c⋆⋆δ + C̃δp ∥un+1(t)∥r1⋆,∞ ≤ c⋆⋆δ + C̃δp for all t > 0, where C̃ = [2c⋆⋆] pmax{c1c3 + c2c4, c2c5 + c2}. Arguing similarly it is possible to show that there exists a constant C, independent of n, δ, and t, such that t N (2−α)r2⋆ ∥vn+1(t)∥∞ ≤ c⋆⋆δ + C(c⋆⋆δ + C̃δp)q, ∥vn+1(t)∥r2⋆,∞ ≤ c⋆⋆δ + C(c⋆⋆δ + C̃δp)q. Since q > 1 and pq > 1, we have t N (2−α)r2⋆ ∥vn+1(t)∥∞ ≤ c⋆⋆δ + 2q−1Ccq⋆⋆δ q + 2q−1C̃q+1δpq ≤ 2c⋆⋆δ EJDE-2024/67 COUPLED PARABOLIC SYSTEMS WITH DEGENERATE COEFFICIENTS 17 for δ > 0 sufficiently small. Analogously, ∥vn+1(t)∥r2⋆,∞ ≤ 2c⋆⋆δ for δ > 0 possibly smaller. Therefore, (un+1, vn+1) satisfies the estimates of (4.11) for all n ∈ N∪{0}, and the induction process is finalized. From the estimates given in (4.11) we see that there exists a global-in-time solution of (1.1) such that (u, v) = (limn→∞ uN , limn→∞ vN ) and ∥u(t)∥∞ ≤ Ct − N (2−α)r1⋆ , ∥u(t)∥r1⋆,∞ ≤ C, ∥v(t)∥∞ ≤ Ct − N (2−α)r2⋆ , ∥v(t)∥r2⋆,∞ ≤ C, for some constant C > 0 (this solution is unique when p > 1 and q > 1). Moreover, by Theorem 4.2, ∥u(t)∥∞ ≤ C(t+ 1) − N (2−α)r1⋆ and ∥v(t)∥∞ ≤ C(t+ 1) − N (2−α)r2⋆ , for t > 0 and some constant C > 0. From this and (2.4), we have ∥u(t)∥µ,∞ ≤ ∥u(t)∥ r1⋆ µ r1⋆,∞∥u(t)∥1− r1⋆ µ ∞ ≤ C(t+ 1)− N 2−α ( 1 r1⋆ − 1 µ ), ∥v(t)∥µ,∞ ≤ ∥v(t)∥ r2⋆ µ r2⋆,∞∥v(t)∥1− r2⋆ µ ∞ ≤ C(t+ 1)− N 2−α ( 1 r2⋆ − 1 µ ), for all µ such that max{r1⋆, r2⋆} < µ ≤ ∞, t > 0 and some constant C > 0. Thus the proof is complete. Acknowledgments. We would like to express our heartfelt gratitude to the re- viewers for their thoughtful and constructive comments that have significantly im- proved the quality of our work. This work was supported by the ANID-FONDECYT project No.11220152 (Ricardo Castillo) and Universidad Nacional Mayor de San Marcos under grant B21141451 (Maŕıa Zegarra). O. Guzmán-Rea was supported by CNPq/Brazil, 166685/2020-8. M. Loayza was partially supported by CAPES- PRINT, 88881.311964/2018-01, MATHAMSUD, 88881.520205/2020-01, 21-MATH- 03. References [1] D. G. Aronson, H. F. Weinberger; Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30(1) (1978), 33-76. [2] L. 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Raizer; Physics of Shock Waves and High-temperature Hydrodynamic Phe- nomena, Academic Press, New York, 1966. [41] W. P. Ziemer; Weakly Differentiable Functions, Springer, New York, 1989. Ricardo Castillo Departamento de Matemática, Facultad de Ciencias, Universidad del B́ıo-B́ıo, Avenida Collao 1202, Casilla 5-C, Concepción, B́ıo-B́ıo, Chile Email address: rcastillo@ubiobio.cl Omar Guzmán-Rea Universidad Tecnológica de Perú, Ica, Peru Email address: c29064@utp.edu.pe Miguel Loayza Departamento de Matemática, Universidade Federal de Pernambuco, Av. Jornalista Ańıbal Fernandes, Cidade Universitáa, Recife, Pernambuco, Brasil Email address: miguel.loayza@ufpe.br Maŕıa Zegarra Departamento de Matemática, Universidad Nacional Mayor de San Marcos, Av. Venezuela cuadra 34, Lima 1, Lima, Perú Email address: mzegarrag@unmsm.edu.pe 1. Introduction 2. Preliminaries and technical results 3. Nonglobal existence 4. Global Existence 4.1. Local existence 4.2. Global existence: proof of Theorem 1.2-(ii) Acknowledgments References