Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 21, pp. 1–18. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.21 CRITICAL FUJITA EXPONENTS FOR A CLASS OF QUASILINEAR COUPLED PARABOLIC EQUATIONS YUANYUAN NIE, YAN LENG, XU ZHAO, QIAN ZHOU Abstract. This article concerns the critical Fujita exponents for a class of quasilinear coupled parabolic equations. Using energy estimates, suitable su- persolutions, and the comparison principle, the blow-up theorem of Fujita type is established, and the critical Fujita exponent is obtained. Furthermore, we show that the critical case belongs to the blow-up case. 1. Introduction In this article, we study the critical Fujita exponent for the Cauchy problem of quasilinear coupled parabolic equations ∂u ∂t = ∆um + (|x|+ 1)λvp, x ∈ Rn, t > 0, (1.1) ∂v ∂t = ∆vm + (|x|+ 1)µuq, x ∈ Rn, t > 0, (1.2) u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Rn, (1.3) where p, q > m > 1, λ ≥ 0, µ = λ(q −m) + 2(q − p) p−m ≥ 0 (1.4) and 0 ≤ u0, v0 ∈ C0(Rn) are nontrivial. The earliest research on critical exponents of parabolic equations was published in 1966 by Fujita [5]. It was demonstrated that the Cauchy problem of the heat equation ∂u ∂t = ∆u+ up, x ∈ Rn, t > 0 admits no nontrivial nonnegative global solution when 1 < p < pc = 1 + 2/n, otherwise, it admits both nontrivial global (with small initial data) and nonglobal nonnegative (with large initial data) solutions when p > pc. Subsequently, in [9, 13, 31], it was proved that any solution blows up in the critical case p = pc. Herein, pc is termed the critical Fujita exponent, and the corresponding results constitute the blow-up theorem of Fujita type. Fujita’s famous work reveals the relationship between the asymptotic behavior of the solutions to nonlinear partial differential equations and the exponents of the 2020 Mathematics Subject Classification. 93B05, 93C20, 35K67. Key words and phrases. Coupled parabolic equations; critical Fujita exponent; critical case. ©2025. This work is licensed under a CC BY 4.0 license. Submitted December 6, 2024. Published February 27, 2025. 1 2 Y. NIE, Y. LENG, X. ZHAO, Q. ZHOU EJDE-2025/21 nonlinear internal sources. Since then, many important results have been obtained, including different types of equations and systems in various geometries with or without degeneracies or singularities, different boundary conditions, and different extension directions. For more detail, we refer the reader to works [1, 2, 3, 8, 9, 12, 15, 16, 17, 18, 11, 25, 35, 21, 23, 24, 26, 30, 31, 34, 36, 37, 38, 39, 40] and the references therein. For the Cauchy problem, Galaktionov et al. [6, 7] studied the single slow diffusion equation ∂u ∂t = ∆um + up, x ∈ Rn, t > 0, where p > m > 1. It was proved that the critical Fujita exponent is pc = m+ 2/n. The Cauchy problem of the equation |x|λ1 ∂u ∂t = ∆um + |x|λ2up, x ∈ Rn, t > 0 with p > m and 0 ≤ λ1 ≤ λ2 < p(λ1 + 1) − 1 was formulated as pc = m + (2 + λ2)/(n+ λ1) in [29]. Furthermore, the authors proved that the critical case p = pc is also a blow-up case. For the Cauchy problem of the following coupled semilinear parabolic system, ∂u ∂t = ∆u+ tβ1 |x|α1vp, ∂v ∂t = ∆v + tβ2 |x|α2uq, x ∈ Rn, t > 0, where α1, α2, β1, β2 ≥ 0, and p, q ≥ 1. Escobedo and Herrero in [4] considered this Cauchy problem with α1 = α2 = β1 = β2 = 0, and they proved that the critical Fujita curve is (pq)c = 1 + 2 n max{p+ 1, q + 1}. More general, if β1 = β2 = 0, Mochizuki and Huang ([19]) proved that the critical Fujita curve is (pq)c = 1 + 1 n max { (α1 + 2) + (α2 + 2)p, (α2 + 2) + (α1 + 2)q } with 0 < α1 < n(p − 1) and 0 < α2 < n(q − 1). If α1 = α2 = 0, it was proved in [27] that the critical Fujita curve is (pq)c = 1 + 2 n max{(β2 + 1)p+ β1 + 1, (β1 + 1)q + β2 + 1} with pq > 1. There are also some studies on the Cauchy problem of the coupled porous medium systems with fast diffusion ∂u ∂t = ∆um1 + vp, ∂v ∂t = ∆vm2 + uq, x ∈ Rn, t > 0, (1.5) where 0 < m1, m2 < 1, p, q ≥ 1 and pq > 1. Qi and Levine ([22]) proved that the critical Fujita curve of the Cauchy problem of (1.5) is (pq)c = m1m2 + 2 n max{m2 + p,m1 + q}, and proved that any nontrivial solution to the Cauchy problem of (1.5) must blow up in finite time if pq < (pq)c, whereas both nonnegative nontrivial global and blowing-up solutions exist if pq > (pq)c with m1 = m2. They pointed out that the method of constructing global supersolutions fails when m1 ̸= m2 because of the different propagation rates of the two types of EJDE-2025/21 CRITICAL FUJITA EXPONENTS 3 diffusion. Later in [10], it was shown that for the “very fast diffusions” case that 0 < m1, m2 < (n − 2)+/n, the Cauchy problem of (1.5) admits nontrivial global solutions if the initial data is small enough althoughm1 is not equal tom2. Recently, the problem (1.1)–(1.3) with the special case λ = 0 was considered in [14], and it was shown that the critical Fujita exponent is pc = m + 2/n. However, the result for the critical case p = pc remains unknown. In this paper, we prove that the critical Fujita exponent of problem (1.1)–(1.3) is pc = m+ λ+ 2 n . (1.6) As in [20, 14, 21, 28, 29], we study the blowing-up properties of solutions by the integral estimates, and global existence of solutions by constructing a pair of suit- able self-similar supersolutions. Note that the choice of µ is used to ensure that self-similar supersolutions have the same support set. This is still open to the other µ. Furthermore, we prove that the critical case p = pc can be classified as a blow-up case by analyzing the asymptotic behavior of the solutions. This article consists of three sections. In §2, we introduce several basic definitions and theorems. Subsequently, in §3, we prove the blow-up theorems of Fujita type for the problem (1.1)–(1.3). Subsequently, the critical case p = pc is considered in §4. 2. Preliminaries In this section, we introduce some basic definitions and relevant lemmas. Definition 2.1. Assume that 0 < T ≤ +∞ and u, v are two nonnegative functions. If u, v ∈ C([0, T ), Lm loc(Rn)) ∩ L∞ loc((0, T ); L ∞(Rn)) and for any 0 ≤ φ, ψ ∈ C2,1(Rn × [0, T )) vanishing when t near T or |x| being sufficiently large, the integral inequalities∫ T 0 ∫ Rn u(x, t) ∂φ ∂t (x, t)dxdt+ ∫ T 0 ∫ Rn um(x, t)∆φ(x, t)dxdt + ∫ T 0 ∫ Rn (|x|+ 1)λvp(x, t)φ(x, t)dxdt+ ∫ Rn u0(x)φ(x, 0)dx ≤ (≥) 0,∫ T 0 ∫ Rn v(x, t) ∂ψ ∂t (x, t)dxdt+ ∫ T 0 ∫ Rn vm(x, t)∆ψ(x, t)dxdt + ∫ T 0 ∫ Rn (|x|+ 1)µuq(x, t)ψ(x, t)dxdt+ ∫ Rn v0(x)ψ(x, 0)dx ≤ (≥) 0 hold, then (u, v) is said to be a super (sub) solution to (1.1)–(1.3) in (0, T ). Fur- thermore, (u, v) is said to be a solution to (1.1)–(1.3) in (0, T ) if it is both a supersolution and a subsolution. Definition 2.2. Assume that (u, v) is a nontrivial solution to (1.1)–(1.3). If for some 0 < T < +∞, ∥u(·, t)∥L∞(Rn) + ∥v(·, t)∥L∞(Rn) → +∞ as t→ T−, then (u, v) is said to blow up in the finite time T . Otherwise, (u, v) is said to be a global solution. 4 Y. NIE, Y. LENG, X. ZHAO, Q. ZHOU EJDE-2025/21 Based on the classical theory of quasilinear parabolic equations (see [32, 33]), we have the following result. Lemma 2.3. For any 0 ≤ u0, v0 ∈ L1 loc(Rn)∩L∞(Rn), there is at least one solution to (1.1)–(1.3) locally in time. Theorem 2.4. Assume that (u1, v1) and (u2, v2) are two solutions to (1.1)–(1.3) in (0, T ) with nonnegative initial data (u0,1(x), v0,1(x)) and (u0,2(x), v0,2(x)), re- spectively. If (u0,1, v0,1) ≤ (u0,2, v0,2) a.e. in Rn, then (u1, v1) ≤ (u2, v2) a.e. in Rn × (0, T ). 3. Blow-up theorems of Fujita type In this section, we prove blow-up theorems of Fujita type for problem (1.1)–(1.3). Theorem 3.1. If m < p < pc, any nontrivial solution to (1.1)–(1.3) blows up in finite time. Proof. We denote Br as a ball in Rn with radius r centered at the origin. Assume that (u, v) is a nontrivial solution to (1.1)–(1.3). Set wl(t) = ∫ Rn ( u(x, t) + lθv(x, t) ) ψl(x)dx, t > 0, (3.1) where θ is a constant to be determined, l > 1, ψl(x) =  1, 0 ≤ |x| ≤ l, 1 2 [ 1 + cos π(|x|−l) l ] , l < |x| < 2l, 0, |x| ≥ 2l, x ∈ Rn. Then for x ∈ B2l \Bl, we have |∇ψl(x)| ≤ C1l −1, |∆ψl(x)| ≤ C1l −2, ∆ψl(x) ≥ −C1l −2ψl(x). where C1 is a constant that depends only on n but is independent of l. By (1.1) and (1.2), we obtain dwl(t) dt = ∫ B2l um(x, t)∆ψl(x)dx+ ∫ Rn (|x|+ 1)λvp(x, t)ψl(x)dx + lθ ∫ B2l vm(x, t)∆ψl(x)dx+ lθ ∫ Rn (|x|+ 1)µuq(x, t)ψl(x)dx. (3.2) From the Hölder inequality we obtain∫ B2l um(x, t)∆ψl(x)dx = ∫ B2l\Bl um(x, t)∆ψl(x)dx ≥ −C1l −2 ∫ B2l\Bl um(x, t)ψl(x)dx ≥ −C2l n−2−(n+µ)m/q (∫ Rn (|x|+ 1)µuq(x, t)ψl(x)dx )m/q , EJDE-2025/21 CRITICAL FUJITA EXPONENTS 5∫ B2l vm(x, t)∆ψl(x)dx = ∫ B2l\Bl vm(x, t)∆ψl(x)dx ≥ −C1l −2 ∫ B2l\Bl vm(x, t)ψl(x)dx ≤ −C2l n−2−(n+λ)m/p (∫ Rn (|x|+ 1)λvp(x, t)ψl(x)dx )m/p , where C2 > 0 is a constant independent of l. From these two estimates and (3.2), one can obtain dwl(t) dt ≥ (∫ Rn (|x|+ 1)µuq(x, t)ψl(x)dx )m/q × ( lθ (∫ Rn (|x|+ 1)µuq(x, t)ψl(x)dx )(q−m)/q − C2l n−2−(n+µ)m/q ) + (∫ Rn (|x|+ 1)λvp(x, t)ψl(x)dx )m/p × ((∫ Rn (|x|+ 1)λvp(x, t)ψl(x)dx )(p−m)/p − C2l n−2−(λ+n)m/p+θ ) . (3.3) It follows from the Hölder inequality that∫ Rn u(x, t)ψl(x)dx ≤ (∫ Rn (|x|+ 1)−µ/(q−1)ψl(x)dx )(q−1)/q(∫ Rn (|x|+ 1)µuq(x, t)ψl(x)dx )1/q ,∫ Rn v(x, t)ψl(x)dx ≤ (∫ Rn (|x|+ 1)−λ/(p−1)ψl(x)dx )(p−1)/p(∫ Rn (|x|+ 1)λvp(x, t)ψl(x)dx )1/p , which indicate that∫ Rn (|x|+ 1)µuq(x, t)ψl(x)dx ≥  C3 ( ∫ Rn u(x, t)ψl(x)dx )q l−qn+n+µ, if − qn+ n+ µ < 0, C3 ( ∫ Rn u(x, t)ψl(x)dx )q (ln l)1−q, if − qn+ n+ µ = 0, C3 ( ∫ Rn u(x, t)ψl(x)dx )q , if − qn+ n+ µ > 0, (3.4) ∫ Rn (|x|+ 1)λvp(x, t)ψl(x)dx ≥  C3 ( ∫ Rn v(x, t)ψl(x)dx )p l−pn+n+λ, if − pn+ n+ λ < 0, C3 ( ∫ Rn v(x, t)ψl(x)dx )p (ln l)1−p, if − pn+ n+ λ = 0, C3 ( ∫ Rn v(x, t)ψl(x)dx )p , if − pn+ n+ λ > 0, (3.5) where C3 is a constant independent of l. 6 Y. NIE, Y. LENG, X. ZHAO, Q. ZHOU EJDE-2025/21 First, consider the case in which −qn + n + µ < 0 and −pn + n + λ < 0. It follows from (3.3)–(3.5) that dwl(t) dt ≥ C m/q 3 l(−qn+n+µ)m/q (∫ Rn u(x, t)ψl(x)dx )m × [ C 1−m/q 3 l(−qn+n+µ)(q−m)/q+θ (∫ Rn u(x, t)ψl(x)dx )q−m − C2l n−2−(n+µ)m/q ] + C m/p 3 l(−pn+n+λ)m/p (∫ Rn v(x, t)ψl(x)dx )m × [ C 1−m/p 3 l(−pn+n+λ)(p−m)/p (∫ Rn v(x, t)ψl(x)dx )p−m − C2l n−2−(n+λ)m/p+θ ] ≥ −C4l κ(θ)wm l (t) + C3l −qn+n+µ+θ (∫ Rn u(x, t)ψl(x)dx )q + C3l −pn+n+λ−pθ (∫ Rn lθv(x, t)ψl(x)dx )p , (3.6) where C4 = max{C2C m/q 3 , C2C m/p 3 }, κ(θ) = max { −mn+n−2, −mn+n−2−(m−1)θ } . We choose θ = q − p p+ 1 ( n− λ+ 2 p−m ) . It is clear that −qn+ n+ µ+ θ = −np+ n+ λ− pθ = Θ with Θ = −p2qn− p2n+ pqmn+ pmn+ (λ+ 2)(pq − p2) (p+ 1)(p−m) + λ+ n. Then we obtain dwl(t) dt ≥ −C4l κ(θ)wm l (t) + C3l Θ [( ∫ Rn u(x, t)ψl(x)dx )q + (∫ Rn lθv(x, t)ψl(x)dx )p] ≥ wm l (t) [ −C4l κ(θ) + 2−(p+q)C3l Θ ·min { wp−m l (t), wq−m l (t) }] . (3.7) It follows from p < pc that κ(θ) < Θ. Because wl(0) is nondecreasing with respect to l ∈ (0,+∞) with sup { wl(0) : l ∈ (0,+∞) } > 0, there exists l1 > 1 such that C4l κ(θ) 1 ≤ 2−(p+q+1)C3l Θ 1 min { wp−m l1 (0), wq−m l1 (0) } . (3.8) From (3.7) and (3.8), we obtain dwl1(t) dt ≥ 2−(p+q+1)C3l Θ 1 min { wp l1 (t), wq l1 (t) } , t > 0. Due to p, q > m > 1, there exists a constant T∗ ∈ (0,+∞) such that wl1(t) = ∫ Rn ( u(x, t) + lθ1v(x, t) ) ψl1(x)dx→ +∞ as t→ T− ∗ . EJDE-2025/21 CRITICAL FUJITA EXPONENTS 7 Since suppψl1(x) = B2l1 , we have ∥u(·, t)∥L∞(Rn) + ∥v(·, t)∥L∞(Rn) → +∞ as t→ T− ∗ . That is, (u, v) blows up in finite time. Let us consider the case in which −qn + n + µ = 0 and −pn + n + λ < 0. It is assumed that θ = 0. It follows from (3.3)–(3.5) that dwl(t) dt ≥ C m/q 3 (ln l)(1−q)m/q (∫ Rn u(x, t)ψl(x)dx )m × [ C 1−m/q 3 (ln l)(1−q)(q−m)/q (∫ Rn u(x, t)ψl(x)dx )q−m − C2l n−2−m(n+µ)/q ] + C m/p 3 l(−pn+n+λ)m/p (∫ Rn v(x, t)ψl(x)dx )m × ( C (p−m)/p 3 l(−pn+n+λ)(p−m)/p (∫ Rn v(x, t)ψl(x)dx )p−m − C2l n−2−m(n+λ)/p ) . (3.9) Thanks to n− 2−m(n+ µ)/q < 0, and n− 2−m(n+ λ)/p < (−np+ n+ λ)(p−m)/p, there exists sufficiently large l2 > 1, such that dwl2(t) dt ≥ C m/q 3 (ln l2) m(1−q)/q (∫ Rn u(x, t)ψl2(x)dx )m × 1 2 C (q−m)/q 3 (ln l2) (1−q)(q−m)/q (∫ Rn u(x, t)ψl2(x)dx )q−m + C m/p 3 l −mn+m(n+λ)/p 2 (∫ Rn v(x, t)ψl2(x)dx )m × 1 2 C (p−m)/p 3 l (−np+n+λ)(p−m)/p 2 (∫ Rn v(x, t)ψl2(x)dx )p−m ≥ C5 ((∫ Rn u(x, t)ψl2(x)dx )q + (∫ Rn v(x, t)ψl2(x)dx )p) ≥ 2−(p+q)C5 min{wp l2 (t), wq l2 (t)}, where C5 denotes a constant that depends only on l2. The same discussion as above shows that (u, v) blows up in finite time. For the other cases, we still assume that θ = 0. Similar to the second case, we can prove that (u, v) blows up in finite time. □ Now we construct self-similar supersolutions to (1.1) and (1.2). Set u(x, t) = U((t+ 1)−β(|x|+ 1)) (t+ 1)α , v(x, t) = V ((t+ 1)−β(|x|+ 1)) (t+ 1)α , (3.10) for (x, t) ∈ Rn × [0,+∞), where U, V ∈ C1([0,+∞)) with Um, V m ∈ C1([0,+∞)) and α = λ+ 2 2(p− 1) + λ(m− 1) , β = (p−m)α λ+ 2 . 8 Y. NIE, Y. LENG, X. ZHAO, Q. ZHOU EJDE-2025/21 If the following inequalities (Um)′′(r) + n− 1 r · |x|+ 1 |x| (Um)′(r) + βrU ′(r) + αU(r) + rλV p(r) ≤ 0, (3.11) (V m)′′(r) + n− 1 r · |x|+ 1 |x| (V m)′(r) + βrV ′(r) + αV (r) + rµUq(r) ≤ 0 (3.12) hold for r > 0 and x ∈ Rn, then (u, v) given in (3.10) is a supersolution to (1.1) and (1.2). Lemma 3.2. Assume that U(r) = V (r) = ( η −Ar2 )1/(m−1) + , r > 0, (3.13) where s+ = max{0, s}, η > 0 and A = (m− 1)(p−m)α mn(p+ pc − 2m) . Then, when p > pc, there exists sufficiently small η > 0, such that (u, v) defined by (3.10) and (3.13) is a supersolution to (1.1) and (1.2). Proof. It is not difficult to cheek that U, V ∈ C1([0,+∞)) satisfy Um, V m ∈ C1([0,+∞)) and (3.11), (3.12) hold for r ≥ (η/A)1/2 and x ∈ Rn. For 0 < r < ( η A )1/2, following from direction calculations, it yields that (Um)′′(r) + n− 1 r (Um)′(r) + βrU ′(r) + αU(r) = ( 2A m− 1 ( 2Am m− 1 − β ) r2U1−m(r) + ( α− 2Amn m− 1 )) U(r) and (V m)′′(r) + n− 1 r (V m)′(r) + βrV ′(r) + αV (r) = ( 2A m− 1 ( 2Am m− 1 − β ) r2V 1−m(r) + ( α− 2Amn m− 1 )) V (r). It follows from 2Am m−1 < β that there exists some η1 > 0 such that for any 0 < η < η1, (Um)′′(r) + n− 1 r (Um)′(r) + βrU ′(r) + αU(r) < − (p− pc)α 2(p+ pc − 2m) U(r) and (V m)′′(r) + n− 1 r (V m)′(r) + βrV ′(r) + αV (r) < − (p− pc)α 2(p+ pc − 2m) V (r). Owing to (Um)′(r) = − 2Amr m−1 U(r) ≤ 0 and (V m)′(r) = − 2Amr m−1 V (r) ≤ 0, we obtain that for any 0 < η < η1, it holds (Um)′′(r) + n− 1 r · |x|+ 1 |x| (Um)′(r) + βrU ′(r) + αU(r) < − (p− pc)α 2(p+ pc − 2m) U(r), (3.14) (V m)′′(r) + n− 1 r · |x|+ 1 |x| (V m)′(r) + βrV ′(r) + αV (r) < − (p− pc)α 2(p+ pc − 2m) V (r). (3.15) EJDE-2025/21 CRITICAL FUJITA EXPONENTS 9 From the definition of U , V and λ, µ ≥ 0, there exists η2 ∈ (0, η1) such that for any 0 < η < η2, rµUq−1(r) = rλV p−1(r) ≤ A−λ/2η(p−1)/(m−1)+λ/2 < (p− pc)α 2(p+ pc − 2m) , which together with (3.14) and (3.15), shows that for and 0 < η < η2, (3.11) and (3.12) hold for 0 < r < ( η A )1/2 and x ∈ Rn. □ Theorem 3.3. For p > pc, there exist both nontrivial global and blow-up solutions to (1.1)–(1.3). Proof. It follows from Lemma 2.4 and Lemma 3.2 that (1.1)–(1.3) has a nontrivial global solution with small initial data. Below, we prove the blowing-up properties of solutions to (1.1)–(1.3) with large initial data. Set w̃l(t) = ∫ Rn ( u(x, t) + v(x, t) ) ψl(x)dx, t ≥ 0, where l > 1 and (u, v) is a solution to (1.1)–(1.3). According to (3.3) and the Hölder inequality, we have dw̃(t) dt ≥ (∫ Rn u(x, t)ψl(x)dx )m(∫ Rn (|x|+ 1)µ/(1−q)ψl(x)dx )m(1−q)/q × [( ∫ Rn u(x, t)ψl(x)dx )q−m(∫ Rn (|x|+ 1)µ/(1−q)ψl(x)dx )(1−q)(q−m)/q − C2l n−2−m(n+µ)/q ] + (∫ Rn v(x, t)ψl(x)dx )m(∫ Rn (|x|+ 1)λ/(1−p)ψl(x)dx )m(1−p)/p × [( ∫ Rn v(x, t)ψl(x)dx )p−m(∫ Rn (|x|+ 1)λ/(1−p)ψl(x)dx )(1−p)(p−m)/p − C2l n−2−m(n+λ)/p ] ≥ w̃m l (t) ( − C2C6 + 2−(p+q)C7 min { w̃p−m l (t), w̃q−m l (t) }) , t ≥ 0, (3.16) where C6 = max { ln−2−m(n+µ)/q (∫ Rn (|x|+ 1)µ/(1−q)ψl(x)dx )m(1−q)/q , ln−2−m(n+λ)/p (∫ Rn (|x|+ 1)λ/(1−p)ψl(x)dx )m(1−p)/p} , C7 = min {(∫ Rn (|x|+ 1)µ/(1−q)ψl(x)dx )1−q , (∫ Rn (|x|+ 1)λ/(1−p)ψl(x)dx )1−p} are positive constants depending on λ, m, n, p, q and l. If (u0, v0) is so large that C2C6 ≤ 2−(p+q+1)C7 min { w̃p−m l (0), w̃q−m l (0) } , then (3.16) indicates that dw̃l(t) dt ≥ 2−(p+q+1)C7 min { w̃p l (t), w̃ q l (t) } , t > 0. It can be proven by the same process as in Theorem 3.1 that (u, v) blows up in finite time. □ 10 Y. NIE, Y. LENG, X. ZHAO, Q. ZHOU EJDE-2025/21 4. Critical case In this section, we consider the critical case p = pc = m+ 2 + λ n . (4.1) For this case, one obtains −qn+ n+ µ = −pcn+ n+ λ = −mn+ n− 2 < 0, (4.2) and (3.6) and (3.7) still hold. The proof is based on the following three lemmas. Lemma 4.1. Assume that (u, v) is a nontrivial global solution to (1.1)–(1.3) with p = pc, then there exists a constant M0 > 0 independent of t, such that∫ Rn ( u(x, t) + v(x, t) ) dx ≤M0, t > 0. (4.3) Furthermore, it holds that d dt ∫ Rn ( u(x, t) + v(x, t) ) dx ≥ 1 2 ∫ Rn ( (|x|+ 1)µuq(x, t) + (|x|+ 1)λvpc(x, t) ) dx. (4.4) Proof. For any sufficiently large l > 1, (3.7) implies dwl(t) dt ≥ wm l (t)l−mn+n−2 ( − C4 + 2−(pc+q)C3 min { wpc−m l (t), wq−m l (t) }) , where wl is defined by (3.1) with θ = 0. Similar to the end of the proof of Theorem 3.1, there exists some l3 > 1, such that for any l > l3, 2−(pc+q+1)C3 min { wpc−m l (t), wq−m l (t) } ≤ C4, i.e. wl(t) ≤ max {( 2pc+q+1C−1 3 C4 )1/(pc−m) , ( 2pc+q+1C−1 3 C4 )1/(q−m)} . (4.5) Let l → +∞ in (4.5); then, we can obtain (4.3). From the Hölder inequality and (4.3), we obtain that for any l ≥ 1,∫ B2l um(x, t)∆ψl(x)dx = ∫ B2l\Bl um(x, t)∆ψl(x)dx ≥ −C1 l2 ∫ B2l\Bl um(x, t)ψl(x)dx ≥ −C1 l2 (∫ B2l\Bl uq(x, t)ψl(x)dx )(m−1)/(q−1)(∫ B2l\Bl u(x, t)ψl(x)dx )(q−m)/(q−1) ≥ −C1 l2 (∫ Rn (|x|+ 1)µuq(x, t)ψl(x)dx )(m−1)/(q−1) × (∫ Rn ( u(x, t) + v(x, t) ) dx )(q−m)/(q−1) ≥ −C8 l2 (∫ Rn (|x|+ 1)µuq(x, t)ψl(x)dx )(m−1)/(q−1) and∫ B2l vm(x, t)|∆ψl(x)|dx ≥ −C8 l2 (∫ Rn (|x|+ 1)λvpc(x, t)ψl(x)dx )(m−1)/(pc−1) , EJDE-2025/21 CRITICAL FUJITA EXPONENTS 11 where C8 > 0 is a constant independent of l. Substituting the above two inequalities into (3.2) yields that for t > 0, dwl(t) dt ≥ (∫ Rn (|x|+ 1)µuq(x, t)ψl(x)dx )(m−1)/(q−1) × ( − C8 l2 + (∫ Rn (|x|+ 1)µuq(x, t)ψl(x)dx )(q−m)/(q−1)) + (∫ Rn (|x|+ 1)λvpc(x, t)ψl(x)dx )(m−1)/(pc−1) × ( − C8 l2 + (∫ Rn (|x|+ 1)λvpc(x, t)ψl(x)dx )(pc−m)/(pc−1)) . By letting l → +∞, (4.4) is obtained. □ Lemma 4.2. Under the assumption of Lemma 4.1, there exist three positive con- stants M1, M2, M3 > 0 independent of l and t, such that for any sufficiently large l > 1, dwl(t) dt ≥Mm−τ 1 l−mn+n−2wm−τ l (t) ( −M2 (∫ B2l\Bl (u(x, t) + v(x, t))ψldx )τ +M −(m−τ) 1 M3 ·min { wpc−m+τ l (t), wq−m+τ l (t) }) , (4.6) where 0 < τ < min {pc −m pc − 1 , q −m q − 1 } . Proof. It is easy to verify that n− 2− m(n+ µ) q + τ(µ− (q − 1)n) q = (−qn+ n+ µ)(q −m+ τ) q , (4.7) n− 2− m(n+ λ) pc + τ(λ− (pc − 1)n) pc = (−pcn+ n+ λ)(pc −m+ τ) pc . (4.8) Using the Hölder inequality, we obtain that for any sufficiently large l > 1,∫ B2l um(x, t)∆ψl(x)dx = ∫ B2l\Bl um(x, t)∆ψl(x)dx ≥ −C1l −2 ∫ B2l\Bl um(x, t)ψl(x)dx ≥ −C1l −2 (∫ B2l\Bl (|x|+ 1)−(m−τ)µ/(q−m−(q−1)τ)dx )(q−m−(q−1)τ)/q × (∫ B2l\Bl (|x|+ 1)µuq(x, t)ψl(x)dx )(m−τ)/q(∫ B2l\Bl u(x, t)ψl(x)dx )τ ≥ −C9l n−2−(n+µ)m/q+τ(µ−(q−1)n)/q (∫ Rn (|x|+ 1)µuq(x, t)ψl(x)dx )(m−τ)/q × (∫ B2l\Bl u(x, t)ψl(x)dx )τ ,∫ B2l vm(x, t)|∆ψl(x)|dx 12 Y. NIE, Y. LENG, X. ZHAO, Q. ZHOU EJDE-2025/21 ≥ −C9l n−2−(n+λ)m/pc+τ(λ−(pc−1)n)/pc (∫ Rn (|x|+ 1)λvpc(x, t)ψl(x)dx )(m−τ)/pc × (∫ B2l\Bl v(x, t)ψl(x)dx )τ , where C9 > 0 is a constant independent of l. Substituting the above two inequalities into (3.2) with θ = 0, then using (3.4), (3.5), (4.2), (4.7) and (4.8), one gets that dwl(t) dt ≥ −C(m−τ)/q 3 C9l −qn+n+µ (∫ Rn u(x, t)ψl(x)dx )m−τ(∫ B2l\Bl u(x, t)ψl(x)dx )τ − C (m−τ)/pc 3 C9l −pcn+n+λ (∫ Rn v(x, t)ψl(x)dx )m−τ(∫ B2l\Bl v(x, t)ψl(x)dx )τ + C3l −qn+n+µ (∫ Rn u(x, t)ψl(x)dx )q + C3l −pcn+n+λ (∫ Rn v(x, t)ψl(x)dx )pc ≥ −Mm−τ 1 C9l −mn+n−2wm−τ l (t) (∫ B2l\Bl ( u(x, t) + v(x, t) ) ψl(x)dx )τ + 2−(pc+q)C3l −mn+n−2 min{wpc l (t), wq l (t)}, t > 0. Thus, (4.6) holds forM1 = max { C 1/pc 3 , C 1/q 3 } ,M2 = C9 andM3 = 2−(pc+q)C3. □ Lemma 4.3. Under the assumption of Lemma 4.1, there exists a constant M4 > 0 independent of l and t, such that for any sufficiently large l > 1, d dt ∫ Rn ( u(x, t) + v(x, t) ) ϕl(x)dx ≤ 2 ∫ Rn ( (|x|+ 1)µuq(x, t) + (|x|+ 1)λvpc(x, t) ) dx +M4l (pc(n−2)−m(n+λ1))/(pc−m), (4.9) where ϕl(x) = 1− ψl(x), x ∈ Rn. Proof. Let ξ ∈ C∞ 0 (Rn) satisfy 0 ≤ ξ(x) ≤ 1 for x ∈ Rn, ξ(x) = 1 if |x| < 2 and ξ(x) = 0 if |x| > 3. For k ≥ l > 0, denote ξk(x) = ξ (x k ) , x ∈ Rn. Obviously, |∇ϕl(x)| ≤ C10 l , |∆ϕl(x)| ≤ C10 l2 , |∆ξk(x)| ≤ C10 k2 , x ∈ Rn, EJDE-2025/21 CRITICAL FUJITA EXPONENTS 13 where C10 > 0 is a constant independent of l and k. From Definition 2.1, we obtain d dt ∫ Rn ( u(x, t) + v(x, t) ) ϕl(x)ξk(x)dx ≤ ∫ Rn\Bl um(x, t)∆(ϕl(x)ξk(x))dx+ ∫ Rn (|x|+ 1)µuq(x, t)ϕl(x)ξk(x)dx + ∫ Rn\Bl vm(x, t)∆(ϕl(x)ξk(x))dx+ ∫ Rn (|x|+ 1)λvpc(x, t)ϕl(x)ξk(x)dx ≤ ∫ B2l\Bl um(x, t)|∆ϕl(x)|dx+ ∫ B3k\B2k um(x, t)|∆ξk(x)|dx + ∫ Rn (|x|+ 1)µuq(x, t)dx+ ∫ B2l\Bl vm(x, t)|∆ϕl(x)|dx + ∫ B3k\B2k vm(x, t)|∆ξk(x)|dx+ ∫ Rn (|x|+ 1)λvpc(x, t)dx. (4.10) Owing to the Hölder inequality, one obtains∫ B2l\Bl um(x, t)|∆ϕl(x)|dx ≤ (∫ B2l\Bl (|x|+ 1)−mµ/(q−m)|∆ϕl(x)|q/(q−m)dx )(q−m)/q × (∫ B2l\Bl (|x|+ 1)µuq(x, t)dx )m/q ≤ C11l n−2−m(n+µ)/q (∫ B2l\Bl (|x|+ 1)µuq(x, t)dx )m/q , (4.11) ∫ B2l\Bl vm(x, t)|∆ϕl(x)|dx ≤ C11l n−2−m(n+λ)/pc (∫ B2l\Bl (|x|+ 1)λvpc(x, t)dx )m/pc , (4.12) where C11 > 0 is a constant independent of l and k. Similarly, one gets∫ B3k\B2k um(x, t)|∆ξk(x)|dx ≤ C10 k2 ∫ B3k\B2k um(x, t)dx ≤ C10 k2 (∫ B3k\B2k uq(x, t)dx )(m−1)/(q−1)(∫ B3k\B2k u(x, t)dx )(q−m)/(q−1) ≤ C10 k2 (∫ Rn (|x|+ 1)µuq(x, t)dx )(m−1)/(q−1)(∫ Rn u(x, t)ϕl(x)dx )(q−m)/(q−1) , (4.13)∫ B3k\B2k vm(x, t)|∆ξk(x)|dx ≤ C10 k2 (∫ Rn (|x|+ 1)λvpc(x, t)dx )(m−1)/(pc−1)(∫ Rn v(x, t)ϕl(x)dx )(pc−m)/(pc−1) . (4.14) 14 Y. NIE, Y. LENG, X. ZHAO, Q. ZHOU EJDE-2025/21 Substituting (4.11)–(4.14) into (4.10) implies that d dt ∫ Rn (u(x, t) + v(x, t))ϕl(x)ξk(x)dx ≤ C11l n−2−m(n+µ)/q (∫ B2l\Bl (|x|+ 1)µuq(x, t)dx )m/q + C10 k2 (∫ Rn (|x|+ 1)µuq(x, t)dx )(m−1)/(q−1)(∫ Rn u(x, t)ϕl(x)dx )(q−m)/(q−1) + ∫ Rn (|x|+ 1)µuq(x, t)dx + C11l n−2−m(n+λ)/pc (∫ B2l\Bl (|x|+ 1)λvpc(x, t)dx )m/pc + C10 k2 (∫ Rn (|x|+ 1)λvpc(x, t)dx )(m−1)/(pc−1)(∫ Rn v(x, t)ϕl(x)dx )(pc−m)/(pc−1) + ∫ Rn (|x|+ 1)λvpc(x, t)dx. Letting k → +∞ in the above inequality yields d dt ∫ Rn ( u(x, t) + v(x, t) ) ϕl(x)dx ≤ C11l n−2−m(n+µ)/q (∫ B2l\Bl (|x|+ 1)µuq(x, t)dx )m/q + ∫ Rn (|x|+ 1)µuq(x, t)dx + C11l n−2−m(n+λ)/pc (∫ B2l\Bl (|x|+ 1)λvpc(x, t)dx )m/pc + ∫ Rn (|x|+ 1)λvpc(x, t)dx. From the Young inequality and (q(n− 2)−m(n+ µ))/(q −m) = (pc(n− 2)−m(n+ λ))/(pc −m), we obtain that d dt ∫ Rn ( u(x, t) + v(x, t) ) ϕl(x)dx ≤ m q ∫ B2l\Bl (|x|+ 1)µuq(x, t)dx+ q −m q C q/(q−m) 11 l(q(n−2)−m(n+µ))/(q−m) + ∫ Rn (|x|+ 1)µuq(x, t)dx+ m pc ∫ B2l\Bl (|x|+ 1)λvpc(x, t)dx + pc −m pc C pc/(pc−m) 11 l(pc(n−2)−m(n+λ))/(pc−m) + ∫ Rn (|x|+ 1)λvpc(x, t)dx ≤ 2 ∫ Rn ( (|x|+ 1)µuq(x, t) + (|x|+ 1)λvpc(x, t) ) dx+M4l (pc(n−2)−m(n+λ))/(pc−m), where M4 = max {q −m q C q/(q−m) 11 , pc −m pc C pc/(pc−m) 11 } . □ EJDE-2025/21 CRITICAL FUJITA EXPONENTS 15 Theorem 4.4. The nontrivial solution to (1.1)–(1.3) with p = pc blows up in finite time. Proof. We prove this theorem by contradiction. Assume that (u, v) is a nontrivial global solution to (1.1)–(1.3) with p = pc. Set Λ = sup l>1,t>0 wl(t) = sup t>0 ∫ Rn ( u(x, t) + v(x, t) ) dx. (4.15) From (4.3) and the nontriviality of (u, v), 0 < Λ < +∞. Owing to (4.15), for any 0 < ε < Λ, there exist l0 > 1 and t0 > 0 such that wl0/2(t0) ≥ Λ− ε. From (4.4) and (4.15), we obtain 1 2 ∫ +∞ t0 ∫ Rn ( (|x|+ 1)µuq(x, t) + (|x|+ 1)λvpc(x, t) ) dxdt ≤ ∫ Rn ( u(x, t) + v(x, t) ) dx ∣∣∣t=+∞ t=t0 ≤ Λ− wl0/2(t0) ≤ ε. For any s ≥ t0, (4.9) gives∫ Rn ( u(x, s) + v(x, s) ) ϕl0/2(x)dx ≤ 2 ∫ s t0 ∫ Rn ( (|x|+ 1)µuq(x, t) + (|x|+ 1)λvpc(x, t) ) dxdt +M4 (1 2 l0 )(pc(n−2)−m(n+λ))/(pc−m) (s− t0) + ∫ Rn (u(x, t0) + v(x, t0))ϕl0/2(x)dx ≤ 4ε+M4 (1 2 l0 )(pc(n−2)−m(n+λ))/(pc−m) (s− t0) + ∫ Rn ( u(x, t0) + v(x, t0) ) dx− wl0/2(t0). ≤ 5ε+M4 (1 2 l0 )(pc(n−2)−m(n+λ))/(pc−m) (s− t0). Letting l = l0 in (4.6) and combining it with the above inequality, we have dwl0(t) dt ≥Mm−τ 1 l−mn+n−2 0 wm−τ l0 (t) × ( −M2 ( 5ε+M4 (1 2 l0 )(pc(n−2)−m(n+λ))/(pc−m) (s− t0) )τ +M −(m−τ) 1 M3 ·min { wpc−m+τ l0 (t), wq−m+τ l0 (t) }) . Taking ε0 ∈ (0,Λ) and M5 > 0 such that M2(5ε0 +M5) τ ≤ 1 2 M −(m−τ) 1 M3 min { (Λ− ε0) pc−m+τ , (Λ− ε0) q−m+τ } with 0 < τ < min { pc−m pc−1 , q−m q−1 } , one obtains dwl0(t) dt ≥ 1 2 M3l −mn+n−2 0 ·min { wpc l0 (t), wq l0 (t) } , t0 < t < t1, (4.16) 16 Y. NIE, Y. LENG, X. ZHAO, Q. ZHOU EJDE-2025/21 where t1 = t0 + M5 M4 (1 2 l0 )−(pc(n−2)−m(n+λ))/(pc−m) . Integrating (4.16) over (t0, t1) with respect to t leads to wl0(t1) ≥ wl0(t0) + 1 2 M3l −mn+n−2 0 min { (Λ− ε0) pc , (Λ− ε0) q } (t1 − t0) ≥ wl0(t0) + 2(pc(n−2)−m(n+λ))/(pc−m) min { (Λ− ε0) pc , (Λ− ε0) q } × M3M5 2M4 l −mn+n−2−(pc(n−2)−m(n+λ))/(pc−m) 0 . Noting that −mn+ n− 2− (pc(n− 2)−m(n+ λ))/(pc −m) = 0, one has ∫ Rn ( u(x, t1) + v(x, t1) ) dx ≥ wl0(t1) ≥ wl0(t0) + δ0 ≥ Λ− ε0 + δ0 with δ0 = M3M5 2M4 min { (Λ− ε0) pc , (Λ− ε0) q } 2(pc(n−2)−m(n+λ))/(pc−m) being a positive constant independent of l0. Obviously, w(2l0)/2(t1) = wl0(t1) ≥ Λ− ε0 + δ0 ≥ Λ− ε0. The same argument leads to∫ Rn ( u(x, t2) + v(x, t2) ) dx ≥ w2l0(t2) ≥ w2l0(t1) + δ0 ≥ Λ− ε0 + 2δ0, where t2 = t1 + M5 M4 (l0) −(pc(n−2)−m(n+λ))/(pc−m). Repeating the above process, one gets that for any positive integer i,∫ Rn ( u(x, ti) + v(x, ti) ) dx ≥ w2i−1l0(ti) ≥ w2i−1l0(ti−1) + δ0 ≥ Λ− ε0 + iδ0 with ti = ti−1 + M5 M4 (2i−2l0) −(pc(n−2)−m(n+λ))/(pc−m). Letting i→ +∞ implies sup t>0 ∫ Rn ( u(x, t) + v(x, t) ) dx = +∞, and this contradicts (4.3). □ Acknowledgments. This research was supported by Science and Technology De- velopment Project of Jilin Province (No. 20240101324JC), and by the Graduate Innovation Fund of Jilin University (No. 2022070). EJDE-2025/21 CRITICAL FUJITA EXPONENTS 17 References [1] J. Aguirre, E. Escobedo; On the blow-up of solutions of a convective reaction diffusion equa- tion, Proc. Roy. Soc. Edinburgh Sect. A, 123(3) (1993), 433–460. [2] D. Andreucci, G. Cirmi, S. Leonardi, A. Tedeev; Large time behavior of solutions to the Neumann problem for a quasilinear second order degenerate parabolic equation in domains with noncompact boundary, J. Differential Equations, 174(2) (2001), 253–288. [3] K. Deng, H. A. Levine; The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl., 243 (2000), 85–126. [4] M. Escobedo, M. A. Herrero; Boundedness and blow up for a semilinear reaction-diffusion system, J. Differential Equations, 89 (1991), 176–202. [5] H. Fujita; On the blowing up of solutions of the Cauchy problem for ut = ∆u + u1+α, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109–124. [6] V. A. Galaktionov; Blow-up for quasilinear heat equations with critical Fujita’s exponents, Proc. Roy. Soc. Edinburgh Sect. A, 124(3) (1994), 517–525. [7] V. A. Galaktionov, S. P. Kurdjumov, A. P. Mikhailov, A. A. Samarskii; On unbounded solutions of the Cauchy problem for the parabolic equation ut = ∇(uσ∇u)+uβ , Dokl. Akad. Nauk SSSR, 252(6) (1980), 1362–1364. [8] G. Girardi; Critical non-linearity for some evolution equations with Fujita-type critical expo- nent, NoDEA Nonlinear Differential Equations Appl., 32(1) (2025), Paper No. 3. [9] K. Hayakawa; On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad., 49 (1973), 503–505. [10] Q. Huang, K. Mochizuki; A note on the global solutions of a degenerate parabolic system, Tokyo J. Math., 20(1) (1997), 63–66. [11] X. X. Jing, Y. Y. Nie, C. P. Wang; Asymptotic behavior of soluions to coupled semilinear par- abolic systems with boundary degeneracy, Electron. J. Differential Equations, 2021 (2021), Paper No. 67, 17 pp. [12] T. Kawakami, Y. Sire, J. N. Wang; Fujita exponent for the global-in-time solutions to a semilinear heat equation with non-homogeneous weights, J. Evol. Equ., 24(2) (2024), Paper No. 44, 24 pp. [13] K. Kobayashi, T. Siaro, H. Tanaka; On the growing up problem for semilinear heat equations, J. Math. Soc. Japan, 29(3) (1977), 407–424. [14] Y. Leng, Y. Y. Nie, Q. Zhou; Asymptotic behavior of solutions to a class of coupled nonlinear parabolic systems, Bound. Value Probl., 2019, Paper No. 68, 11 pp. [15] H. Levine; The role of critical exponents in blow-up theorems, SIAM Review, 32(2) (1990), 262–288. [16] F. J. Li, J. Q. Liu; Asymptotic results and critical Fujita exponent in parabolic equations with nonlocal nonlinearity, Appl. Anal., 101(18) (2022), 6411–6434. [17] H. L. Li, X. Y. Wang, Y. Y. Nie, H. He; Asymptotic behavior of solutions to a degenerate quasilinear parabolic equation with a gradient term, Electron. J. Differential Equations, 2015 (2015), Paper No. 298, 12 pp. [18] Y. P. Li, Z. B. Fang; Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term, Adv. Nonlinear Anal., 12(1) (2023), Paper No. 20220303, 28 pp. [19] K. Mochizuki, Q. Huang; Existence and behavior of solutions for a weakly coupled system of reaction-diffusion equations, Methods Appl. Anal., 5(2) (1998), 109–124. [20] Y. Na, Y. Y. Nie, X. Zhou; Asymptotic behavior of solutions to a class of coupled semilinear parabolic systems with gradient terms, J. Nonlinear Sci. Appl., 10(11) (2017), 5813–5824. [21] Y. W. Qi; The critical exponents of parabolic equations and blow-up in Rn, Proc. Roy. Soc. Edinburgh Sect. A, 128(1) (1998), 123–136. [22] Y. W. Qi, H. A. Levine; The critical exponent of degenerate parabolic systems, Z. Angew. Math. Phys., 44(2) (1993), 249–265. [23] F. Quirós, J. D. Rossi; Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions, Indiana Univ. Math. J., 50(1) (2001), 629–654. [24] P. Quittner, Ph. Souplet; Superlinear parabolic problems: Blow-up, global existence and steady states, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007. 18 Y. NIE, Y. LENG, X. ZHAO, Q. ZHOU EJDE-2025/21 [25] X. Z. Sun, B. C. Liu, F. J. Li; Fujita exponents for an inhomogeneous parabolic equation with variable coefficients, Math. Methods Appl. Sci., 45(11) (2022), 7058—7071. [26] G. H. Todorova, B. Ts. Yordanov; Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174(2) (2001), 464–489. [27] Y. Uda; The critical exponent for a weakly coupled system of the generalized Fujita type reaction-diffusion equations, Z. Angew. Math. Phys., 46(3) (1995), 366–383. [28] C. P. Wang; Asymptotic behavior of solutions to a class of semilinear parabolic equations with boundary degeneracy, Proc. Amer. Math. Soc., 141(9) (2013), 3125–3140. [29] C. P. Wang, S. N. Zheng; Critical Fujita exponents of degenerate and singular parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 136(2) (2006), 415–430. [30] C. P. Wang, S. N. Zheng, Z. J. Wang; Critical Fujita exponents for a class of quasilinear equations with homogeneous Neumann boundary data, Nonlinearity, 20(6) (2007), 1343– 1359. [31] F. B. Weissler; Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38(1-2) (1981), 29–40. [32] Zh. Q. Wu; Degenerate quasilinear parabolic equations, Adv. in Math. (Chinese), 16(2) (1987), 121–158. [33] Zh. Q. Wu, J. N. Zhao, J. X. Yin, H. L. Li; Nonlinear diffusion equations, World Scientific Publishing Co. Inc., River Edge, NJ, 2001. [34] Q. S. Zhang; Blow-up results for nonlinear parabolic equations on manifolds, Duke Math. J., 97(3) (1999), 515–539. [35] X. T. Zhao, M. J. Zhou, X. X. Jing; Asymptotic behavior of solutions to porous medium equations with boundary degeneracy, Electron. J. Differential Equations, 2021 (2021), Paper No. 96, 19 pp. [36] S. N. Zheng; Global existence and global non-existence of solutions to a reaction-diffusion system, Nonlinear Anal., 39(3) (2000), 327–340. [37] S. N. Zheng, X. F. Song, Z. X. Jiang; Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl., 298(1) (2004), 308– 324. [38] S. N. Zheng, C. P. Wang; Large time behaviour of solutions to a class of quasilinear parabolic equations with convection terms, Nonlinearity, 21(9) (2008), 2179–2200. [39] M. J. Zhou, H. L. Li, W. Guo, X. Zhou; Critical Fujita exponents to a class of non-Newtonian filtration equations with fast diffusion, Bound. Value Probl., 2016, Paper No. 146, 16 pp. [40] Q. Zhou, Y. Y. Nie, X. Y. Han; Large time behavior of solutions to semilinear parabolic equations with gradient, J. Dyn. Control Syst., 22(1) (2016), 191–205. Yuanyuan Nie School of Mathematics, Jilin University, Changchun 130012, China Email address: nieyy@jlu.edu.cn Yan Leng School of Science, Northeast Electric Power University, Jilin 132012, China Email address: 1678949428@qq.com Xu Zhao School of Mathematics, Jilin University, Changchun 130012, China Email address: zhaoxu21@mails.jlu.edu.cn Qian Zhou (corresponding author) School of Mathematics, Jilin University, Changchun 130012, China Email address: zhouqian@jlu.edu.cn 1. Introduction 2. Preliminaries 3. Blow-up theorems of Fujita type 4. Critical case Acknowledgments References