Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 73, pp. 1–19. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO COUPLED POROUS MEDIUM SYSTEMS WITH BOUNDARY DEGENERACY XUTONG ZHAO, MINGJUN ZHOU, QIAN ZHOU Abstract. This article concerns the asymptotic behavior of solutions of one- dimensional porous medium systems with boundary degeneracy in bounded and unbounded intervals. It is shown that the degree of the boundary de- generacy and the exponent of the nonlinear diffusion determine asymptotic behaviors of solutions. For the problem in a bounded interval, if the degener- acy is not strong, the problem admits both nontrivial global and blowing-up solutions, while if the degeneracy is strong enough, any nontrivial solution to the problem must blow up in a finite time. For the problem in an unbounded interval, the Fujita type blowing-up theorems are established and the critical Fujita exponent is formulated by the degree of the boundary degeneracy and the exponent of nonlinear diffusion. 1. Introduction The semilinear parabolic equation ∂u ∂t − ∂ ∂x ( xλ ∂u ∂x ) = f(x, t, u), x ∈ R, t > 0, (λ > 0) (1.1) is a typical parabolic equation degenerating on the boundary. It is clear that (1.1) becomes degenerate at x = 0, a portion of the lateral boundary. Equations with such degeneracy as (1.1) can be used to describe some models, such as the Budyko-Sellers climate model [23], the Black-Scholes model coming from the option pricing problem [3], and a simplified Crocco-type equation coming from the velocity field of a laminar flow on a flat plate [5]. In recent decades, semilinear equations with boundary degeneracy have received a lot of attentions from mathematicians. Among them, it was proved that for the control system ∂u ∂t − ∂ ∂x ( xλ ∂u ∂x ) = h(x, t)χω, (x, t) ∈ (0, 1)× (0, T ), u(0, t) = 0, t ∈ (0, T ) if 0 < λ < 1,( xλ ∂u ∂x ) (0, t) = 0, t ∈ (0, T ) if λ ≥ 1, u(1, t) = 0, t ∈ (0, T ), u(x, T ) = u0(x), x ∈ (0, 1), 2020 Mathematics Subject Classification. 35K59, 35B33, 35K65. Key words and phrases. Porous medium equation; boundary degeneracy; asymptotic behavior. ©2022. This work is licensed under a CC BY 4.0 license. Submitted April 4, 2022. Published October 28, 2022. 1 2 X. ZHAO, M. ZHOU, Q. ZHOU EJDE-2022/73 there is a threshold λ = 2 in the sense that the system is null controllable if 0 < λ < 2 [1, 5, 6, 20], while not if λ ≥ 2 [4], where u0 ∈ L2((0, 1)), h is the control function, ω is a subinterval of (0, 1), and χω is the characteristic function of ω. In addition, the null controllability of other control systems with boundary degeneracy were studied, see, e.g., [7, 10, 11, 28, 31, 32, 9, 26]. For another instance, the quenching phenomenon of solutions to the problem ∂u ∂t − ∂ ∂x ( xλ ∂u ∂x ) = f(u), (x, t) ∈ (0, a)× (0, T ),( xλ ∂u ∂x ) (0, t) = u(a, t) = 0, t ∈ (0, T ), u(x, 0) = 0, x ∈ (0, a) was studied in [37], where a > 0 and f ∈ C2([0, c)) with a constant c > 0 satisfies f(0) > 0, f ′(0) > 0, f ′′(s) ≥ 0 for 0 < s < c, lim s→c− f(s) = +∞. It was shown that λ = 2 is also a threshold in the sense that the critical length satisfies a∗ { > 0, if 0 < λ < 2, = 0, if λ ≥ 2. That is to say, in the case that 0 < λ < 2, there is a critical length a∗ > 0 such that the solution exists globally in time if a < a∗, while quenches in a finite time if a > a∗. As to the case that λ ≥ 2, the solution must quench in a finite time for each a > 0. In [27] and [33], the asymptotic behavior of solutions to the following problem in a bounded interval was studied ∂u ∂t − ∂ ∂x ( xλ ∂um ∂x ) = up, 0 < x < 1, t > 0, (1.2)( xλ ∂um ∂x ) (0, t) = u(1, t) = 0, t > 0, (1.3) u(x, 0) = u0(x), 0 < x < 1 (1.4) where λ > 0, p > m ≥ 1 and u0 ∈ L∞((0, 1)) is a nonnegative function. For the problem (1.2)–(1.4), it was proved that there exist both nontrivial global and blowing-up solutions if the degeneracy is not strong such that λ < 2, while any nontrivial solution must blow up in a finite time if the degeneracy is so strong that λ ≥ 2. Furthermore, the blowing-up theorems of Fujita type were also established in [27] and [33] for the following problem in an unbounded interval ∂u ∂t − ∂ ∂x ( xλ ∂um ∂x ) = up, x > 0, t > 0, (1.5)( xλ ∂um ∂x ) (0, t) = 0, t > 0, (1.6) u(x, 0) = u0(x), x > 0, (1.7) where λ > 0, p > m ≥ 1 and and u0 ∈ L∞((0, 1)) is a nonnegative function. It was proved that the critical Fujita exponent can be formulated as pc = { m+ 2− λ, if 0 < λ < 2, +∞, if λ ≥ 2. EJDE-2022/73 COUPLED POROUS MEDIUM SYSTEMS 3 In 1966, Fujita [13] showed that for the Cauchy problem of ∂u ∂t = ∆u+ up, x ∈ RN , t > 0, any nontrivial solution must blow up in a finite time if 1 < p < 1 + 2/N , whereas there exist both nontrivial global and blowing-up solutions if p > 1 + 2/N . Then, the critical case p = 1+2/N , which is well known as the Fujita exponent, was proved to be the blowing-up case in [14, 17]. Such results revealed the relationship between the asymptotic behavior of the solutions to nonlinear partial differential equations and the exponents of nonlinear internal sources. Different extension directions, such as different types of parabolic equations and systems in various of geometries with or without degeneracies or singularities, have been obtained since then, see the survey papers [8, 18] and also the recent papers [2, 19, 22, 29, 30, 34, 35, 36, 38]. Among them, the Cauchy problem of the following coupled semilinear parabolic system was studied ∂u ∂t −∆u = tµ1 |x|ν1vp, ∂v ∂t −∆v = tµ2 |x|ν2uq, x ∈ RN , t > 0, where µ1, µ2, ν1, ν2 ≥ 0, and p, q ≥ 1. Escobedo and Herrero in [12] considered this Cauchy problem with µ1 = µ2 = ν1 = ν2 = 0, and they proved that the critical Fujita curve is (pq)c = 1 + 2 N max{p+ 1, q + 1}. In [25] and [21], it was proved that the critical Fujita curve is (pq)c = 1 + 2 N max{(µ2 + 1)p+ µ1 + 1, (µ1 + 1)q + µ2 + 1} if ν1 = ν2 = 0, while is (pq)c = 1 + 1 N max{(ν2 + 2)p+ ν1 + 2, (ν1 + 2)q + ν2 + 2} if µ1 = µ2 = 0. There are also some studies on the Cauchy problem of the coupled porous medium systems with fast diffusion ∂u ∂t −∆um = vp, ∂v ∂t −∆vn = uq, x ∈ RN , t > 0, (1.8) where 0 < m, n < 1, p, q ≥ 1 and pq > 1. Qi et al [24] proved that the critical Fujita curve of the Cauchy problem of (1.8) is (pq)c = mn+ 2 N max{p+ n, q +m}, and proved that any nontrivial solution to the Cauchy problem of (1.8) must blow up in a finite time if pq < (pq)c, whereas there exist both nonnegative nontrivial global and blowing-up solutions if pq > (pq)c with m = n. They pointed out that the method of constructing the global supersolutions fails for the case that m 6= n because of the different propagating rates for the two kinds of diffusions. Later in [15], it was shown that for the “very fast diffusions” case that 0 < m, n < (N − 2)+/N , the Cauchy problem of (1.8) admits nontrivial global solutions if the initial data is small enough even though m is not equal to n. 4 X. ZHAO, M. ZHOU, Q. ZHOU EJDE-2022/73 In this article, we study the asymptotic behavior of the solutions to the following two coupled porous medium systems with the boundary degeneracy ∂u ∂t = ∂ ∂x ( xλ ∂um ∂x ) + vp, 0 < x < 1, t > 0, (1.9) ∂v ∂t = ∂ ∂x ( xλ ∂vm ∂x ) + uq, 0 < x < 1, t > 0, (1.10)( xλ ∂um ∂x ) (0, t) = ( xλ ∂vm ∂x ) (0, t) = 0, t > 0, (1.11) u(1, t) = v(1, t) = 0, t > 0, (1.12) u(x, 0) = u0(x), v(x, 0) = v0(x), 0 < x < 1, (1.13) and ∂u ∂t = ∂ ∂x ( xλ ∂um ∂x ) + vp, x > 0, t > 0, (1.14) ∂v ∂t = ∂ ∂x ( xλ ∂vm ∂x ) + uq, x > 0, t > 0, (1.15)( xλ ∂um ∂x ) (0, t) = ( xλ ∂vm ∂x ) (0, t) = 0, t > 0, (1.16) u(x, 0) = u0(x), v(x, 0) = v0(x), x > 0, (1.17) where 0 < m < 1, λ > 0 and p, q > 1. In [16], the semilinear case with m = 1 was considered. It was shown that for the problem (1.9)–(1.13) with m = 1, there exist both nontrivial global and blowing-up solutions if λ < 2, while any nontrivial solution must blow up in a finite time if λ ≥ 2; for the problem (1.14)–(1.17) with m = 1, the critical Fujita curve is (pq)c = { 1 + (2− λ) max{p+ 1, q + 1}, if 0 < λ < 2, +∞, if λ ≥ 2. Compared with the semilinear case with m = 1, the quasilinear case with 0 < m < 1 processes both the degeneracy and the singularity. Indeed, (1.9) and (1.14) are degenerate at x = 0 and are singular at points where u = 0, while (1.10) and (1.15) are degenerate at x = 0 and are singular at points where v = 0. For the problem (1.9)–(1.13) in a bounded interval, we prove that λ = 2 is a threshold in the sense that there exist both nontrivial global and blowing-up solutions to problem (1.9)– (1.13) if λ < 2, while any nontrivial solution to problem (1.9)–(1.13) must blow up in a finite time if λ ≥ 2. For problem (1.14)–(1.17) in an unbounded interval, it is proved that λ = 2 is also a threshold in the sense that the critical Fujita exponent is finite if λ < 2, while infinite if λ ≥ 2. More precisely, it is proved that the critical Fujita exponent is (pq)c = { m2 + (2− λ) max{p+m, q +m}, if 0 < λ < 2, +∞, if λ ≥ 2. That is to say, in the case 0 < λ < 2, any nontrivial solution to the problem (1.14)– (1.17) must blow up in a finite time if pq < (pq)c, while the problem (1.14)–(1.17) admits both nontrivial global and blowing-up solutions if pq > (pq)c. Whereas in the case λ ≥ 2, any nontrivial solution to the problem (1.14)–(1.17) must blow up in a finite time. The methods used in this paper are mainly inspired by [27, 33, 16, 24]. But the discussions and estimates are much more complicated than the semilinear EJDE-2022/73 COUPLED POROUS MEDIUM SYSTEMS 5 case with m = 1 in [16]. To prove the blowing-up of the solution to the problem (1.9)–(1.13) with λ > 0 and (1.14)–(1.17) with λ ≥ 2, we analyze the interactions between the nonlinear degenerate diffusions and the sources through estimating energy integrals constructed by choosing appropriate weight functions, and show the blowing-up of the energy integrals instead of constructing blowing-up subsolutions. As for the blowing-up of the solutions to the problem (1.14)–(1.17) with 0 < λ < 2, we establish the ordinary differential inequalities satisfied by the two energy integrals, and prove the blowing-up of the energy integrals by using the theory of invariant region for ordinary differential systems. For the global existence of the solutions to the problem (1.14)–(1.17) with 0 < λ < 2, we construct the self-similar global supersolutions for the case that 1 + m < λ < 2, while use the nontrivial explicit solution to ∂ω ∂t − ∂ ∂x ( xλ ∂ωm ∂x ) = 0, x > 0, t > 0,( xλ ∂ωm ∂x ) (0, t) = 0, t > 0 to construct global supersolutions for the case that 0 < λ ≤ 1 +m. This article is organized as follows. Some preliminaries, including the definition of solutions, comparison principles and well-posedness, are stated in Section 2. The asymptotic behavior of solutions to the problem (1.9)–(1.13) and (1.14)–(1.17) are studied in Section 3 and Section 4, respectively. 2. Preliminaries The subsolutions, supersolutions, and solutions to problems (1.9)–(1.13) and (1.14)–(1.17) are defined as follows. Definition 2.1. Let 0 < T ≤ +∞. A pair of nonnegative functions (u, v) in L∞((0, 1)×(0, T )) is called a subsolution (supersolution, solution) to problem (1.9)– (1.13) in (0, T ), if (i) For any 0 < τ < T , xλ/2 ∂u m ∂x , xλ/2 ∂v m ∂x ∈ L 2((0, 1)× (0, τ)). (ii) For any 0 < τ < T and any nonnegative functions ϕ, ψ ∈ C1([0, 1]× [0, τ ]) vanishing at t = τ and x = 1, it hold∫ τ 0 ∫ 1 0 ( − u(x, t) ∂ϕ ∂t (x, t) + xλ ∂um ∂x (x, t) ∂ϕ ∂x (x, t) ) dxdt ≤ (≥, =) ∫ τ 0 ∫ 1 0 vp(x, t)ϕ(x, t)dxdt, and ∫ τ 0 ∫ 1 0 ( − v(x, t) ∂ψ ∂t (x, t) + xλ ∂vm ∂x (x, t) ∂ψ ∂x (x, t) ) dxdt ≤ (≥, =) ∫ τ 0 ∫ 1 0 uq(x, t)ψ(x, t)dxdt. Definition 2.2. Let 0 < T ≤ +∞. A pair of nonnegative functions (u, v) in L∞((0,+∞) × (0, T )) is called a subsolution (supersolution, solution) to problem (1.14)–(1.17) in (0, T ), if (i) For any 0 < τ < T , xλ/2 ∂u m ∂x , xλ/2 ∂v m ∂x ∈ L 2((0,+∞)× (0, τ)). 6 X. ZHAO, M. ZHOU, Q. ZHOU EJDE-2022/73 (ii) For any 0 < τ < T and any nonnegative ϕ,ψ ∈ C1([0,+∞) × [0, τ ]) van- ishing at t = τ and for large x, it holds∫ τ 0 ∫ +∞ 0 ( − u(x, t) ∂ϕ ∂t (x, t) + xλ ∂um ∂x (x, t) ∂ϕ ∂x (x, t) ) dxdt ≤ (≥, =) ∫ τ 0 ∫ +∞ 0 vp(x, t)ϕ(x, t)dxdt, and ∫ τ 0 ∫ +∞ 0 ( − v(x, t) ∂ψ ∂t (x, t) + xλ ∂vm ∂x (x, t) ∂ψ ∂x (x, t) ) dxdt ≤ (≥,=) ∫ τ 0 ∫ +∞ 0 uq(x, t)φ(x, t)dxdt. If (u, v) is a solution to (1.9)–(1.13) (or to (1.14)–(1.17)) in (0,+∞), then (u, v) is called a global solution in time. Otherwise, there exists T > 0 such that (u, v) is a solution in (0, T ) and ‖u(·, t)‖L∞((0,1)) + ‖v(·, t)‖L∞((0,1)) → +∞, as t→ T−, (or ‖u(·, t)‖L∞((0,+∞)) + ‖v(·, t)‖L∞((0,+∞)) → +∞, as t→ T−), and we say that (u, v) blows up in a finite time. Similarly to [27] and [33], one can establish the well-posedness and the compar- ison principles for problems (1.9)–(1.13) and (1.14)–(1.17). Proposition 2.3. (i) For any 0 ≤ u0, v0 ∈ L∞((0, 1)), the problem (1.9)–(1.13) admits at least one solution locally in time. (ii) Assume that (û, v̂) and (ǔ, v̌) are a supersolution and a subsolution to the problem (1.9)–(1.13) in (0, T ), respectively. Then (ǔ, v̌) ≤ (û, v̂) in (0, 1)× (0, T ). Proposition 2.4. (i) For any 0 ≤ u0, v0 ∈ L∞((0,+∞)) ∩ L1((0,+∞)), the prob- lem (1.14)–(1.17) admits at least one solution locally in time. (ii) Assume that (û, v̂) and (ǔ, v̌) are a supersolution and a subsolution to the problem (1.14)–(1.17) in (0, T ), respectively. Then (ǔ, v̌) ≤ (û, v̂) in (0,+∞) × (0, T ). 3. Problem in a bounded domain In this section, we investigate the blowing-up and global existence of the solution to the problem (1.9)–(1.13). Theorem 3.1. Assume that λ ≥ 2, 0 < m < 1 and p, q > 1. Then for any nontrivial 0 ≤ u0, v0 ∈ L∞((0, 1)), the solution to problem (1.9)–(1.13) must blow up in a finite time. Proof. Without loss of generality, we assume that p ≥ q. For 0 < δ < 1, set ξδ(x) = { λ−1 δ 2λ−1−δx−δ − λ−1−δ δ 2λ−1 − 1, 0 < x < 1/2, x1−λ − 1, 1/2 ≤ x ≤ 1. It is not hard to check that 0 ≤ ξδ(x) ∈ C1,1((0, 1]) satisfies (xλξ′δ(x))′ ≥ −M1δξδ(x), 0 < x < 1; (3.1) EJDE-2022/73 COUPLED POROUS MEDIUM SYSTEMS 7∫ 1 0 ξδ(x)dx ≤M2, (3.2) where M1,M2 > 0 is a constant depending only on λ but independent of δ. Assume that (u, v) is a solution to (1.9)–(1.13) in (0,+∞), and denote wδ(t) = ∫ 1 0 (u(x, t) + v(x, t))ξδ(x)dx, t ≥ 0. By a similar argument as in [33, Theorem 3.1] and Hölder inequality, one can obtain w′δ(t) = ∫ 1 0 (um(x, t) + vm(x, t))(xλξ′δ(x))′dx + ∫ 1 0 vp(x, t)ξδ(x)dx+ ∫ 1 0 uq(x, t)ξδ(x)dx ≥ −M1δ ∫ 1 0 (um(x, t) + vm(x, t))ξδ(x)dx + (∫ 1 0 ξδ(x)dx )1−q(∫ 1 0 u(x, t)ξδ(x)dx )q + (∫ 1 0 ξδ(x)dx )1−p(∫ 1 0 v(x, t)ξδ(x)dx )p ≥ −M1δ (∫ 1 0 ξδ(x)dx )1−m × [( ∫ 1 0 u(x, t)ξδ(x)dx )m + (∫ 1 0 v(x, t)ξδ(x)dx )m] + min{M1−p 2 , M1−q 2 } × [( ∫ 1 0 u(x, t)ξδ(x)dx )q + (∫ 1 0 v(x, t)ξδ(x)dx )p] ≥  −2M1M 1−m 2 δwmδ (t) + 2−2p min{M1−p 2 , M1−q 2 }wpδ (t), if wδ(t) ≤ 2, t > 0, −2M1M 1−m 2 δwmδ (t) + 2−p min{M1−p 2 , M1−q 2 }wqδ(t), if wδ(t) > 2 t > 0, (3.3) Owing to infδ∈(0,1) wδ(0) > 0, there exists a sufficient small constant δ0 ∈ (0, 1) such that 2M1M 1−m 2 δ0 ≤ 2−2p−1 min{M1−p 2 , M1−q 2 }wp−mδ0 (0), (3.4) 2M1M 1−m 2 δ0 ≤ 2−p−1 min{M1−p 2 , M1−q 2 }wq−mδ0 (0). (3.5) If wδ0(0) < 2, we claim that there exists T̃ > 0 such that wδ0(T̃ ) ≥ 2. Otherwise, wδ0(t) < 2, t ∈ [0,+∞). (3.6) It follows from (3.3), (3.4) and (3.6) that w′δ0(t) ≥ 2−2p−1 min{M1−p 2 , M1−q 2 }wpδ (t), t > 0. Since p > 1, there exists T̂ > 0 such that limt→T̂− wδ0(t) = +∞, which contradicts (3.6). Therefore, one can assume that wδ0(0) ≥ 2. Then from (3.3) and (3.5) one 8 X. ZHAO, M. ZHOU, Q. ZHOU EJDE-2022/73 obtains w′δ0(t) ≥ 2−p−1 min{M1−p 2 , M1−q 2 }wqδ0(t), t > 0. Since q > 1, there exists some T > 0 such that limt→T− wδ0(t) = +∞, which leads to ‖u(·, t)‖L∞((0,1)) + ‖v(·, t)‖L∞((0,1)) → +∞, as t→ T−. That is to say, (u, v) must blow up in a finite time. � Theorem 3.2. Assume that 0 < λ < 2, 0 < m < 1, p, q > 1. Then the solution to problem (1.9)–(1.13) exists globally in time if (u0, v0) ∈ L∞((0, 1)) is suitably small, while blows up in a finite time if (u0, v0) ∈ L∞((0, 1)) is large enough, where 0 ≤ u0, v0 ∈ L∞((0, 1)). Proof. First we consider the global existence case. To show the existence of a global solution to (1.9)–(1.13), we try to find a nonnegative nontrivial self-similar supersolutions to the problem (1.9)–(1.13) of the form u(x, t) = (t+ τ)−α1U((t+ τ)−βx), 0 ≤ x ≤ 1, t ≥ 0, (3.7) v(x, t) = (t+ τ)−α2V ((t+ τ)−βx), 0 ≤ x ≤ 1, t ≥ 0, (3.8) where α1 = p+ 1 pq − 1 , α2 = q + 1 pq − 1 , β = 1 2− λ , and τ ≥ 1 will be determined below. If 0 ≤ U, V ∈ C0,1((0, τ−β)) with Um, V m ∈ C1,1((0, τ−β)) satisfies (t+ τ)(1−m)α1+1(rλ(Um(r))′)′ + α1U(r) + βrU ′(r) + V p(r) ≤ 0, 0 < r < τ−β , (t+ τ)(1−m)α2+1(rλ(V m(r))′)′ + α2V (r) + βrV ′(r) + Uq(r) ≤ 0, 0 < r < τ−β , then (u, v) given by (3.7) and (3.8) is a supersolution to (1.9) and (1.10). Set U(r) = V (r) = 1 (2− λ)1/m (1 τ − r2−λ )1/m , 0 ≤ r ≤ τ−β . For 0 < r < τ−β , a direct calculation shows that (t+ τ)(1−m)α1+1(rλ(Um(r))′)′ + α1U(r) + βrU ′(r) + V p(r) = −(t+ τ)(1−m)α1+1 + α1U(r)− β m r1−mU1−m(r) + Up(r) ≤ −1 + α1U(r) + Up(r) ≤ −1 + 1 τ1/m ( α1 (2− λ)1/m + 1 (2− λ)p/m ) , and (t+ τ)(1−m)α2+1(rλ(V m(r))′)′ + α2V (r) + βrV ′(r) + Uq(r) = −(t+ τ)(1−m)α2+1 + α2V (r)− β m r1−mV 1−m(r) + V q(r) ≤ −1 + α2V (r) + V q(r) ≤ −1 + 1 τ1/m ( α2 (2− λ)1/m + 1 (2− λ)q/m ) . EJDE-2022/73 COUPLED POROUS MEDIUM SYSTEMS 9 Hence (u, v) is a supersolution to (1.9) and (1.10) for each τ ≥ τ0 = 2m (αm1 + αm2 2− λ + 1 (2− λ)p ) + 1. It is noted that lim r→0+ rλ(Um(r))′ = lim r→0+ rλ(V m(r))′ = 0. Therefore, (u, v) is a supersolution to the problem (1.9)–(1.13) if u0(x) ≤ u(x, 0), v0(x) ≤ v(x, 0), 0 < x < 1 (3.9) for some τ ≥ τ0. Thanks to Proposition 2.3 (ii), one obtains that the solution to (1.9)–(1.13) exists globally in time if (u0, v0) satisfies (3.9). Now we turn to the blowing-up case. Without loss of generality, it is still assumed that p ≥ q. Set η(x) = { 2, 0 ≤ x ≤ 1/2, 1 + cos(2x− 1)π, 1/2 < x ≤ 1. It is easy to verify that η ∈ C1,1([0, 1]) satisfies (xλη(x))′ ≥ −4π2η(x), 1/2 < x < 1. (3.10) Assume that (u, v) is a solution to (1.9)–(1.13) in (0,+∞), and denote w(t) = ∫ 1 0 (u(x, t) + v(x, t))η(x)dx, t ≥ 0. It follows from Definition 2.1, Hölder’s inequality and (3.10) that w′(t) = ∫ 1 0 (um(x, t) + vm(x, t))(xλη′(x))′dx + ∫ 1 0 vp(x, t)η(x)dx+ ∫ 1 0 uq(x, t)η(x)dx ≥ −4π2 ∫ 1 0 (um(x, t) + vm(x, t))η(x)dx + (∫ 1 0 η(x)dx )1−q(∫ 1 0 u(x, t)η(x)dx )q + (∫ 1 0 η(x)dx )1−p(∫ 1 0 v(x, t)η(x)dx )p ≥ −4π2 (∫ 1 0 η(x)dx )1−m × [( ∫ 1 0 u(x, t)η(x)dx )m + (∫ 1 0 v(x, t)η(x)dx )m] + 21−p [( ∫ 1 0 u(x, t)η(x)dx )q + (∫ 1 0 v(x, t)η(x)dx )p] ≥ { −16π2wm(t) + 21−3pwp(t), if w(t) ≤ 2, t > 0 −16π2wm(t) + 21−2pwq(t), if w(t) > 2, t > 0 (3.11) If (u0, v0) is sufficiently large such that w(0) > 2, wq−m(0) ≥ 24+2pπ2, (3.12) 10 X. ZHAO, M. ZHOU, Q. ZHOU EJDE-2022/73 from (3.11) and (3.12) one obtains w′(t) ≥ 2−2pwq(t), t > 0. Since q > 1, there exists T > 0 such that limt→T− w(t) = +∞, which leads to ‖u(·, t)‖L∞((0,1)) + ‖v(·, t)‖L∞((0,1)) → +∞, as t→ T−. That is, if (u0, v0) satisfies (3.12), then (u, v) must blow up in a finite time. � 4. Problem in unbounded domains In this section, we formulate the critical Fujita exponent for problem (1.14)– (1.17) and establish a Fujita-type blowing-up theorem. Theorem 4.1. Assume that 0 < λ < 2, 0 < m < 1, p, q > 1 and pq < (pq)c = m2 + (2− λ) max{p+m, q +m}. Then for any nontrivial 0 ≤ u0, v0 ∈ L∞((0,+∞)) ∩ L1((0,+∞)), the solution to (1.14)–(1.17) must blow up in a finite time. Proof. Without loss of generality, we assume that p ≥ q. For R > 0, we set ζR(x) =  1, x ∈ [0, R], 1 2 ( 1 + cos (x−R)π R ) , x ∈ (R, 2R), 0, x ∈ [2R,+∞). (4.1) It follows from the proof of [27, Lemma 2.1] that ηR ∈ C1,1([0,+∞)) satisfies (xλζR(x))′ ≥ −2λπ2Rλ−2ζR(x), x > 0. (4.2) Assume that (u, v) is a solution to problem (1.14)–(1.17) in (0,+∞), and denote FR(t) = 1 2R ∫ +∞ 0 u(x, t)ζR(x)dx, GR(t) = 1 2R ∫ +∞ 0 v(x, t)ζR(x)dx, t ≥ 0. From Definition 2.2, Hölder’s inequality, and (4.2) one gets F ′R(t) = 1 2R ∫ +∞ 0 um(x, t)(xλζ ′R(x))′dx+ 1 2R ∫ +∞ 0 vp(x, t)ζR(x)dx ≥ −2λ−1π2Rλ−3 ∫ +∞ 0 um(x, t)ζR(x)dx + 1 2R (∫ +∞ 0 ζR(x)dx )1−p(∫ +∞ 0 v(x, t)ζR(x)dx )p ≥ −2λ−1π2Rλ−3 (∫ +∞ 0 ζR(x)dx )1−m(∫ +∞ 0 u(x, t)ζR(x)dx )m +GpR(t) ≥ −2λπ2Rλ−2FmR (t) +GpR(t), t > 0. (4.3) Similarly, G′R(t) ≥ −2λπ2Rλ−2GmR (t) + F qR(t), t > 0. (4.4) If pq < (pq)c then either (λ−2)(p+m)/(pq−m2) < −1 or (λ−2)(q+m)/(pq−m2) < −1, which, together with the non-negativity and non-triviality of (u0, v0), yields either lim R→+∞ FR(0) R(λ−2)(p+m)/(pq−m2) = lim R→+∞ 1 2R ∫ +∞ 0 u0(x)ζR(x)dx R(λ−2)(p+m)/(pq−m2) = +∞ EJDE-2022/73 COUPLED POROUS MEDIUM SYSTEMS 11 or lim R→+∞ GR(0) R(λ−2)(q+m)/(pq−m2) = lim R→+∞ 1 2R ∫ +∞ 0 v0(x)ζR(x)dx R(λ−2)(q+m)/(pq−m2) = +∞. It is known from (4.3), (4.4) and Corollary 2 in [24] that (FR(t), GR(t)) blows up in a finite time for sufficient large R > 0. Therefore, (u, v) must blow up in a finite time. � Theorem 4.2. Assume that 0 < λ < 2, 0 < m < 1, p, q > 1 and pq > (pq)c = m2 + (2− λ) max{p+m, q +m}. Then the solution to (1.14)–(1.17) exists globally in time if (u0, v0) is suitably small, while blows up in a finite time if (u0, v0) is large enough, where 0 ≤ u0, v0 ∈ L∞((0,+∞)) ∩ L1((0,+∞)). Proof. Thanks to Theorem 3.2 and Proposition 2.4, the solution to (1.14)–(1.17) blows up in a finite time if (u0, v0) is large enough. Below we prove that there exists a nonnegative nontrivial global solution to (1.14)–(1.17) if (u0, v0) is suitably small by constructing proper nonnegative nontrivial global supersolution. Without loss of generality, we still assume that p ≥ q in the remaining part of proof. Then p2 ≥ pq > (pq)c = m2 + (2− λ)(p+m), which implies that p > m+ 2− λ. The discussion will be divided into three cases. Case 1. 1 +m < λ < 2. Set û(x, t) = (t+ 1)−γ1Φ((t+ 1)−βx), x ≥ 0, t ≥ 0, (4.5) v̂(x, t) = (t+ 1)−γ2Ψ((t+ 1)−βx), x ≥ 0, t ≥ 0, (4.6) where γ1 = p+m pq −m2 , γ2 = q +m pq −m2 , β = 1 2− λ . If 0 ≤ Φ, Ψ ∈ C0,1((0,+∞)) with Φm, Ψm ∈ C1,1((0,+∞)) satisfies (t+ 1)(1−m)γ1(rλ(Φm(r))′)′ + γ1Φ(r) + βrΦ′(r) + (t+ 1)(1−m)γ1Ψp(r) ≤ 0, (t+ 1)(1−m)γ2(rλ(Ψm(r))′)′ + γ2Ψ(r) + βrΨ ′(r) + (t+ 1)(1−m)γ2Φq(r) ≤ 0, for r > 0, then (û, v̂) given by (4.5) and (4.6) is a supersolution to (1.14) and (1.15). We take Φ(r) = Ψ(r) = (1 +Ar2−λ)1/(m−1), r ≥ 0, (4.7) where A = (1−m)2(γ1 + γ2 + 1) m(2− λ)(λ− 1−m) . (4.8) For r > 0, by direct calculations we have (t+ 1)(1−m)γ1(rλ(Φm(r))′)′ + γ1Φ(r) + βrΦ′(r) + (t+ 1)(1−m)γ1Ψp(r) = Φ(r) [ − m(2− λ)A 1−m ( 1− (2− λ)A 1−m r2−λΦ1−m(r) ) (t+ 1)(1−m)γ1 + γ1 − A 1−m r2−λΦ1−m(r) + (t+ 1)(1−m)γ1Φp−1(r) ] = Φ(r) [ − m(2− λ)A 1−m ( 1− 2− λ 1−m ) (t+ 1)(1−m)γ1 + γ1 + (t+ 1)(1−m)γ1Φp−1(r) ] 12 X. ZHAO, M. ZHOU, Q. ZHOU EJDE-2022/73 ≤ (t+ 1)(1−m)γ1Φ(r) [ − m(2− λ)(λ− 1−m)A (1−m)2 + γ1 + 1 ] = −γ2(t+ 1)(1−m)γ1Φ(r) ≤ 0, and similarly, (t+ 1)(1−m)γ2(rλ(Ψm(r))′)′ + γ2Ψ(r) + βrΨ ′(r) + (t+ 1)(1−m)γ2Φq(r) ≤ −γ1(t+ 1)(1−m)γ2Ψ(r) ≤ 0. Hence (û, v̂) given by (4.5)–(4.8) is a supersolution to (1.14) and (1.15). Addition- ally, lim r→0+ rλ(Φm(r))′ = lim r→0+ rλ(Ψm(r))′ = 0. Therefore, (û, v̂) is a supersolution to the problem (1.14)–(1.17) if u0(x) ≤ û(x, 0), v0(x) ≤ v̂(x, 0), x > 0. (4.9) Case 2. λ = 1 +m. It is easy to check that ω1+m(x, t) = e−t(1 + e−(1−m)tx1−m)1/(m−1), x ≥ 0, t ≥ 0 (4.10) solves ∂ω ∂t − ∂ ∂x ( xλ ∂ωm ∂x ) = 0, x > 0, t > 0, (4.11)( xλ ∂ωm ∂x ) (0, t) = 0, t > 0 (4.12) with λ = 1 +m. Set û(x, t) = v̂(x, t) = Θ 1/m 0 (t)ω1+m ( x, ∫ t 0 Θ (m−1)/m 0 (s)ds ) , x ≥ 0, t ≥ 0. (4.13) If 0 ≤ Θ0 ∈ C1([0,+∞)) satisfies e−tΘ0(t) ∈ L∞([0,+∞)), Θ′0(t) ≥ 0 for t > 0, (4.14) and Θ′0(t) ≥ Θ1+(p−1)/m 0 (t) exp { − (p− 1) ∫ t 0 Θ (m−1)/m 0 (s)ds } , t > 0, (4.15) Θ′0(t) ≥ Θ1+(q−1)/m 0 (t) exp { − (q − 1) ∫ t 0 Θ (m−1)/m 0 (s)ds } , t > 0, (4.16) then (û, v̂) given by (4.10) and (4.13) is a supersolution to (1.14) and (1.15). We suppose that 0 < θ 2 ≤ Θ0(t) ≤ θ < 1 (4.17) holds for some constant θ ∈ (0, 1) to be determined. Then (4.15) and (4.16) hold provided that Θ′0(t) ≥ θ1+(q−1)/me−(q−1)t, t > 0. (4.18) We take Θ0(t) = θ 2 + θ1+(q−1)/m q − 1 (1− e−(q−1)t), t ≥ 0, (4.19) where θ = min {1 2 , (q − 1 2 )m/(q−1)} . (4.20) EJDE-2022/73 COUPLED POROUS MEDIUM SYSTEMS 13 Then (4.14), (4.17) and (4.18) hold. Hence (û, v̂) given by (4.10), (4.13), (4.19) and (4.20) is a supersolution to (1.14)–(1.17) if u0(x) ≤ û(x, 0), v0(x) ≤ v̂(x, 0), x > 0. (4.21) Case 3. 0 < λ < 1 +m. One can check that ωλ(x, t) = (t+ 1)−µ ( 1 + (1−m)µ (2− λ)m (t+ 1)−(2−λ)µx2−λ )1/(m−1) , (4.22) for x ≥ 0 and t ≥ 0, solves (4.11) and (4.12) with 0 < λ < 1 + m, where µ = 1/(1 +m− λ). Set û(x, t) = Θ 1/m 1 (t)ωλ ( x, ∫ t 0 Θ (m−1)/m 1 (s)ds ) , x ≥ 0, t ≥ 0, (4.23) v̂(x, t) = Θ 1/m 2 (t)ωλ ( x, ∫ t 0 Θ (m−1)/m 2 (s)ds ) , x ≥ 0, t ≥ 0. (4.24) If 0 ≤ Θ1, Θ2 ∈ C1([0,+∞)) satisfy (t+ 1)−µΘ1(t), (t+ 1)−µΘ2(t) ∈ L∞([0,+∞)), Θ′1(t), Θ′2(t) ≥ 0 for t > 0, (4.25) and Θ′1(t) ≥ mΘ(m−1)/m 1 (t)Θ p/m 2 (t) ωpλ ( x, ∫ t 0 Θ (m−1)/m 2 (s)ds ) ωλ ( x, ∫ t 0 Θ (m−1)/m 1 (s)ds ) , x > 0, t > 0, Θ′2(t) ≥ mΘ(m−1)/m 2 (t)Θ q/m 1 (t) ωqλ ( x, ∫ t 0 Θ (m−1)/m 1 (s)ds ) ωλ ( x, ∫ t 0 Θ (m−1)/m 2 (s)ds ) , x > 0, t > 0, then (û, v̂) given by (4.22)–(4.24) is a supersolution to (1.14) and (1.15). For Θ1(t) ≤ Θ2(t), a directly calculatin shows that ωpλ ( x, ∫ t 0 Θ (m−1)/m 2 (s)ds ) ωλ ( x, ∫ t 0 Θ (m−1)/m 1 (s)ds ) = ( 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )−pµ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )µ × ( 1 + (1−m)µ (2− λ)m ( 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )−(2−λ)µ x2−λ )(p−1)/(m−1) × (1 + (1−m)µ (2−λ)m ( 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )−(2−λ)µ x2−λ 1 + (1−m)µ (2−λ)m ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )−(2−λ)µ x2−λ )1/(m−1) ≤ ( 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )−pµ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )µ , x ≥ 0, t ≥ 0, 14 X. ZHAO, M. ZHOU, Q. ZHOU EJDE-2022/73 and similarly, ωqλ ( x, ∫ t 0 Θ (m−1)/m 1 (s)ds ) ωλ ( x, ∫ t 0 Θ (m−1)/m 2 (s)ds ) ≤ ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )−(q−1)µ+1/(1−m)( 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )1/(m−1) for x ≥ 0 and t ≥ 0. Therefore (û, v̂) are a supersolution to (1.14) and (1.15) if Θ′1(t) ≥ Θ(m−1)/m 1 (t)Θ p/m 2 (t) ( 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )−pµ × ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )µ , t > 0, (4.26) Θ′2(t) ≥ Θ(m−1)/m 2 (t)Θ q/m 1 (t) ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )−(q−1)µ+1/(1−m) × ( 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )1/(m−1) , t > 0. (4.27) Case 3.1. p ≥ q > m+ 2− λ. We assume in advance that 0 < θ0 2 ≤ Θ1(t) ≤ θ0 ≤ Θ2(t) ≤ 2θ0 < 1 (4.28) holds for some constant θ0 ∈ (0, 1/2) to be determined. It is not hard to check that( 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )−pµ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )µ = ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )−(p−1)µ(1 + ∫ t 0 Θ (m−1)/m 1 (s)ds 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )pµ ≤ (1 + t)−(p−1)µ (1 + (θ1/2)(m−1)/mt 1 + (2θ1)(m−1)/mt )pµ/m ≤ 4pµ/m(1 + t)−(p−1)µ, t > 0, and ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )−(q−1)µ+1/(1−m)( 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )1/(m−1) = ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )−(q−1)µ(1 + ∫ t 0 Θ (m−1)/m 1 (s)ds 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )1/(1−m) ≤ 41/m(1 + t)−(q−1)µ, t > 0. In order that (4.26) and (4.27) hold, it is sufficient that Θ′1(t) ≥ 4p(µ+2)/mθ 1+(p−1)/m 0 (1 + t)−(p−1)µ, t > 0, (4.29) Θ′2(t) ≥ 41/mθ 1+(q−1)/m 0 (1 + t)−(q−1)µ, t > 0. (4.30) We take Θ1(t) = θ0 2 + 4p(µ+2)/mθ 1+(p−1)/m 0 (p− 1)µ− 1 ( 1− (1 + t)1−(p−1)µ ) , t ≥ 0, (4.31) EJDE-2022/73 COUPLED POROUS MEDIUM SYSTEMS 15 Θ2(t) = θ0 + 41/mθ 1+(q−1)/m 0 (q − 1)µ− 1 ( 1− (1 + t)1−(q−1)µ ) , t ≥ 0, (4.32) where θ0 = min {1 2 , ( (p− 1)µ− 1 41+p(µ+2)/m )m/(p−1) , ( (q − 1)µ− 1 41/m )m/(q−1)} . (4.33) It follows from 1− (p− 1)µ ≤ 1− (q − 1)µ < 0 that (4.25) and (4.28)–(4.30) hold. Hence (û, v̂) given by (4.22)–(4.24) and (4.31)– (4.33) is a supersolution to the problem (1.14)–(1.17) if (4.9) holds. Case 3.2. p > m+ 2− λ = q > 1. We assume in advance that 0 < θ1 2 ≤ Θ1(t) ≤ θ1 ≤ Θ2(t) = θ1(1 + t)mσ1 (4.34) holds for some constant θ1 and σ1 ∈ (0, 1) to be determined. One can show that( 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )−pµ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )µ = ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )−(p−1)µ(1 + ∫ t 0 Θ (m−1)/m 1 (s)ds 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )pµ ≤ (1 + t)−(p−1)µ ( 1 + ( θ1 2 )(m−1)/m t 1 + θ (m−1)/m 1 1−(1−m)σ1 ( (1 + t)1−(1−m)σ1 − 1 ))pµ ≤ (1 + t)−(p−1)µ (1 + (θ1/2)(m−1)/mt 1 + θ (m−1)/m 1 t )pµ ≤ 2pµ/m(1 + t)−(p−1)µ, t > 0, and ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )−(q−1)µ+1/(1−m)( 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )1/(m−1) ≤ ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )−1+σ1 (1 + ∫ t 0 Θ (m−1)/m 1 (s)ds 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )1/(1−m) ≤ 21/m(1 + t)−1+σ1 , t > 0. In order that (4.26) and (4.27) hold, it is sufficient that Θ′1(t) ≥ 2(pµ+1)/mθ 1+(p−1)/m 1 (1 + t)−(p−1)µ+pσ1 , t > 0, (4.35) Θ′2(t) = mσ1θ1(1 + t)mσ1−1 ≥ 21/mθ 1+(q−1)/m 1 (1 + t)mσ1−1, t > 0. (4.36) We take Θ1(t) = θ1 2 + 2(pµ+1)/mθ 1+(p−1)/m 1 (p− 1)µ− pσ1 − 1 ( 1− (1 + t)1−(p−1)µ+pσ1 ) , t ≥ 0, (4.37) Θ2(t) = θ1(1 + t)mσ1 , t ≥ 0, (4.38) where σ1 = min {1 2 , (p− 1)µ− 1 2p } , (4.39) 16 X. ZHAO, M. ZHOU, Q. ZHOU EJDE-2022/73 θ1 = min {1 2 , ( (p− 1)µ− pσ1 − 1 21+(pµ+1)/m )m/(p−1) , (mσ1)m/(q−1) 21/(q−1) } . (4.40) It follows from 1− (p− 1)µ+ pσ1 < 0 that σ1 < (p− 1)µ− 1 p = µ− µ+ 1 p < µ, and hence (4.25) and (4.34)–(4.36) hold. Therefore, (û, v̂) given by (4.22)–(4.24) and (4.37)–(4.40) is a supersolution to the problem (1.14)–(1.17) if (4.9) holds. Case 3.3. p > m+ 2− λ > q > 1. We assume in advance that 0 < θ2 2 ≤ Θ1(t) ≤ θ2 ≤ Θ2(t) = θ2(1 + t)mσ2 (4.41) holds for some constants θ2 and σ2 ∈ (0, 1) to be determined. One can show that( 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )−pµ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )µ = ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )−(p−1)µ(1 + ∫ t 0 Θ (m−1)/m 1 (s)ds 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )pµ ≤ (1 + t)−(p−1)µ ( 1 + ( θ2 2 )(m−1)/m t 1 + θ (m−1)/m 2 1−(1−m)σ2 ( (1 + t)1−(1−m)σ2 − 1 ))pµ ≤ (1 + t)−(p−1)µ (1 + (θ2/2)(m−1)/mt 1 + θ (m−1)/m 2 t )pµ ≤ 2pµ/m(1 + t)−(p−1)µ, t > 0, and ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )−(q−1)µ+1/(1−m) × ( 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )1/(m−1) = ( 1 + ∫ t 0 Θ (m−1)/m 1 (s)ds )−(q−1)µ(1 + ∫ t 0 Θ (m−1)/m 1 (s)ds 1 + ∫ t 0 Θ (m−1)/m 2 (s)ds )1/(1−m) ≤ 21/m(1 + t)−(q−1)µ, t > 0. In order that (4.26) and (4.27) hold, it is sufficient that Θ′1(t) ≥ 2(pµ+1)/mθ 1+(p−1)/m 2 (1 + t)−(p−1)µ+pσ2 , (4.42) Θ′2(t) = mσ2θ2(1 + t)mσ2−1 ≥ 21/mθ 1+(q−1)/m 2 (1 + t)(m−1)σ2−(q−1)µ, (4.43) for t > 0. We take Θ1(t) = θ2 2 + 2(pµ+1)/mθ 1+(p−1)/m 2 (p− 1)µ− pσ2 − 1 ( 1− (1 + t)1−(p−1)µ+pσ2 ) , t ≥ 0, (4.44) Θ2(t) = θ2(1 + t)mσ2 , t ≥ 0, (4.45) EJDE-2022/73 COUPLED POROUS MEDIUM SYSTEMS 17 where σ2 = 1− (q − 1)µ, θ2 = min {1 2 , ( (p− 1)µ− pσ2 − 1 21+(pµ+1)/m )m/(p−1) , (mσ2)m/(q−1) 21/(q−1) } . (4.46) From pq > (pq)c = m2 + (2− λ)(p+m) and p > m+ 2− λ it follows that 1− (p− 1)µ+ pσ2 = (p+ 1)(m+ 1− λ) + 1− pq m+ 1− λ ≤ (p+ 1)(m+ 1− λ) + 1−m2 − (2− λ)(p+m) m+ 1− λ = (m− 1)(p−m− 2− λ) m+ 1− λ ≤ 0, and σ2 − µ = 1− qµ < 1− 1 m+ 1− λ (m(m+ 2− λ) p + 2− λ ) < 1− 2− λ m+ 1− λ = m− 1 m+ 1− λ < 0. Then (4.25) and (4.34)–(4.36) hold. Therefore, (û, v̂) given by (4.22)–(4.24) and (4.37)–(4.40) is a supersolution to the problem (1.14)–(1.17) if (4.9) holds. � Acknowledgments. This research was supported by National Key R & D Pro- gram of China (No. 2020YFA0714101), and by the National Natural Science Foun- dation of China (Nos. 11925105 and 12001227). X. 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EJDE-2022/73 COUPLED POROUS MEDIUM SYSTEMS 19 [36] S. Zheng, C. Wang; Large time behaviour of solutions to a class of quasilinear parabolic equations with convection terms, Nonlinearity, 21 (9) (2008), 2179–2200. [37] M. Zhou, C. Wang, Y. Nie; Quenching of solutions to a class of semilinear parabolic equa- tions with boundary degeneracy, Journal of Mathematical Analysis and Applications, 421 (1) (2015), 59–74. [38] Q. Zhou, Y. Nie, X. Han; Large time behavior of solutions to semilinear parabolic equations with gradient, Journal of Dynamical and Control Systems, 22 (1) (2016), 191–205. Xutong Zhao School of Mathematics, Jilin University, Changchun 130012, China Email address: 847692570@qq.com Mingjun Zhou (corresponding author) School of Mathematics, Jilin University, Changchun 130012, China Email address: zhoumingjun@jlu.edu.cn Qian Zhou School of Mathematics, Jilin University, Changchun 130012, China Email address: zhouqian@jlu.edu.cn 1. Introduction 2. Preliminaries 3. Problem in a bounded domain 4. Problem in unbounded domains Acknowledgments References