Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 67, pp. 1–17. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO COUPLED SEMILINEAR PARABOLIC SYSTEMS WITH BOUNDARY DEGENERACY XINXIN JING, YUANYUAN NIE, CHUNPENG WANG Abstract. This article concerns the asymptotic behavior of solutions to cou- pled semilinear parabolic systems with boundary degeneracy. For the problem in a bounded domain, it is proved that there exist both nontrivial global and blowing-up solutions if the degeneracy is not strong, while any nontrivial so- lution must blow up in a finite time if the degeneracy is enough strong. For the problem in an unbounded domain, blowing-up theorems of Fujita type are established. It is shown that the critical Fujita curve is determined by the strength of degeneracy. In particular, it is infinite if the degeneracy is enough strong. 1. Introduction As a typical parabolic equation with boundary degeneracy, ∂u ∂t = ∂ ∂x ( xλ ∂u ∂x ) + f(x, t, u), 0 < x < 1, t > 0, (λ > 0) (1.1) is degenerate at x = 0, a portion of the lateral boundary. It is well known that (1.1) can be used to describe some models, such as the Budyko-Sellers climate model ([18]), the Black-Scholes model coming from the option pricing problem [3], and a simplified Crocco-type equation coming from the study on the velocity field of a laminar flow on a flat plate ([4]). In recent years, the null controllability of the control system governed by (1.1) was studied in [1, 4, 5, 6, 8, 9, 16, 21, 24, 25], and it was shown that the null controllability depends on the degenerate exponent. In particular, for the control system governed by ∂u ∂t = ∂ ∂x ( xλ ∂u ∂x ) + h(x, t)χω, (x, t) ∈ (0, 1)× (0, T ), it was proved that the control system is null controllable if 0 < λ < 2, while not if λ ≥ 2, where h is the control function, ω is a subinterval of (0, 1), and χω is the characteristic function of ω. Although the system is not null controllable for λ ≥ 2, it was shown in [19] that the system is approximately controllable in L2(0, 1) for any λ > 0. In [20], the author proved that the asymptotic behavior of solutions 2010 Mathematics Subject Classification. 35K65, 35D30, 35B33. Key words and phrases. Asymptotic behavior; boundary degeneracy. c©2021. This work is licensed under a CC BY 4.0 license. Submitted March 19, 2021. Published August 10, 2021. 1 2 X. JING, Y. NIE, C. WANG EJDE-2021/67 to (1.1) depends on the degenerate exponent. Precisely, the following problem was studied ∂u ∂t = ∂ ∂x ( xλ ∂u ∂x ) + up, 0 < x < 1, t > 0, (1.2)( xλ ∂u ∂x ) (0, t) = 0, u(1, t) = 0, t > 0, (1.3) u(x, 0) = u0(x), 0 < x < 1, (1.4) where λ > 0, p > 1, and u0 is a nonnegative function. For problem (1.2)–(1.4), it was proved that there exist both nontrivial global and blowing-up solutions if the degeneracy is not strong that λ < 2, while the 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 [20] for the following problem in an unbounded domain ∂u ∂t = ∂ ∂x ( xλ ∂u ∂x ) + up, x > 0, t > 0, (1.5)( xλ ∂u ∂x ) (0, t) = 0, t > 0, (1.6) u(x, 0) = u0(x), x > 0. (1.7) It was shown that the critical Fujita exponent is pc = { 3− λ, 0 < λ < 2, +∞, λ ≥ 2. That is to say, in the case 0 < λ < 2, any nontrivial solution to problem (1.5)–(1.7) must blow up in a finite time if 1 < p ≤ 3−λ, while there are both nontrivial global and blowing-up solutions to problem (1.5)–(1.7) if p > 3− λ. As to the case λ ≥ 2, any nontrivial solution to (1.5)–(1.7) must blow up in a finite time for p > 1. In 1966, Fujita [11] proved that for the Cauchy problem of the semilinear equation ∂u ∂t = ∆u+ up, x ∈ Rn, t > 0, any nontrivial solution must blow up in a finite time if 1 < p < 1 + n/2, whereas there exist both nontrivial global and blowing-up solutions when p > 1 + n/2. For this problem, pc = 1 + n/2 is called the critical Fujita exponent, and the critical case p = pc was proved to belong to the blowing-up case in [12, 13]. Fujita revealed an important topic of nonlinear partial differential equations. And there have been a great number of extensions of Fujita’s results in several directions since then, including similar results for numerous of quasilinear parabolic equations and systems in various of geometries with nonlinear sources or nonhomogeneous boundary conditions, see the survey papers [7, 14] and also the recent papers [2, 15, 17, 22, 23, 26, 27, 28, 29]. In particular, Escobedo and Herrero in [10] investigated the Cauchy problem of the coupled semilinear parabolic system ∂u ∂t = ∆u+ vp, ∂v ∂t = ∆v + uq, x ∈ Rn, t > 0, (p, q > 1), and proved that the critical Fujita curve is (pq)c = 1 + 2 n max{p+ 1, q + 1}. EJDE-2021/67 ASYMPTOTIC BEHAVIOR OF SOLUTIONS 3 That is to say, any nontrivial solution must blow up in a finite time if pq ≤ 1 + 2/nmax{p + 1, q + 1}, whereas there exist both nontrivial global and blowing-up solutions when pq > 1 + 2/nmax{p+ 1, q + 1}. A natural question of [20] is that how about the parabolic systems with the boundary degeneracy, which is solved in this paper. More precisely, in this paper we study the asymptotic behavior of solutions to the following two coupled parabolic systems with the boundary degeneracy ∂u ∂t = ∂ ∂x ( xλ ∂u ∂x ) + vp, 0 < x < 1, t > 0, (1.8) ∂v ∂t = ∂ ∂x ( xλ ∂v ∂x ) + uq, 0 < x < 1, t > 0, (1.9)( xλ ∂u ∂x ) (0, t) = ( xλ ∂v ∂x ) (0, t) = 0, t > 0, (1.10) u(1, t) = v(1, t) = 0, t > 0, (1.11) u(x, 0) = u0(x), v(x, 0) = v0(x), 0 < x < 1, (1.12) and ∂u ∂t = ∂ ∂x ( xλ ∂u ∂x ) + vp, x > 0, t > 0, (1.13) ∂v ∂t = ∂ ∂x ( xλ ∂v ∂x ) + uq, x > 0, t > 0, (1.14)( xλ ∂u ∂x ) (0, t) = ( xλ ∂v ∂x ) (0, t) = 0, t > 0, (1.15) u(x, 0) = u0(x), v(x, 0) = v0(x), x > 0, (1.16) where p, q > 1 and λ > 0. For problem (1.8)–(1.12) in a bounded domain, it is shown that λ = 2 is a threshold in the sense that there exist both nontrivial global and blowing-up solutions if λ < 2, while the nontrivial solution must blow up in a finite time if λ ≥ 2. As for problem (1.13)–(1.16) in an unbounded domain, it is shown that λ = 2 is also a threshold such that the critical Fujita curve is (pq)c = { 1 + (2− λ) max{p+ 1, q + 1}, 0 < λ < 2, +∞, λ ≥ 2. Furthermore, the critical curve pq = (pq)c belongs to the blowing-up case if 0 < λ < 2. That is to say, if the degeneracy is not strong that 0 < λ < 2, any nontrivial solution to (1.13)–(1.16) blows up in a finite time when pq ≤ (pq)c, whereas there exist both nontrivial global solution and blowing-up solution when pq > (pq)c. If the degeneracy is so strong that λ ≥ 2, any nontrivial solution to (1.13)–(1.16) must blow up. The methods for the system in this paper are similar to the ones in [20] for the single equation. For the blowing-up of solutions to (1.8)–(1.12) in a bounded domain and problem (1.13)–(1.16) in an unbounded domain, we apply the methods of weighted energy estimates to determine the interaction of the degenerate diffusions and the reactions, and the key is to choose appropriate weights. To prove the global existence of nontrivial solutions, we construct suitable self-similar supersolutions. Since the problems in this paper are on coupled parabolic systems with boundary degeneracy, some complicated estimates are needed. In particular, for the critical case pq = (pq)c when 0 < λ < 2, it is shown that it belongs to the blowing-up case by a series of elaborate energy estimates. 4 X. JING, Y. NIE, C. WANG EJDE-2021/67 This article is organized as follows. Some preliminaries and main results are stated in Section 2. Problem (1.8)–(1.12) in a bounded domain and problem (1.13)– (1.16) in an unbounded domain are studied in Section 3 and Section 4, respectively. 2. Preliminaries and main results The subsolutions, supersolutions, as well as solutions to problems (1.8)–(1.12) and (1.13)–(1.16) are defined as follows. Definition 2.1. Let 0 < T ≤ +∞. A pair of nonnegative functions (u, v) is called a subsolution (supersolution, solution) to problem (1.8)–(1.12) in (0, T ), if (i) For any 0 < T̃ < T , u, v ∈ L∞((0, 1)×(0, T̃ )), and ∂u ∂t , ∂v ∂t , x λ/2 ∂u ∂x , x λ/2 ∂v ∂x ∈ L2((0, 1)× (0, T̃ )). (ii) For any 0 < T̃ < T and any nonnegative functions ϕ, φ ∈ C1([0, 1]× [0, T̃ ]) vanishing at x = 1,∫ T̃ 0 ∫ 1 0 (∂u ∂t (x, t)ϕ(x, t) + xλ ∂u ∂x (x, t) ∂ϕ ∂x (x, t) ) dx dt ≤ (≥,=) ∫ T̃ 0 ∫ 1 0 vp(x, t)ϕ(x, t) dxdt, and ∫ T̃ 0 ∫ 1 0 (∂v ∂t (x, t)φ(x, t) + xλ ∂v ∂x (x, t) ∂φ ∂x (x, t) ) dxdt ≤ (≥,=) ∫ T̃ 0 ∫ 1 0 uq(x, t)φ(x, t) dxdt. (iii) u(1, ·), v(1, ·) ≤ (≥,=)0 in (0, T ) and u(·, 0) ≤ (≥,=)u0(·), v(·, 0) ≤ (≥,= )v0(·) in (0, 1) in the sense of trace. Definition 2.2. Let 0 < T ≤ +∞. A pair of nonnegative functions (u, v) is called a subsolution (supersolution, solution) to problem (1.13)–(1.16) in (0, T ), if (i) For any 0 < T̃ < T , u, v ∈ L∞((0,+∞) × (0, T̃ )), and ∂u ∂t , ∂v ∂t , xλ/2 ∂u∂x , xλ/2 ∂v∂x in L2((0, R)× (0, T̃ )) for each R > 0. (ii) For any 0 < T̃ < T and any nonnegative ϕ, φ ∈ C1([0,+∞) × [0, T̃ ]) vanishing when x is large,∫ T̃ 0 ∫ +∞ 0 (∂u ∂t (x, t)ϕ(x, t) + xλ ∂u ∂x (x, t) ∂ϕ ∂x (x, t) ) dxdt ≤ (≥,=) ∫ T̃ 0 ∫ +∞ 0 vp(x, t)ϕ(x, t) dx dt, and ∫ T̃ 0 ∫ +∞ 0 (∂v ∂t (x, t)φ(x, t) + xλ ∂v ∂x (x, t) ∂φ ∂x (x, t) ) dxdt ≤ (≥,=) ∫ T̃ 0 ∫ +∞ 0 uq(x, t)φ(x, t) dxdt. (iii) u(·, 0) ≤ (≥,=)u0(·), v(·, 0) ≤ (≥,=)v0(·) in (0,+∞) in the sense of trace. EJDE-2021/67 ASYMPTOTIC BEHAVIOR OF SOLUTIONS 5 If (u, v) is a solution to (1.8)–(1.12) (or to (1.13)–(1.16)) in (0,+∞), it is said that (u, v) is a global solution in time. Otherwise, there exists T > 0 such that (u, v) is a solution in (0, T ) and satisfies ‖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 it is said that (u, v) blows up in a finite time. Similarly to [20], one can establish the well-posedness and the comparison prin- ciples for problems (1.8)–(1.12) and (1.13)–(1.16). Proposition 2.3. (i) For any 0 ≤ u0, v0 ∈ L∞(0, 1) with xλ/2u′0, x λ/2v′0 ∈ L2(0, 1), problem (1.8)–(1.12) admits at least one solution locally in time. (ii) Assume that (û, v̂) and (ǔ, v̌) are a supersolution and a subsolution to problem (1.8)–(1.12) 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,+∞) with xλ/2u′0, x λ/2v′0 ∈ L2((0, R)) for each R > 0, problem (1.13)–(1.16) admits at least one solution locally in time. (ii) Assume that (û, v̂) and (ǔ, v̌) are a supersolution and a subsolution to problem (1.13)–(1.16) in (0, T ), respectively. Then (ǔ, v̌) ≤ (û, v̂) in (0,+∞)× (0, T ). The main results of the paper are as follows. Theorem 2.5. Assume that 0 < λ < 2, p, q > 1 and 0 ≤ u0, v0 ∈ L∞(0, 1) with xλ/2u′0, x λ/2v′0 ∈ L2(0, 1). Then the solution to (1.8)–(1.12) exists globally in time if (u0, v0) is suitably small, while blows up in a finite time if (u0, v0) is suitably large. Theorem 2.6. Assume that λ ≥ 2 and p, q > 1. Then for any nontrivial 0 ≤ u0, v0 ∈ L∞(0, 1) with xλu′0, x λv′0 ∈ L2(0, 1), the solution to (1.8)–(1.12) must blow up in a finite time. Theorem 2.7. Assume that 0 < λ < 2, 0 ≤ u0, v0 ∈ L∞(0,+∞) with xλ/2u′0, xλ/2v′0 ∈ L2((0, R)) for each R > 0, and (u0, v0) is nontrivial. (i) If p, q > 1 and pq < 1 + (2 − λ) max{p + 1, q + 1}, then the solution to (1.13)–(1.16) must blow up in a finite time. (ii) If p, q > 1 and pq = 1 + (2 − λ) max{p + 1, q + 1}, then the solution to t (1.13)–(1.16) must blow up in a finite time. (iii) If p, q > 1 and pq > 1 + (2 − λ) max{p + 1, q + 1}, then the solution to (1.13)–(1.16) exists globally in time if (u0, v0) is suitably small, while blows up in a finite time if (u0, v0) is suitably large. Theorem 2.8. Assume that λ ≥ 2. Then for a nontrivial 0 ≤ u0, v0 ∈ L∞(0,+∞) with xλ/2u′0, x λ/2v′0 ∈ L2((0, R)) for each R > 0, the solution to (1.13)–(1.16) must blow up in a finite time. 3. Problem in a bounded domain In this section, we prove Theorems 2.5 and 2.6 for problem (1.8)–(1.12) in a bounded domain. 6 X. JING, Y. NIE, C. WANG EJDE-2021/67 Proof of Theorem 2.5. First consider the global case. To show the existence of a global solution to (1.8)–(1.12), we study self-similar supersolutions to (1.8) and (1.9) of the form û(x, t) = (t+ τ)−(p+1)/(pq−1)U((t+ τ)−1/(2−λ)x), x ∈ [0, 1], t ≥ 0, (3.1) v̂(x, t) = (t+ τ)−(q+1)/(pq−1)V ((t+ τ)−1/(2−λ)x), x ∈ [0, 1], t ≥ 0, (3.2) where τ > 0 will be determined later. If U, V ∈ C2((0, τ−1/(2−λ))) solve (rλU ′(r))′ + 1 2− λ rU ′(r) + p+ 1 pq − 1 U(r) + V p(r) ≤ 0, 0 < r < τ−1/(2−λ), (3.3) (rλV ′(r))′ + 1 2− λ rV ′(r) + q + 1 pq − 1 V (r) + Uq(r) ≤ 0, 0 < r < τ−1/(2−λ), (3.4) then (û, v̂) given by (3.1) and (3.2) is a supersolution to (1.8) and (1.9). Set U(r) = V (r) = 1 2− λ (1 τ − r 2−λ ) , 0 ≤ r ≤ τ−1/(2−λ). For 0 < r < τ1/(2−λ), from direct calculations we have (rλU ′(r))′ + 1 2− λ rU ′(r) + p+ 1 pq − 1 U(r) + V p(r) = −1− 1 2− λ r2−λ + p+ 1 (pq − 1)(2− λ) ( 1 τ − r2−λ) + 1 (2− λ)p (1 τ − r2−λ )p , (rλV ′(r))′ + 1 2− λ rV ′(r) + q + 1 pq − 1 V (r) + Uq(r) = −1− 1 2− λ r2−λ + q + 1 (pq − 1)(2− λ) ( 1 τ − r2−λ) + 1 (2− λ)q (1 τ − r2−λ )q . Hence (û, v̂) is a supersolution to (1.8) and (1.9) for each τ ≥ τ0, where τ0 = max { p+ 1 (pq − 1)(2− λ) + 1 (2− λ)p + 1, q + 1 (pq − 1)(2− λ) + 1 (2− λ)q + 1 } . It is noted that ( xλ ∂û ∂x ) (0, t) = 0, û(1, t) ≥ 0, t > 0,( xλ ∂v̂ ∂x ) (0, t) = 0, v̂(1, t) ≥ 0, t > 0. Therefore, (û, v̂) is a supersolution to (1.8)–(1.12) if u0(x) ≤ û(x, 0), v0(x) ≤ v̂(x, 0), 0 < x < 1 (τ ≥ τ0). (3.5) Thanks to Proposition 2.3 (ii), one gets that the solution to (1.8)–(1.12) exists globally in time if (u0, v0) satisfies (3.5). Now we turn to the blowing-up case. Without loss of generality, it is assumed that p ≥ q > 1. Set ζ(x) = { 2, 0 < x ≤ 1 2 , 1 + cos(2x− 1)π, 1 2 < x ≤ 1. It is clear that ζ ∈ C1 ( [0, 1] ) is piecewise smooth and satisfies ζ ′(0) = 0, ζ(1) = 0, and for 1/2 < x ≤ 1,( xλζ ′(x) )′ = −2λπxλ−1 sin(2x− 1)π − 4π2xλ cos(2x− 1)π ≥ −4π2ζ(x). EJDE-2021/67 ASYMPTOTIC BEHAVIOR OF SOLUTIONS 7 Assume that (u, v) is a solution to (1.8)–(1.12) in (0,+∞). It follows from Definition 2.1 that (u, v) satisfies d dt ∫ 1 0 ( u(x, t) + v(x, t) ) ζ(x) dx = ∫ 1 0 ( u(x, t) + v(x, t) ) (xλζ ′(x))′ dx+ ∫ 1 0 vp(x, t)ζ(x) dx+ ∫ 1 0 uq(x, t)ζ(x) dx ≥ −4π2 ∫ 1 0 ( u(x, t) + v(x, t) ) ζ(x) dx+ ∫ 1 0 vp(x, t)ζ(x) dx + ∫ 1 0 uq(x, t)ζ(x) dx, t > 0. (3.6) Denote ω(t) = ∫ 1 0 (u(x, t) + v(x, t))ζ(x) dx, t ≥ 0. It follows from (3.6) and Hölder’s inequality that d dt ω(t) ≥ −4π2ω(t) + ∫ 1 0 uq(x, t)ζ(x) dx+ ∫ 1 0 vp(x, t)ζ(x) dx ≥ −4π2ω(t) + (∫ 1 0 ζ(x) dx )1−q(∫ 1 0 u(x, t)ζ(x) dx )q + (∫ 1 0 ζ(x) dx )1−p(∫ 1 0 v(x, t)ζ(x) dx )p ≥ −4π2ω(t) + 21−p {(∫ 1 0 u(x, t)ζ(x) dx )q + (∫ 1 0 v(x, t)ζ(x) dx )p} . (3.7) If ω(t) ≥ 2, then∫ 1 0 u(x, t)ζ(x) dx ≥ 1 or ∫ 1 0 v(x, t)ζ(x) dx ≥ 1. It follows from [17, Lemma 3.6] that(∫ 1 0 u(x, t)ζ(x) dx )q + (∫ 1 0 v(x, t)ζ(x) dx )p ≥ 2−pωq(t), if ω(t) ≥ 2. (3.8) If (u0, v0) is sufficiently large such that ω(0) ≥ 2, ωq−1(0) ≥ 22+2pπ2, (3.9) one gets from (3.7) and (3.8) that d dt ω(t) ≥ 2−2pωq−1(t), t > 0. Since p ≥ q > 1, there exists T > 0 such that ω(t) = ∫ 1 0 (u(x, t) + v(x, t))ζ(x) dx→ +∞, as t→ T−, which leads to ‖u(·, t)‖L∞(0,1) + ‖v(·, t)‖L∞(0,1) → +∞, as t→ T−. 8 X. JING, Y. NIE, C. WANG EJDE-2021/67 That is to see, if (u0, v0) satisfies (3.9), then (u, v) blows up in a finite time. The proof of Theorem 2.5 is complete. � Proof of Theorem 2.6. Without loss of generality, it is assumed that p ≥ q > 1. For 0 < δ < 1, set ζδ(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 clear that ζδ(x) ∈ C1 ( (0, 1] ) is piecewise smooth, and∫ 1 0 ζδ(x) dx ≤M1 (3.10) ( xλζ ′δ(x) )′ = { −(λ− 1)(λ− 1− δ)2λ−1−δxλ−2−δ, 0 < x < 1/2, 0, 1/2 ≤ x ≤ 1, where M1 > 0 is a constant depending only on λ but independent of δ. Therefore, there exists a constant M2 > 0 depending only on λ but independent of δ such that( xλζ ′δ(x) )′ ≥ −M2δζδ(x), 0 < x < 1. (3.11) Assume that (u, v) is a solution to (1.8)–(1.12) in (0,+∞). By the similar argument in [20, Theorem 2.2], one can get that (u, v) satisfies d dt ∫ 1 0 ( u(x, t) + v(x, t) ) ζδ(x) dx = ∫ 1 0 ( u(x, t) + v(x, t) ) (xλζ ′δ(x))′ dx+ ∫ 1 0 vp(x, t)ζδ(x) dx + ∫ 1 0 uq(x, t)ζδ(x) dx ≥ −M2δ ∫ 1 0 ( u(x, t) + v(x, t) ) ζδ(x) dx+ ∫ 1 0 vp(x, t)ζδ(x) dx + ∫ 1 0 uq(x, t)ζδ(x) dx, t > 0. (3.12) We denote ω(t) = ∫ 1 0 (u(x, t) + v(x, t))ζδ(x) dx, t > 0. It follows from (3.12) and Hölder’s inequality that d dt ω(t) ≥ −M2δω(t) + ∫ 1 0 uq(x, t)ζδ(x) dx+ ∫ 1 0 vp(x, t)ζδ(x) dx ≥ −M2δω(t) + (∫ 1 0 ζδ(x) dx )1−q(∫ 1 0 u(x, t)ζδ(x) dx )q + (∫ 1 0 ζδ(x) dx )1−p(∫ 1 0 v(x, t)ζδ(x) dx )p ≥ −M2δω(t) +M1−q 1 (∫ 1 0 u(x, t)ζδ(x) dx )q +M1−p 1 (∫ 1 0 v(x, t)ζδ(x) dx )p EJDE-2021/67 ASYMPTOTIC BEHAVIOR OF SOLUTIONS 9 ≥ −M2δω(t) +M1−p 1 {(∫ 1 0 u(x, t)ζδ(x) dx )q + (∫ 1 0 v(x, t)ζδ(x) dx )p} . (3.13) If ω(t) < 2, then ∫ 1 0 u(x, t)ζδ(x) dx < 2, ∫ 1 0 v(x, t)ζδ(x) dx < 2. It follows from [17, Lemma 3.6] that(∫ 1 0 u(x, t)ζδ(x) dx )q + (∫ 1 0 v(x, t)ζδ(x) dx )p ≥ 2−2pωp(t), if ω(t) < 2. (3.14) Similarly to the proof of (3.8), one can get(∫ 1 0 u(x, t)ζδ(x) dx )q + (∫ 1 0 v(x, t)ζδ(x) dx )p ≥ 2−pωq(t), if ω(t) ≥ 2. (3.15) Owing to inf0<δ<1{ω(0)} > 0, there exists sufficiently small 0 < δ < 1 such that M2δ ≤ 2−2p−1M1−p 1 ωp−1(0), M2δ ≤ 2−p−1M1−p 1 ωq−1(0). (3.16) If ω(0) < 2, we claim that there exists T̃ > 0 such that ω(T̃ ) ≥ 2. Otherwise, ω(t) < 2, t ∈ [0,+∞). (3.17) It follows from (3.13), (3.14), (3.16) and (3.17) that d dt ω(t) ≥ 2−2p−1M1−p 1 ωp(t), t > 0. Since p ≥ q > 1, there exists T̂ > 0 such that ω(t) = ∫ 1 0 (u(x, t) + v(x, t))ζδ(x) dx→ +∞, as t→ T̂−, which contradicts (3.17). Therefore, one can assume that ω(0) ≥ 2. Then one gets from (3.13), (3.15) and (3.16) that d dt ω(t) ≥ 2−p−1M1−p 1 ωq(t), t > 0. Since p ≥ q > 1, there exists T > 0 such that ω(t) = ∫ 1 0 (u(x, t) + v(x, t))ζδ(x) dx→ +∞, as t→ 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. The proof of Theorem 2.6 is complete. � 10 X. JING, Y. NIE, C. WANG EJDE-2021/67 4. Problem in an unbounded domain In this section, we prove Theorems 2.7 and 2.8 for problem (1.13)–(1.16) in an unbounded domain. It is clear that Theorem 2.8 follows from Theorem 2.6, Propositions 2.4 and 2.3. We only prove Theorem 2.7. Proof of Theorem 2.7 (i) and (iii). Without loss of generality, it is assumed that p ≥ q > 1. For R > 0, set ζR(x) =  1, 0 < x ≤ R, 1 2 ( 1 + cos (x−R)π R ) , R < x < 2R, 0, x ≥ 2R. (4.1) It is clear that ζR ∈ C1 ( [0,+∞) ) is piecewise smooth, and (xλζ ′R(x))′ =− λπ 2R xλ−1 sin (x−R)π R − π2 2R2 xλ cos (x−R)π R ≥− 2λπ2Rλ−2ζR(x), R < x < 2R. Assume that (u, v) is a solution to (1.13)–(1.16) in (0,+∞). It follows from Defi- nition 2.2 that (u, v) satisfies d dt ∫ +∞ 0 ( u(x, t) +Rσv(x, t) ) ζR(x) dx = ∫ +∞ 0 ( u(x, t) +Rσv(x, t) ) (xλζ ′R(x))′ dx+Rσ ∫ +∞ 0 uq(x, t)ζR(x) dx + ∫ +∞ 0 vp(x, t)ζR(x) dx ≥ −2λπ2Rλ−2 ∫ +∞ 0 ( u(x, t) +Rσv(x, t) ) ζR(x) dx+Rσ ∫ +∞ 0 uq(x, t)ζR(x) dx + ∫ +∞ 0 vp(x, t)ζR(x) dx, t > 0, (4.2) where σ = q − p p+ 1 . We denote ωR(t) = ∫ +∞ 0 (u(x, t) +Rσv(x, t))ζR(x) dx, t > 0. It is noted that 1− q + σ = 1− p− σp = −pq − 1 p+ 1 . EJDE-2021/67 ASYMPTOTIC BEHAVIOR OF SOLUTIONS 11 It follows from (4.2) and Hölder’s inequality that d dt ωR(t) ≥− 2λπ2Rλ−2ωR(t) +Rσ ∫ +∞ 0 uq(x, t)ζR(x) dx+ ∫ +∞ 0 vp(x, t)ζR(x) dx ≥− 2λπ2Rλ−2ωR(t) +Rσ (∫ +∞ 0 ζR(x) dx )1−q(∫ +∞ 0 u(x, t)ζR(x) dx )q + (∫ +∞ 0 ζR(x) dx )1−p(∫ +∞ 0 v(x, t)ζR(x) dx )p ≥− 2λπ2Rλ−2ωR(t) + 21−qR1−q+σ (∫ +∞ 0 u(x, t)ζR(x) dx )q + 21−pR1−p−σp (∫ +∞ 0 Rσv(x, t)ζR(x) dx )p ≥− 2λπ2Rλ−2ωR(t) + 21−pR−(pq−1)/(p+1) ((∫ +∞ 0 u(x, t)ζR(x) dx )q + (∫ +∞ 0 Rσv(x, t)ζR(x) dx )p) , t > 0. (4.3) Similarly to the proof of (3.14) and (3.15), one obtains(∫ +∞ 0 u(x, t)ζR(x) dx )q + (∫ +∞ 0 Rσv(x, t)ζR(x) dx )p ≥ 2−2pωpR(t), if ωR(t) < 2, (4.4)(∫ +∞ 0 u(x, t)ζR(x) dx )q + (∫ +∞ 0 Rσv(x, t)ζR(x) dx )p ≥ 2−pωqR(t), if ωR(t) ≥ 2. (4.5) Since pq < (pq)c, it holds that −pq − 1 p+ 1 > λ− 2. Thanks to infR>0 ωR(0) > 0, there exists R > 0 sufficiently large such that 2λπ2Rλ−2 ≤ 2−3pR−(pq−1)/(p+1)ωp−1R (0), (4.6) 2λπ2Rλ−2 ≤ 2−2pR−(pq−1)/(p+1)ωq−1R (0). (4.7) If ωR(0) < 2, we claim that there exists T̃ > 0 such that ωR(T̃ ) ≥ 2. Otherwise, ωR(t) < 2, t ∈ [0,+∞). (4.8) It follows from (4.3), (4.4), (4.6) and (4.8) that d dt ωR(t) ≥ 2−3pR−(pq−1)/(p+1)ωpR(t), t > 0. Since p ≥ q > 1, there exists T̂ > 0 such that ωR(t) = ∫ +∞ 0 (u(x, t) +Rσv(x, t))ζR(x) dx→ +∞, as t→ T̂−, 12 X. JING, Y. NIE, C. WANG EJDE-2021/67 which contradicts (4.8). Therefore, one can assume that ωR(0) ≥ 2 without loss of generality. Then one gets from (4.3), (4.5) and (4.7) that d dt ωR(t) ≥ 2−2pR−(pq−1)/(p+1)ωqR(t), t > 0. Since p ≥ q > 1, there exists T > 0 such that ωR(t) = ∫ +∞ 0 (u(x, t) +Rσv(x, t))ζR(x) dx→ +∞, as t→ T−, which leads to ‖u(·, t)‖L∞(0,+∞) + ‖v(·, t)‖L∞(0,+∞) → +∞, as t→ T−. That is to say, (u, v) must blow up in a finite time. We turn to case (iii) that pq > (pq)c. Thanks to Theorem 2.5 and Propositions 2.4 and 2.3, the solution to (1.13)–(1.16) blows up in a finite time if (u0, v0) is suitably large. Below we prove that there exists a nontrivial global solution to (1.13)–(1.16) if (u0, v0) is suitably small. Set û(x, t) = (t+ 1)−(p+1)/(pq−1)U((t+ 1)−1/(2−λ)x), x ≥ 0, t ≥ 0, (4.9) v̂(x, t) = (t+ 1)−(q+1)/(pq−1)V ((t+ 1)−1/(2−λ)x), x ≥ 0, t ≥ 0. (4.10) If U, V ∈ C2((0,+∞)) satisfy (rλU ′(r))′ + 1 2− λ rU ′(r) + p+ 1 pq − 1 U(r) + V p(r) ≤ 0, r > 0, (4.11) (rλV ′(r))′ + 1 2− λ rV ′(r) + q + 1 pq − 1 V (r) + Uq(r) ≤ 0, r > 0, (4.12) then, (û, v̂) given by (4.9) and (4.10) is a supersolution to (1.13) and (1.14). We take U(r) = V (r) = εe−Ar 2−λ , r ≥ 0, where ε > 0 will be determined and A is a constant such that p+ 1 (2− λ)(pq − 1) < A < 1 (2− λ)2 . (4.13) For r > 0, one gets from direct calculations that (rλU ′(r))′ + 1 2− λ rU ′(r) + p+ 1 pq − 1 U(r) + V p(r) = U(r) ( A(2− λ) ( A(2− λ)− 1 2− λ ) r2−λ + ( p+ 1 pq − 1 −A(2− λ) ) + εp−1e−A(p−1)r2−λ ) ≤ U(r) ( A(2− λ) ( A(2− λ)− 1 2− λ ) r2−λ + ( p+ 1 pq − 1 −A(2− λ) ) + εp−1 ) , (rλV ′(r))′ + 1 2− λ rV ′(r) + q + 1 pq − 1 V (r) + Uq(r) = V (r) ( A(2− λ) ( A(2− λ)− 1 2− λ ) r2−λ + ( q + 1 pq − 1 −A(2− λ) ) + εq−1e−A(q−1)r2−λ ) EJDE-2021/67 ASYMPTOTIC BEHAVIOR OF SOLUTIONS 13 ≤ V (r) ( A(2− λ) ( A(2− λ)− 1 2− λ ) r2−λ + ( q + 1 pq − 1 −A(2− λ) ) + εq−1 ) . It follows from (4.13) and p ≥ q > 1 that A(2− λ) < 1 2− λ , p+ 1 pq − 1 < A(2− λ), q + 1 pq − 1 < A(2− λ). Therefore, (û, v̂) is a supersolution to (1.13) and (1.14) for each 0 < ε ≤ ε0, where ε0 = min {( A(2− λ)− p+ 1 pq − 1 )1/(p−1) , ( A(2− λ)− q + 1 pq − 1 )1/(q−1)} . Note that ( xλ ∂û ∂x ) (0, t) = 0, ( xλ ∂v̂ ∂x ) (0, t) = 0, t > 0. Therefore, (û, v̂) is a supersolution to (1.13)–(1.16) if 0 < ε ≤ ε0 and u0(x) ≤ û(x, 0), v0(x) ≤ v̂(x, 0), x > 0. (4.14) Thanks to Proposition 2.4 (ii), the solution to (1.13)–(1.16) exists globally in time if (u0, v0) satisfies (4.14) and 0 < ε ≤ ε0. The proof of Theorem 2.7 (i) and (iii) is complete. � To prove Theorem 2.7 (ii), we need the following lemma. Lemma 4.1. Let 0 < λ < 2, p ≥ q > 1 and pq = (pq)c. Assume that (u, v) is a solution to (1.13)–(1.16) in (0,+∞). Then for any R > 0, ωR(t) ≤ N, t > 0, (4.15) d dt ωR(t) ≥ −p− 1 p 2(2p+λp−1)/(p−1)Np/(p−1)π2p/(p−1)Rλ−2, t > 0, (4.16) d dt ωR(t) ≥Rλ−2ω1/2 R (t) ( − 2λπ2 (∫ 2R R (u(x, t) +Rσv(x, t))ζR(x) dx )1/2 + 21−2pN−pω (2p−1)/2 R (t) ) , t > 0, (4.17) where ωR(t) = ∫ +∞ 0 (u(x, t) +Rσv(x, t))ζR(x) dx, t > 0, N = 2(2p+λ)/(q−1)π2/(q−1), σ = q − p p+ 1 , and ζR(x) is the function defined by (4.1). Proof. Since p ≥ q > 1, it holds that N ≥ 2. It follows from (4.3) and (4.5) that d dt ωR(t) ≥ −2λπ2Rλ−2ωR(t) + 21−2pR−(pq−1)/(p+1)ωqR(t), if ωR(t) ≥ 2. (4.18) Since pq = (pq)c, one obtains −pq − 1 p+ 1 = λ− 2. Let us prove (4.15) by a contradiction. If (4.15) is not true, there exists t0 > 0 such that ωR(t0) > N ≥ 2, 2λπ2 ≤ 2−2pωq−1R (t0). (4.19) 14 X. JING, Y. NIE, C. WANG EJDE-2021/67 Then, from (4.18) and (4.19) we obtain d dt ωR(t) ≥ 2−2pRλ−2ωqR(t), t > t0. (4.20) Since p ≥ q > 1, there exists T > t0 such that ωR(t) = ∫ +∞ 0 (u(x, t) +Rσv(x, t))ζR(x) dx→ +∞, as t→ T−, which leads to ‖u(·, t)‖L∞(0,+∞) + ‖v(·, t)‖L∞(0,+∞) → +∞, as t→ T−. Hence (u, v) must blow up in a finite time, which is a contradiction. Thus (4.15) is proved. Now we turn to (4.16). It follows from (4.3) that d dt ωR(t) ≥ −2λπ2Rλ−2ωR(t) + 21−pRλ−2 ((∫ +∞ 0 u(x, t)ζR(x) dx )q + (∫ +∞ 0 Rσv(x, t)ζR(x) dx )p) , t > 0. (4.21) Owing to (4.15), it holds that∫ +∞ 0 u(x, t)ζR(x) dx ≤ N, ∫ +∞ 0 Rσv(x, t)ζR(x) dx ≤ N, t > 0. Hence from [17, Lemma 3.6] we obtain(∫ 1 0 u(x, t)ζδ(x) dx )q + (∫ 1 0 Rσv(x, t)ζδ(x) dx )p ≥ (2N)−pωpR(t), t > 0. (4.22) It follows from (4.21), (4.22) and Hölder’s inequality that d dt ωR(t) ≥− 2λπ2Rλ−2ωR(t) + 21−2pN−pRλ−2ωpR(t) ≥21−2pN−pRλ−2 ( − 22p+λ−1Npπ2ωR(t) + ωpR(t) ) ≥21−2pN−pRλ−2 ( − p− 1 p 2p(2p+λ−1)/(p−1)Np2/(p−1)π2p/(p−1) − 1 p ωpR(t) + ωpR(t) ) ≥− p− 1 p 2(2p+λp−1)/(p−1)Np/(p−1)π2p/(p−1)Rλ−2, t > 0. Thus (4.16) is proved. Next we prove (4.17). It follows from (4.2) and Hölder’s inequality that d dt ωR(t) = ∫ 2R R ( u(x, t) +Rσv(x, t) ) (xλζ ′R(x))′ dx +Rσ ∫ +∞ 0 uq(x, t)ζR(x) dx+ ∫ +∞ 0 vp(x, t)ζR(x) dx ≥− 2λπ2Rλ−2 ∫ 2R R ( u(x, t) +Rσv(x, t) ) ζR(x) dx +Rσ ∫ +∞ 0 uq(x, t)ζR(x) dx+ ∫ +∞ 0 vp(x, t)ζR(x) dx EJDE-2021/67 ASYMPTOTIC BEHAVIOR OF SOLUTIONS 15 ≥− 2λπ2Rλ−2 ∫ 2R R ( u(x, t) +Rσv(x, t) ) ζR(x) dx +Rσ (∫ +∞ 0 ζR(x) dx )1−q(∫ +∞ 0 u(x, t)ζR(x) dx )q +R−σp (∫ +∞ 0 ζR(x) dx )1−p(∫ +∞ 0 Rσv(x, t)ζR(x) dx )p ≥− 2λπ2Rλ−2 ∫ 2R R ( u(x, t) +Rσv(x, t) ) ζR(x) dx + 21−qR1−q+σ (∫ +∞ 0 u(x, t)ζR(x) dx )q + 21−pR1−p−σp (∫ +∞ 0 Rσv(x, t)ζR(x) dx )p ≥− 2λπ2Rλ−2 ∫ 2R R (u(x, t) +Rσv(x, t))ζR(x) dx + 21−2pN−pRλ−2ωpR(t), t > 0, which leads to (4.17). � Proof of Theorem 2.7 (ii). Assuming that (u, v) is a solution to (1.13)–(1.16) in (0,+∞), we set Γ = sup R>0,t>0 ωR(t) = sup t>0 ∫ +∞ 0 (u(x, t) +Rσv(x, t)) dx. (4.23) From (4.15) and the nontriviality of (u0, v0), it holds that 0 < Γ < +∞. For ε0 > 0, there exists t1 ≥ 0 and R0 > 0 such that ωR0(t1) ≥ Γ− ε0, (4.24) where ε0 will be determined below. For any t ≥ t1, it follows from (4.16) with R = R0 and (4.24) that ωR0(t) ≥ωR0(t1)− p− 1 p 2(2p+λp−1)/(p−1)Np/(p−1)π2p/(p−1)Rλ−20 (t− t1) ≥Γ− ε0 − p− 1 p 2(2p+λp−1)/(p−1)Np/(p−1)π2p/(p−1)Rλ−20 (t− t1), which, together with (4.23), leads to∫ 4R0 2R0 (u(x, t) +Rσv(x, t))ζ2R0(x) ≤ ∫ +∞ 0 (u(x, t) +Rσv(x, t)) dx− ωR0(t) ≤ ε0 + p− 1 p 2(2p+λp−1)/(p−1)Np/(p−1)π2p/(p−1)Rλ−20 (t− t1). (4.25) Taking R = 2R0 in (4.17), we obtain d dt ω2R0 (t) 16 X. JING, Y. NIE, C. WANG EJDE-2021/67 ≥ (2R0)λ−2ω 1/2 2R0 (t) ( − 2λπ2 (∫ 4R0 2R0 (u(x, t) +Rσv(x, t))ζ2R0(x) dx )1/2 + 21−2pN−pω (2p−1)/2 2R0 (t) ) , t > t1. Fix ε0 ∈ (0,Γ) and M0 > 0 such that 2λπ2(ε0 +M0)1/2 ≤ 2−2pN−p(Γ− ε0)(2p−1)/2. Owing to (4.23)–(4.25), it holds that d dt ω2R0 (t) ≥ 2λ−2−2pN−pRλ−20 (Γ− ε0)p, t1 < t < t2, (4.26) where t2 = t1 + p p− 1 2(2p+λp−1)/(1−p)Np/(1−p)π2p/(1−p)M0R 2−λ 0 . 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Xinxin Jing School of Mathematics, Jilin University, Changchun 130012, China Email address: 1776043712@qq.com Yuanyuan Nie School of Mathematics, Jilin University, Changchun 130012, China Email address: nieyy@jlu.edu.cn Chunpeng Wang (corresponding author) School of Mathematics, Jilin University, Changchun 130012, China Email address: wangcp@jlu.edu.cn 1. Introduction 2. Preliminaries and main results 3. Problem in a bounded domain 4. Problem in an unbounded domain Acknowledgments References