Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 96, pp. 1–19. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO POROUS MEDIUM EQUATIONS WITH BOUNDARY DEGENERACY XUTONG ZHAO, MINGJUN ZHOU, XINXIN JING Abstract. This article concerns the asymptotic behavior of solutions to a class of one-dimensional porous medium equations with boundary degeneracy on bounded and unbounded intervals. It is proved that the degree of degener- acy, the exponents of the nonlinear diffusion, and the nonlinear source affect the asymptotic behavior of solutions. It is shown that on a bounded inter- val, the problem admits both nontrivial global and blowing-up solutions if the degeneracy is not strong; while any nontrivial solution must blow up if the degeneracy is strong enough. For the problem on an unbounded interval, the blowing-up theorems of Fujita type are established. The critical Fujita expo- nent is finite if the degeneracy is not strong, while infinite if the degeneracy is strong enough. Furthermore, the critical case is proved to be the blowing-up case if it is finite. 1. Introduction In this article, we consider the asymptotic behavior of solutions to the following two problems ∂u ∂t − ∂ ∂x ( xλ ∂um ∂x ) = up, (x, t) ∈ (0, 1)× (0, T ), (1.1)( xλ ∂um ∂x ) (0, t) = u(1, t) = 0, t ∈ (0, T ), (1.2) u(x, 0) = u0(x), x ∈ (0, 1), (1.3) and ∂u ∂t − ∂ ∂x ( xλ ∂um ∂x ) = up, (x, t) ∈ (0,+∞)× (0, T ), (1.4)( xλ ∂um ∂x ) (0, t) = 0, t ∈ (0, T ), (1.5) u(x, 0) = u0(x), x ∈ (0,+∞), (1.6) where p > m > 1, λ > 0 and 0 < T ≤ +∞. If λ = 0, both (1.1) and (1.4) are porous medium equations, which have been studied extensively (see, e.g., Chapter 1 in [27]). If m = 1, both (1.1) and (1.4) are semilinear equations which are degenerate at a portion of the lateral boundary x = 0. Semilinear equations with 2010 Mathematics Subject Classification. 35K59, 35B33, 35K65. Key words and phrases. Critical Fujita exponent; porous medium equation; boundary degeneracy. ©2021. This work is licensed under a CC BY 4.0 license. Submitted May 29, 2021. Published December 3, 2021. 1 2 X. ZHAO, M. ZHOU, X. JING EJDE-2021/96 such degeneracy appear in many physical or economic models, such as the Budyko- Sellers climate model [20], a simplified Crocco-type equation coming from the study on the velocity field of a laminar flow on a flat plate [5, 7], and the Black-Scholes model coming from the option pricing problem [4]. For m > 1 and λ > 0, both (1.1) and (1.4) admit two kinds of degeneracy. That is to say, (1.1) and (1.4) are degenerate not only at points where u = 0 but also at a portion of the lateral boundary x = 0. In recent years, semilinear equations with boundary degeneracy have attracted many attentions, and it was shown that the boundary degeneracy causes many essential differences. For instance, the null controllability of the system governed by ∂u ∂t − ∂ ∂x ( xλ ∂u ∂x ) = h(x, t)χω, (x, t) ∈ (0, 1)× (0, T ), and related problems were studied in [2, 5, 6, 8, 10, 11, 19, 22, 26, 28], where h is the control function, ω is a subinterval of (0, 1), and χω is the characteristic function of ω. It was proved that λ = 2 is a threshold in the sense that the system is null controllable if 0 < λ < 2, while not if λ ≥ 2. For another instance, the quenching phenomenon of solutions to 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 [29], where a > 0 and f ∈ C2([0, c)) with 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 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 λ ≥ 2, the solution must quench in a finite time for each a > 0. In [23], it was shown that the boundary degeneracy also affects the asymptotic behavior of solutions to the semilinear problems (1.1)–(1.3) and (1.4)–(1.6) in the case m = 1. More precisely, for problem (1.1)–(1.3) in the case 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. As to problem (1.4)–(1.6) in the case m = 1, the critical Fujita exponent is pc = { 3− λ, if 0 < λ < 2, +∞, if λ ≥ 2. That is to say, in the case 0 < λ < 2, any nontrivial solution must blow up in a finite time if 1 < p < 3− λ, while there are both nontrivial global and blowing-up solutions if p > 3 − λ. Whereas in the case λ ≥ 2, any nontrivial solution must blow up in a finite time for p > 1. EJDE-2021/96 POROUS MEDIUM EQUATIONS 3 The blowing-up phenomenon of solutions to nonlinear diffusion equations was first introduced in 1966 by Fujita, who proved in [13] that pc = 1 + 2/n, which is called the critical Fujita exponent, is critical for the Cauchy problem of ∂u ∂t −∆u = up, (x, t) ∈ Rn × (0,+∞) in the sense that any nontrivial solution blows up in a finite time if 1 < p < pc, whereas there exist both nontrivial global and blowing-up solutions if p > pc. The critical case p = pc was proved to belong to the blowing-up case in [14, 16]. 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 [9, 17] and the references therein, and more recent works, e.g. [1, 3, 12, 15, 18, 21, 23, 24, 25]. In this paper, we study the asymptotic behavior of solutions to the quasilinear problems (1.1)–(1.3) and (1.4)–(1.6). As mentioned above, both (1.1) and (1.4) admit two kinds of degeneracy. They are degenerate not only at points where u = 0 but also at a portion of the lateral boundary x = 0. For problem (1.1)– (1.3) in a bounded interval, it is shown that λ = 2 is a threshold in the sense that there exist both nontrivial global and blowing-up solutions to problem (1.1)–(1.3) if λ < 2, while any nontrivial solution to problem (1.1)–(1.3) must blow up in a finite time if λ ≥ 2. For problem (1.4)–(1.6) in an unbounded interval, 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 pc = { m+ 2− λ, if 0 < λ < 2, +∞, if λ ≥ 2. That is to say, in the case 0 < λ < 2, any nontrivial solution to problem (1.4)– (1.6) must blow up in a finite time if m < p < pc = m + 2 − λ, while there are both nontrivial global and blowing-up solutions to problem (1.4)–(1.6) if p > pc = m+2−λ. Furthermore, the critical case p = pc = m+2−λ belongs to the blowing- up case. Whereas in the case λ ≥ 2, any nontrivial solution to problem (1.4)–(1.6) must blow up in a finite time for p > m. The methods used in this paper are mainly inspired by [23]. For the blowing-up of solutions to problem (1.1)–(1.3) in a bounded interval and problem (1.4)–(1.6) in an unbounded interval, we apply the methods of weighted energy estimates instead of constructing blowing-up subsolutions. The key is to choose appropriate weights and to estimate the interaction of the nonlinear degenerate diffusions and the sources. To prove the global existence of nontrivial solutions, we construct suitable self-similar supersolutions. Since (1.1) and (1.4) admit two kinds of degeneracy, some complicated estimates are needed. This article is organized as follows. Comparison principles and well-posedness for problems (1.1)–(1.3) and (1.4)–(1.6) are established in Section 2. The asymptotic behavior of solutions to problem (1.1)–(1.3) in a bounded interval and problem (1.4)–(1.6) in an unbounded interval are studied in Section 3 and Section 4, respec- tively. 4 X. ZHAO, M. ZHOU, X. JING EJDE-2021/96 2. Comparison principles and well-posedness In this section, we establish comparison principles and well-posedness for prob- lems (1.1)–(1.3) and (1.4)–(1.6). Since (1.1) and (1.4) admit two kinds of degeneracy, we consider weak solutions defined as follows. Definition 2.1. Let 0 < T ≤ +∞. A nonnegative function u ∈ L∞((0, 1)× (0, T )) is called a subsolution (supersolution, solution) to problem (1.1)–(1.3) in (0, T ) if for any 0 < τ < T , xλ/2 ∂u m ∂x ∈ L 2((0, 1)× (0, τ)), and∫ 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 up(x, t)ϕ(x, t) dxdt+ ∫ 1 0 u0(x)ϕ(x, 0) dx holds for any 0 ≤ ϕ ∈ C1([0, 1]× [0, τ ]) with ϕ(τ, ·) ∣∣ (0,1) = ϕ(1, ·) ∣∣ (0,τ) = 0. Definition 2.2. Let 0 < T ≤ +∞. A nonnegative function u ∈ L∞((0,+∞) × (0, T )) is called a subsolution (supersolution, solution) to problem (1.4)–(1.6) in (0, T ) if for any 0 < τ < T , xλ/2 ∂u m ∂x ∈ L 2((0,+∞)× (0, τ)), and∫ τ 0 ∫ +∞ 0 ( −u(x, t) ∂ϕ ∂t (x, t) + xλ ∂um ∂x (x, t) ∂ϕ ∂x (x, t) ) dxdt ≤ (≥,=) ∫ τ 0 ∫ +∞ 0 up(x, t)ϕ(x, t) dxdt+ ∫ +∞ 0 u0(x)ϕ(x, 0) dx holds for any 0 ≤ ϕ ∈ C1([0,+∞)× [0, τ ]) vanishing at t = τ and for large x. If u is a solution to problem (1.1)–(1.3) (or problem (1.4)–(1.6)) in (0,+∞), it is said that u is a global solution in time. Otherwise, there exists T > 0 such that u is a solution in (0, T ) and satisfies ‖u(·, t)‖L∞(0,1) → +∞, as t→ T−, ( or ‖u(·, t)‖L∞(0,+∞) → +∞, as t→ T−), and it is said that u blows up in a finite time. We establish the comparison principle for problem (1.1)–(1.3). Proposition 2.3. Assume that u and u are a subsolution and a supersolution, respectively, to problem (1.1)–(1.3). then u ≤ u a.e. in (0, 1)× (0, T ). Proof. Set u(x, t) = u(x, t)− u(x, t), (x, t) ∈ (0, 1)× (0, T ). Let 0 < τ < T . For any function 0 ≤ ϕ ∈ C1([0, 1] × [0, τ ]) with ϕ(τ, ·) ∣∣ (0,1) = ϕ(1, ·) ∣∣ (0,τ) = 0, it follows from Definition 2.1 that∫ 1 0 u(x, τ)ϕ(x, τ) dx ≤ ∫ τ 0 ∫ 1 0 ( z ∂ϕ ∂t + (um − um) ∂ ∂x ( xλ ∂ϕ ∂x ) + (up − up)ϕ ) dxdt = ∫ τ 0 ∫ 1 0 z (∂ϕ ∂t + a ∂ ∂x ( xλ ∂ϕ ∂x ) + acϕ ) dxdt, (2.1) EJDE-2021/96 POROUS MEDIUM EQUATIONS 5 where a(x, t) = { um(x,t)−um(x,t) u(x,t)−u(x,t) , if u(x, t) 6= u(x, t), mum−1(x, t), if u(x, t) = u(x, t), c(x, t) = { up(x,t)−up(x,t) um(x,t)−um(x,t) , if u(x, t) 6= u(x, t), p mu m−p(x, t), if u(x, t) = u(x, t). It is clear that a, c ∈ L∞((0, 1)× (0, τ)) satisfying 0 ≤ a(x, t) ≤ a0, 0 ≤ c(x, t) ≤ c0, (x, t) ∈ (0, 1)× (0, τ), where a0 and c0 are two positive constants. Choosing {an}∞n=1 and {cn}∞n=1 in C∞([0, 1]× [0, τ ]) such that 1 n ≤ an(x, t) ≤ a0 + 1 n , 0 ≤ cn(x, t) ≤ c0, (x, t) ∈ (0, 1)× (0, τ), (2.2) and ∫ τ 0 ∫ 1 0 (a− an)2 dxdt ≤ 1 + τ n2 , ∫ τ 0 ∫ 1 0 (c− cn)2 dxdt ≤ 1 n2 . (2.3) For any nonnegative function h ∈ C∞0 (0, 1), we consider the problem ∂ϕn ∂t + an ∂ ∂x ( xλ ∂ϕn ∂x ) + ancnϕn = 0, (x, t) ∈ (0, 1)× (0, τ), (2.4)( xλ ∂ϕn ∂x ) (0, t) = ϕn(1, t) = 0, t ∈ (0, τ), (2.5) ϕn(x, τ) = h(x), x ∈ (0, 1). (2.6) The same proof as of [29, Theorem 2.2] yields that problem (2.4)–(2.6) admits a unique solution ϕn ∈ C∞((0, 1)× (0, τ)) ∩ C([0, 1]× [0, τ ]) satisfying 0 ≤ ϕn(x, t) ≤ ‖h‖L∞(0,1). (2.7) Multiplying (2.4) by ∂ ∂x ( xλ ∂ϕn ∂x ) and then integrating over (0, 1)× (0, τ) by parts, one obtains from (2.2), (2.5) and (2.6) that 1 2 ∫ 1 0 xλ ∣∣∣∂ϕn ∂x (x, 0) ∣∣∣2 dx+ ∫ τ 0 ∫ 1 0 an ∣∣∣ ∂ ∂x ( xλ ∂ϕn ∂x )∣∣∣2 dxdt = 1 2 ∫ 1 0 xλ(h′(x))2 dx− ∫ τ 0 ∫ 1 0 ancnϕn ∂ ∂x ( xλ ∂ϕn ∂x ) dxdt ≤ 1 2 ∫ 1 0 xλ(h′(x))2 dx+ 1 2 ∫ τ 0 ∫ 1 0 anc 2 nϕ 2 n dxdt + 1 2 ∫ τ 0 ∫ 1 0 an ∣∣∣ ∂ ∂x ( xλ ∂ϕn ∂x )∣∣∣2 dxdt. (2.8) It follows from (2.8), (2.2) and (2.7) that∫ τ 0 ∫ 1 0 an ∣∣∣ ∂ ∂x ( xλ ∂ϕn ∂x )∣∣∣2 dxdt ≤M1, (2.9) where M1 is a positive constant independent of n. Taking ϕ = ϕn in (2.1), one obtains from (2.2), (2.3), (2.7), (2.9) and the Hölder inequality that∫ 1 0 u(x, τ)h(x) dx 6 X. ZHAO, M. ZHOU, X. JING EJDE-2021/96 ≤ ∫ τ 0 ∫ 1 0 z (∂ϕn ∂t + a ∂ ∂x ( xλ ∂ϕn ∂x ) + acϕn ) dxdt = ∫ τ 0 ∫ 1 0 u(a− an) ∂ ∂x ( xλ ∂ϕn ∂x ) dxdt+ ∫ τ 0 ∫ 1 0 u(ac− ancn)ϕn dxdt ≤ ‖u‖L∞((0,1)×(0,τ)) (∫ τ 0 ∫ 1 0 (a− an)2 an dxdt )1/2 × (∫ τ 0 ∫ 1 0 an ∣∣∣ ∂ ∂x ( xλ ∂ϕn ∂x )∣∣∣2 dx dt )1/2 + ‖u‖L∞((0,1)×(0,τ)) (∫ τ 0 ∫ 1 0 ϕ2 n dx dt )1/2(∫ τ 0 ∫ 1 0 (ac− ancn)2 dxdt )1/2 ≤M2 √ n (∫ τ 0 ∫ 1 0 (a− an)2 dxdt )1/2 + M2 n ≤ M3√ n , where M2 and M3 are positive constants independent of n. Letting n → ∞, one obtains that ∫ 1 0 u(x, τ)h(x) dx ≤ 0. Thanks to the arbitrariness of 0 ≤ h ∈ C∞0 (0, 1) and τ ∈ (0, T ), it holds that u ≤ 0 a.e. in (0, 1)× (0, T ). That is, u ≤ u a.e. in (0, 1)× (0, T ). � We turn to the local well-posedness of problem (1.1)–(1.3). Proposition 2.4. Assume that 0 ≤ u0 ∈ L∞(0, 1). There exists a constant T > 0 such that problem (1.1)–(1.3) admits a unique solution. Furthermore, ∂um ∂t in L2((0, 1)× (τ, T )) for any τ ∈ (0, T ). Proof. The uniqueness follows from Proposition 2.3. Let us prove the local exis- tence. For each integer n ≥ 1, consider the problem ∂un ∂t − ∂ ∂x (( x+ 1 n )λ ∂umn ∂x ) = upn, (x, t) ∈ (0, 1)× (0, Tn), (2.10) ∂umn ∂x (0, t) = 0, un(1, t) = 1 n , t ∈ (0, Tn), (2.11) un(x, 0) = u0,n(x) + 1 n , x ∈ (0, 1), (2.12) where u0,n ∈ C∞0 (0, 1) satisfies 0 ≤ u0,n(x) ≤ ‖u0‖L∞(0,1), x ∈ [0, 1], (2.13) lim n→∞ ‖u0,n − u0‖L∞(0,1) = 0. (2.14) We set un(x, t) = 0, un(x, t) = (2M0 − (p− 1)t)1/(1−p), (x, t) ∈ [0, 1]× [0, T ], where M0 = 1 2 (‖u0‖L∞(0,1) + 1)1−p, T = M0 p− 1 . It is clear that un and un are a subsolution and a supersolution, respectively, to problem (2.10)–(2.12) for Tn = T . Thanks to the classical theory on par- abolic equations, problem (2.10)–(2.12) with Tn = T admits a unique solution EJDE-2021/96 POROUS MEDIUM EQUATIONS 7 un ∈ C∞([0, 1]× [0, T ]), and 0 = un(x, t) ≤ un(x, t) ≤ un(x, t) ≤M1/(1−p) 0 , (x, t) ∈ [0, 1]× [0, T ]. (2.15) Multiplying (2.10) by umn and then integrating over (0, 1)× (0, T ) by parts, one obtains 1 m+ 1 ∫ 1 0 um+1 n (x, T ) dx+ ∫ T 0 ∫ 1 0 ( x+ 1 n )λ∣∣∣∂umn ∂x ∣∣∣2 dxdt = 1 m+ 1 ∫ 1 0 um+1 0 dx+ ∫ T 0 ∫ 1 0 up+mn dx dt. (2.16) It follows from (2.13) and (2.16) that∫ T 0 ∫ 1 0 ( x+ 1 n )λ∣∣∣∂umn ∂x ∣∣∣2 dxdt ≤M1, (2.17) where M1 is a positive constant depending only on ‖u0‖L∞(0,1), m and p. Thanks to (2.17), for τ ∈ (0, T ), there exists τ̃ ∈ (0, τ) such that∫ 1 0 ( x+ 1 n )λ∣∣∣∂umn ∂x (x, τ̃) ∣∣∣2 dx ≤ M1 τ . (2.18) Multiplying (2.10) by ∂um n ∂t and then integrating over (0, 1) × (τ̃ , T ) by parts, one obtains from the Hölder inequality that 4m (m+ 1)2 ∫ T τ̃ ∫ 1 0 ∣∣∣∂u(m+1)/2 n ∂t ∣∣∣2 dxdt = 1 2 ∫ 1 0 ( x+ 1 n )λ∣∣∣∂umn ∂x (x, τ̃) ∣∣∣2 dx− 1 2 ∫ 1 0 ( x+ 1 n )λ∣∣∣∂umn ∂x (x, T ) ∣∣∣2 dx + 2m m+ 1 ∫ T τ̃ ∫ 1 0 up+(m−1)/2 n ∂u (m+1)/2 n ∂t dxdt ≤ 1 2 ∫ 1 0 ( x+ 1 n )λ∣∣∣∂umn ∂x (x, τ̃) ∣∣∣2 dx+ m 2 ∫ T τ̃ ∫ 1 0 u2p+m−1n dx + 2m (m+ 1)2 ∫ T τ̃ ∫ 1 0 ∣∣∣∂u(m+1)/2 n ∂t ∣∣∣2 dx dt. (2.19) It follows from (2.15), (2.18) and (2.19) that∫ T τ̃ ∫ 1 0 ∣∣∣∂u(m+1)/2 n ∂t ∣∣∣2 dxdt ≤M2, which, together with (2.15), leads to∫ T τ̃ ∫ 1 0 ∣∣∣∂umn ∂t ∣∣∣2 dxdt = 4m2 (m+ 1)2 ∫ T τ̃ ∫ 1 0 um−1n ∣∣∣∂u(m+1)/2 n ∂t ∣∣∣2 dxdt ≤M3, (2.20) where M2 and M3 are positive constants depending only on ‖u0‖L∞(0,1), m, p, and τ . Thanks to (2.15), (2.17) and (2.20), there exists a subsequence of {un}∞n=1, de- noted by itself for convenience, and three functions u, ξ, ζ ∈ L∞((0, 1)×(0, T )) such that 0 ≤ u(x, t) ≤M1/(1−p) 0 , (x, t) ∈ (0, 1)× (0, T ), (2.21) 8 X. ZHAO, M. ZHOU, X. JING EJDE-2021/96∫ T 0 ∫ 1 0 xλ ∣∣∣∣ ∂ξ∂x ∣∣∣∣2 dxdt ≤M1, (2.22)∫ T τ ∫ 1 0 ∣∣∣∂ξ ∂t ∣∣∣2 dxdt ≤M3, τ ∈ (0, T ), (2.23) and un ⇀ u, umn ⇀ ξ and upn ⇀ ζ weakly ∗ in L∞((0, 1)× (0, T )) as n→∞, (2.24) umn → ξ in L2((ε, 1)× (ε, T )) as n→∞ for any 0 < ε < min{1, T}, (2.25)( x+ 1 n )λ/2 ∂umn ∂x ⇀ xλ/2 ∂ξ ∂x weakly in L2((0, 1)× (0, T )) as n→∞, (2.26) ∂umn ∂t ⇀ ∂ξ ∂t weakly in L2((0, 1)× (ε, T )) as n→∞ for any 0 < ε < T. (2.27) It follows from (2.24) and (2.25) that ξ(x, t) = um(x, t), ζ(x, t) = up(x, t), (x, t) ∈ (0, 1)× (0, T ). (2.28) Owing to (2.21)–(2.28), one can prove that u is a solution to problem (1.1)–(1.3), and ∂um ∂t ∈ L 2((0, 1)× (τ, T )) for any τ ∈ (0, T ). � For problem (1.4)–(1.6) in an unbounded interval, the comparison principle can be proved similarly to Proposition 2.3, and one needs some estimates in an un- bounded interval (see, e.g., [23]). As to the local existence of solutions to problem (1.4)–(1.6), one can use the problems in bounded intervals to approximate the problem in an unbounded interval (see, e.g., [23]). We state these results without proof. Proposition 2.5. Assume that u and u are a subsolution and a supersolution, respectively, to problem (1.4)–(1.6). then u ≤ u a.e. in (0,+∞)× (0, T ). Proposition 2.6. Assume that 0 ≤ u0 ∈ L∞(0,+∞) ∩ L1(0,+∞). There exists a constant T > 0 such that problem (1.4)–(1.6) admits a unique solution. Further- more, ∂um ∂t ∈ L 2((0,+∞)× (τ, T )) for any τ ∈ (0, T ). 3. Problems in a bounded interval In this section, we investigate the asymptotic behavior of solutions to problem (1.1)–(1.3) in a bounded interval. Theorem 3.1. Let λ ≥ 2. For any nontrivial 0 ≤ u0 ∈ L∞(0, 1), the solution to problem (1.1)–(1.3) must blow up in a finite time. Proof. For 0 < δ < 1, we set ηδ(x) = { λ−1 δ 2λ−1−δx−δ − λ−1−δ δ 2λ−1 − 1, if x ∈ (0, 1/2), x1−λ − 1, if x ∈ [1/2, 1). It follows from the proof of [23, Theorem 2.2] that 0 ≤ ηδ ∈ C1,1([0, 1]), and there exists two constants M1, M2 > 0 depending only on λ but independent of δ such that (xλη′δ(x))′ ≥ −M1δηδ(x), 0 < x < 1, (3.1)∫ 1 0 ηδ(x)dx ≤M2. (3.2) EJDE-2021/96 POROUS MEDIUM EQUATIONS 9 Assume that u is a global solution of problem (1.1)–(1.3), and denote wδ(t) = ∫ 1 0 u(x, t)ηδ(x)dx, t ≥ 0. For 0 < ε < 1/2, choose ξε ∈ C∞([0, 1]) to satisfy ξε(x) = { 0, if x ∈ [0, ε], 1, if x ∈ [2ε, 1], and 0 ≤ ξε(x) ≤ 1, 0 ≤ ξ′ε(x) ≤ 2 ε , |ξ′′ε (x)| ≤ 4 ε2 , 0 ≤ x ≤ 1. Thanks to Definition 2.1 and Proposition 2.4, one obtains d dt ∫ 1 0 u(x, t)ξε(x)ηδ(x)dx = ∫ 1 0 um(x, t)(xλ(ξε(x)ηδ(x))′)′dx+ ∫ 1 0 up(x, t)ξε(x)ηδ(x)dx = ∫ 1 0 um(x, t)ξε(x)(xλη′δ(x))′dx + ∫ 2ε ε um(x, t)ξ′ε(x)(2xλη′δ(x) + λxλ−1ηδ(x))dx + ∫ 2ε ε um(x, t)xλξ′′ε (x)ηδ(x)dx+ ∫ 1 0 up(x, t)ξε(x)ηδ(x)dx, t > 0. (3.3) Letting ε→ 0+ in (3.3), it follows from λ ≥ 2 and (3.1) that w′δ(t) = d dt ∫ 1 0 u(x, t)ηδ(x)dx = ∫ 1 0 um(x, t)(xλη′δ(x))′dx+ ∫ 1 0 up(x, t)ηδ(x)dx ≥ −M1δ ∫ 1 0 um(x, t)ηδ(x)dx+ ∫ 1 0 up(x, t)ηδ(x)dx, t > 0. (3.4) Since p > m > 1, it follows from the Hölder inequality and (3.2) that∫ 1 0 um(x, t)ηδ(x)dx ≤ (∫ 1 0 ηδ(x)dx )(p−m)/p(∫ 1 0 up(x, t)ηδ(x)dx )m/p ≤M (p−m)/p 2 (∫ 1 0 up(x, t)ηδ(x)dx )m/p , t > 0, (3.5) and ∫ 1 0 u(x, t)ηδ(x)dx ≤ (∫ 1 0 ηδ(x)dx )(p−1)/p(∫ 1 0 up(x, t)ηδ(x)dx )1/p ≤M (p−1)/p 2 (∫ 1 0 up(x, t)ηδ(x)dx )1/p , t > 0, (3.6) 10 X. ZHAO, M. ZHOU, X. JING EJDE-2021/96 Substituting (3.5) and (3.6) into (3.4) we obtain w′δ(t) ≥ −M1δ (∫ 1 0 ηδ(x)dx )(p−m)/p(∫ 1 0 up(x, t)ηδ(x)dx )m/p + ∫ 1 0 up(x, t)ηδ(x)dx ≥ (∫ 1 0 up(x, t)ηδ(x)dx )m/p( −M1M (p−m)/p 2 δ + (∫ 1 0 up(x, t)ηδ(x)dx )(p−m)/p) ≥M−m(p−1)/p 2 wm(t) ( −M1M (p−m)/p 2 δ +M −(p−1)(p−m)/p 2 wp−m(t) ) , (3.7) for t > 0. Owing to the nontriviality of u0, it holds that µ = inf 0<δ<1 wδ(0) = inf 0<δ<1 ∫ 1 0 u0(x)ηδ(x)dx > 0. We take δ = 1 2 min{µp−mM−11 M −(p−m) 2 , 1}. Then M1M (p−m)/p 2 δ ≤ 1 2 M −(p−1)(p−m)/p 2 wp−mδ (0). Hence one obtains from (3.7) that w′δ(t) ≥ 1 2 M −(p−1) 2 wpδ (t), t > 0. Since p > 1, there exists T∗ > 0 such that wδ(t) = ∫ 1 0 u(x, t)ηδ(x)dx→ +∞ as t→ T−∗ , which leads to lim t→T−∗ ‖u(·, t)‖L∞(0,1) → +∞. That is to say, u blows up in a finite time. � Theorem 3.2. Let 0 < λ < 2. There exist both nontrivial global and blowing-up solutions to problem (1.1)–(1.3). Proof. First, we prove the existence of the global solution for sufficiently small 0 ≤ u0 ∈ L∞(0, 1). We set u(x, t) = (t+ τ)−αU((t+ τ)−βx), 0 ≤ x ≤ 1, t ≥ 0, (3.8) where α = 1 p− 1 , β = p−m (2− λ)(p− 1) , and τ > 1 will be determined later. One can verify that if 0 ≤ U ∈ C0,1(0, τ−β) with Um ∈ C1,1(0, τ−β) satisfies (rλ(Um)′(r))′ + αU(r) + βrU ′(r) + Up(r) ≤ 0, 0 < r < τ−β , EJDE-2021/96 POROUS MEDIUM EQUATIONS 11 then u given by (3.8) is a supersolution to (1.1). We take U(r) = 1 (2− λ)1/m ( 1 τ (p−m)/(p−1) − r 2−λ )1/m , 0 ≤ r ≤ τ−β . (3.9) A direct calculation shows that (rλ(Um)′(r))′ + αU(r) + βrU ′(r) + Up(r) = −1 + αU(r)− β m r1−mU1−m(r) + Up(r) ≤ −1 + αU(r) + Up(r) ≤ −1 + 1 τ1/m ( α (2− λ)1/m + 1 (2− λ)p/m ) . Therefore, u is a supersolution to (1.1) for τ ≥ τ0, where τ0 = 2m−1 ( αm (2− λ) + 1 (2− λ)p ) + 1. It is noted that limr→0+ r λ(Um)′(r) = 0. Hence u is a supersolution to problem (1.1)–(1.3) if u0(x) ≤ u(x, 0), 0 < x < 1 (3.10) for some τ ≥ τ0. It follows from Proposition 2.3 that problem (1.1)–(1.3) admits a global solution if 0 ≤ u0 ∈ L∞(0, 1) satisfies (3.10) for some τ ≥ τ0. Below we prove that the solution to problem (1.1)–(1.3) must blow up if 0 ≤ u0 ∈ L∞(0, 1) is sufficiently large. We set η(x) = { 2, 0 < x ≤ 1/2, 1 + cos(2x− 1)π, 1/2 < x ≤ 1. It is clear that η ∈ C1,1([0, 1]) satisfies η(1) = 0, η′(0) = 0, and (xλη(x))′ ≥ −4π2η(x), 1/2 < x ≤ 1. Assume that u is a global solution to problem (1.1)–(1.3). We denote w(t) = ∫ 1 0 u(x, t)η(x)dx, t ≥ 0. It follows from the Hölder inequality that∫ 1 0 um(x, t)η(x)dx ≤ (∫ 1 0 η(x)dx )(p−m)/p(∫ 1 0 up(x, t)η(x)dx )m/p ≤ 2 (∫ 1 0 up(x, t)η(x)dx )m/p , t > 0, (3.11) and w(t) ≤ (∫ 1 0 η(x)dx )(p−1)/p(∫ 1 0 up(x, t)η(x)dx )1/p ≤ 2 (∫ 1 0 up(x, t)η(x)dx )1/p , t > 0. (3.12) 12 X. ZHAO, M. ZHOU, X. JING EJDE-2021/96 It follows from Definition 2.1, (3.11) and (3.12) that w′(t) = ∫ 1 0 um(x, t)(xλη′(x))′dx+ ∫ 1 0 up(x, t)η(x)dx ≥ −4π2 ∫ 1 0 um(x, t)η(x)dx+ ∫ 1 0 up(x, t)η(x)dx ≥ −8π2 (∫ 1 0 up(x, t)η(x)dx )m/p + ∫ 1 0 up(x, t)η(x)dx ≥ (∫ 1 0 up(x, t)η(x)dx )m/p( − 8π2 + (∫ 1 0 up(x, t)η(x)dx )(p−m)/p) ≥ 2−mwm(t) ( − 8π2 + 2−(p−m)wp−m(t) ) , t > 0. (3.13) If u0 is sufficiently large so that wp−m(0) = (∫ 1 0 u0(x)η(x)dx )p−m ≥ 2p−m+4π2, (3.14) from (3.13) one obtains w′(t) ≥ 2−(p+1)wp(t), t > 0. Since p > 1, there exists T∗ > 0 such that w(t) = ∫ 1 0 u(x, t)η(x)dx→ +∞ as t→ T−∗ , which leads to lim t→T−∗ ‖u(·, t)‖L∞(0,1) → +∞. That is to say, u blows up in a finite time. � 4. Problems in an unbounded interval In this section, we investigate the asymptotic behavior of solutions to problem (1.4)–(1.6) in an unbounded interval. As a corollary of Theorem 3.1 and Proposition 2.3, one obtains the following result. Theorem 4.1. Let λ ≥ 2. For any nontrivial 0 ≤ u0 ∈ L∞(0,+∞) ∩ L1(0,+∞), the solution to problem (1.4)–(1.6) must blow up in a finite time. Below we deal with the case that 0 < λ < 2. Theorem 4.2. Let 0 < λ < 2 and m < p < pc = m + 2 − λ. For any nontrivial 0 ≤ u0 ∈ L∞(0,+∞) ∩ L1(0,+∞), the solution to problem (1.4)–(1.6) must blow up in a finite time. Proof. For R > 0, set ηR(x) =  1, 0 < x ≤ R, 1 2 ( 1 + cos (x−R)π R ) , R < x < 2R, 0, x ≥ 2R. It follows from the proof of [23, Lemma 2.1 ] that ηR ∈ C1,1([0,+∞) satisfies (xληR(x))′ ≥ −2λπ2Rλ−2ηR(x), x > 0. EJDE-2021/96 POROUS MEDIUM EQUATIONS 13 Assume that u is a global solution to problem (1.4)–(1.6). We denote wR(t) = ∫ +∞ 0 u(x, t)ηR(x)dx, t ≥ 0. (4.1) Thanks to Definition 2.2 and Proposition 2.6, one obtains that u satisfies w′R(t) = ∫ +∞ 0 um(x, t)(xλη′R(x))′dx+ ∫ +∞ 0 up(x, t)ηR(x)dx ≥ −2λπ2Rλ−2 ∫ +∞ 0 um(x, t)ηR(x)dx+ ∫ +∞ 0 up(x, t)ηR(x)dx, (4.2) for t > 0. It follows from the Hölder inequality that∫ +∞ 0 um(x, t)ηR(x)dx ≤ (∫ +∞ 0 ηR(x)dx )(p−m)/p(∫ +∞ 0 up(x, t)ηR(x)dx )m/p ≤ (2R)(p−m)/p (∫ +∞ 0 up(x, t)ηR(x)dx )m/p , t > 0, (4.3) and ∫ +∞ 0 u(x, t)ηR(x)dx ≤ (∫ +∞ 0 ηR(x)dx )(p−1)/p(∫ +∞ 0 up(x, t)ηR(x)dx )1/p ≤ 2R(p−1)/p (∫ +∞ 0 up(x, t)ηR(x)dx )1/p , t > 0. (4.4) We substitute (4.3) and (4.4) into (4.2) to obtain w′R(t) ≥ −2λπ2Rλ−2 (∫ +∞ 0 ηR(x)dx )(p−m)/p(∫ +∞ 0 up(x, t)ηR(x)dx )m/p + ∫ +∞ 0 up(x, t)ηR(x)dx ≥ (∫ +∞ 0 up(x, t)ηR(x)dx )m/p( − 2λπ2Rλ−1−m/p + (∫ +∞ 0 up(x, t)ηR(x)dx )(p−m)/p) ≥ 2−mR−m(p−1)/pwmR (t) ( − 2λπ2Rλ−1−m/p + 2−(p−m)R−(p−1)(p−m)/pwp−mR (t) ) , t > 0. (4.5) It is noted that p < m+2−λ, wR(0) is nondecreasing with respect to R ∈ (0,+∞), and supR>0 wR(0) > 0. Therefore, there exists R > 0 suitably large such that 2p−m+λπ2Rp−(m+2−λ) ≤ 1 2 wp−mR (0). (4.6) It follows from that (4.4)–(4.6) that w′R(t) ≥ 2−(p+1)R−(p−1)wpR(t), t > 0. 14 X. ZHAO, M. ZHOU, X. JING EJDE-2021/96 Since p > 1, there exists T∗ > 0 such that wR(t) = ∫ +∞ 0 u(x, t)ηR(x)dx→ +∞ as t→ T−∗ , which leads to ‖u(·, t)‖L∞(Rn) → +∞ as t→ T−∗ . That is to say, u blows up in a finite time. � Theorem 4.3. Assume that 0 < λ < 2 and p > pc = m+ 2− λ. There exist both nontrivial global and blowing-up solutions to problem (1.4)–(1.6). Proof. Owing to Theorem 3.2 and Proposition 2.3, problem (1.4)–(1.6) admits blowing-up solutions. Below we prove the existence of nontrivial global solutions to problem (1.4)–(1.6). We set u(x, t) = (t+ 1)−αU((t+ 1)−βx), x ≥ 0, t ≥ 0, (4.7) where α = 1 p− 1 , β = p−m (2− λ)(p− 1) . It follows from p > m+2−λ that α < β. One can verify that if 0 ≤ U ∈ C0,1(0,+∞) with Um ∈ C1,1(0,+∞) satisfies (rλ(Um)′(r))′ + αU(r) + βrU ′(r) + Up(r) ≤ 0, r > 0, then u given by (4.7) is a supersolution to (1.4). We take U(r) = (l(2−λ)/2 −Ar2−λ) 1/(m−1) + , r ≥ 0, (4.8) where (m− 1)α m(2− λ) < A < (m− 1)β m(2− λ) , 0 < l < l0 with l0 satisfying l (2−λ)(p−1)/(2m−2) 0 = m(2− λ)A m− 1 − α. (4.9) For 0 < r < A−1/(2−λ)l, one has (rλ(Um)′(r))′ + αU(r) + βrU ′(r) + Up(r) = m(2− λ)2A2 (m− 1)2 r2−λU2−m(r)− m(2− λ)A m− 1 U(r) + αU(r) − (2− λ)Aβ m− 1 r2−λU2−m(r) + Up(r) = (2− λ)A m− 1 (m(2− λ)A m− 1 − β ) r2−λU2−m(r) + ( α− m(2− λ)A m− 1 ) U(r) + Up(r) ≤ U(r) ( α− m(2− λ)A m− 1 + (l(2−λ)/2 −Ar2−λ)(p−1)/(m−1) ) ≤ U(r) ( α− m(2− λ)A m− 1 + l(2−λ)(p−1)/(2m−2) ) , 0 < r < A−1/(2−λ)l. (4.10) EJDE-2021/96 POROUS MEDIUM EQUATIONS 15 Thanks to (4.9) and (4.10), ū is a supersolution to (1.4) if 0 < l < l0. It is noted that limr→0+ r λ(Um)′(r) = 0. Hence u is a supersolution to problem (1.4)–(1.6) if u0(x) ≤ u(x, 0), x > 0 for some 0 < l < l0. Thanks to Proposition 2.5, problem (1.4)–(1.6) admits non- trivial global solutions. � Below we investigate the critical case p = pc = m+ 2− λ. Lemma 4.4. Assume that 0 < λ < 2, p = pc = m+ 2− λ and 0 < θ < p−m p−1 . Let u be a nontrivial global solution to problem (1.4)–(1.6). Then for any R > 0, wR(t) ≤ 23/(2−λ)π2/(2−λ), t > 0, (4.11) w′R(t) ≥ −2λ−m+2m/(2−λ)π2+2m/(2−λ)Rλ−m−1, t > 0, (4.12) w′R(t) ≥ Rλ−m−1wm−θR (t) ( − 2λ−m+θπ2 (∫ 2R R u(x, t)ηR(x)dx )θ + 2λ−m−2w2−λ+θ R (t) ) , t > 0, (4.13) where wR(t) is the function defined by (4.1). Proof. Since p = pc = m+ 2− λ, one obtains from (4.5) that w′R(t) ≥ 2−mR−m(pc−1)/pcwmR (t) ( − 2λπ2Rλ−1−m/pc + 2−(pc−m)R−(pc−1)(pc−m)/pcwpc−mR (t) ) = −2λ−mπ2Rλ−m−1wmR (t) + 2λ−m−2Rλ−m−1wm+2−λ R (t), t > 0. (4.14) If (4.11) is not true, there exists t0 > 0 such that 2λ−m+1π2 ≤ 2λ−m−2w2−λ R (t0). Then, it follows from (4.14) that w′R(t) ≥ 2λ−m−3Rλ−m−1wm+2−λ R (t), t > t0, which leads to that u blows up in a finite time as the discussion at the end of the proof of Theorem 4.2. Therefore, (4.11) holds. It follows from (4.14) and the Hölder inequality that w′R(t) ≥ 2λ−m−2Rλ−m−1 ( − 4π2wmR (t) + wm+2−λ R (t) ) ≥ 2λ−m−2Rλ−m−1 ( − m m+ 2− λ wm+2−λ R (t) − 2− λ m+ 2− λ (4π2)(m+2−λ)/(2−λ) + wm+2−λ R (t) ) ≥ −2λ−m+2m/(2−λ)π2+2m/(2−λ)Rλ−m−1, t > 0, which is (4.12). Below we prove (4.13). For t > 0, it follows from the Hölder inequality that∫ 2R R um(x, t)ηR(x)dx ≤ (∫ 2R R ηR(x)dx )(pc−m−θ(pc−1))/pc(∫ 2R R u(x, t)ηR(x)dx )θ 16 X. ZHAO, M. ZHOU, X. JING EJDE-2021/96 × (∫ 2R R upc(x, t)ηR(x)dx )(m−θ)/pc ≤ R(pc−m−θ(pc−1))/pc (∫ 2R R u(x, t)ηR(x)dx )θ(∫ +∞ 0 upc(x, t)ηR(x)dx )(m−θ)/pc , which, together with (4.2), yield w′R(t) ≥ (∫ +∞ 0 upc(x, t)ηR(x)dx )(m−θ)/pc × ( − 2λπ2Rλ−2+(pc−m−θ(pc−1))/pc (∫ 2R R u(x, t)ηR(x)dx )θ + (∫ +∞ 0 upc(x, t)ηR(x)dx )(pc−m+θ)/pc) . (4.15) Since pc = m+ 2− λ, it follows that λ− 2 + pc −m− θ(pc − 1) pc = − (m+ 1− λ)(2− λ+ θ) m+ 2− λ . Then from (4.4) and (4.15) it follows that w′R(t) ≥ 2θ−mR−(m+1−λ)(m−θ)/(m+2−λ)wm−θR × ( − 2λπ2R−(m+1−λ)(2−λ+θ)/(m+2−λ) (∫ 2R R u(x, t)ηR(x)dx )θ + 2λ−2−θR−(m+1−λ)(2−λ+θ)/(m+2−λ)w2−λ+θ R ) ≥ Rλ−m−1wm−θR (t) ( − 2λ−m+θπ2 (∫ 2R R u(x, t)ηR(x)dx )θ + 2λ−m−2w2−λ+θ R (t) ) , t > 0. This is (4.13). � Theorem 4.5. Assume that 0 < λ < 2 and p = pc = m+2−λ. For any nontrivial 0 ≤ u0 ∈ L∞(0,+∞) ∩ L1(0,+∞), the solution to problem (1.4)–(1.6) must blow up in a finite time. Proof. Assumed that u is a global solution to problem (1.4)–(1.6). We denote Λ = sup R>0, t>0 wR(t) = sup t>0 ∫ +∞ 0 u(x, t)dx. (4.16) The nontriviality of u and (4.11) yield 0 < Λ < +∞. Fix ε0 ∈ (0, Λ) and M0 > 0 such that 2θ+3π2(ε0 +M0)θ ≤ (Λ− ε0)pc−m+θ, (4.17) where 0 < θ < p−m p−1 is a constant. By (4.16), there exist t0 > 0 and R0 > 0 such that wR0 (t0) ≥ Λ− ε0. (4.18) For any t ≥ t0, it follows from (4.12) with R = R0 and (4.18) that wR0(t) ≥ wR0(t0)− 2λ−m+2m/(2−λ)π2+2m/(2−λ)Rλ−m−10 (t− t0) ≥ Λ− ε0 − 2λ−m+2m/(2−λ)π2+2m/(2−λ)Rλ−m−10 (t− t0), t > t0, EJDE-2021/96 POROUS MEDIUM EQUATIONS 17 which, together with (4.16), leads to∫ 4R0 2R0 u(x, t)η2R0 (x)dx ≤ ∫ +∞ 0 u(x, t)dx− wR0 (t) ≤ ε0 + 2λ−m+2m/(2−λ)π2+2m/(2−λ)Rλ−m−10 (t− t0), t ≥ t0. (4.19) Choosing R = 2R0 in (4.13) yields w′2R0 (t) ≥ (2R0)λ−m−1wm−θ2R0 (t) ( − 2λ−m+θπ2 (∫ 4R0 2R0 u(x, t)ηR(x)dx )θ + 2λ−m−2w2−λ+θ 2R0 (t) ) , t > t0, which together with (4.16)–(4.19) implies w′2R0 (t) ≥ 22(λ−m−2)Rλ−m−10 (Λ− ε0)m+2−λ, t0 < t < t1, (4.20) where t1 = t0 + 2m−λ−2m/(2−λ)π−2−2m/(2−λ)M0R 1+m−λ 0 . It follows from (4.18) and(4.20) that w2R0 (t1) ≥ w2R0 (t0) + 22(λ−m−2)Rλ−m−10 (Λ− ε0)m+2−λ(t1 − t0) ≥ Λ− ε0 + σ0, (4.21) where σ0 = 2λ−m−4−2m/(2−λ)π−2−2m/(2−λ)M0(Λ− ε0)m+2−λ. Applying (4.12) with R = 2R0 and (4.21), one obtains w2R0 (t) ≥ w2R0 (t1)− 2λ−m+2m/(2−λ)π2+2m/(2−λ)(2R0)λ−m−1(t− t1) ≥ Λ− ε0 − 2λ−m+2m/(2−λ)π2+2m/(2−λ)(2R0)λ−m−1(t− t1), t > t1, which, together with (4.16) with R = 2R0, leads to∫ 8R0 4R0 u(x, t)η4R0 (x)dx ≤ ∫ +∞ 0 u(x, t)dx− w2R0(t) ≤ ε0 + 2λ−m+2m/(2−λ)π2+2m/(2−λ)(2R0)λ−m−1(t− t1), t ≥ t1. (4.22) Taking R = 4R0 in (4.13), one obtains w′4R0 (t) ≥ (4R0)λ−m−1wm−θ4R0 (t) ( − 2λ−m+θπ2 (∫ 8R0 4R0 u(x, t)ηR(x)dx )θ + 2λ−m−2w2−λ+θ 4R0 (t) ) , t > t1, (4.23) Thanks to (4.12)–(4.23), one obtains w′4R0 (t) ≥ 22(λ−m−2)(2R0)λ−m−1(Λ− ε0)m+2−λ, t1 < t < t2, (4.24) where t2 = t1 + 2m−λ−2m/(2−λ)π−2−2m/(2−λ)M0(2R0)1+m−λ. 18 X. ZHAO, M. ZHOU, X. JING EJDE-2021/96 It follows from (4.12) and (4.24) that w4R0 (t2) ≥ w4R0 (t1) + σ0 ≥ w2R0 (t1) + σ0 ≥ Λ− ε0 + 2σ0. 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Wu, J. Zhao, J. Yin, H. Li; Nonlinear Diffusion Equations World Scientific, 2001. [28] J. Xu, C. Wang, Y. Nie; Carleman estimate and null controllability of a cascade degenerate parabolic system with general convection terms, Electron. J. Differential Equations, 2018 (2018), no. 195, 20 pp. [29] M. Zhou, C. Wang, Y. Nie; Quenching of solutions to a class of semilinear parabolic equations with boundary degeneracy, Journal of Mathematical Analysis and Applications, 421(1) (2015), 59–74. 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 Xinxin Jing School of Mathematics, Jilin University, Changchun 130012, China Email address: 1776043712@qq.com 1. Introduction 2. Comparison principles and well-posedness 3. Problems in a bounded interval 4. Problems in an unbounded interval Acknowledgments References