EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 3, 2020, 645-662 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Some results on blow-up phenomenon for nonlinear porous medium equations with weighted source Huafei Di1,∗, Lin Chen2, Zefang Song3 1 School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, P R China 2 College of Management and Economics, Tianjin University, Tianjin 300072, P R China 3 School of Economics and Statistics, Guangzhou University, Guangzhou 510006, P R China Abstract. This paper deals with the blow-up phenomena for a type of nonlinear porous medium equations with weighted source ut−4um = a(x)f(u) subject to Dirichlet (or Neumann) boundary conditions. Based on the auxiliary functions and differential-integral inequalities, the blow-up criterions which ensure that u cannot exist all time are given under two different assumptions, and the corresponding estimates on the upper bounds for blow-up time and blow-up rate are derived respectively. Moreover, we use three different methods to determine the lower bounds for blow-up time and blow-up rate estimates if blow-up does occurs. 2020 Mathematics Subject Classifications: 35A01, 35B44, 35K20, 35K61 Key Words and Phrases: Porous medium equation, Upper and lower bounds, Blow-up rate, Weighted source 1. Introduction In this paper, we deal with the blow-up time and blow-up rate estimates of the solutions to the following problem: ut −4um = a(x)f(u), x ∈ Ω, t > 0, (1.1) u(x, t) = 0 or ∂u ∂ν = 0, x ∈ ∂Ω, t > 0, (1.2) u(x, 0) = g(x) ≥ 0, x ∈ Ω, (1.3) where m > 1 and Ω ⊂ Rn (n ≥ 3) is a smooth bounded domain, ν is the outward normal vector, g(x) is a continuous nonnegative function and satisfies the compatible condition. Here, the nonlinear function f satisfies ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v13i3.3768 Email addresses: dihuafei@yeah.net (H.F. Di), chenlinalbert@126.com (L. Chen), song−zefang@163.com (Z.F. Song). https://www.ejpam.com 645 c© 2020 EJPAM All rights reserved. H.F. Di, L. Chen and Z.F. Song / Eur. J. Pure Appl. Math, 13 (3) (2020), 645-662 646 (f1): f(s) ≥ 0 for all s ≥ 0; And the weighted function a(x) ∈ C1(Ω) ∩ C0(Ω̄) satisfies (a1): a(x) ≥ C > 0 on Ω̄ for some constant C or (a2): a(x) > 0 in Ω and a(x) = 0 on ∂Ω. It is well known that the porous medium equations have extensive physical background and rich theoretical connotation. They have been used to model the processes involving the chemical reaction, heat transfer or diffusion, population dynamics and so on. We refer readers to see [1, 26] and references therein, where a series of physical application of Eq.(1.1) are also summarized. For instance, the nonlinear term f(u) of Eq.(1.1) describes the nonlinear source in the diffusion phenomena, and it is called to be “heat source”. If the “heat source” occurs, the solutions of Eq.(1.1) might be unbounded at finite time, namely, the solutions might be blowing up in finite time. The Eq.(1.1) includes many important physical models. If the exponent m = 1 and weighted function a(x) ≡ 1, the model (1.1) reduces to the semilinear heat equations ut −4u = f(u), x ∈ Ω, t > 0. (1.4) About this model, many results about the blow-up phenomenon of the solutions have been obtained, we refer to see [13, 16–19, 23, 25] and references therein. In [18, 19], Payne and Schaefer obtained a lower bound on blow-up time of the solutions to the Eq.(1.4) under null Dirichlet boundary condition and homogeneous Neumann boundary condition, respectively. Later, Payne et al. [16, 17] studied the blow-up phenomenon of the solutions for Eq.(1.4) with nonlinear boundary conditions. When the nonlinear source term f(u) = ∫ Ω u qdx− kus, Song [23] obtained the lower bounds for blow-up time of the solutions with either homogeneous Dirichlet or homogeneous Neumann boundary conditions in three dimensional space. Afterwards, Liu [13] studied the lower bounds for blow-up time under nonlinear boundary conditions in three dimensional space. In [25], Tang et al. extended the results of literature [13] in higher dimensional space. When the exponent m 6= 1 and weighted function a(x) ≡ 1, the model (1.1) becomes the following porous medium equations ut −4um = f(u), x ∈ Ω, t > 0. (1.5) This type of equations appears in several branches of applied mathematics [10, 12]. There is large body of literature on the study of the Eq.(1.5), such as the existence and uniqueness in [4, 5, 9, 11, 26], blow-up in [5, 7–10, 26], asymptotic behavior in [3, 20, 21, 26] and other interesting results in [2, 4, 12, 26] and references therein. For instance, in the case of nonlinear source f(u) = up, Galaktionov et al. [5] obtained the finite time blow-up of the solutions for 1 < p < m + 2 n , and proved the global existence of the solutions for p > m+ 2 n . For the critical case, Galaktionov and Levine [6], Kawanago [9] revealed that all nonnegative nontrivial mild solutions blow up in finite time. Jiang, Zheng and Song [8] gave some sufficient or necessary blow-up conditions, and the blow-up rate estimates, where f(u) = up, m > 1, p > 1. H.F. Di, L. Chen and Z.F. Song / Eur. J. Pure Appl. Math, 13 (3) (2020), 645-662 647 If the exponent m = 1 and weighted function a(x) 6≡ 1, the model (1.1) reduces to the semilinear parabolic equations with weighted source ut −4u = a(x)f(u), x ∈ Ω, t > 0. (1.6) Recently, the studying on the blow-up phenomenon had some new development, where more attention was paid on the parabolic equations with weighted source. These models can be used to illustrate the processes of heat transfer arising in physical and engineering applications, such as a model of phase separation in binary alloys [22]. The existence and nonexistence of global solutions, bounds for blow-up time, blow-up rate, blow-up sets and asymptotic behavior for this type of equations were investigated by many authors. We refer the reader to see [14, 15, 24] and papers cited therein. For example, Song and Lv [14, 24] studied the initial boundary value problem for the above equations with nonlinear Neumann boundary condition, and they derived the upper and lower bounds for blow-up time in three dimensional space [14]. In [24], they further investigated the estimates of blow-up rate and the bounds for blow-up time in higher dimensional space. Ma and Fang [15] changed the diffusion term 4u into ∑N i,j=1 ( ai,j(x)uxi ) xj in Eq.(1.6), where the upper and lower bounds for the blow-up time were derived in higher dimensional space. In the present work, we main study the blow-up phenomena for the porous medium equations with weighted nonlinear source. As far as we known, there is little information on the blow-up results of the solutions for problem (1.1)-(1.3). Obviously, the existence and uniqueness of local solutions for this problem can be obtained by applying the classical Faedo-Galerkin method or Contraction Mapping Principle. Naturally, we would like to study the estimates of blow-up rate and the bounds for blow-up time of the solutions in any smooth bounded domain Ω ⊂ Rn (n ≥ 3). Here, the appearance of the diffusion term 4um and weighted nonlinear source a(x)f(u) cause some difficulties in dealing with the qualitative properties of problem (1.1)-(1.3). Hence, we shall use some modified auxiliary functions and differential-integral inequality skills to over these difficulties. In detail, this paper is organized as follows: the blow-up criterions are given under two different assumptions, and the corresponding estimates on the upper bounds for blow-up time and blow-up rate are derived in Subsection 2.1 and 2.2. In section 3, we will use three methods to give the lower bounds for blow-up time and blow-up rate of the solutions if the blow-up does occurs. 2. Upper estimates for blow-up time and blow-up rate The purpose of this section is to establish some estimates about the upper bounds for blow-up time and blow-up rate of the solutions to problem (1.1)-(1.3) under two different assumptions, respectively. 2.1. The first method To obtain the results of this subsection, we first assume that H.F. Di, L. Chen and Z.F. Song / Eur. J. Pure Appl. Math, 13 (3) (2020), 645-662 648 (f2): there exists a positive constant C1 > 2 such that∫ Ω a(x)smf(s)dx ≥ C1 ∫ Ω a(x)F (s)dx, for any function s(x) ≥ 0, where F (s) = m ∫ s 0 θ m−1f(θ)dθ; (g1): the initial data g(x) satisfies∫ Ω |∇gm|2dx < 2 ∫ Ω a(x)F (g)dx. Then, inspired by Payne et al. [16, 17], we further define the following auxiliary function ϕ(t) = ∫ Ω um+1dx. (2.1) Theorem 1. Assume that the conditions (f1), (f2), (g1), (a1), (a2) hold, and u is a nonnegative solution of problem (1.1)-(1.3). Then, we conclude that the solution u becomes unbounded in Lm+1−norm at t = t∗. Moreover, an upper bound for blow-up time t∗ is given by t∗ ≤ (m+ 1)ϕ(0) (m− 1)φ(0) , (2.2) and the upper estimate of blow-up rate can be given by ‖u‖m+1 ≤ ( (m+ 1)ϕ(0) 2m m+1 (m− 1)φ(0) ) 1 m−1 (t∗ − t)− 1 m−1 , (2.3) where ϕ(0) = ‖g‖m+1 m+1 and φ(0) = −(m+ 1) ∫ Ω |∇g m|2dx+ 2(m+ 1) ∫ Ω a(x)F (g)dx > 0. Proof. Firstly, differentiating (2.1) with respect to t and using Eq.(1.1), then we have ϕ′(t) = (m+ 1) ∫ Ω umutdx = (m+ 1) ∫ Ω um ( 4um + a(x)f(u) ) dx = −(m+ 1) ∫ Ω |∇um|2dx+ (m+ 1) ∫ Ω a(x)umf(u)dx. (2.4) By the combination of (2.4) and condition (f2), we obtain ϕ′(t) ≥ −(m+ 1) ∫ Ω |∇um|2dx+ C1(m+ 1) ∫ Ω a(x)F (u)dx > φ(t), (2.5) where φ(t) = −(m+ 1) ∫ Ω |∇um|2dx+ 2(m+ 1) ∫ Ω a(x)F (u)dx. (2.6) H.F. Di, L. Chen and Z.F. Song / Eur. J. Pure Appl. Math, 13 (3) (2020), 645-662 649 On the other hand, a simple computation yields φ′(t) = −2(m+ 1) ∫ Ω ∇um · (∇um)tdx + 2m(m+ 1) ∫ Ω a(x)um−1utf(u)dx = 2m(m+ 1) ∫ Ω um−1ut ( 4um + a(x)f(u) ) dx = 2m(m+ 1) ∫ Ω um−1u2 tdx ≥ 0. (2.7) Here, we have used the fact that u(x, t) = 0 (or ∂u ∂ν = 0) on ∂Ω. Using Schwarz’s inequality, we get (∫ Ω umutdx )2 ≤ ∫ Ω um+1dx ∫ Ω um−1u2 tdx. (2.8) Hence, multiplying ϕ(t) by φ′(t), it follows from (2.5) that ϕ(t)φ′(t) = 2m(m+ 1) ∫ Ω um+1dx ∫ Ω um−1u2 tdx ≥ 2m(m+ 1) (∫ Ω umutdx )2 = 2m m+ 1 [ϕ′(t)]2 ≥ 2m m+ 1 ϕ′(t)φ(t). (2.9) Thus, the above inequality implies that( φ(t) [ϕ(t)]− 2m m+1 )′ = [ϕ(t)]− 3m+1 m+1 { ϕ(t)φ′(t)− 2m m+ 1 ϕ′(t)φ(t) } ≥ 0. (2.10) Utilizing the assumption (g1) and (2.1), (2.7), we know that ϕ(0) = ‖g‖m+1 m+1 > 0, (2.11) and φ(t) ≥ φ(0) = −(m+ 1) ∫ Ω |∇gm|2dx+ 2(m+ 1) ∫ Ω a(x)F (g)dx > 0. (2.12) Integrating (2.10) from 0 to t, we obtain φ(t) [ϕ(t)]− 2m m+1 ≥ φ(0) [ϕ(0)]− 2m m+1 = M > 0. (2.13) By (2.5) and (2.13), we get m+ 1 1−m ( [ϕ(t)] 1−m m+1 )′ = ϕ′(t) [ϕ(t)]− 2m m+1 ≥ φ(t) [ϕ(t)]− 2m m+1 > 0. (2.14) H.F. Di, L. Chen and Z.F. Song / Eur. J. Pure Appl. Math, 13 (3) (2020), 645-662 650 Integrating (2.14) from 0 to t, we have [ϕ(t)] 1−m m+1 ≤ [ϕ(0)] 1−m m+1 − m− 1 m+ 1 Mt. (2.15) Clearly, the above inequality cannot hold for all t > 0. Consequently, u blows up at some finite time t∗ and t∗ ≤ m+ 1ϕ(0) (m− 1)φ(0) . Furthermore, from (2.5) and (2.13) again, we have ϕ′(t) ≥ φ(t) ≥ φ(0) [ϕ(0)]− 2m m+1 [ϕ(t)] 2m m+1 . (2.16) Integrating (2.16) from t to t∗, we obtain ϕ(t) ≤ ( (m+ 1)ϕ(0) 2m m+1 (m− 1)φ(0) )m+1 m−1 (t∗ − t)− m+1 m−1 , (2.17) which implies that the upper estimate of blow-up rate is given by (2.3). 2.2. The second method We first assume that (f3): there exists a positive function G(θ) such that∫ Ω a(x)f(s)dx ≥ C2G [∫ Ω sdx ] with ∫ +∞ 0 dθ G(θ) < +∞, for any function s(x) ≥ 0. Then, we define the following auxiliary function ϕ1(t) = ∫ Ω udx. (2.18) Theorem 2. Assume that the conditions (f1), (f3), (a1), (a2) hold, and u is a nonnegative solution of problem (1.1)-(1.3). Then, we conclude that the solution u becomes unbounded in L1−norm at t = t∗. Moreover, an upper bound for blow-up time t∗ is given by t∗ ≤ ∫ +∞ ϕ1(0) dθ G(θ) < +∞, (2.19) and the upper estimate of blow-up rate can be given by ‖u‖L1 ≤ Y −1(t∗ − t), (2.20) where the function Y (s) := ∫ +∞ s dθ G(θ) for any function s(x) ≥ 0, and ϕ1(0) = ∫ Ω gdx. H.F. Di, L. Chen and Z.F. Song / Eur. J. Pure Appl. Math, 13 (3) (2020), 645-662 651 Proof. Integrating the Eq.(1.1) by parts, from the condition (f3) and (2.18) we have∫ Ω utdx = ∫ Ω a(x)f(u)dx ≥ C2G [∫ Ω udx ] , (2.21) which means that ϕ′1(t) ≥ C2G [ϕ1(t)] > 0. (2.22) Here, we have used the fact that u(x, t) = 0 ( or ∂u ∂ν = 0) on ∂Ω. It then follows from (2.22) that ϕ1(t) is a increasing function, so we have ϕ1(t) > ϕ1(0) = ∫ Ω g(x)dx ≥ 0. (2.23) Integrating (2.22) from 0 to t and using (2.23), (f3), we discover t ≤ ∫ ϕ1(t) ϕ1(0) dθ G(θ) ≤ ∫ +∞ ϕ1(0) dθ G(θ) < +∞. (2.24) Obviously, (2.24) cannot hold for all time t. Consequently, we can derive an upper bound t∗ such that t∗ ≤ ∫ +∞ ϕ1(0) dθ G(θ) < +∞, and lim t→t∗ ϕ1(t) = +∞, (2.25) where (0, t∗) is the interval of existence of the solutions u in L1−norm. In fact, if the equality (2.25) doesn’t hold, then there exists a time t1 > t∗ such that ϕ1(t∗) < ϕ1(t1) < +∞ and t1 satisfies the inequality (2.23),(2.24), which contradict the maximum existence of t∗. Furthermore, integrating (2.22) from t to t∗, it follows that t∗ − t ≤ ∫ +∞ ϕ1(t) dθ G(θ) := Y (ϕ1(t)). (2.26) We note that Y is a decreasing function, which means its inverse function Y −1 exists and is also a decreasing function. Therefore, we have ϕ1(t) ≤ Y −1(t∗ − t), (2.27) which implies that the estimate (2.20) of blow-up rate holds. Remark 1. This result can be generalized to the case of problem (1.1)-(1.3) subject to ∂u ∂ν = b(x, t) ≥ 0. In this case, it follows that ϕ′1(t) ≥ m ∫ ∂Ω um−1b(x, t)ds+ C2G [ϕ1(t)] > 0. (2.28) We also can obtain the inequalities (2.19) and (2.20). H.F. Di, L. Chen and Z.F. Song / Eur. J. Pure Appl. Math, 13 (3) (2020), 645-662 652 3. Lower estimates for blow-up time and blow-up rate In this section, we will give three methods to establish the lower bounds for blow-up time and blow-up rate of the solution to problem (1.1)-(1.3). 3.1. The first method Firstly, let us assume that (f4): there exists positive constants C3, C4 such that a(x)f(s) ≤ C3 + C4s l+1, for any function s(x) ≥ 0, where 0 < l ≤ 2nm−(n−2)(m+1) n . And then we introduce the auxiliary function ϕ(t) = ∫ Ω u m+1dx as (2.1). Next, we shall state and prove the main results of this subsection as follows: Theorem 3. Assume that the conditions (f1), (f4), (a1), (a2) hold, and u is a nonnegative solution of problem (1.1)-(1.3) which becomes unbounded in Lm+1−norm at t = t∗. Then, we conclude that a lower bound for blow-up time t∗ is given by t∗ ≥ ∫ +∞ ϕ(0) dη k1η + k2η 1+ 2 nε2 . (3.1) and the lower estimate of blow-up rate is ‖u‖m+1 ≥ (4k2 nε2 )− nε2 2(m+1) (t∗ − t)− nε2 2(m+1) , (3.2) where k1, k2 and ε2 are positive constants which will be given later. Proof. Differentiating (2.1) with respect to t and using the condition (f4) and (1.1), we have ϕ′(t) = (m+ 1) ∫ Ω umutdx = (m+ 1) ∫ Ω um ( 4um + a(x)f(u) ) dx = −(m+ 1) ∫ Ω |∇um|2dx+ (m+ 1) ∫ Ω a(x)umf(u)dx ≤ −(m+ 1) ∫ Ω |∇um|2dx+ (m+ 1)C3 ∫ Ω umdx + (m+ 1)C4 ∫ Ω um+l+1dx. (3.3) From Hölder’s inequality, Young’s inequality and condition (f4), we have∫ Ω umdx ≤ m m+ l + 1 ∫ Ω um+l+1dx+ l + 1 m+ l + 1 |Ω|, (3.4) H.F. Di, L. Chen and Z.F. Song / Eur. J. Pure Appl. Math, 13 (3) (2020), 645-662 653 and ∫ Ω um+l+1dx = ∫ Ω um+1−ε1ul−ε1dx ≤ ( ∫ Ω um+1dx )m+1−ε1 m+1 ( ∫ Ω u (l+ε1)(m+1) ε1 dx ) ε1 m+1 = ( ∫ Ω um+1dx )m+1−ε1 m+1 ( ∫ Ω u 2nm n−2 dx ) ε1 m+1 ≤ ε2C(ε) 1 + ε2 ( ∫ Ω um+1dx ) (m+1−ε1)(ε2+1) (m+1)ε2 + ε 1 + ε2 ( ∫ Ω u 2nm n−2 dx ) ε1(1+ε2) m+1 , (3.5) where ε1 = (n−2)(m+1)l 2nm−(n−2)(m+1) > 0, ε2 = n(2m−l)−(n−2)(m+1) nl > 0, and ε will be determined later. Noting that ∫ Ω u 2nm n−2 dx ≤ C 2n n−2 5 ( ∫ Ω |∇um|2 ) n n−2 , (3.6) where C5 is the optimal constant of the Sobolev embedding H1(Ω) ↪→ L 2n n−2 (Ω). Furthermore, from the choice of ε1 and ε2, it is easy to see that ε1(1+ε2)n (m+1)(n−2) = 1. Inserting (3.4)-(3.6) into (3.3), it follows that ϕ′(t) ≤ −(m+ 1) ∫ Ω |∇um|2dx+ (m+ 1)C3(l + 1) m+ l + 1 |Ω| + [m(m+ 1)C3 m+ l + 1 + (m+ 1)C4 ] ∫ Ω um+l+1dx ≤ − [ m+ 1− (m+ 1)(mC3 +mC4 + lC4 + C4)C2 5ε (1 + ε2)(m+ l + 1) ] ∫ Ω |∇um|2dx + (m+ 1)(mC3 +mC4 + lC4 + C4)ε2C(ε) (1 + ε2)(m+ l + 1) ( ∫ Ω um+1dx ) (m+1−ε1)(ε2+1) (m+1)ε2 + (m+ 1)C3(l + 1) m+ l + 1 |Ω|. (3.7) Taking ε small enough such that m+ 1− (m+ 1)(mC3 +mC4 + lC4 + C4)C2 5ε (1 + ε2)(m+ l + 1) > 0. (3.8) Hence, we have ϕ′(t) ≤ k1 + k2[ϕ(t)] (m+1−ε1)(1+ε2) (m+1)ε2 = k1 + k2[ϕ(t)] 1+ 2 nε2 , (3.9) where k1 = (m+1)C3(l+1) m+l+1 |Ω| and k2 = (m+1)(mC3+mC4+lC4+C4)ε2C(ε) (1+ε2)(m+l+1) . H.F. Di, L. Chen and Z.F. Song / Eur. J. Pure Appl. Math, 13 (3) (2020), 645-662 654 Integrating (3.9) from 0 to t, we get∫ ϕ(t) ϕ(0) dη k1η + k2η 1+ 2 nε2 ≤ t. (3.10) If u blows up in the measure ϕ(t) as t→ t∗, then we can obtain the lower bound t∗ ≥ ∫ +∞ ϕ(0) dη k1η + k2η 1+ 2 nε2 . Moreover, integrating the inequality (3.9) from t to t∗, we obtain t∗ − t ≥ ∫ +∞ ϕ(t) dη k1η + k2η 1+ 2 nε2 := Y1(ϕ(t)). (3.11) We note that Y1 is a decreasing function, which means its inverse function Y −1 1 exists and is also a decreasing function. Therefore, we have ϕ(t) ≥ Y −1 1 (t∗ − t), (3.12) which gives the lower estimate of blow-up rate. In fact, if t closes t∗ enough such that ϕ(t)� 1 and k2η 1+ 2 nε2 > k1η in the inequality (3.11), then we have t∗ − t ≥ nε2 4k2 [ ϕ(t) ]− 2 nε2 , (3.13) which means that ϕ(t) ≥ (4k2 nε2 )−nε2 2 (t∗ − t)− nε2 2 . (3.14) Thus, the estimate (3.2) of blow-up rate also holds. 3.2. The second method Firstly, we need the following assumption: (f5): there exists positive constants C6, C7, q and Q such that a(x)f(s) ≤ C6 + C7s p (∫ Ω sq+1dx )Q , for any function s(x) ≥ 0. (e1): we also assume that 0 ≤ p ≤ 1, 0 ≤ q ≤ m, and (q + 1)Q+ p > 1. To obtain the main results, we define the auxiliary function ϕ(t) = ∫ Ω u m+1dx again. Next, we will state our results below: H.F. Di, L. Chen and Z.F. Song / Eur. J. Pure Appl. Math, 13 (3) (2020), 645-662 655 Theorem 4. Assume that the conditions (f1), (f5), (e1), (a1), (a2) hold, and u is a nonnegative solution of problem (1.1)-(1.3) which becomes unbounded in Lm+1−norm at t = t∗. Then, we conclude that a lower bound for blow-up time t∗ is given by t∗ ≥ ∫ +∞ ϕ(0) dη k3η m m+1 + k4η m+p+(q+1)Q m+1 , (3.15) and the lower estimate of blow-up rate is ‖u‖m+1 ≥ [2k4((q + 1)Q+ p− 1) m+ 1 ]− 1 (q+1)Q+p−1 (t∗ − t)− 1 (q+1)Q+p−1 , (3.16) where k3 = C6(m+ 1)|Ω| 1 m+1 , k4 = C7(m+ 1)|Ω| 1−p+(m−q)Q m+1 . Proof. Under the assumption condition (f5), we have from (1.1) and (2.1) that ϕ′(t) = (m+ 1) ∫ Ω umutdx = (m+ 1) ∫ Ω um ( 4um + a(x)f(u) ) dx = −(m+ 1) ∫ Ω |∇um|2dx+ (m+ 1) ∫ Ω a(x)umf(u)dx ≤ C6(m+ 1) ∫ Ω umdx+ C7(m+ 1) ∫ Ω um+pdx (∫ Ω uq+1dx )Q . (3.17) Applying Hölder’s inequality and condition (e1), we know that∫ Ω umdx ≤ (∫ Ω um+1dx ) m m+1 |Ω| 1 m+1 , (3.18) ∫ Ω um+pdx ≤ (∫ Ω um+1dx )m+p m+1 |Ω| 1−p m+1 , (3.19) and ∫ Ω uq+1dx ≤ (∫ Ω um+1dx ) q+1 m+1 |Ω| m−q m+1 . (3.20) Inserting (3.18)-(3.20) into (3.17), it follows that ϕ′(t) ≤ C6(m+ 1)|Ω| 1 m+1 (∫ Ω um+1dx ) m m+1 + C7(m+ 1)|Ω| 1−p+(m−q)Q m+1 (∫ Ω um+1dx )m+p+(q+1)Q m+1 , (3.21) H.F. Di, L. Chen and Z.F. Song / Eur. J. Pure Appl. Math, 13 (3) (2020), 645-662 656 where from the condition (e1), it is easy to see that m+p+(q+1)Q m+1 > 1. Then, integrating the above inequality from 0 to t yields that∫ ϕ(t) ϕ(0) dη k3η m m+1 + k4η m+p+(q+1)Q m+1 ≤ t, where k3 = C6(m+ 1)|Ω| 1 m+1 , k4 = C7(m+ 1)|Ω| 1−p+(m−q)Q m+1 . If u blows up in the measure ϕ(t) as t→ t∗, then we can obtain the lower bound t∗ ≥ ∫ +∞ ϕ(0) dη k3η m m+1 + k4η m+p+(q+1)Q m+1 . Furthermore, integrating the inequality (3.21) from t to t∗, we obtain t∗ − t ≥ ∫ +∞ ϕ(t) dη k3η m m+1 + k4η m+p+(q+1)Q m+1 := Y2(ϕ(t)). (3.22) We note that Y2 is a decreasing function, which means its inverse function Y −1 2 exists and it is also a decreasing function. Therefore, we have ϕ(t) ≥ Y −1 2 (t∗ − t), (3.23) which gives the lower estimate of blow-up rate. In fact, the auxiliary function ϕ(t) becomes unbounded at time t = t∗, so we know that ϕ(t)� 1 and the inequality k4η m+p+(q+1)Q m+1 > k3η m m+1 as t→ t∗−. Hence, when t is close to t∗, inserting the above inequality into (3.22) and then a direct calculation yields that t∗ − t ≥ m+ 1 2k4[(q + 1)Q+ p− 1] [ ϕ(t) ]− (q+1)Q+p−1 m+1 , (3.24) which means that ϕ(t) ≥ [2k4((q + 1)Q+ p− 1) m+ 1 ]− m+1 (q+1)Q+p−1 (t∗ − t)− m+1 (q+1)Q+p−1 . (3.25) Hence, the estimate (3.16) of blow-up rate holds. 3.3. The third method This subsection is devoted to the estimates of the lower bounds for blow-up time and blow-up rate of the solutions to problem (1.1)-(1.3) by utilizing the method appearing in [18, 19]. For this purpose, we need to assume that (f6): sm−1 ≥ α (∫ +∞ s dη f(η) )−γ , for any function s(x) ≥ 0, where α, γ are positive constants and 0 < γ < 1; H.F. Di, L. Chen and Z.F. Song / Eur. J. Pure Appl. Math, 13 (3) (2020), 645-662 657 (f7): there exist positive constants k and β such that k > 4(n−2)−nγ 2n , f(s) (∫ +∞ s dη f(η) )nk+1 → +∞, as s→ 0+, and f ′(s) ∫ +∞ s dη f(η) ≤ nk + 1− β, for any function s(x) ≥ 0. Then, we define the following auxiliary function ϕ2(t) = ∫ Ω V nk(u)dx, V (u) = (∫ +∞ u dη f(η) )−1 . (3.26) Next, we will state our results below: Theorem 5. Assume that the conditions (f1), (f6), (f7), (a1), (a2) hold, and u is a nonnegative solution of problem (1.1)-(1.3) which becomes unbounded in ϕ2(t)-form at t = t∗. Then, we conclude that a lower bound for blow-up time t∗ is given by t∗ ≥ ∫ +∞ ϕ2(0) dη k5 + k6η 3n−6 3n−8 , (3.27) and the lower estimate of blow-up rate can be given by ϕ2(t) ≥ ( 4k6 3n− 8 )− 3n−8 2 (t∗ − t)− 3n−8 2 , (3.28) where k5, k6 will be given later, and ϕ2(0) = ∫ Ω [∫ +∞ g dη f(η) ]−nk dx. Proof. Under the assumptions (f7), we have from (1.1) and (3.26) that ϕ′2(t) = nk ∫ Ω V nk+1[f(u)]−1utdx = nk ∫ Ω V nk+1[f(u)]−1[4um + a(x)f(u)]dx = −mnk(nk + 1) ∫ Ω V nk+2[f(u)]−2|∇u|2um−1dx+ nk ∫ Ω V nk+1a(x)dx +mnk ∫ Ω V nk+1[f(u)]−2f ′(u)um−1|∇u|2dx. ≤ −mnk(nk + 1) ∫ Ω V nk+2[f(u)]−2|∇u|2um−1dx+ nk ∫ Ω V nk+1a(x)dx +mnk(nk + 1− β) ∫ Ω V nk+2[f(u)]−2um−1|∇u|2dx H.F. Di, L. Chen and Z.F. Song / Eur. J. Pure Appl. Math, 13 (3) (2020), 645-662 658 = nk ∫ Ω V nk+1a(x)dx−mnkβ ∫ Ω V nk+2[f(u)]−2um−1|∇u|2dx. (3.29) Since |∇V nk+γ 2 |2 = ( nk + γ 2 )2 V nk+γ+2[f(u)]−2|∇u|2. (3.30) In view of assumptions (f6), (3.29) and (3.30), we discover that ϕ′2(t) ≤ nk ∫ Ω V nk+1a(x)dx− 4nkαβ (nk + γ)2 ∫ Ω |∇V nk+γ 2 |2dx. (3.31) For convenience, we denote ε3 = 3nk(n− 2) + n(nk + γ)− 4(n− 2)(nk + 1) 3nk(n− 2) + n(nk + γ) , ε4 = 4(n− 2)(nk + 1) 3nk(n− 2) + n(nk + γ) , where the assumptions (f7) implies that ε3 > 0, ε4 > 0 and ε3 + ε4 = 1. So by Hölder’s inequality, we have∫ Ω V nk+1a(x)dx ≤ (∫ Ω V 3nk(n−2)+n(nk+γ) 4(n−2) dx )ε4 (∫ Ω a(x) 1 ε3 dx )ε3 . (3.32) Using Hölder’s inequality again, it follows that∫ Ω V 3nk(n−2)+n(nk+γ) 4(n−2) dx ≤ (∫ Ω V nkdx ) 3 4 (∫ Ω V n(nk+γ) n−2 dx ) 1 4 . (3.33) Furthermore, applying the Sobolev’s inequality, we obtain that(∫ Ω V n(nk+γ) n−2 dx )n−2 2n = (∫ Ω (V nk+γ 2 ) 2n n−2dx )n−2 2n ≤ C5 (∫ Ω |∇V nk+γ 2 |2dx ) 1 2 , (3.34) where C5 is the optimal Sobolev’s embedding constant defined as (3.6). Substituting (3.34) into (3.33) yields that∫ Ω V 3nk(n−2)+n(nk+γ) 4(n−2) dx ≤ C n 2(n−2) 5 (∫ Ω V nkdx ) 3 4 (∫ Ω |∇V nk+γ 2 |2dx ) n 4(n−2) . (3.35) By Young’s inequality, (3.32) and (3.35), we obtain∫ Ω V nk+1a(x)dx H.F. Di, L. Chen and Z.F. Song / Eur. J. Pure Appl. Math, 13 (3) (2020), 645-662 659 ≤ ( C n 2(n−2) 5 (∫ Ω V nkdx ) 3 4 (∫ Ω |∇V nk+γ 2 |2dx ) n 4(n−2) )ε4 (∫ Ω a(x) 1 ε3 dx )ε3 ≤ ε4C n 2(n−2) 5 (∫ Ω V nkdx ) 3 4 (∫ Ω |∇V nk+γ 2 |2dx ) n 4(n−2) + ε3 ∫ Ω a(x) 1 ε3 dx. (3.36) Applying Young’s inequality again, we have(∫ Ω V nkdx ) 3 4 (∫ Ω |∇V nk+γ 2 |2dx ) n 4(n−2) ≤ nδ 4(n− 2) ∫ Ω |∇V nk+γ 2 |2dx+ (3n− 8)C(δ) 4(n− 2) (∫ Ω V nkdx ) 3n−6 3n−8 , (3.37) where n 4(n−2) > 1, 3n−8 4(n−2) > 1 with n 4(n−2) + 3n−8 4(n−2) = 1. Consequently, it follows that∫ Ω V nk+1a(x)dx ≤ ε3 ∫ Ω a(x) 1 ε3 dx+ ε4nδ 4(n− 2) ∫ Ω |∇V nk+γ 2 |2dx + ε4(3n− 8)C(δ) 4(n− 2) (∫ Ω V nkdx ) 3n−6 3n−8 . (3.38) Substituting (3.38) into (3.29), we get ϕ′2(t) ≤ ε3 ∫ Ω a(x) 1 ε3 dx+ ε4nk(3n− 8)C(δ) 4(n− 2) (∫ Ω V nkdx ) 3n−6 3n−8 − [ 4nkαβ (nk + γ)2 − ε4n 2kδ 4(n− 2) ] ∫ Ω |∇V nk+γ 2 |2dx. (3.39) Now, we can choose δ small enough to make the coefficient 4nkαβ (nk+γ)2 − ε4n2kδ 4(n−2) > 0. Hence, we have ϕ′2(t) ≤ k5 + k 3n−6 3n−8 6 , (3.40) where k5 = ε3 ∫ Ω a(x) 1 ε3 dx and k6 = ε4nk(3n−8)C(δ) 4(n−2) . Then, integrating (3.40) from 0 to t yields that∫ ϕ2(t) ϕ2(0) dη k5 + k6η 3n−6 3n−8 ≤ t. (3.41) If u blows up in the measure ϕ2(t) as t→ t∗, then we can obtain the lower bound t∗ ≥ ∫ +∞ ϕ2(0) dη k5 + k6η 3n−6 3n−8 . REFERENCES 660 Similar to the above derivation of (3.14) and (3.25), it is easy to get ϕ2(t) ≥ ( 4k6 3n− 8 )− 3n−8 2 (t∗ − t)− 3n−8 2 . This completes the proof of Theorem 5. Acknowledgements The authors would like to express their thanks to the editor and the referees for their helpful comments. 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