Blow-Up Analysis for a Quasilinear Degenerate Parabolic Equation with Strongly Nonlinear Source

Abstract

We investigate the blow-up properties of the positive solution of the Cauchy problem for a quasilinear degenerate parabolic equation with strongly nonlinear source 𝑢_𝑡=𝑑𝑖𝑣(|∇𝑢^𝑚|^𝑝−2∇𝑢^𝑙)+𝑢^𝑞,(𝑥,𝑡)∈𝑅^𝑁×(0,𝑇), where 𝑁≥1, 𝑝>2 , and 𝑚, 𝑙,𝑞>1, and give a secondary critical exponent on the decay asymptotic behavior of an initial value at infinity for the existence and nonexistence of global solutions of the Cauchy problem. Moreover, under some suitable conditions we prove single-point blow-up for a large class of radial decreasing solutions.

1. Introduction

In this paper, we consider the following Cauchy problem to a quasilinear degenerate parabolic equation with strongly nonlinear source 𝑢_𝑡=div∇𝑢^𝑚𝑝−2∇𝑢^𝑙+𝑢^𝑞,(𝑥,𝑡)∈𝑅^𝑁×(),𝑢()0,𝑇𝑥,0=𝑢₀(𝑥),𝑥∈𝑅^𝑁, (1.1) where 𝑁≥1, 𝑝>2, ?𝑚?,??𝑙?,??𝑞>1, and the initial data 𝑢₀(𝑥) is nonnegative bounded and continuous.

Equation (1.1) has been suggested as a mathematical model for a variety of physical problems (see [1]). For instance, it appears in the non-Newtonian fluids and is a nonlinear form of heat equation. Moreover, it can also be used to model the nonlinear heat propagation in a reaction medium (see [2]).

One of the particular features of problem (1.1) is that the equation is degenerate at points where 𝑢=0 or ∇𝑢=0. Hence, there is no classical solution in general and we introduce the following definition of weak solution (see [3, 4]).

Definition 1.1. A nonnegative measurable function 𝑢(𝑥,𝑡) defined in 𝑅^𝑁×(0,𝑇) is called a weak solution of the Cauchy problem (1.1) if for every bounded open set Ω with smooth boundary 𝜕Ω, 𝑢∈𝐶_loc(Ω×(0,𝑇)), 𝑢^𝑚,𝑢^𝑙∈𝐿^𝑝_loc(0,𝑇;𝑊^1,𝑝(Ω)), and _Ω𝑢𝜑𝑑𝑥+^𝑡_𝑡0Ω−𝑢𝜑_𝑡+∇𝑢^𝑚𝑝−2∇𝑢^𝑙⋅∇𝜑𝑑𝑥𝑑𝑡=^𝑡_𝑡0Ω𝑢^𝑞𝜑𝑑𝑥𝑑𝑡+_Ω𝑢𝑥,𝑡₀𝜑𝑥,𝑡₀𝑑𝑥 (1.2) for all 0≤𝑡₀≤𝑡≤𝑇 and all test functions 𝜑∈𝐶¹₀(Ω×(0,𝑇)). Moreover, lim_{𝑡→𝑡0Ω}𝑢(𝑥,𝑡)𝜂(𝑥)𝑑𝑥=_Ω𝑢𝑥,𝑡₀𝜂(𝑥)𝑑𝑥 (1.3) for any 𝜂(𝑥)∈𝐶¹₀(Ω).

Under some suitable assumptions, the existence, uniqueness and regularity of a weak solution to the Cauchy problem (1.1) and their variants have been extensively investigated by many authors (see [5–7] and the references therein).

The first goal of this paper is to study the blow-up behavior of solution 𝑢(𝑥,𝑡) of (1.1) when the initial data 𝑢₀(𝑥) has slow decay near 𝑥=∞. For instance, in the following case 𝑢₀(𝑥)≅𝑀|𝑥|^−𝑎with𝑀>0,𝑎≥0, (1.4) we investigate the existence of global and nonglobal solutions for the Cauchy problem (1.1) in terms of 𝑀 and 𝑎. In recent years, many authors have studied the properties of solutions to the Cauchy problem (1.1) and their variants (see [8–17] and the references therein). In particular, J.-S. Guo and Y. Y. Guo [18] obtained the secondary critical exponent for the following porous medium type equation in high dimensions: 𝑢_𝑡=Δ𝑢^𝑚+𝑢^𝑝,(𝑥,𝑡)∈𝑅^𝑁×(),𝑢()0,𝑇𝑥,0=𝑢₀(𝑥),𝑥∈𝑅^𝑁, (1.5) where 𝑝>1, 𝑚>1 or max{0,1−(2/𝑁)}<𝑚<1, 𝑢₀(𝑥) is nonnegative bounded and continuous, and proved that for 𝑝>𝑝^∗_𝑚=𝑚+(2/𝑁), there exists a secondary critical exponent 𝑎^∗=2/(𝑝−𝑚) such that the solution 𝑢(𝑥,𝑡) of (1.5) blows up in finite time for the initial data 𝑢₀(𝑥), which behaves like |𝑥|^−𝑎 at 𝑥=∞ if 𝑎∈(0,𝑎^∗), and there exists a global solution for the initial data 𝑢₀(𝑥), which behaves like |𝑥|^−𝑎 at 𝑥=∞ if 𝑎∈(𝑎^∗,𝑁). Here, we say that the solution blows up in finite time; it means that there exists 𝑇∈(0,+∞) such that ‖𝑢(⋅,𝑡)‖_𝐿∞<∞ for all 𝑡∈[0,𝑇), but lim_{𝑡→𝑇−}sup‖𝑢(.,𝑡)‖_𝐿∞=∞.

Mu et al. [19] studied the secondary critical exponent for the following 𝑝-Laplacian equation with slow decay initial values: 𝑢_𝑡=div|∇𝑢|^𝑝−2∇𝑢+𝑢^𝑞,(𝑥,𝑡)∈𝑅^𝑁×(),𝑢()0,𝑇𝑥,0=𝑢₀(𝑥),𝑥∈𝑅^𝑁, (1.6) where 𝑝>2, 𝑞>1, and showed that, for 𝑞>𝑞^∗_𝑐=𝑝−1+(𝑝/𝑁), there exists a secondary critical exponent 𝑎^∗_𝑐=(𝑝/(𝑞+1−𝑝)) such that the solution 𝑢(𝑥,𝑡) of (1.6) blows up in finite time for the initial data 𝑢₀(𝑥) which behaves like |𝑥|^−𝑎 at 𝑥=∞ if 𝑎∈(0,𝑎^∗_𝑐), and there exists a global solution for the initial data 𝑢₀(𝑥), which behaves like |𝑥|^−𝑎 at 𝑥=∞ if 𝑎∈(𝑎^∗_𝑐,𝑁).

Recently, Mu et al. [20] also investigated the secondary critical exponent for the doubly degenerate parabolic equation with slow decay initial values and obtained similar results.

On the other hand, in this paper, we will also consider single-point blow-up for the Cauchy problem (1.1). It is interesting to study the set of blow-up points and the behavior of the solution 𝑢(𝑥,𝑡) at the blow-up point.

In order to investigate single-point blow-up for the Cauchy problem (1.1), we introduce the concept of the blow-up point.

Definition 1.2. A point 𝑥∈Ω is called a blow-up point if there exists a sequence (𝑥_𝑛,𝑡_𝑛) such that 𝑥_𝑛→𝑥, 𝑡_𝑛→𝑇⁻ and 𝑢(𝑥_𝑛,𝑡_𝑛)→∞ as 𝑛→∞, where 𝑇 is blow-up time.

In recent years, some authors also studied single-point blow-up for the Cauchy problem to nonlinear parabolic equations (see [21, 22] and the references therein) by different methods. In particular, when 𝑝=2, 𝑙=1 and 𝑁=1, the Cauchy problem (1.1) has been investigated by Weissler in [23], and the author obtained that the solution blows up only at a single point. Galaktionov and Posashkov [24] studied the single-point blow-up and gave the upper and lower bound near the blow-up point for the Cauchy problem (1.1) when 𝑝>2 and 𝑚=𝑙=1. Recently, when 𝑝>2 and 𝑚=𝑙, Mu and Zeng [25] extended Galaktionov's results to the doubly degenerate parabolic equation. For more works about single-point blow-up, we refer to [26, 27], where the parabolic systems have been considered.

Motivated by the above works, based on a modification of the energy methods, comparison principle, and regularization methods used in [15, 19, 21, 24], we investigate the secondary critical exponent and single-point blow-up for the Cauchy problem (1.1). Before stating the results of the secondary critical exponent, we start with some notations as follows.

Let 𝐶_𝑏(𝑅^𝑁) be the space of all bounded continuous functions in 𝑅^𝑁. For 𝑎≥0, we define Φ^𝑎=𝜙(𝑥)∈𝐶_𝑏𝑅^𝑁∣𝜙(𝑥)≥0,lim_|𝑥|→∞sup|𝑥|^𝑎𝜙(𝑥),Φ<∞_𝑎=𝜙(𝑥)∈𝐶_𝑏𝑅^𝑁∣𝜙(𝑥)≥0,lim_|𝑥|→∞inf|𝑥|^𝑎𝜙(𝑥)>0. (1.7) Moreover, we denote 𝑞^∗_𝑐+𝑝=𝑙+𝑚𝑝−2𝑁,𝑎^∗_𝑐=𝑝.𝑞−𝑙−𝑚𝑝−2 (1.8) Our main results of this paper are stated as follows.

Theorem 1.3. For 𝑁≥2, 𝑝>2, 𝑚>1, 𝑙>1, and 𝑞>𝑞^∗_𝑐=𝑙+𝑚(𝑝−2)+(𝑝/𝑁), suppose that 𝑢₀(𝑥)∈Φ_𝑎 for some 𝑎∈(0,𝑎^∗_𝑐); then the solution 𝑢(𝑥,𝑡) of the Cauchy problem (1.1) blows up in finite time.

Theorem 1.4. For 𝑁≥2, 𝑝>2, 𝑚>1, 𝑙>1, and 𝑞>𝑞^∗_𝑐=𝑙+𝑚(𝑝−2)+(𝑝/𝑁), suppose that 𝑢₀(𝑥)=𝜆𝜑(𝑥) for some 𝜆>0 and 𝜑(𝑥)∈Φ^𝑎 for some 𝑎∈(𝑎^∗_𝑐,𝑁); then there is 𝜆₀=𝜆₀(𝜑)>0 such that the solution 𝑢(𝑥,𝑡) of the Cauchy problem (1.1) exists globally for all 𝑡>0, and if 𝜆<𝜆₀, one has ||𝑢(𝑥,𝑡)||_∞≤𝐶𝑡^−𝑎𝛽,∀𝑡>0, (1.9) where 𝛽=1/(𝑎(𝑙+𝑚(𝑝−2)−1)+𝑝), 𝐶>0.

Remark 1.5. When 𝑝>2, 𝑁≥2 and 𝑞>𝑞^∗_𝑐, we have 𝑞^∗_𝑐>1 and 0<𝑎^∗_𝑐<𝑁.

Remark 1.6. It follows from Theorems 1.3 and 1.4 that the number 𝑎^∗_𝑐=𝑝/(𝑞−𝑙−𝑚(𝑝−2)) gives another cut-off between the blow-up case and the global existence case. Therefore, the number 𝑎^∗_𝑐 is a new secondary critical exponent of the Cauchy problem (1.1). Unfortunately, in the critical case 𝑎=𝑎^∗_𝑐, we do not know whether the solution of (1.1) exists globally or blows up in finite time.

Remark 1.7. When 𝑚=𝑙=1 or 𝑚=𝑙>1, the results of Theorems 1.3 and 1.4 are consistent with those in [19, 20], respectively.

Remark 1.8. In [28], Afanas’eva and Tedeev also established the Fujita type results for (1.1) with 𝑚=𝑙. In particular, if 𝑢₀(𝑥)∼|𝑥|^−𝑎, 0<𝑎<𝑁, they obtained that if 𝑞<𝑚(𝑝−1)+(𝑝/𝑎), then every nontrivial solution blows up in finite time, and if 𝑞>𝑚(𝑝−1)+(𝑝/𝑎), then the solution exists globally for a small initial data 𝑢₀(𝑥). We note that when 𝑚=𝑙 in (1.1), if 𝑞>𝑚(𝑝−1)+(𝑝/𝑁) and 0<𝑎<𝑝/(𝑞−𝑚(𝑝−1)), then 0<𝑎<𝑁 and 𝑞<𝑚(𝑝−1)+(𝑝/𝑎), while if 𝑞>𝑚(𝑝−1)+(𝑝/𝑁) and 𝑝/(𝑞−𝑚(𝑝−1))<𝑎<𝑁, then 𝑞>𝑚(𝑝−1)+(𝑝/𝑎). Therefore, the results of Theorems 1.3 and 1.4 coincide with those in [28].

Finally, we also consider single-point blow-up for a large number of radial decreasing solutions of the Cauchy problem (1.1) and give upper bound of the radial solution 𝑢(𝑟,𝑡) in a small neighborhood of the point (𝑥,𝑡), where 𝑥=0, 𝑡=𝑇. We assume that the initial data 𝑢₀(𝑥)=𝑢₀(𝑟) satisfies the following condition:(H)𝑢₀(𝑥)=𝑢₀(𝑟)≥0 for 𝑟>0, 𝑢₀(0)>0, and 𝑢₀(𝑟)∈𝐶¹(𝑅¹₊), 𝑢^′₀(0)=0, and 𝑢^′₀(𝑟)≤0 for 𝑟>0, 𝑀₀=sup𝑢₀(𝑟)<+∞, 𝐾₀=sup|𝑢^′₀(𝑟)|<+∞.

Theorem 1.9. Let 𝑁≥1,𝑝>2,𝑚>1,𝑙>1,and𝑞>𝑙+𝑚(𝑝−2), and let condition (H) hold. In addition, assume that the initial function 𝑢₀(𝑥)=𝑢₀(𝑟) satisfies 𝑢^𝑞₀(𝑟)⋅𝑟^𝑁(1)𝜆=𝑜as𝑟⟶∞,₀=inf_{𝑟>0,𝑢0(𝑟)>0}−𝑢^𝑚₀^′𝑝−2𝑢^𝑙₀^′𝑟𝑢^𝑞₀∈0,𝑝−2𝑝−2(𝑁+1)+2. (1.10)(1.11) Let 𝑇 be the blow-up time; then one has 𝑢(𝑟,𝑡)≤𝐶𝑟^{−𝑝/(𝑞−𝑙−𝑚(𝑝−2))},(𝑟,𝑡)∈𝑅¹₊×(0,𝑇), (1.12) where 𝐶=𝑞−𝑙−𝑚𝑝−2(𝑙𝑚^𝑝−2)^1/(𝑝−1)𝑝𝜆₀^{1/(𝑝−1)−((𝑝−1)/(𝑞−𝑙−𝑚(𝑝−2)))}>0, (1.13) that is, there is single-point blow-up at point 𝑥=0.

Remark 1.10. By (1.11), the best upper estimate (1.12) obtained by our method has the following form: 𝑢(𝑟,𝑡)≤𝑞−𝑙−𝑚𝑝−2𝑝𝑙𝑚^{𝑝−21/(𝑝−1)}𝑝−2𝑝−2(𝑁+1)+2^{1/(𝑝−1)−((𝑝−1)/(𝑞−𝑙−𝑚(𝑝−2)))}⋅𝑟^{−(𝑝/(𝑞−𝑙−𝑚(𝑝−2)))} (1.14) in 𝑅¹₊×(0,𝑇). But, we do not give the lower bound estimate of the radial solution 𝑢(𝑟,𝑡) in a small neighborhood of the point (𝑥,𝑡), where 𝑥=0, 𝑡=𝑇.

Remark 1.11. When 𝑚=𝑙=1 or 𝑚=𝑙>1, the results of Theorem 1.9 are consistent with those in [24, 25], respectively. For 1<𝑞<𝑙+𝑚(𝑝−2), in [29], the authors obtained the results of global blow-up to arbitrary compactly supported initial data.

Remark 1.12. From Theorem 1.9, we obtain the same decay exponent as that of Theorem 1.2 in [29] by different methods. Moreover, it is interesting to see that the decay exponent of the upper estimate of Theorem 1.9 is also the same as the secondary critical exponent of Theorems 1.3 and 1.4.

This paper is organized as follows. In Section 2, by using the energy method, we will obtain a blow-up condition and prove Theorem 1.3. In Section 3, using the comparison principle, we can construct a global supersolution to prove Theorem 1.4. Finally, we consider the single-point blow-up under some suitable conditions and prove Theorem 1.9 in Section 4.

2. Blow-Up Case

By using the energy method, we will obtain a blow-up condition corresponding to (1.1). Therefore, we need to select a suitable test function as follows: 𝜓_𝜀(𝑥)=𝐴_𝜀𝑒^{−𝜀|𝑥|},𝐴_𝜀=1∫_𝑅𝑁𝑒^{−𝜀|𝑥|}=𝜀𝑑𝑥∫_𝜔𝑁∫^∞₀𝑒^𝑁−𝑟𝑟^𝑁−1𝑑𝑟𝑑𝑠,∇𝜓_𝜀(𝑥)=−𝜀𝜓_𝜀𝑥|𝑥|. (2.1)

Proof of Theorem 1.3. Suppose that 𝑢(𝑥,𝑡) is the solution of the Cauchy problem (1.1) and 𝑇 is the blow-up time. Let 𝐸(𝑡)=1𝑠_𝑅^𝑁𝑢^𝑠(𝑥,𝑡)𝜓_𝜀(𝑥)𝑑𝑥,𝑡∈[0,𝑇), (2.2) where 0<𝑠<1/𝑝,𝑝>2, Then, 𝐸(𝑡)∈𝐶[0,𝑇)⋂𝐶¹(0,𝑇) and 𝐸^′(𝑡)=_𝑅𝑁𝑢^𝑠−1𝜓_𝜀𝑢_𝑡𝑑𝑥=_𝑅𝑁𝑢^𝑠−1𝜓_𝜀div∇𝑢^𝑚𝑝−2∇𝑢^𝑙𝑑𝑥+_𝑅𝑁𝑢^{𝑞+𝑠−1}𝜓_𝜀()𝑑𝑥=−𝑙𝑠−1_𝑅𝑁𝑢^{𝑠+𝑙−3}𝜓_𝜀|∇𝑢^𝑚|^𝑝−2|∇𝑢|²𝑑𝑥+𝜀_𝑅𝑁𝑢^𝑠−1𝜓_𝜀|∇𝑢^𝑚|^𝑝−2∇𝑢^𝑙𝑥|𝑥|+𝑑𝑥_𝑅𝑁𝑢^{𝑞+𝑠−1}𝜓_𝜀𝑑𝑥≥𝑙𝑚^𝑝−2(1−𝑠)_𝑅𝑁𝑢^{𝑙+𝑚(𝑝−2)+𝑠−𝑝−1}𝜓_𝜀|∇𝑢|^𝑝𝑑𝑥−𝑙𝑚^𝑝−2𝜀_𝑅𝑁𝑢^{𝑙+𝑚(𝑝−2)+𝑠−𝑝}𝜓_𝜀|∇𝑢|^𝑝−1𝑑𝑥+_𝑅𝑁𝑢^{𝑞+𝑠−1}𝜓_𝜀𝑑𝑥. (2.3) Using Young's inequality, we have 𝜀_𝑅𝑁𝑢^{𝑙+𝑚(𝑝−2)+𝑠−𝑝}𝜓_𝜀|∇𝑢|^𝑝−1𝑑𝑥≤𝑝−1𝑝_𝑅^𝑁𝑢^{𝑙+𝑚(𝑝−2)+𝑠−𝑝−1}𝜓_𝜀|∇𝑢|^𝑝+𝜀𝑑𝑥_𝑝𝑝_𝑅^𝑁𝑢^{𝑙+𝑚(𝑝−2)+𝑠−1}𝜓_𝜀𝑑𝑥. (2.4) Since 0<𝑠<(1/𝑝), it follows from (2.3) and (2.4) that 𝐸^′(𝑡)≥_𝑅𝑁𝑢^{𝑞+𝑠−1}𝜓_𝜀𝑑𝑥−𝑙𝑚^𝑝−2𝜀^𝑝𝑝_𝑅^𝑁𝑢^{𝑙+𝑚(𝑝−2)+𝑠−1}𝜓_𝜀𝑑𝑥. (2.5) By 𝑞>𝑞^∗_𝑐=𝑙+𝑚(𝑝−2)+(𝑝/𝑁)>𝑙+𝑚(𝑝−2)>1, ∫_𝑅𝑁𝜓_𝜀(𝑥)𝑑𝑥=1, and Hölder’s inequality, we obtain _𝑅𝑁𝑢^{𝑙+𝑚(𝑝−2)+𝑠−1}𝜓_𝜀𝑑𝑥=_𝑅𝑁𝑢^{𝑙+𝑚(𝑝−2)+𝑠−1}𝜓_𝜀^{(𝑙+𝑚(𝑝−2)+𝑠−1)/(𝑞+𝑠−1)}𝜓_𝜀^{𝑞−[𝑙+𝑚(𝑝−2)]/(𝑞+𝑠−1)}≤𝑑𝑥_𝑅𝑁𝑢^{𝑞+𝑠−1}𝜓_𝜀𝑑𝑥^{(𝑙+𝑚(𝑝−2)+𝑠−1)/(𝑞+𝑠−1)}. (2.6) Therefore, by (2.5) and (2.6), we have 𝑑𝐸≥𝑑𝑡_𝑅^𝑁𝑢^{𝑞+𝑠−1}𝜓_𝜀𝑑𝑥^{(𝑙+𝑚(𝑝−2)+𝑠−1)/(𝑞+𝑠−1)}×_𝑅𝑁𝑢^{𝑞+𝑠−1}𝜓_𝜀𝑑𝑥^{(𝑞−𝑙−𝑚(𝑝−2))/(𝑞+𝑠−1)}−𝑙𝑚^𝑝−2𝜀^𝑝𝑝. (2.7) Applying Jensen’s inequality, we obtain _𝑅𝑁𝑢^{𝑞+𝑠−1}𝜓_𝜀𝑑𝑥^{(𝑞−𝑙−𝑚(𝑝−2))/(𝑞+𝑠−1)}≥_𝑅𝑁𝑢^𝑠𝜓_𝜀𝑑𝑥^{(𝑞−𝑙−𝑚(𝑝−2))/𝑠}. (2.8) Thus, it follows from (2.7) and (2.8) that 𝑑𝐸≥1𝑑𝑡2_𝑅^𝑁𝑢^𝑠𝜓_𝜀𝑑𝑥^{(𝑞+𝑠−1)/𝑠}=12𝑠^{(𝑞+𝑠−1)/𝑠}𝐸^{(𝑞+𝑠−1)/𝑠}(𝑡) (2.9) as long as 𝐸(𝑡)≥1𝑠2𝑙𝑚^𝑝−2𝜀^𝑝𝑝^{𝑠/(𝑞−𝑙−𝑚(𝑝−2))}∀𝑡∈[0,𝑇). (2.10) Hence, if 𝐸(0) satisfies 𝐸(0)≥1𝑠2𝑙𝑚^𝑝−2𝜀^𝑝𝑝^{𝑠/(𝑞−𝑙−𝑚(𝑝−2))}=𝐶₀, (2.11) then 𝐸(𝑡) increases and remains below 𝐶₀ for all 𝑡∈[0,𝑇).
And by (2.9) we have 𝐸(𝑡)≥𝐸^{𝑞−1/𝑠}(0)−𝐶₁𝑡^{−𝑠/(𝑞−1)},where𝐶₁=𝑞−12𝑠^{(𝑞−1)/𝑠}>0. (2.12) Therefore, from (2.11) and (2.12), we obtain that 𝑢(𝑥,𝑡) blows up in finite time 𝑇=(1/𝐶₁)𝐸^{𝑞−1/𝑠}(0): and get an estimate on the blow-up time 𝑇 of the solution 𝑢(𝑥,𝑡) as follows: 2𝑇≤𝑞−12𝑙𝑚^𝑝−2𝜀^𝑝𝑝^{(1−𝑞)/(𝑞−𝑙−𝑚(𝑝−2))}. (2.13) Finally, it remains to verify the blow-up condition (2.11). Since 𝑢₀(𝑥)∈Φ_𝑎 for some 𝑎∈(0,𝑎^∗_𝑐), there exist two positive constants 𝑀 and 𝑅₀>1 such that 𝑢₀(𝑥)≥𝑀|𝑥|^−𝑎 for all |𝑥|≥𝑅₀, and we have 𝐸(0)=1𝑠_𝑅^𝑁𝑢^𝑠₀(𝑥)𝜓_𝜀(𝑥)>𝑀𝑑𝑥_𝑠𝐴_𝜀𝑠_{|𝑥|≥𝑅0}|𝑥|^−𝑎𝑠𝑒^{−𝜀|𝑥|}=𝑀𝑑𝑥_𝑠𝐴_𝜀𝑠𝜀^{−𝑁+𝑎𝑠}_{|𝑦|≥𝜀𝑅0}𝑦^𝑎𝑠𝑒^−|𝑦|𝑑𝑦. (2.14) By the definition of 𝐴_𝜀, 0<𝑎<𝑎^∗_𝑐, we can choose 0<𝜀≤(1/𝑅₀) so small such that (2.11) holds. The proof of Theorem 1.3 is complete.

3. Global Existence

In this section, we shall prove Theorem 1.4 by constructing a global supersolution. To do this, we introduce the radially symmetric self-similar solution 𝑈_𝑀,𝑎(𝑥,𝑡) to the following Cauchy problem: 𝑢_𝑡=div∇𝑢^𝑚𝑝−2∇𝑢^𝑙,(𝑥,𝑡)∈𝑅^𝑁×(),𝑢()0,+∞𝑥,0=𝑢₀(𝑥)=𝑀|𝑥|^−𝑎,𝑥∈𝑅^𝑁. (3.1)(3.2) It is well known that the existence and uniqueness of the solution of (3.1) have been well established (see [7]). By symmetric properties of (3.1), the solution 𝑈_𝑀,𝑎(𝑥,𝑡) is given by the following form 𝑈_𝑀,𝑎(𝑥,𝑡)=𝑡^−𝑎𝛽𝑓_𝑀(𝑟)|𝑥|,with𝑟=𝑡^𝛽1,𝛽=𝑎,𝑙+𝑚𝑝−2−1+𝑝 (3.3) where the positive function 𝑓_𝑀 is the solution of the problem 𝑓^𝑚_𝑀^′𝑝−2𝑓^𝑙_𝑀^′′+𝑁−1𝑟𝑓^𝑚_𝑀^′𝑝−2𝑓^𝑙_𝑀^′(𝑟)+𝛽𝑟𝑓^′_𝑀(𝑟)+𝑎𝛽𝑓_𝑀(𝑟)𝑓=0,𝑟>0,_𝑀(𝑟)≥0,𝑟≥0,𝑓^′_𝑀(0)=0,lim_𝑟→+∞𝑟^𝑎𝑓_𝑀(𝑟)=𝑀. (3.4) We shall prove the existence of solution 𝑓_𝑀(𝑟) to (3.4) by the following ordinary differential equation, and furthermore we obtain the nonincreasing property of the solution 𝑓_𝑀(𝑟).

Firstly, given a fixed 𝜂>0, we consider the following Cauchy problem: 𝑔^{𝑚′𝑝−2}𝑔^𝑙′′+𝑁−1𝑟𝑔^{𝑚′𝑝−2}𝑔^𝑙′(𝑟)+𝛽𝑟𝑔^′(𝑟)(𝑟)𝑔(0)+𝑎𝛽𝑔=0,𝑟>0,=𝜂,𝑔^′(0)=0. (3.5) According to the standard of the Cauchy problem for ODE and the methods used in [7, 30], we can obtain that the solution 𝑔(𝑟) of the Cauchy problem (3.5) is positive, and 𝑔(𝑟)→0 as 𝑟→∞; moreover, lim_𝑟→+∞𝑟^𝑎𝑔(𝑟)=𝑀 (3.6) for some 𝑀=𝑀(𝜂)>0.

Secondly, we shall prove that there exists a one-to-one correspondence between 𝑀∈(0,+∞) and 𝜂∈(0,+∞). Indeed, this can be seen from the following relation: 𝑔_𝜂(𝑟)=𝜂𝑔₁𝜂^𝜎𝑟,𝜎=1−𝑙−𝑚𝑝−2𝑝, (3.7) where 𝑔₁(𝑟) is the solution of (3.5) for 𝜂=1. Then, 𝑀𝜂=𝜂^1−𝑎𝜎𝑀(1),with𝑀(1)=lim_𝑟→+∞𝑟^𝑎𝑔₁(𝑟). (3.8) Therefore, we can deduce that, for each 𝑀>0, there exists a positive, bounded, and global solution 𝑓_𝑀(𝑟) satisfying (3.4).

Finally, we shall prove that the solution 𝑔(𝑟) is non-increasing, that is, 𝑓_𝑀(𝑟) is also non-increasing. To do this, we need the following lemmas.

Lemma 3.1. Let 𝑔(𝑟) be the solution of (3.5); then lim_𝑟→0𝑁𝑔^𝑚′(𝑟)^𝑝−2𝑔^𝑙′(𝑟)𝑟=−𝑎𝛽𝑔(0). (3.9)

Proof. Integrating the (3.5) over (0,𝜀) with 𝜀>0, we have 𝑔^{𝑚′𝑝−2}𝑔^𝑙′(𝜀)+^𝜀₀𝑁−1𝑟𝑔^{𝑚′𝑝−2}𝑔^𝑙′𝑑𝑟+^𝜀₀𝛽𝑟𝑔^′𝑑𝑟+^𝜀₀𝑎𝛽𝑔𝑑𝑟=0. (3.10) Dividing by 𝜀 and taking 𝜀→0 in (3.10), we obtain lim_𝜀→0𝑔^𝑚′(𝜀)^𝑝−2𝑔^𝑙′(𝜀)𝜀+𝑁−1𝜀𝑔^𝑚′(𝜀)^𝑝−2𝑔^𝑙′(𝜀)=−lim_𝜀→0𝑎𝛽𝑔(𝜀), (3.11) which implies that (3.9) holds. The proof of Lemma 3.1 is complete.

Lemma 3.2. If there exists 𝑟₀∈[0,+∞) such that 𝑔(𝑟₀)=0, then 𝑔(𝑟)=0 for all 𝑟≥𝑟₀.

Proof. We shall prove by contradiction. Assuming that Lemma 3.2 does not hold, it is easy to see that there exists 𝜀>0 such that 𝑔(𝑟)>0,𝑔^′(𝑟)>0in𝑟₀,𝑟₀+𝜀. (3.12) Multiplying (3.5) by 𝑟^𝑁−1 and integrating over (𝑟₀,𝑟) with 𝑟∈(𝑟₀,𝑟₀+𝜀), we obtain 𝑟^𝑁−1𝑔^{𝑚′𝑝−2}𝑔^𝑙′+𝛽𝑟^𝑁𝑔(𝑟)=^𝑟_𝑟0𝑁𝛽𝑟^𝑁−1𝑔(𝑟)𝑑𝑟−^𝑟_𝑟0𝑎𝛽𝑟^𝑁−1𝑔(𝑟)𝑑𝑟. (3.13) It follows from (3.12) and (3.13) that 𝛽𝑟^𝑁𝑔(𝑟)≤^𝑟_𝑟0𝑁𝛽𝑟^𝑁−1𝑔(𝑟)𝑑𝑟−^𝑟_𝑟0𝑎𝛽𝑟^𝑁−1𝑔(𝑟)()𝛽𝑑𝑟≤𝑁−𝑎𝑁𝑔(𝑟)𝑟^𝑁−𝑟^𝑁₀, (3.14) equivalently, 1≤𝑁−𝑎𝑁𝑟1−⁰𝑟^𝑁. (3.15) Letting 𝑟→𝑟₀ in (3.15), we obtain the inequality 1≤0, which is a contradiction. The proof of Lemma 3.2 is complete.

Lemma 3.3. The solution 𝑔(𝑟) of (3.5) is monotone nonincreasing in [0,+∞).

Proof. Our method is based on the contradiction argument. Suppose that, for some 𝑟₀>0, 𝑔^′(𝑟₀)>0, by Lemma 3.1, there exists 𝑟₁∈(0,𝑟₀) such that 𝑔^′𝑟₁=0,𝑔^{𝑚′𝑝−2}𝑔^𝑙′′𝑟₁≥0. (3.16) By Lemma 3.2, we have 𝑔(𝑟₁)>0. Using the similar argument in Lemma 3.1, we obtain lim_𝑟→𝑟1𝑁𝑔^𝑚′(𝑟)^𝑝−2𝑔^𝑙′(𝑟)𝑟−𝑟₁=−𝑎𝛽𝑔𝑟₁<0, (3.17) which is a contradiction with (3.14). The proof of Lemma 3.3 is complete.

Next, we apply the monotone properties of 𝑓_𝑀(𝑟) to obtain the condition on the global existence of the solution to (1.1).

Proof of Theorem 1.4. We prove Theorem 1.4 by the following steps.
Step 1. Since 𝜑(𝑥)∈Φ^𝑎, there exists a constant 𝐾>0 such that 𝜑(𝑥)≤𝐾(1+|𝑥|)^−𝑎∀𝑥∈𝑅^𝑁. (3.18) Taking 𝑀>𝐾 and the self-similar solution 𝑈_𝑀,𝑎(𝑥,𝑡) of (3.1) defined as (3.3), since lim_𝑟→+∞𝑟^𝑎𝑓_𝑀(𝑟)=𝑀>𝐾, there exists a positive constant 𝑅₀ such that 𝑟^𝑎𝑓_𝑀(𝑟)>𝐾for𝑟≥𝑅₀. (3.19) Setting 𝛾=𝑓_𝑀(𝑅0)=min{𝑓_𝑀(𝑟)∣𝑟∈[0,𝑅₀]}>0, it is easy to verify that 𝜑(𝑥)≤𝑈_𝑀,𝑎(𝑥,𝑡) for all 𝑥∈𝑅^𝑁, where 𝑡₀∈(0,1) and 𝑡₀^−𝑎𝛽𝛾>‖𝜑‖_∞.
Let 𝜆>0; then 𝑤(𝑥,𝑡)=𝜆𝑈_𝑀,𝑎(𝑥,𝜆^{𝑙+𝑚(𝑝−2)−1}𝑡+𝑡₀) is the solution of the following problem 𝑤_𝑡=div∇𝑤^𝑚𝑝−2∇𝑤^𝑙,(𝑥,𝑡)∈𝑅^𝑁×(),𝑤()0,+∞𝑥,0=𝜆𝑈_𝑀,𝑎𝑥,𝑡₀≥𝜆𝜑(𝑥),𝑥∈𝑅^𝑁. (3.20) Taking 𝜂=𝑓_𝑀(0) and noting that 𝑓_𝑀(𝑟) is non-increasing, we have ‖𝑤(𝑥,𝑡)‖_∞=𝜂𝜆𝜆^{𝑙+𝑚(𝑝−2)−1}𝑡+𝑡₀^−𝑎𝛽. (3.21)
Step 2. Set 𝑣(𝑥,𝑡)=𝐴(𝑡)𝑤(𝑥,𝐵(𝑡)), where 𝐴(𝑡) and 𝐵(𝑡) are solutions of the following problem: 𝐴^′(𝑡)=𝜂^𝑞−1𝜆^𝑞−1𝜆^{𝑙+𝑚(𝑝−2)−1}𝐵(𝑡)+𝑡₀^{−(𝑎(𝑞−1)/(𝑎(𝑙+𝑚(𝑝−2)−1)+𝑝))}𝐴^𝑞(𝑡)(),𝐵,𝑡∈0,+∞_′(𝑡)=𝐴^{𝑙+𝑚(𝑝−2)−1}(𝑡)(),𝐴(0)(0)𝑡∈0,+∞=1,𝐵=0. (3.22) By a direct calculation, we obtain that 𝑣(𝑥,𝑡) satisfies 𝑣_𝑡≥div∇𝑣^𝑚𝑝−2∇𝑣^𝑙+𝑣^𝑞,(𝑥,𝑡)∈𝑅^𝑁×(),𝑣()()0,+∞𝑥,0=𝑤𝑥,0=𝜆𝑈_𝑀,𝑎𝑥,𝑡₀≥𝜆𝜑(𝑥),𝑥∈𝑅^𝑁. (3.23)
Step??3. We shall prove that there exists a positive constant 𝜆₀=𝜆₀(𝜑) such that the problem (3.22) has a global solution (𝐴(𝑡),𝐵(𝑡)) with 𝐴(𝑡) bounded in [0,𝑇) if 𝜆∈[0,𝜆₀). According to the standard theory of ODE, the local existence and uniqueness of solution (𝐴(𝑡),𝐵(𝑡)) of (3.22) hold. By (3.22), we have 𝐴^′(𝑡)>0,𝐴(𝑡)>1 for 𝑡>0; furthermore, the solution is continuous as long as the solution exists and 𝐴(𝑡) is finite.
From (3.22), when 𝐴(𝑡) exists in [0,𝑡], then 𝐵(𝑡) is uniquely defined by 𝐵(𝑡)=^𝑡₀𝐴^{𝑙+𝑚(𝑝−2)−1}(𝑠)𝑑𝑠. (3.24) Since 𝑝>2 and 𝐴(𝑡) is increasing, we obtain 𝐵(𝑠)=^𝑠₀𝐴^{𝑙+𝑚(𝑝−2)−1}(𝜏)𝑑𝜏≥𝐴^{𝑙+𝑚(𝑝−2)−1}(0)𝑠=𝑠∀𝑠∈[0,𝑡]. (3.25) By (3.22), (3.25), and 𝑎>𝑎^∗_𝑐=𝑝/(𝑞−𝑙−𝑚(𝑝−2)), it follows that 1−𝐴^1−𝑞(𝑡)=𝑞−1𝜂^𝑞−1𝜆^𝑞−1𝑡₀𝜆^{𝑙+𝑚(𝑝−2)−1}𝐵(𝑠)+𝑡₀^{−(𝑎(𝑞−1)/(𝑎(𝑙+𝑚(𝑝−2)−1)+𝑝))}≤𝜂𝑑𝑠𝑞−1^𝑞−1𝜆^𝑞−1𝑡₀𝜆^{𝑙+𝑚(𝑝−2)−1}𝑠+𝑡₀^{−(𝑎(𝑞−1)/(𝑎(𝑙+𝑚(𝑝−2)−1)+𝑝))}≤𝜂𝑑𝑠𝑞−1^𝑞−1𝜆^{𝑞−𝑙−𝑚(𝑝−2)}𝑎/𝑎𝑡𝑞−1𝑙+𝑚𝑝−2−1+𝑝−1^{01−(𝑎(𝑞−1)/(𝑎(𝑙+𝑚(𝑝−2)−1)+𝑝))}. (3.26) Let 𝜆₀=𝜆₀(𝜑) be a positive constant defined by 𝑞−1𝑎𝑙+𝑚𝑝−2−1+𝑝𝑎𝜂𝑞−𝑙−𝑚𝑝−2−𝑝^𝑞−1𝜆₀^{𝑞−𝑙−𝑚(𝑝−2)}𝑡₀^{1−(𝑎(𝑞−1)/(𝑎(𝑙+𝑚(𝑝−2)−1)+𝑝))}=12. (3.27) Then from (3.26), 𝑞>𝑞^∗_𝑐>𝑙+𝑚(𝑝−2)>1 and 𝑎>𝑎^∗_𝑐=𝑝/(𝑞−𝑙−𝑚(𝑝−2)), we have 1≤𝐴(𝑡)≤2^1/𝑞−1 for any 𝜆∈(0,𝜆₀], as long as 𝐴(𝑡) exists globally.
On the other hand, by (3.22) and (3.25), we have 𝑡≤𝐵(𝑡)≤2^{(𝑝−2)/(𝑞−1)}𝑡∀𝑡≥0. (3.28) Therefore, 𝐵(𝑡) is also global.
Step??4. For any 𝜆∈(0,𝜆₀], where 𝜆₀=𝜆₀(𝜑) is defined as (3.27), the solution 𝑢(𝑥,𝑡) of (1.1) with initial value 𝑢₀(𝑥)=𝜆𝜑(𝑥) exists globally, and 𝑢(𝑥,𝑡)≤𝑣(𝑥,𝑡) in 𝑅^𝑁×(0,+∞). Moreover, there exists a positive constant 𝐶 such that ‖𝑢(⋅,𝑡)‖_∞≤‖𝑣(.,𝑡)‖_∞≤2^1/(𝑞−1)𝜂𝜆𝜆^{𝑙+𝑚(𝑝−2)−1}𝐵(𝑡)+𝑡₀^−𝑎𝛽≤𝐶𝑡^−𝑎𝛽∀𝑡>0. (3.29) The proof of Theorem 1.4 is complete.

4. Single Point Blow-Up

In this section, under some suitable assumptions, we shall prove that the blow-up set consists of the single point 𝑥=0. Moreover, we also give the upper estimate of the solution 𝑢(𝑥,𝑡) in a small neighborhood of the point (𝑥,𝑡), where 𝑥=0, 𝑡=𝑇.

First, we suppose that the solution is radially symmetric, that is, depending only on 𝑟=|𝑥| at a given time 𝑡>0. Therefore, we study the following problem: 𝑢_𝑡=𝑟^{−(𝑁−1)}𝑟^𝑁−1𝑢^𝑚_𝑟^𝑝−2𝑢^𝑙_𝑟𝑟+𝑢^𝑞,()∈()×(),𝑢()𝑟,𝑡0,+∞0,+∞𝑟,0=𝑢₀(𝑟)(),𝑟,𝑟∈0,+∞_𝑁−1𝑢^𝑚_𝑟^𝑝−2𝑢^𝑙_𝑟=0,(𝑟,𝑡)∈{0}×[0,+∞). (4.1)

Proof of Theorem 1.9. It is based on the method in [21]. The main idea is to apply the maximum principle to the auxiliary function 𝜔_𝑖(𝑟,𝑡), which is defined in (4.7), and to show that 𝜔_𝑖 is small enough in (0,𝑖)×(0,𝑇). Then by integrating the obtained inequality and taking limit as 𝑖→∞, one can get upper bound of the solution 𝑢(𝑟,𝑡). Therefore, we divide the proof into the following steps.
Step 1. Since problem (1.1) has no classical solution, we will construct the weak solution by means of regularization of the degenerate equation.
Now define a strictly monotone sequence {𝜀_𝑖}, 𝜀_𝑖>0 for all 𝑖=1,2,3,…, such that 𝜀_𝑖⟶0,as𝑖⟶+∞. (4.2) Then, the weak solution 𝑢(𝑟,𝑡) is the limit function of the solution of the following regularized problem (see [31]): 𝑢_𝑖𝑡=𝑟^{−(𝑁−1)}𝑟^𝑁−1𝑢^𝑚_𝑖𝑟²+𝜀²_𝑖^(𝑝−2)/2𝑢^𝑙_𝑖𝑟+𝜀_𝑖𝑟+𝑢^𝑞_𝑖,()∈()×(),𝑢𝑟,𝑡0,𝑖0,𝑇_𝑖(𝑟,0)=𝑢₀(𝑟)(),𝑢,𝑟∈0,𝑖_𝑖(𝑖,𝑡)=𝑢₀(𝑖)(),𝑟,𝑡∈0,𝑇_𝑁−1𝑢^𝑚_𝑖𝑟²+𝜀²_𝑖^(𝑝−2)/2𝑢^𝑙_𝑖𝑟+𝜀_𝑖=0,(𝑟,𝑡)∈{0}×(0,𝑇). (4.3)
By the standard methods used in [1, 32], the uniform estimates for the passage to the limit which do not depend on 𝜀_𝑖 are established. Therefore, for any fixed 𝜀_𝑖>0, we may assume that, for all sufficiently large 𝑖, the function 𝑢_𝑖(𝑟,𝑡) satisfies the following conditions: 𝑢_𝑖≤𝑀₁,𝑢_𝑖𝑟≤𝑀₂in𝑄_𝑖,𝑇=(0,𝑖)×(0,𝑇), (4.4) where 𝑀₁, 𝑀₂ do not depend on 𝑖, and 𝑢_𝑖(𝑟,𝑡)_𝑟∣_𝑟=0=0∀𝑡∈(0,𝑇). (4.5) Moreover, by using condition (H) and the maximum principle in [33], we have 𝑢_𝑖(𝑟,𝑡)_𝑟≤0in𝑄_𝑖,𝑇. (4.6)
Step 2. Set 𝑤_𝑖(𝑟,𝑡)=𝑟^𝑁−1(𝑢^𝑚_𝑖𝑟)²+𝜀²_𝑖^(𝑝−2)/2𝑢^𝑙_𝑖𝑟+𝜆₀𝑟^𝑁𝑢^𝑞_𝑖, (4.7) where 𝜆₀ is given in (1.11).
By a direct calculation, we find that 𝑤_𝑖(𝑟,𝑡) satisfies the following parabolic equation: 𝑤_𝑖𝑡=𝑎_𝑖𝑤_𝑖𝑟𝑟+𝑏_𝑖𝑤_𝑖𝑟+𝑐_𝑖𝑤_𝑖+𝑑_𝑖+𝑒_𝑖, (4.8) where 𝑎_𝑖=𝑙𝑢_𝑖^𝑙−1𝑢^𝑚_𝑖𝑟²+𝜀²_𝑖^(𝑝−2)/2+𝑚𝑝−2𝑢_𝑖^𝑚−1𝑢^𝑚_𝑖𝑟²+𝜀²_𝑖^(𝑝−4)/2𝑢^𝑙_𝑖𝑟𝑢^𝑚_𝑖𝑟,𝑏_𝑖=𝑢^𝑚_𝑖𝑟²+𝜀²_𝑖^(𝑝−2)/2𝑙(1−𝑁)𝑢_𝑖^𝑙−1𝑟+𝑙(𝑙−1)𝑢_𝑖^𝑙−2𝑢_𝑖𝑟+𝑝−2𝑢^𝑙_𝑖𝑟𝑢^𝑚_𝑖𝑟⋅𝑢^𝑚_𝑖𝑟²+𝜀²_𝑖^(𝑝−4)/2𝑚(1−𝑁)𝑢_𝑖^𝑚−1𝑟+𝑚(𝑚−1)𝑢_𝑖^𝑚−2𝑢_𝑖𝑟,𝑐_𝑖=𝑞𝑢_𝑖^𝑞−1𝑝−1−𝜆₀(),𝑑𝑝−2𝑁+1+2_𝑖=𝐴𝑞𝑟^𝑁−11−𝜆₀(𝑁+1)𝑢_𝑖^𝑞−1,with𝐴≡−𝑝−2𝜀²_𝑖𝑢^𝑙_𝑖𝑟𝑢^𝑚_𝑖𝑟²+𝜀²_𝑖^(𝑝−4)/2,𝑒_𝑖=𝑝−2𝑟^𝑁−1𝑢^𝑚_𝑖𝑟²+𝜀²_𝑖^(𝑝−4)/2𝑢^𝑙_𝑖𝑟𝑢^𝑚_𝑖𝑟×−𝑚+𝑞−2𝑚𝑞𝑟𝜆₀𝑢_𝑖^{𝑞+𝑚−3}𝑢_𝑖𝑟²+(𝑚−1)1−𝜆₀𝑁+𝑞1−𝜆₀(𝑁+1)𝑚𝑢^{𝑞+𝑚−2}𝑢_𝑖𝑟+𝑟^𝑁−1𝑢^𝑚_𝑖𝑟²+𝜀²_𝑖^(𝑝−2)/2−𝑙+𝑞−2𝑙𝑞𝑟𝜆₀𝑢_𝑖^{𝑞+𝑙−3}𝑢_𝑖𝑟²+(𝑙−1)1−𝜆₀𝑁+𝑞1−𝜆₀(𝑁+1)𝑙𝑢^{𝑞+𝑙−2}𝑢_𝑖𝑟+𝜆₀𝑞𝜆₀𝑝−2(𝑁+1)+2−𝑝−2𝑟^𝑁𝑢_𝑖^2𝑞−1. (4.9)(4.10)(4.11)(4.12)(4.13) Since 𝑁≥1,𝑝>2,𝑚>1,𝑙>1,and𝑞>𝑙+𝑚(𝑝−2), it follows from (1.11) and (4.6) that 𝑒_𝑖≤0.
Now we consider the coefficients 𝑐_𝑖 and 𝑑_𝑖 in 𝑄_𝑖,𝑇. By (4.6) and 𝑝>2, we have 𝐴=−𝑝−2𝜀²_𝑖𝑢^𝑙_𝑖𝑟𝑢^𝑚_𝑖𝑟²+𝜀²_𝑖^(𝑝−4)/2≥0. (4.14) It follows from (4.4) that |𝐴|=𝑂(𝜀^𝜈_𝑖) as 𝑖→∞, where 𝜈=min{2,𝑝−1}, and we obtain the following estimate: sup𝑐_𝑖≤𝑀₃,sup𝑑_𝑖≤𝑀₄𝑖^𝑁−1𝜀^𝜈_𝑖in𝑄_𝑖,𝑇, (4.15) where 𝑀₃, 𝑀₄ are positive constants, which are independent of 𝑖.
Therefore, we have the parabolic differential inequality 𝑤_𝑖𝑡≤𝑎_𝑖𝑤_𝑖𝑟𝑟+𝑏_𝑖𝑤_𝑖𝑟+𝑐_𝑖𝑤_𝑖+𝑑_𝑖,in𝑄_𝑖,𝑇, (4.16) where 𝑐_𝑖, 𝑑_𝑖 satisfy (4.15).
Next, we consider the function 𝑤_𝑖(𝑟,𝑡) on the parabolic boundary of 𝑄_𝑖,𝑇. At first, it is easy to see that 𝑤_𝑖(0,𝑡)=0 for all 𝑡∈(0,𝑇). By (1.10), we have 𝑤_𝑖(𝑖,𝑡)≤𝜆₀𝑖^𝑁𝑢^𝑞₀(𝑖)=𝑜(1), as 𝑖→∞ for all 𝑡∈(0,𝑇). Finally, it follows from (1.11) that 𝑤_𝑖(𝑟,0)=𝑟^𝑁−1𝑢^𝑚_0𝑟²+𝜀²_𝑖^(𝑝−2)/2𝑢^𝑙_0𝑟+𝜆₀𝑟^𝑁𝑢^𝑞₀≤𝑟^𝑁−1𝑢^𝑚_0𝑟^𝑝−2𝑢^𝑙_0𝑟+𝜆₀𝑟𝑢^𝑞₀≤0∀𝑟∈[0,𝑖]. (4.17)
Hence, for all sufficiently large 𝑖, there exists 𝛾_𝑖=sup𝑤_𝑖>0 on the parabolic boundary of 𝑄_𝑖,𝑇 and 𝛾_𝑖=𝑜(1) as 𝑖→∞.
In order to estimate 𝑤_𝑖(𝑟,𝑡) in 𝑄_𝑖,𝑇, we study the following ODE: 𝑑𝑤_𝑖𝑑𝑡=𝑀₃𝑤_𝑖+𝑀₄𝑖^𝑁−1𝜀^𝜈_𝑖,𝑡>0,𝑤_𝑖(0)=𝛾_𝑖, (4.18) which has the solution 𝑤_𝑖(𝑡)=𝑀₄𝑖^𝑁−1𝜀^𝜈_𝑖𝑀₃+𝛾_𝑖𝑒^𝑀3𝑡−𝑀₄𝑖^𝑁−1𝑀₃𝜀^𝜈_𝑖,𝑡>0. (4.19) Taking the sequence 𝜀_𝑖 such that 𝑖^𝑁−1𝜀^𝜈_𝑖⟶0as𝑖⟶∞, (4.20) it is obvious that 𝑤_𝑖(𝑡)≤𝑤_𝑖(𝑇)=𝛼_𝑖∀𝑡∈(0,𝑇), (4.21) where 𝛼_𝑖⟶0as𝑖⟶∞. (4.22) Setting 𝑧(𝑟,𝑡)=𝑤_𝑖(𝑟,𝑡)−𝑤_𝑖(𝑡), (4.23) we have 𝑧(𝑟,𝑡)≤0 on the parabolic boundary, and 𝑧(𝑟,𝑡) satisfies the following parabolic inequality: 𝑧_𝑡≤𝑎_𝑖𝑧_𝑟𝑟+𝑏_𝑖𝑧_𝑟+𝑐_𝑖𝑧−𝑀₃−𝑐_𝑖𝑤_𝑖+𝑑_𝑖−𝑀₄𝑖^𝑁−1𝜀^𝜈_𝑖,in𝑄_𝑖,𝑇. (4.24) It follows from (4.15) that 𝑧_𝑡≤𝑎_𝑖𝑧_𝑟𝑟+𝑏_𝑖𝑧_𝑟+𝑐_𝑖𝑧,in𝑄_𝑖,𝑇. (4.25) By the maximum principle (Chapter II, [33]), we obtain that 𝑧(𝑟,𝑡)≤0 in 𝑄_𝑖,𝑇, that is, 𝑤_𝑖(𝑟,𝑡)≤𝑤_𝑖(𝑡)≤𝑤_𝑖(𝑇)=𝛼_𝑖in𝑄_𝑖,𝑇. (4.26)
Step 3. For large 𝑖 and (𝑢_𝑖(𝑟,𝑡))_𝑟∈[−𝑀₂,0], we have the following estimate 𝑢^𝑚_𝑖𝑟²+𝜀²_𝑖^(𝑝−2)/2𝑢^𝑙_𝑖𝑟≥𝑢^𝑚_𝑖𝑟^𝑝−2𝑢^𝑙_𝑖𝑟−𝛿_𝑖, (4.27) where 0<𝛿_𝑖=𝑂𝜀^𝜈_𝑖⟶0as𝑖⟶∞. (4.28) By (4.7) and (4.26), we have 𝑢^𝑚_𝑖𝑟^𝑝−2𝑢^𝑙_𝑖𝑟𝑢^𝑞_𝑖≤𝑟^1−𝑁𝛼_𝑖+𝛿_𝑖𝑢^𝑞_𝑖−𝜆₀𝑟, (4.29) namely, 𝑙𝑚^𝑝−2𝑢_𝑖^{(𝑙−1)+(𝑚−1)(𝑝−2)−𝑞}𝑢_𝑖𝑟𝑢_𝑖𝑟^𝑝−2≤𝑟^1−𝑁𝛼_𝑖+𝛿_𝑖𝑢^𝑞_𝑖−𝜆₀𝑟. (4.30) Letting 𝐹(𝑏)=𝑏|𝑏|^{−((𝑝−2)/(𝑝−1))}, we deduce that 𝑙𝑚^{𝑝−21/(𝑝−1)}𝑢_𝑖^{((𝑙+𝑚(𝑝−2)−𝑞−𝑝+1)/(𝑝−1))}𝑢_𝑖𝑟𝑟≤𝐹^1−𝑁𝛼_𝑖+𝛿_𝑖𝑢^𝑞_𝑖−𝜆₀𝑟. (4.31) For arbitrary 𝑡₀∈(0,𝑇), 𝑟₀∈suppu(𝑟,𝑡₀)≡{𝑟>0∣𝑢(𝑟,𝑡₀)>0}, we denote 𝜇₀=𝑢(𝑥₀,𝑡₀)>0. By (4.6) and the uniform convergence 𝑢_𝑖(𝑟,𝑡₀)→𝑢(𝑟,𝑡₀) as 𝑖→∞, 𝑟∈[0,𝑟₀], we have 𝑢_𝑖(𝑟,𝑡₀)≥(𝜇₀/2),?? 𝑟∈[0,𝑟₀] for all sufficiently large 𝑖≥𝑖₀. Therefore, from (4.31) we obtainly for 𝑡=𝑡₀,𝑖>𝑖₀, 𝑙𝑚^{𝑝−21/(𝑝−1)}𝑢_𝑖^{((𝑙+𝑚(𝑝−2)−𝑞−𝑝+1)/(𝑝−1))}𝑢_𝑖𝑟2≤𝐹^𝑞𝑟^1−𝑁𝛼_𝑖+𝛿_𝑖𝜇^𝑞₀−𝜆₀𝑟. (4.32) Integrating the above inequality over interval [𝑟₁,𝑟₀], where 𝑟₁>0, we obtain −𝑙𝑚^{𝑝−21/(𝑝−1)}𝑝−1𝑢𝑞−𝑙−𝑚𝑝−2^{𝑖−((𝑞−𝑙−𝑚(𝑝−2))/(𝑝−1))}|^𝑟0_𝑟1≤^𝑟0_𝑟1𝐹2^𝑞𝑟^1−𝑁𝛼_𝑖+𝛿_𝑖𝜇^𝑞₀−𝜆₀𝑟𝑑𝑟. (4.33) Letting 𝑖→∞, by (4.22), (4.28), and the uniform convergence 𝑢_𝑖(𝑟,𝑡₀)→𝑢(𝑟,𝑡₀) as 𝑖→∞, 𝑟∈[0,𝑟₀], we have the following estimate: −𝑙𝑚^{𝑝−21/(𝑝−1)}𝑝−1𝑢𝑞−𝑙−𝑚𝑝−2^{−((𝑞−𝑙−𝑚(𝑝−2))/𝑝−1)}𝑟₀,𝑡₀−𝑢^{−((𝑞−𝑙−𝑚(𝑝−2))/𝑝−1)}𝑟₁,𝑡₀≤lim_𝑖→∞^𝑟0_𝑟1𝐹2^𝑞𝑟^1−𝑁𝛼_𝑖+𝛿_𝑖𝜇^𝑞₀−𝜆₀𝑟𝑑𝑟=−𝜆₀^1/(𝑝−1)𝑝−1𝑝𝑟₀^{𝑝/(𝑝−1)}−𝑟₁^{𝑝/(𝑝−1)}. (4.34) Setting 𝑟₁→0, from (4.34), we obtain the upper estimate 𝑢𝑟₀,𝑡₀≤𝑞−𝑙−𝑚𝑝−2𝑝𝑙𝑚^{𝑝−21/𝑝−1}𝜆₀^{1/𝑝−1−((𝑝−1)/(𝑞−𝑙−𝑚(𝑝−2)))}𝑟₀^{−(𝑝/(𝑞−𝑙−𝑚(𝑝−2)))}, (4.35) where (𝑟₀,𝑡₀)∈𝑅¹₊×(0,𝑇). The proof of Theorem 1.9 is complete.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments on this work. The second author is supported in part by NSF of China (11071266) and in part by the Natural Science Foundation Project of CQ CSTC(2010BB9218). This work is supported by the Fundamental Research Funds for the Central University, Project No CDJXS 11 10 00 40.