On the Blow-Up Set for Non-Newtonian Equation with a Nonlinear Boundary Condition

Abstract

We identify the blow-up set of solutions to the problem 𝑢_𝑡=(|𝑢_𝑥|^𝑝−2𝑢_𝑥)_𝑥, 𝑥>0, 𝑡∈(0,𝑇), −|𝑢_𝑥|^𝑝−2𝑢_𝑥(0,𝑡)=𝑢^𝑝−1(0,𝑡), 𝑡∈(0,𝑇), and 𝑢(𝑥,0)=𝑢₀(𝑥)≥0, 𝑥>0, where 𝑝>2. We obtain that the blow up set 𝐵(𝑢) satisfies [0,𝑝(𝑝−1)/(𝑝−2))⊂𝐵(𝑢)⊂[0,𝑝(𝑝−1)/(𝑝−2)]. The proof is based on the analysis of the asymptotic behavior of self-similar representation and on the comparison methods.

1. Introduction

Consider a one-dimensional process of diffusion in a medium that occupies the half space {𝑥∶𝑥≥0}; that is, 𝑢_𝑡=(|𝑢_𝑥|^𝑝−2𝑢_𝑥)_𝑥,𝑥>0,𝑡∈(0,𝑇),−|𝑢_𝑥|^𝑝−2𝑢_𝑥(0,𝑡)=𝑢^𝑞(0,𝑡),𝑡∈(0,𝑇),𝑢(𝑥,0)=𝑢₀(𝑥)≥0,𝑥>0, (1.1) where 𝑝>2,𝑞>0 and 𝑢₀(𝑥) is an appropriately smooth function with some compatibility conditions. Problem (1.1) describes the non-Newtonian fluid with a power dependence of the tangential stress on the velocity of the displacement under nonlinear condition. It has many applications and has been intensively studied; see [1–3] and the references cited therein. For the local in time existence, we refer to [4]. Also it is known that (1.1) has no classical solution in general due to the possible degeneration at 𝑢_𝑥=0. So we usually understand the weak solution defined in the following sense.

Definition 1.1. A nonnegative function 𝑢∈𝐶([0,+∞)×[0,𝑡)) with 𝑢_𝑥∈𝐿^∞([0,+∞)×[0,𝑡)) is said to be a weak solution of (1.1), if the integral identity ^𝑡₀₀^+∞(𝑢𝜙_𝑡−|𝑢_𝑥|^𝑝−2𝑢_𝑥𝜙_𝑥)𝑑𝑥𝑑𝑠+₀^+∞𝑢₀(𝑥)𝜙(𝑥,0)𝑑𝑥+^𝑡₀𝑢^𝑞(0,𝑠)𝜙(0,𝑠)𝑑𝑠=0 (1.2) is fulfilled for all 𝜙∈𝐶^∞₀([0,+∞)×[0,𝑡)).

An interesting phenomenon is that, due to the boundary effect, the solution of (1.1) may exist for 𝑡∈[0,𝑇) and becomes unbounded as 𝑡→𝑇 for some 𝑇<∞. Namely, the solutionoccurs blow-upphenomenon. In this connection, Galaktionov and Levine proved in [5] that the solutions are global in time when 0<𝑞≤2(𝑝−1)/𝑝 but occur blow-up for the range 2(𝑝−1)/𝑝<𝑞≤2(𝑝−1), while for 𝑞>2(𝑝−1) blow-up happens or not depending on the size of the initial data. The main concern in this work is on the set of points at which solutions becomes unbounded, that is, the blow-up set, which is defined as 𝐵(𝑢)={𝑥∶∃𝑥_𝑛→𝑥,𝑡_𝑛→𝑇−,suchthat𝑢(𝑥_𝑛,𝑡_𝑛)→∞as𝑛→∞}. (1.3) A problem which has attracted a lot of attention in the literature is the identification of possible blow-up sets. Numerical analysis hints that the blow-up set should be a single point (single point blow-up) when 𝑞>𝑝−1, a proper subset of the spatial domain (regional blow-up) when 𝑞=𝑝−1, and the whole half line (global blow-up) when 0<𝑞<𝑝−1. In fact, based on the Gilding and Herrero's work in [6], Quirós and Rossi considered the porous medium type equation 𝑢_𝑡=(𝑢^𝑚)_𝑥𝑥,𝑥>0,𝑡∈(0,𝑇),−(𝑢^𝑚)_𝑥(0,𝑡)=𝑢^𝑞(0,𝑡),𝑡∈(0,𝑇),𝑢(𝑥,0)=𝑢₀(𝑥)≥0,𝑥>0. (1.4) They proved in [1]: the blow-up set 𝐵(𝑢)={0} if 𝑞>𝑚 but 𝐵(𝑢)=[0,+∞) if 𝑞<𝑚, however, the blow-up set is regional in case of 𝑞=𝑚, namely, 0<𝐵(𝑢)<+∞. Afterwards, Cortázar et al. given a detailed description on the regional blow-up set. They proved in [7] that if 𝑞=𝑚, then the blow-up set satisfies [0,2𝑚/𝑚−1)⊂𝐵(𝑢)⊂[0,2𝑚/𝑚−1].

In the light of previous works, we discuss the blow-up set of solutions for the 𝑝-Laplacian equation (1.1). In the current paper, we identify the set of blow-up points in case of 𝑞=𝑝−1. So, in the following we consider 𝑢_𝑡=(|𝑢_𝑥|^𝑝−2𝑢_𝑥)_𝑥,𝑥>0,𝑡∈(0,𝑇),−|𝑢_𝑥|^𝑝−2𝑢_𝑥(0,𝑡)=𝑢^𝑝−1(0,𝑡),𝑡∈(0,𝑇),𝑢(𝑥,0)=𝑢₀(𝑥)≥0,𝑥>0, (1.5) where 𝑝>2. We take 𝑢₀(𝑥) to be a 𝐶¹, nonincreasing and compactly supported function with some compatibility conditions. The below theorem is our main result.

Theorem 1.2. Assume that 𝑢₀(𝑥)∈𝐶¹([0,+∞)) is a nonnegative, nonincreasing and compactly supported function, then all the nontrivial solutions 𝑢(𝑥,𝑡) of problem (1.5) occur blow-up; moreover, the blow-up set 𝐵(𝑢) satisfies [0,𝑝(𝑝−1)𝑝−2)⊂𝐵(𝑢)⊂[0,𝑝(𝑝−1)𝑝−2]. (1.6)

Remark 1.3. The nonincreasing assumption on 𝑢₀ makes the proof much simpler (see also [7]).

Remark 1.4. One could expect that if the solutions of (1.1) occur blow-up, then the blow-up set 𝐵(𝑢)=[0,+∞) if 0<𝑞<𝑝−1, but 𝐵(𝑢)={0} if 𝑞>𝑝−1.

2. Proof of Theorem 1.2

In order to study the solution 𝑢 of (1.5) near the blow-up time 𝑇, as in [8] we introduce the rescaled function 𝑣(𝑥,𝜏)=(𝑇−𝑡)^1/(𝑝−2)𝑢(𝑥,𝑡), (2.1) where 𝜏=−ln((𝑇−𝑡)/𝑇)∈[0,+∞). Hence, we obtain the below equations in terms of 𝑣 by substituting (2.1) into (1.5), 𝜕𝑣𝜕𝜏=(|𝑣_𝑥|^𝑝−2𝑣_𝑥)_𝑥−1𝑝−2𝑣,𝑥>0,𝜏>0,−|𝑣_𝑥|^𝑝−2𝑣_𝑥(0,𝜏)=𝑣^𝑝−1(0,𝜏),𝜏>0,𝑣(𝑥,0)=𝑇^1/(𝑝−2)𝑢₀(𝑥)≥0,𝑥>0. (2.2)

If 𝑣 is 𝜏 independent, then 𝑢 is called a self-similar solution. Let 𝑤(𝑥)=(𝑇−𝑡)^1/(𝑝−2)𝑢(𝑥,𝑡), we have (|𝑤_𝑥|^𝑝−2𝑤_𝑥)_𝑥−1𝑝−2𝑤=0,𝑥>0,−|𝑤_𝑥|^𝑝−2𝑤_𝑥(0)=𝑤^𝑝−1(0). (2.3) A direct integration shows that the non-negative solution of (2.3) is 𝑤≡0 or 𝑤(𝑥)=(𝑝−2𝑝)^{(𝑝−1)/(𝑝−2)}(1)2(𝑝−1)^1/(𝑝−2)(𝑝(𝑝−1)𝑝−2−𝑥)₊^{𝑝/(𝑝−2)}, (2.4) where 𝑎₊=max{0,𝑎}.

Theorem 1.2 is a direct consequence of the following two propositions.

Proposition 2.1. Let 𝑣(𝑥,𝜏) be a solution of (2.2), then there exists a function 𝑤(𝑥), such that the limit holds uniformly on [0,+∞),𝑣(⋅,𝜏)→𝑤(⋅),a𝑠𝜏→+∞. (2.5) Moreover, 𝑤(𝑥) has the explicit form (2.4) and solves problem (2.3).

Proposition 2.1 and the formula (2.1) generate [0,𝑝(𝑝−1)𝑝−2)⊂𝐵(𝑢). (2.6)

Proposition 2.2. Suppose that the constants 𝑎,𝑏 satisfy 𝑝(𝑝−1)/(𝑝−2)<𝑎<𝑏<+∞, then there exists a large time point 𝜏₀ such that for all 𝜏>𝜏₀𝑣(𝑥,𝜏)≤𝐶𝑒^{−𝜏/(𝑝−2)},𝑥∈[𝑎,𝑏], (2.7) where constant 𝐶 depends on 𝑎,𝑏 and 𝑢₀.

Proposition 2.2, along with (2.1), shows that 𝑢(𝑥,𝑡) is uniformly bounded for all 𝑡∈(0,𝑇) if 𝑥>𝑝(𝑝−1)/𝑝−2. This claims 𝐵(𝑢)⊂[0,𝑝(𝑝−1)𝑝−2]. (2.8)

Hence, Theorem 1.2 immediately follows from (2.6) and (2.8). So the remaining task in this paper is to prove the validity of Propositions 2.1 and 2.2. This will be discussed in Section 3.

3. Proof of Propositions 2.1 and 2.2

In this section we prove Propositions 2.1 and 2.2. The argument contains several lemmas.

Lemma 3.1. All the nontrivial solutions of (1.5) occur blow-up.

The proof is available in [5], (see also [3]).

Lemma 3.2. If the initial function 𝑢₀ is nonincreasing, then so does the solution 𝑢(𝑥,𝑡) for all existent times, that is, 𝑢_𝑥≤0,𝑎.𝑒.(𝑥,𝑡)∈(0,∞)×(0,𝑇).

Proof. Consider the regularized problem 𝜕𝑢𝜕𝑡=((𝑢²_𝑥+𝜖)^(𝑝−2)/2𝑢_𝑥)_𝑥,𝑥>0,𝑡∈(0,𝑇),−(𝑢²_𝑥+𝜖)^(𝑝−2)/2𝑢_𝑥(0,𝑡)=𝑢^𝑝−1(0,𝑡),𝑡∈(0,𝑇),𝑢(𝑥,0)=𝑢_0𝜖(𝑥)≥0,𝑥>0, (3.1) where 𝜖>0,𝑢_0𝜖(𝑥)∈𝐶^∞₀([0,+∞)) is nonincreasing and converges to 𝑢₀(𝑥) uniformly as 𝜖→0+. By uniqueness, the solution of (1.5) should be the limit function of that of (3.1). Setting 𝑧(𝑥,𝑡)=(𝑢²_𝑥+𝜖)^(𝑝−2)/2𝑢_𝑥 and then differentiating the equation in 𝑡 to obtain 𝑧_𝑡=(𝑢²_𝑥+𝜖)^(𝑝−2)/2𝑢_𝑥𝑡+(𝑝−2)(𝑢²_𝑥+𝜖)^{(𝑝−2)/2−1}𝑢²_𝑥𝑢_𝑥𝑡=[(𝑢²_𝑥+𝜖)^(𝑝−2)/2+(𝑝−2)(𝑢²_𝑥+𝜖)^{(𝑝−2)/2−1}𝑢²_𝑥]𝑧_𝑥𝑥. (3.2) Clearly, (3.2) is a linear parabolic equation with respect to 𝑧. On account of the boundary conditions 𝑧(0,𝑡)=−𝑢(0,𝑡)^𝑝−1≤0 and 𝑧(𝑥,0)=(𝑢²_0𝜖𝑥+𝜖)^(𝑝−2)/2𝑢_0𝜖𝑥≤0, we deduce 𝑧(𝑥,𝑡)≤0 via comparison theorem. The proof is completed.

Remark 3.3. Using the maximum principle for problem (3.1), we obtain max_{𝑥≥0,0≤𝑡≤𝑇}{−(𝑢²_𝑥+𝜖)^(𝑝−2)/2𝑢_𝑥}=max{𝑢^𝑝−1(0,𝑡),−(𝑢²_0𝜖𝑥+𝜖)^(𝑝−2)/2𝑢_0𝜖𝑥}. (3.3) By sending 𝜖→0+, it follows that max_{𝑥≥0,0≤𝑡≤𝑇}{−|𝑢_𝑥|^𝑝−2𝑢_𝑥}=max{𝑢^𝑝−1(0,𝑡),−|𝑢_0𝑥|^𝑝−2𝑢_0𝑥}. (3.4)

Lemma 3.4. There exists a constant 𝐶=𝐶(𝑝,𝑢₀) such that ^𝑡₀𝑢^𝑝−1(0,𝑠)𝑑𝑠≤𝐶(𝑇−𝑡)^{−1/(𝑝−2)},∀𝑡∈[0,𝑇). (3.5)

Proof. Integrating (1.5)₁, the first equation of (1.5), over [0,+∞)×[0,𝑡) yields ^∞₀𝑢(𝑥,𝑡)𝑑𝑥−^∞₀𝑢₀𝑑𝑥=^𝑡₀𝑢^𝑝−1(0,𝑠)𝑑𝑠. (3.6) Now multiplying (1.5)₁ by 𝑥 and integrating the resulting expression once more, we have ^∞₀𝑥𝑢(𝑥,𝑡)𝑑𝑥−^∞₀𝑥𝑢₀𝑑𝑥=^𝑡₀^∞₀𝑥(|𝑢_𝑥|^𝑝−2𝑢_𝑥)_𝑥=𝑑𝑥𝑑𝑠_𝑡0∞0(𝑥|𝑢_𝑥|^𝑝−2𝑢_𝑥)_𝑥𝑑𝑥𝑑𝑠−^𝑡₀^∞₀|𝑢_𝑥|^𝑝−2𝑢_𝑥𝑑𝑥𝑑𝑠=−^𝑡₀^∞₀|𝑢_𝑥|^𝑝−2𝑢_𝑥𝑑𝑥𝑑𝑠. (3.7) Thanks to Remark 3.3, we estimate −^𝑡₀^∞₀|𝑢_𝑥|^𝑝−2𝑢_𝑥𝑑𝑥𝑑𝑠≤^𝑡₀|𝑢_𝑥(0,𝑠)|^𝑝−2∞₀(−𝑢_𝑥)𝑑𝑥𝑑𝑠+^𝑡₀^∞₀|𝑢_0𝑥|^𝑝−1=^𝑡₀|𝑢_𝑥(0,𝑠)|^𝑝−2𝑢(0,𝑠)𝑑𝑠+𝐶(𝑇,𝑢_0𝑥)=^𝑡₀𝑢^𝑝−1(0,𝑠)𝑑𝑠+𝐶(𝑇,𝑢_0𝑥), (3.8) the last sign comes from the boundary condition. Thus, (3.7) becomes ^∞₀𝑥𝑢(𝑥,𝑡)𝑑𝑥−^∞₀𝑥𝑢₀𝑑𝑥≤^𝑡₀𝑢^𝑝−1(0,𝑠)𝑑𝑠+𝐶(𝑇,𝑢_0𝑥). (3.9) Multiplying (3.6) by a constant 𝐾>1 and then subtracting (3.9) produces ^∞₀(𝐾−𝑥)𝑢(𝑥,𝑡)𝑑𝑥−^∞₀(𝐾−𝑥)𝑢₀𝑑𝑥≥(𝐾−1)^𝑡₀𝑢^𝑝−1(0,𝑠)𝑑𝑠−𝐶(𝑇,𝑢_0𝑥). (3.10) If we choose 𝐾>1 so large such that supp𝑢₀⊂[0,𝐾], then ^∞₀(𝐾−𝑥)𝑢(𝑥,𝑡)𝑑𝑥−^∞₀(𝐾−𝑥)𝑢₀𝑑𝑥≤^𝐾₀(𝐾−𝑥)𝑢(𝑥,𝑡)𝑑𝑥≤𝐾²𝑢(0,𝑡). (3.11) Inequalities (3.10) and (3.11) lead to 𝐾²𝑢(0,𝑡)≥(𝐾−1)^𝑡₀𝑢^𝑝−1(0,𝑠)𝑑𝑠−𝐶(𝑇,𝑢_0𝑥). (3.12) Putting 𝐹(𝑡)=∫^𝑡₀𝑢^𝑝−1(0,𝑠)𝑑𝑠 and integrating above inequality over (𝑡,𝑇) brings 𝐹(𝑡)≤𝐶(𝑇−𝑡)^{−1/(𝑝−2)}+𝐶(𝑇,𝑢_0𝑥)≤𝐶₁(𝑇−𝑡)^{−1/(𝑝−2)},(𝑡closeto𝑇). (3.13) Because 𝑢 is bounded when time 𝑡 is away from the blow-up time 𝑇, (3.13) and thus (3.5) is valid for all 𝑡∈[0,𝑇).

The next lemma proves that 𝑢(𝑥,𝑡) is localized. Namely, the support of 𝑢(⋅,𝑡) is uniformly bounded for all 𝑡∈([0,𝑇).

Lemma 3.5. There exists a constant 𝑀 depending on 𝑝 and 𝑢₀ such that if 𝑥>𝑀, then 𝑢(𝑥,𝑡)≡0,∀𝑡∈[0,𝑇). (3.14)

Proof. Let us introduce the function 𝑞(𝑥,𝑡)=𝐷(𝑇−𝑡)^{−1/(𝑝−2)}(𝐿−𝑥)₊^{𝑝/(𝑝−2)}, (3.15) where 𝐷 satisfies where 𝐷^𝑝−2=(1/2(𝑝−1))(𝑝−2/𝑝)^𝑝−1 and 𝐿 is an arbitrary positive constant. It is easy to check that 𝑞(𝑥,𝑡) satisfies (1.5)₁. By differentiating (3.5), we have 𝑢(0,𝑡)≤𝐶(𝑇−𝑡)^{−1/𝑝−2}. This, together with the monotonicity of 𝑢, deduces 𝑢(𝑥,𝑡)≤𝐶(𝑇−𝑡)^{−1/(𝑝−2)},∀𝑥≥0,𝑡∈(0,𝑇). (3.16) Because the initial support is bounded, we choose a suitably large point 𝑥₀<+∞ to guarantee supp𝑢₀⊂[0,𝑥₀]. Setting 𝑢(𝑥,𝑡)=𝑢(𝑥+𝑥₀,𝑡), it has 𝑢(𝑥,0)=𝑢(𝑥+𝑥₀,0)≡0≤𝑞(𝑥,0),∀𝑥>0. (3.17) By virtue of (3.16), there exists some large constant 𝐿₀ such that for all 𝐿>𝐿₀, it holds 𝑢(0,𝑡)=𝑢(𝑥₀,𝑡)≤𝐶(𝑇−𝑡)^{−1/(𝑝−2)}≤𝐷𝐿^{𝑝/(𝑝−1)}(𝑇−𝑡)^{−1/(𝑝−2)}=𝑞(0,𝑡),∀𝑡∈(0,𝑇). (3.18) So comparison theorem concludes 𝑢(𝑥,𝑡)≤𝑞(𝑥,𝑡),∀𝑥≥0,𝑡∈(0,𝑇). (3.19) Hence, for every constant 𝐿 satisfying 𝐿>𝐿₀, (3.19) ensures supp𝑢(⋅,𝑡)⊂[0,𝐿]. Thus supp𝑢(⋅,𝑡)⊂[0,𝑥₀+𝐿]. Denoted by 𝑀=𝑥₀+𝐿, it follows that for all 𝑥>𝑀,𝑢(𝑥,𝑡)≡0,∀𝑡∈(0,𝑇). (3.20) This completes the proof.

Corollary 3.6. Lemma 3.5 and (2.1) claim that 𝑣(𝑥,𝜏) is localized as well. Moreover, (2.1) and (3.16) guarantee the boundedness of 𝑣(𝑥,𝜏) for all 𝑥≥0 and 𝜏≥0. By a similar argument as that in Lemma 3.2, one can prove that |𝑣_𝑥|^𝑝−2𝑣_𝑥 also is bounded.

In what follows, we analyze the large time behavior of the solution 𝑣(𝑥,𝜏) of problem (2.2).

Multiplying (2.2) by 𝑣 and then integrating by parts, we deduce that ^∞₀^𝜏2_𝜏1𝑣²_𝜏𝑑𝑥𝑑𝑠+𝐽[𝑣](𝜏₂)≤𝐽[𝑣](𝜏₁),a.e.,0≤𝜏₁<𝜏₂<∞, (3.21) where 𝐽[𝑣](𝜏) is the Liapunov functional and 1𝐽[𝑣](𝜏)=𝑝^∞₀|𝑣_𝑥|^𝑝1𝑑𝑥−𝑝^∞₀𝑣^𝑝1(0,𝜏)+2(𝑝−2)^∞₀𝑣²(𝑥,𝜏)𝑑𝑥. (3.22) Based on the estimates 𝑣,𝑣_𝑥∈𝐿^∞([0,+∞)×[0,+∞)),𝑣_𝜏∈𝐿²([0,+∞)×[0,+∞)), (3.23) the 𝜔-limit set𝜔(𝑣₀)={𝑤(𝑥)∶∃𝜏_𝑛→∞,s.t.𝑣(⋅,𝜏_𝑛)→𝑤(𝑥)in𝐿²} (3.24) consists of solutions of the problem (2.3). Indeed, using the estimates (3.23) in passing to the limit in (2.2), we have that, given a monotone sequence 𝜏_𝑛→∞, 𝑣(⋅,𝜏_𝑛+𝑠)→𝑤(𝑥;𝑠)𝑖𝑛𝐿², where 𝑤(𝑥;𝑠) depending on 𝑠 solves (2.3) in a weak sense, and 𝑤(𝑥;0)∈𝜔(𝑣₀). It follows from (3.23) that uniformly in 𝑠∈[0,1]‖𝑣(⋅,𝜏_𝑛+𝑠)−𝑣(⋅,𝜏_𝑛)‖≤^𝜏𝑛_𝜏^+𝑠_𝑛‖𝑣_𝑡‖²_𝐿2𝑑𝜏→0(𝜏_𝑛→∞). (3.25) This means that the limit function 𝑤 does not rely on 𝑠 and is a weak solution for the (2.3). Finally, the independence of the choice of the sequence 𝜏_𝑛→∞ follows from the nonincreasing of 𝐽[𝑣](𝜏) in time. Furthermore, the convergence is uniform in 𝑥 due to the boundness of 𝑣_𝑥. In conclusion we arrive to the following lemma.

Lemma 3.7. The limit holds as 𝜏→∞𝑣(𝑥,𝜏)→𝑤(𝑥),uniformlyfor𝑥≥0, (3.26) where 𝑤(𝑥) satisfies the problem (2.3).

To finish Proposition 2.1, it remains to confirm that 𝑤(𝑥) is not null and thus to be of the form (2.4). The proof will be given at the end of the paper. Now let us turn to Proposition 2.2.

Consider the initial-boundary-value problem ℎ_𝜏=(|ℎ_𝑥|^𝑝−2ℎ_𝑥)_𝑥−1𝑝−2ℎ,𝑥∈(𝑎−𝛿,𝑏+𝛿),𝜏>0,ℎ(𝑎−𝛿,𝜏)=ℎ(𝑏+𝛿,𝜏)=𝜖,𝜏>0,ℎ(𝑥,0)=𝜖,𝑥∈(𝑎−𝛿,𝑏+𝛿), (3.27) where 𝑎,𝑏,𝜖,𝛿 are positive constants and 𝜖,𝛿 are suitably small. We have the following.

Lemma 3.8. Let 𝐷𝛿^{𝑝/(𝑝−2)}=𝜖 suitably small with 𝐷 that satisfies 𝐷^𝑝−2=(1/2(𝑝−1))((𝑝−1)/𝑝)^𝑝−1. Then there exists a constant 𝐶>0 such that for all 𝜏≥0, ℎ(𝑥,𝜏)≤𝐶𝑒^{−𝜏/(𝑝−2)},∀𝑥∈[𝑎,𝑏]. (3.28)

Proof. It is easy to check that the function of the form 𝑗(𝑥)=𝐷(𝑎−𝑥)^{𝑝/(𝑝−2)}𝛿,𝑥∈[𝑎−𝛿,𝑎−2𝛿),0,𝑥∈[𝑎−2𝛿,𝑏+2],𝐷(𝑥−𝑏)^{𝑝/(𝑝−2)}𝛿,𝑥∈(𝑏+2,𝑏+𝛿]. (3.29) solves 0=(|𝑗_𝑥|^𝑝−2𝑗_𝑥)_𝑥−1𝑝−2𝑗,𝑥∈(𝑎−𝛿,𝑏+𝛿),𝑗(𝑎−𝛿)=𝑗(𝑏+𝛿)=𝐷𝛿^{𝑝/(𝑝−2)}. (3.30) Moreover, it has 𝑗(𝑥)≤𝐷𝛿^{𝑝/(𝑝−2)} since 𝐷𝛿^{𝑝/(𝑝−2)} is a super solution to problem (3.30). Choosing 𝐷𝛿^{𝑝/(𝑝−2)}=𝜖 and subtracting (3.30) from (3.27), we obtain (ℎ−𝑗)_𝜏=(|ℎ_𝑥|^𝑝−2ℎ_𝑥−|𝑗_𝑥|^𝑝−2𝑗_𝑥)_𝑥−1𝑝−2(ℎ−𝑗),𝑥∈(𝑎−𝛿,𝑏+𝛿),𝜏>0,ℎ(𝑎−𝛿,𝜏)−𝑗(𝑎−𝛿)=ℎ(𝑏+𝛿,𝜏)−𝑗(𝑏+𝛿)=0,𝜏>0,ℎ(𝑥,0)−𝑗(𝑥)≥0,𝑥∈(𝑎−𝛿,𝑏+𝛿). (3.31) Integrating (3.31) over (𝑎−𝛿,𝑏+𝛿)×(0,𝜏) gives rise to ^𝑏+𝛿_𝑎−𝛿(ℎ−𝑗)(𝑥,𝜏)𝑑𝑥=^𝑏+𝛿_𝑎−𝛿+(ℎ−𝑗)(𝑥,0)𝑑𝑥_𝜏0(|ℎ_𝑥|^𝑝−2ℎ_𝑥−|𝑗_𝑥|^𝑝−2𝑗_𝑥−)(𝑏+𝛿,𝑠)𝑑𝑠_𝜏0(|ℎ_𝑥|^𝑝−2ℎ_𝑥−|𝑗_𝑥|^𝑝−2𝑗_𝑥−1)(𝑎−𝛿,𝑠)𝑑𝑠𝑝−2^𝜏₀^𝑏+𝛿_𝑎−𝛿(ℎ−𝑗)(𝑥,𝑠)𝑑𝑥𝑑𝑠. (3.32) Noting that (ℎ−𝑗)(𝑥,𝜏) is symmetric with respect to (𝑎+𝑏)/2 and nonincreasing on [𝑎+𝑏/2,𝑏+𝛿], thus we have for all 𝜏>0,sign{(|ℎ_𝑥|^𝑝−2ℎ_𝑥−|𝑗_𝑥|^𝑝−2𝑗_𝑥)(𝑏+𝛿,𝜏)}=sign{(ℎ_𝑥−𝑗_𝑥)(𝑏+𝛿,𝜏)}≤0, (3.33) where sign is the sign function. Then, ^𝜏₀(|ℎ_𝑥|^𝑝−2ℎ_𝑥−|𝑗_𝑥|^𝑝−2𝑗_𝑥)(𝑏+𝛿,𝑠)𝑑𝑠≤0. (3.34) Similarly, ^𝜏₀(|ℎ_𝑥|^𝑝−2ℎ_𝑥−|𝑗_𝑥|^𝑝−2𝑗_𝑥)(𝑎−𝛿,𝑠)𝑑𝑠≥0. (3.35) Therefore, (3.32) becomes ^𝑏+𝛿_𝑎−𝛿(ℎ−𝑗)(𝑥,𝜏)𝑑𝑥≤^𝑏+𝛿_𝑎−𝛿1(ℎ−𝑗)(𝑥,0)𝑑𝑥−𝑝−2^𝜏₀^𝑏+𝛿_𝑎−𝛿(ℎ−𝑗)(𝑥,𝑠)𝑑𝑥𝑑𝑠. (3.36) Applying the Gronwall inequality yields ^𝑏+𝛿_𝑎−𝛿(ℎ−𝑗)(𝑥,𝜏)𝑑𝑥≤C𝑒^{−(1/(𝑝−2))𝜏}. (3.37) On the other hand, since ℎ is symmetric and nondecreasing on [(𝑎+𝑏)/2,𝑏+𝛿] with 𝑥-variable, for all 𝑥∈[𝑎,𝑏] it has 1ℎ(𝑥,𝜏)≤𝛿^𝑏+𝛿/2_{𝑎−𝛿/2}1ℎ(𝑦,𝜏)𝑑𝑦=𝛿^𝑏+𝛿/2_{𝑎−𝛿/2}≤1(ℎ−𝑗)(𝑦,𝜏)𝑑𝑦𝛿^𝑏+𝛿_𝑎−𝛿(ℎ−𝑗)(𝑥,𝜏)𝑑𝑥≤𝐶₁𝑒^{−(1/(𝑝−2))𝜏}, (3.38) where we have used the fact supp𝑗(⋅)⊂[𝑎−𝛿,𝑎−𝛿/2]∪[𝑏+𝛿/2,𝑏+𝛿]. That is, ℎ(𝑥,𝜏)≤𝐶₁𝑒^{−(1/(𝑝−2))𝜏}. (3.39) This completes Lemma 3.8.

We are now in a position to prove Proposition 2.2. From Lemma 3.7 we observe that 𝑣 tends to the limit function 𝑤(𝑥) which is none for all points 𝑥>𝑝(𝑝−1)/(𝑝−2). Taking 𝑎>𝑝(𝑝−1)/(𝑝−2) and a small constant 𝛿>0 to satisfy 𝑎−𝛿>𝑝(𝑝−1)/(𝑝−2). Then, for a given 𝜖>0, the inequality 𝑣(𝑥,𝜏)≤𝜖 for all 𝑥≥𝑎−𝛿 holds provided 𝜏>𝜏₀, where 𝜏₀ is a large point which depends on 𝜖. Using comparison theorem, one has 𝑣(𝑥,𝜏+𝜏₀)≤ℎ(𝑥,𝜏),for𝜏>𝜏₀,𝑥>𝑎−𝛿. (3.40) The proof ends up with (3.40) and Lemma 3.8.

Finally, We prove that the function 𝑤(𝑥) appeared in Lemma 3.7 takes the form (2.4). For this purpose, it is enough to illuminate that 𝑤(𝑥)≡0 is impossible. We argue by contradiction. Assuming that 𝑤(𝑥)≡0 for a moment. Then 𝑢(𝑥,𝑡) blows up only at the boundary point 𝑥=0. By this we can find a sequence 𝑡_𝑛→∞ as 𝑛→∞ such that 𝑢(0,𝑡_𝑛)→∞as𝑛→∞. Moreover, By the maximin principle (see Remark 3.3), one has max_𝑥≥0{−|𝑢_𝑥|^𝑝−2𝑢_𝑥(𝑥,𝑡_𝑛)}=max{−|𝑢_0𝑥|^𝑝−2𝑢_0𝑥,𝑢^𝑝−1(0,𝑡_𝑛)}=𝑢^𝑝−1(0,𝑡_𝑛), (3.41) as long as 𝑛 is chosen large enough. Recalling that 𝑢 is nonincreasing with respect to 𝑥-variable, from (3.41) we conclude −𝑢_𝑥(𝑥,𝑡_𝑛)≤𝑢(0,𝑡_𝑛),∀𝑥≥0. (3.42) Integrating (3.42) gives rise to (1−𝑥)𝑢(0,𝑡_𝑛)≤𝑢(𝑥,𝑡_𝑛). (3.43) This shows that 𝑢 occurs blow-up at least in the interval [0,1), which contradicts with the assumption that blow-up happens only at 𝑥=0.

Now the proof of Theorem 1.2 is completed.