Quenching for a Non-Newtonian Filtration Equation with a Singular Boundary Condition

Abstract

This paper deals with a nonlinear 𝑝-Laplacian equation with singular boundary conditions. Under proper conditions, the solution of this equation quenches in finite time and the only quenching point thatis 𝑥=1 are obtained. Moreover, the quenching rate of this equation is established. Finally, we give an example of an application of our results.

1. Introduction

In this paper, we consider the following problem:(𝜓(𝑢))_𝑡=(|𝑢_𝑥|^𝑝−2𝑢_𝑥)_𝑥𝑢,0<𝑥<1,𝑡>0,_𝑥(0,𝑡)=0,𝑢_𝑥()(𝑢(𝑢()1,𝑡=−𝑔1,𝑡)),𝑡>0,𝑥,0=𝑢₀(𝑥),0≤𝑥≤1, (1.1) where 𝜓(𝑢) is a monotone increasing function with 𝜓(0)=0, 𝑝>1, 𝑔(𝑢)>0, 𝑔^′(𝑢)<0 for 𝑢>0, and lim_𝑢→0+𝑔(𝑢)=+∞. The initial value 𝑢₀(𝑥) is positive and satisfying some compatibility conditions.

If 𝜓(𝑢)=𝑢^1/𝑚 with 𝑚>0, (1.1) becomes the well-known non-Newtonian filtration equation, which is used to describe the non-stationary flow in a porous medium of fluids with a power dependence of the tangential stress on the velocity of the displacement under polytropic conditions (see [1, 2]).

Many papers have been devoted to the study of critical exponents of non-Newtonian filtration equation, see [3–5]. There are many results on the quenching phenomenon, see, for instance [6–12]. By the quenching phenomenon we mean that the solution approaches a constant but its derivative with respect to time variable 𝑡 tends to infinity as (𝑥,𝑡) tends to some point in the spatial-time space. The study of the quenching phenomenon began with the work of Kawarada through the famous initial boundary problem for the reaction-diffusion equation: 𝑢_𝑡=𝑢_𝑥𝑥+1/(1−𝑢) (see [13]).

As an example of the type of results, we wish to obtain, let us recall results for a closely related problem𝑢_𝑡=𝑢_𝑥𝑥𝑢,0<𝑥<1,𝑡>0,_𝑥(0,𝑡)=0,𝑢_𝑥(1,𝑡)=−𝑢^−𝑞()𝑢()1,𝑡,𝑡>0,𝑥,0=𝑢₀(𝑥),0≤𝑥≤1, (1.2) where 𝑞>0. In [14], it was shown that 𝑢 quenches in finite time for all 𝑢₀, and the only quenching point is 𝑥=1. Furthermore, the behavior of 𝑢 near quenching was described there. It is easily seen that (1.2) is a special case of (1.1).

If 𝑝=2, then (1.1) reduces to the following equation:(𝜓(𝑢))_𝑡=𝑢_𝑥𝑥𝑢,0<𝑥<1,𝑡>0,_𝑥(0,𝑡)=0,𝑢_𝑥()(𝑢(𝑢()1,𝑡=−𝑔1,𝑡)),𝑡>0,𝑥,0=𝑢₀(𝑥),0≤𝑥≤1. (1.3) Deng and Xu proved in [15] the finite time quenching for the solution and established results on quenching set and rate for (1.3). If 𝜓(𝑢)=𝑢, then (1.1) reduces to the following equation, see [11],𝑢_𝑡=(|𝑢_𝑥|^𝑝−2𝑢_𝑥)_𝑥𝑢,0<𝑥<1,𝑡>0,_𝑥(0,𝑡)=0,𝑢_𝑥()(𝑢(𝑢()1,𝑡=−𝑔1,𝑡)),𝑡>0,𝑥,0=𝑢₀(𝑥),0≤𝑥≤1, (1.4) and they obtained that the bounds for the quenching rate, and the quenching occurs only at 𝑥=1.

In this paper, we extend the equation 𝑢_𝑡=(|𝑢_𝑥|^𝑝−2𝑢_𝑥)_𝑥, see [11], to a more general form (𝜓(𝑢))_𝑡=(|𝑢_𝑥|^𝑝−2𝑢_𝑥)_𝑥. We prove that quenching occurs only at 𝑥=1. We determine the bounds for the quenching rate, and present an example which shows the applicability of our results.

The main results are stated as follows.

Theorem 1.1. Suppose that the initial data satisfies 𝑢^′₀(𝑥)≤0 and 𝑢₀^′′(𝑥)≤0 for 0≤𝑥≤1, and one of the following conditions holds:(i)𝜓^′′(𝑢)>0 for 𝑢>0,(ii)𝜓^′′(𝑢)<0 for 𝑢>0, limsup_𝑢→0+(𝑔′(𝑢)/𝜓′(𝑢))<0.Then every solution of (1.1) quenches in finite time, and the only quenching point is 𝑥=1.

Next, we deal with the quenching rate. Before we establish upper bounds for the quenching rate, we introduce the following hypothesis:(𝐻₁)𝜓^′′(𝑢)𝑔(𝑢)≥2(𝑝−1)𝜓^′(𝑢)𝑔^′(𝑢), 𝜓^′(𝑢)[(𝑝−2)𝑔^′2(𝑢)+𝑔(𝑢)𝑔^′′(𝑢)]≥𝜓^′′(𝑢)𝑔(𝑢)𝑔^′(𝑢).

Theorem 1.2. Suppose that the conditions of Theorem 1.1 and the hypothesis (𝐻₁) hold. Then there exists a positive constant 𝐶₁ such that ₀^𝑢(1,𝑡)𝜓^′(𝑠)𝑑𝑠−𝑔^𝑝−1(𝑠)𝑔^′(𝑠)≤𝐶₁(𝑇−𝑡). (1.5)

Next, we will give the lower bound on the quenching rate, the derivation of which is in the spirit of [15]. We need the following additional hypotheses: there exists a constant 𝜎(−∞<𝜎≤𝜎₀=min{1,2−1/(𝑝−1)}) such that(𝐻₂)(𝑔^{(𝑝−1)(𝜎−1)}(𝑢)𝑔′(𝑢)/𝜓′(𝑢))^′′<0, (𝐻₃)(5𝑝−2𝜎𝑝−2𝜎−6)(𝑔^{(𝑝−1)(𝜎−1)}(𝑢)𝑔′(𝑢))′𝜓′(𝑢)≥(𝑝−1)(3−𝜎)𝑔^{(𝑝−1)(𝜎−1)}(𝑢)𝑔^′(𝑢)𝜓^′′(𝑢), (𝐻₄)(𝑔^{(𝑝−1)(𝜎−1)−1}(𝑢)𝑔′(𝑢)/𝜓′(𝑢))′<0.

Theorem 1.3. Suppose that the hypotheses of Theorem 1.1 hold. Furthermore, suppose that the hypotheses (𝐻₂)–(𝐻₄) hold. Then there exists a positive constant 𝐶₂ such that 𝜓′(₀^𝑢(1,𝑡)𝑠)𝑑𝑠−𝑔^𝑝−1(𝑠)𝑔′(𝑠)≥𝐶₂(𝑇−𝑡). (1.6)Furthermore, if (𝐻₁) holds, then the quenching rates are 𝐶₁(𝑇−𝑡)≥₀^𝑢(1,𝑡)𝜓′(𝑠)𝑑𝑠−𝑔^𝑝−1(𝑠)𝑔′(𝑠)≥𝐶₂(𝑇−𝑡). (1.7)

Next, as an application of the main results of this paper, we study the following concrete example:(𝑢^𝑚)_𝑡=(|𝑢_𝑥|^𝑝−2𝑢_𝑥)_𝑥𝑢,0<𝑥<1,𝑡>0,_𝑥(0,𝑡)=0,𝑢_𝑥(1,𝑡)=−𝑢^−𝑞()𝑢()1,𝑡,𝑡>0,𝑥,0=𝑢₀(𝑥),0≤𝑥≤1, (1.8) where 𝑞>0, 𝑝>1, and 𝑚>0. We will verify that (1.8) satisfies the hypotheses (𝐻₁)–(𝐻₄), and we give the following theorem.

Theorem 1.4. Suppose that 𝑢^′₀(𝑥)≤0 and 𝑢₀^′′(𝑥)≤0 for 0≤𝑥≤1. Then the solution of (1.8) satisfies 𝐶₄≤𝑢(1,𝑡)(𝑇−𝑡)^{−1/(𝑚+𝑝𝑞+1)}≤𝐶₃, (1.9) where 𝐶₃ and 𝐶₄ are positive constants.

The plan of this paper is as follows. In Section 2, we prove that quenching occurs only at 𝑥=1, that is the proof of Theorem 1.1. In Section 3, we derive the estimates for the quenching rate, that is the proof of Theorems 1.2 and 1.3. In Section 4, we present results for certain 𝜓(𝑢) and 𝑔(𝑢), that is the proof of the Theorem 1.4.

2. Quenching on the Boundary

In this section, we prove finite time quenching. We rewrite problem (1.1) into the following form:𝑢_𝑡=𝑎(𝑢)(|𝑢_𝑥|^𝑝−2𝑢_𝑥)_𝑥𝑢,0<𝑥<1,𝑡>0,_𝑥(0,𝑡)=0,𝑢_𝑥()(𝑢(𝑢()1,𝑡=−𝑔1,𝑡)),𝑡>0,𝑥,0=𝑢₀(𝑥),0≤𝑥≤1, (2.1) where 𝑎(𝑢)=1/(𝜓′(𝑢)). Clearly, 𝜓′(𝑢)≠0 for 𝑢>0.

Lemma 2.1. Assume the solution 𝑢 of problem (2.1) exists in (0,𝑇₀) for some 𝑇₀>0, and 𝑢^′₀(𝑥)≤0, 𝑢₀^′′(𝑥)≤0 for 0≤𝑥≤1. Then 𝑢_𝑥(𝑥,𝑡)<0 and 𝑢_𝑡(𝑥,𝑡)<0 in (0,1]×(0,𝑇₀).

Proof. Let 𝑣(𝑥,𝑡)=𝑢_𝑥(𝑥,𝑡). Then 𝑣(𝑥,𝑡) satisfies 𝑣_𝑡=𝑎(𝑢)(|𝑣|^𝑝−2𝑣)_𝑥𝑥+𝑎^′(𝑢)𝑣(|𝑣|^𝑝−2𝑣)_𝑥,0<𝑥<1,0<𝑡<𝑇₀,𝑣(0,𝑡)=0,𝑣(1,𝑡)=−𝑔(𝑢(1,𝑡)),0<𝑡<𝑇₀,𝑣(𝑥,0)=𝑢^′₀(𝑥),0≤𝑥≤1. (2.2)The maximum principle leads to 𝑣(𝑥,𝑡)<0, and thus 𝑢_𝑥(𝑥,𝑡)<0 in (0,1]×(0,𝑇₀). Then it is easy to see that the problem (2.2) is nondegenerate in (0,1]×(0,𝑇₀). So 𝑢_𝑥(𝑥,𝑡) is a classical solution of (2.2). Similarly, letting 𝑤(𝑥,𝑡)=𝑢_𝑡(𝑥,𝑡), we have 𝑤_𝑡=𝑎^′(𝑢)(|𝑢_𝑥|^𝑝−2𝑢_𝑥)_𝑥𝑤+𝑝−1𝑎(𝑢)(|𝑢_𝑥|^𝑝−2𝑤_𝑥)_𝑥,0<𝑥<1,0<𝑡<𝑇₀,𝑤_𝑥(0,𝑡)=0,𝑤_𝑥(1,𝑡)=−𝑔^′(𝑢(1,𝑡))𝑤(1,𝑡),0<𝑡<𝑇₀,𝑤(𝑥,0)=𝑝−1𝑎𝑢₀(𝑥)𝑢^′₀(𝑥)^𝑝−2𝑢₀^′′(𝑥),0≤𝑥≤1. (2.3)Making use of the maximum principle, we obtain 𝑢_𝑡(𝑥,𝑡)<0 in (0,1]×(0,𝑇₀). Hence, the solutions of problem (2.1) 𝑢∈𝐶^2,1((0,1]×(0,𝑇₀)) with 𝑢_𝑥(𝑥,𝑡)<0 and 𝑢_𝑡(𝑥,𝑡)<0 in (0,1]×(0,𝑇₀).

The Proof of Theorem 1.1
By the maximum principle, we know that 0<𝑢(⋅,𝑡)≤𝑀 for all 𝑡 in the existence interval, where 𝑀=max_0≤𝑥≤1𝑢₀(𝑥). Define 𝐹(𝑡)=∫¹₀𝜓(𝑢(𝑥,𝑡))𝑑𝑥. Then 𝐹(𝑡) satisfies 𝐹^′(𝑡)=¹₀(𝜓(𝑢(𝑥,𝑡)))_𝑡𝑑𝑥=¹₀𝑢_𝑥^𝑝−2𝑢_𝑥𝑥𝑑𝑥=−𝑔(𝑢(1,𝑡))𝑔(𝑢(1,𝑡))^𝑝−2≤−𝑔^𝑝−1(𝑀). (2.4) Thus 𝐹(𝑡)≤𝐹(0)−𝑔^𝑝−1(𝑀)𝑡, which means that 𝐹(𝑡₀)=0 for some 𝑡₀>0. From the fact that 𝜓(𝑢)>0 for 𝑢>0 and 𝑢_𝑥(𝑥,𝑡)<0 for 0<𝑥≤1, we find that there exists a 𝑇(0<𝑡≤𝑇₀) such that lim_{𝑡→𝑇−}𝑢(1,𝑡)=0. By virtue of the singular nonlinearity in the boundary condition, 𝑢 must quench at 𝑥=1. In what follows, we only need to prove that quenching cannot occur in ((1/2),1)×(𝜂,𝑇) for some 𝜂(0<𝜂<𝑇). Consider two cases.

Case 1. 𝜓^′′(𝑢)>0 for 𝑢>0. Let ℎ(𝑥,𝑡)=|𝑢_𝑥|^𝑝−2𝑢_𝑥+𝜀(𝑥−(1/4))𝑔^𝑝−1(𝑀) in ((1/4),1)×(𝜂,𝑇), where 𝜀 is a positive constant. Then ℎ(𝑥,𝑡) satisfies ℎ_𝑡=𝑢_𝑥^𝑝−2𝑢_𝑥𝑡=𝑝−1|𝑢_𝑥|^𝑝−2𝑢_𝑥𝑡=𝑝−1𝑢_𝑥^𝑝−2𝑎(𝑢)|𝑢_𝑥|^𝑝−2𝑢_𝑥𝑥𝑥=𝑝−1|𝑢_𝑥|^𝑝−2𝑎^′(𝑢)𝑢_𝑥𝑢_𝑡𝑎(𝑢)+𝑎(𝑢)ℎ_𝑥𝑥≤𝑝−1𝑎(𝑢)|𝑢_𝑥|^𝑝−2ℎ_𝑥𝑥, (2.5) for (𝑥,𝑡)∈((1/4),1)×(𝜂,𝑇), since (𝑎′(𝑢))/(𝑎(𝑢))=−(𝜓′′(𝑢))/(𝜓′(𝑢))≤0. On the parabolic boundary, ℎ((1/4),𝑡)=|𝑢_𝑥|^𝑝−2𝑢_𝑥((1/4),𝑡)<0 for 𝜂≤𝑡<𝑇; if 𝜀 is sufficiently small, ℎ(1,𝑡)≤𝑔^𝑝−1(𝑀)((3𝜀/4)−1)<0 for 𝜂≤𝑡<𝑇, and ℎ(𝑥,𝜂)≤−|𝑢_𝑥((1/4),𝜂)|^𝑝−1+(3𝜀/4)𝑔^𝑝−1(𝑀)<0 for (1/4)≤𝑥≤1. Thus by the maximum principle, we have ℎ(𝑥,𝑡)≤0 in ((1/4),1)×(𝜂,𝑇), which leads to |𝑢_𝑥|^𝑝−2𝑢_𝑥1+𝜀𝑥−4𝑔^𝑝−1(𝑀)1≤0in4,1×𝜂,𝑇. (2.6) So we have 𝜀1𝑥−4𝑔^𝑝−1(𝑀)^1/(𝑝−1)≤−𝑢_𝑥. (2.7) Integrating (2.7) from 𝑥 to 1, we obtain 𝑢(𝑥,𝑡)≥𝑢(1,𝑡)+¹_𝑥𝜀1𝑥−4𝑔^𝑝−1(𝑀)^1/(𝑝−1)𝑑𝑥>¹_𝑥𝜀1𝑥−4𝑔^𝑝−1(𝑀)^1/(𝑝−1)𝑑𝑥>0. (2.8) It then follows that 𝑢(𝑥,𝑡)>0 if 𝑥<1.

Case 2. 𝜓^′′(𝑢)<0 for 𝑢>0. Let 𝑘(𝑥,𝑡)=𝑢_𝑡−𝜀(𝑥−(1/2))𝑢_𝑥 in ((1/2),1)×(𝜂,𝑇). Then 𝑘(𝑥,𝑡) satisfies 𝑘_𝑡=𝑢_𝑡𝑡1−𝜀𝑥−2𝑢_𝑡𝑥=𝑝−1𝑎(𝑢)𝑢_𝑥^𝑝−2𝑘_𝑥𝑥+𝑝−1𝑝−2𝑎(𝑢)𝑢_𝑥^𝑝−2−𝑢_𝑥𝑥𝑘_𝑥+𝑎^′(𝑢)𝑢_𝑥^𝑝−2𝑢_𝑥𝑥𝑘+𝜀𝑝𝑝−1𝑎(𝑢)𝑢_𝑥^𝑝−2𝑢_𝑥𝑥≤𝑝−1𝑎(𝑢)𝑢_𝑥^𝑝−2𝑘_𝑥𝑥+𝑝−1𝑝−2𝑎(𝑢)𝑢_𝑥^𝑝−2−𝑢_𝑥𝑥𝑘_𝑥+𝑎^′(𝑢)𝑢_𝑥^𝑝−2𝑢_𝑥𝑥𝑘, (2.9) for (𝑥,𝑡)∈((1/2),1)×(𝜂,𝑇). On the boundary, 𝑘((1/2),𝑡)=𝑢_𝑡((1/2),𝑡)<0 for 𝜂≤𝑡<𝑇. Since lim_{𝑡→𝑇−}𝑢(1,𝑡)=0 and limsup_𝑢→0+((𝑔′(𝑢))/(𝜓′(𝑢)))<0, if 𝜂 is close to 𝑇 and 𝜀 is small enough, 𝑔^′(𝑢(1,𝑡))+(𝜀𝜓(𝑢(1,𝑡))^′/2(𝑝−1)𝑔^(𝑝−2)𝑢(1,𝑡))+2≤0 for 𝜂≤𝑡<𝑇. Thus 𝑘_𝑥(1,𝑡)=𝑢_𝑡𝑥()−𝜀1,𝑡2𝑢_𝑥𝑥(1,𝑡)−𝜀𝑢_𝑥()𝑔1,𝑡=−^′(𝑢(1,𝑡))+𝜀𝜓^′(𝑢(1,𝑡))2𝑝−1𝑔^𝑝−2(𝑢(𝑢1,𝑡))+2_𝑡()()().1,𝑡+2𝑘1,𝑡≤2𝑘1,𝑡 (2.10) It is easily seen that 𝑘(𝑥,𝜂)≤0 for (1/2)≤𝑥≤1. Hence, the maximum principle yields that 𝑘(𝑥,𝑡)≤0 in [(1/2),1]×[𝜂,𝑇). In particular, 𝑢_𝑡()+𝜀1,𝑡2𝑔(𝑢(1,𝑡))≤0for𝜂≤𝑡<𝑇. (2.11) Integrating (2.11) from 𝑡 to 𝑇, we obtain ₀^𝑢(1,𝑡)𝑑𝑠𝑔(𝑠)≥𝜀2(𝑇−𝑡)for𝜂≤𝑡<𝑇. (2.12) Define 𝐺(𝑢)=∫^𝑢₀(1/𝑔(𝑠))𝑑𝑠. Since 𝐺^′(𝑢)=1/𝑔(𝑢)>0 for 𝑢>0, the inverse 𝐺⁻¹ exists. In view of (2.12), we can see 𝑢(1,𝑡)=𝐺⁻¹(𝐺(𝑢(1,𝑡)))≥𝐺⁻¹𝜀2(𝑇−𝑡)for𝜂≤𝑡<𝑇. (2.13) Let 𝐻(𝑥,𝑡)=𝑢(𝑥,𝑡)−𝑐₁(1−𝑥²)−𝑐₂𝐺⁻¹((𝜀/2)(𝑇−𝑡)), where 𝑐₁ and 𝑐₂ are positive constants. Since 𝑔′(𝑢)<0, 𝑢_𝑥(𝑥,𝑡)<0, and by (2.13), we have that in (0,1)×(𝜂,𝑇)𝐻_𝑡=𝑝−1𝑎(𝑢)|𝑢_𝑥|^𝑝−2𝑢_𝑥𝑥+𝜀𝑐₂2𝑔𝐺⁻¹𝜀2()≥𝑎(𝑢)𝑇−𝑡𝑝−1|𝑢_𝑥|^𝑝−2𝐻_𝑥𝑥−2𝑐₁𝑝−1𝑎(𝑢)𝑢_𝑥^𝑝−2+𝜀𝑐₂2𝑔(𝑢(≥𝑎(𝑢𝑢1,𝑡))𝑝−1)_𝑥^𝑝−2𝐻_𝑥𝑥−2𝑐₁𝑝−1𝑢_𝑥^𝑝−2(𝑀)+𝜓′𝜀𝑐₂2𝑔(𝑀)≥𝑝−1𝑎(𝑢)𝑢_𝑥^𝑝−2𝐻_𝑥𝑥, (2.14)provided 𝜀𝑐₂𝑔(𝑀)𝜓^′(𝑀)≥4𝑐₁(𝑝−1)|𝑢_𝑥|^𝑝−2, which is true since 0≤|𝑢_𝑥(𝑥,𝑡)|≤𝑔(𝑢(1,𝑡))=𝑔(𝑀₀), where 0<𝑀₀≤𝑀. On the other hand, 𝐻_𝑥(0,𝑡)=0; 𝐻(1,𝑡)=𝑢(1,𝑡)−𝑐₂𝐺⁻¹((𝜀/2)(𝑇−𝑡))≥0 if 𝑐₂≤1 and 𝐻(𝑥,𝜂)≥0 if 𝑐₁ and 𝑐₂ are small enough. Thus by the maximum principle, we find that 𝐻(𝑥,𝑡)≥0 in (0,1)×(𝜂,𝑇), which implies that 𝑢(𝑥,𝑡)≥𝑐₁(1−𝑥²)>0 if 𝑥<1.

3. Bounds for Quenching Rate

In this section, we establish bounds on the quenching rate. We first present the upper bound.

The Proof of Theorem 1.2
We define a function Φ(𝑥,𝑡)=|𝑢_𝑥|^𝑝−2𝑢_𝑥+𝜑^𝑝−1(𝑥)𝑔^𝑝−1(𝑢(𝑥,𝑡)) in (0,1)×(𝜂,𝑇), where 𝜑(𝑥) is given as follows: 𝜑(𝑥)=0,𝑥∈0,𝑥₀,𝑥−𝑥₀^𝑙(1−𝑥₀)^𝑙,𝑥∈𝑥₀,1, (3.1) with some 𝑥₀<1 and 𝑙≥max{3,(1/(𝑝−1))} is chosen so large that 𝜑(𝑥)≤−(𝑢^′₀(𝑥)/𝑔(𝑢₀(𝑥)) for 𝑥₀<𝑥≤1. It is easy to see that Φ(0,𝑡)=Φ(1,𝑡)=0, and Φ(𝑥,0)≤0. On the other hand, in (0,1)×(𝜂,𝑇), Φ satisfies Φ_𝑡=𝑝−1𝑎(𝑢)|𝑢_𝑥|^𝑝−2Φ_𝑥𝑥+𝑝−1𝑎^′(𝑢)𝑢_𝑥|𝑢_𝑥|^𝑝−2Φ_𝑥−𝑅₁(𝑥,𝑡), (3.2) where 𝑅₁(𝑥,𝑡)=𝑝−1²𝑎(𝑢)𝑢_𝑥^𝑝−2𝜑^𝑝−3(𝑥)𝑔^𝑝−1(𝑢)𝑝−2𝜓^′2(𝑥)+𝜑(𝑥)𝜑^′′(𝑥+)𝑝−1²𝑎²(𝑢)𝑢_𝑥^𝑝−2𝑢_𝑥𝜑^𝑝−2(𝑥)𝜑^′(𝑥)𝑔^𝑝−2(𝑢)−𝑔(𝑢)𝜓^′′(𝑢)+2𝑝−1𝑔^′(𝑢)𝜓^′(𝑢+)𝑝−1²𝑎(𝑢)|𝑢_𝑥|^𝑝𝜑^𝑝−1(𝑥)𝑔^𝑝−3(𝑢−𝜓)^′′(𝑢)𝜓^′(𝑢)𝑔(𝑢)𝑔^′(𝑢)+𝑝−2𝑔^′2(𝑢)+𝑔(𝑢)𝑔^′′(𝑢). (3.3) By (𝐻₁) and the definition of 𝜑(𝑥), it follows that Φ_𝑡≤𝑝−1𝑎(𝑢)|𝑢_𝑥|^𝑝−2Φ_𝑥𝑥+𝑝−1𝑎^′(𝑢)𝑢_𝑥|𝑢_𝑥|^𝑝−2Φ_𝑥. (3.4) Thus, the maximum principle yields Φ(𝑥,𝑡)≤0, that is 𝜑(𝑥)𝑔(𝑢(𝑥,𝑡))≤−𝑢_𝑥(𝑥,𝑡),for[0,1]×𝜂,𝑇. (3.5) Moreover, by the definition of the limit, we see that Φ_𝑥(1,𝑡)≥0 since Φ(𝑥,𝑡)≤0. In fact, Φ_𝑥(1,𝑡)=lim_𝑥→1−Φ(𝑥,𝑡)−Φ(1,𝑡)𝑥−1≥0, (3.6) which means 𝑢_𝑡(1,𝑡)≥𝑝−1𝑎(𝑢(1,𝑡))𝑔^𝑝−1(𝑢(1,𝑡))𝑔^′(𝑢(1,𝑡))−𝜑^′(1)≥𝑐₃𝑝−1𝑎(𝑢(1,𝑡))𝑔^𝑝−1(𝑢(1,𝑡))𝑔^′(𝑢(1,𝑡)). (3.7) Integrating (3.7) from 𝑡 to 𝑇, we get ₀^𝑢(1,𝑡)𝜓^′(𝑠)𝑑𝑠−𝑔^𝑝−1(𝑠)𝑔^′(𝑠)≤𝐶₁(𝑇−𝑡). (3.8)
We then give the lower bound.

The Proof of Theorem 1.3
Let 𝑑(𝑢)=𝑎(𝑢)𝑔^{(𝑝−1)(𝜎−1)}(𝑢)𝑔^′(𝑢). Notice that the hypotheses (𝐻₂)–(𝐻₄) are equivalent to(𝐻₂)𝑑^′′(𝑢)≤0, (𝐻₃)𝑑^′(𝑢)(5𝑝−2𝜎𝑝+2𝜎−6)≥(𝑑(𝑢)/𝑎(𝑢))𝑎^′(𝑢)(2𝑝−𝜎𝑝+𝜎−3), (𝐻₄)𝑑(𝑢)𝑔′(𝑢)>𝑑′(𝑢)𝑔(𝑢), respectively. Letting 𝜏 be close to 𝑇, we consider Ψ(𝑥,𝑡)=𝑢_𝑡−𝜀𝑑(𝑢)(−𝑢_𝑥)^{(𝑝−1)(2−𝜎)} in (1−𝑇+𝜏,1)×(𝜏,𝑇), where 𝜀 is a positive constant. Through a fairly complicated calculation, we find that Ψ_𝑡=𝑝−1𝑎(𝑢)|𝑢_𝑥|^𝑝−2Ψ_𝑥𝑥+𝑝−1𝑝−2𝑎(𝑢)|−𝑢_𝑥|^𝑝−3−𝑢_𝑥𝑥Ψ_𝑥+𝐶(𝑥,𝑡)Ψ+𝑅₂(𝑥,𝑡), (3.9) where 𝐶()=𝑎𝑥,𝑡^′(𝑢)𝑎(𝑢)()𝑑(𝑢)+𝜀2−𝜎𝑎(𝑢)2𝑝−𝜎𝑝+𝜎−3−𝑢_𝑥^{𝑝−𝜎𝑝+𝜎−2}𝑢_𝑡+𝜀²()𝑑2−𝜎²(𝑢)𝑎(𝑢)2𝑝−𝜎𝑝+𝜎−3(−𝑢_𝑥)^{3𝑝−2𝜎𝑝+2𝜎−4}+𝜀𝑑^′(𝑢−𝑑(𝑢))5𝑝−2𝜎𝑝+2𝜎−6𝑎(𝑢)𝑎^′(𝑢)2𝑝−𝜎𝑝+𝜎−3(−𝑢_𝑥)^{2(𝑝−1)(2−𝜎)},𝑅₂(𝑥,𝑡)=𝜀³()𝑑2−𝜎³(𝑢)𝑎(𝑢)2𝑝−𝜎𝑝+𝜎−3(−𝑢_𝑥)^{5𝑝−3𝜎𝑝+3𝜎−6}+𝜀²𝑑(𝑢)𝑑^′(𝑢)−𝑑(𝑢)5𝑝−2𝜎𝑝+2𝜎−6𝑎(𝑢)𝑎^′(𝑢)2𝑝−𝜎𝑝+𝜎−3(−𝑢_𝑥)^{2(𝑝−1)(2−𝜎)}+𝜀𝑎(𝑢)𝑝−1𝑑^′′(𝑢)(−𝑢_𝑥)^{(𝑝−1)(3−𝜎)+1}. (3.10) Since the hypotheses (𝐻₂)–(𝐻₃) hold, and 𝑑(𝑢)<0, 𝑎(𝑢)>0, we see that 𝑅₂(𝑥,𝑡)<0. Thus, we have Ψ_𝑡≤𝑝−1𝑎(𝑢)|𝑢_𝑥|^𝑝−2Ψ_𝑥𝑥+𝑝−1𝑝−2𝑎(𝑢)𝑢_𝑥^𝑝−3−𝑢_𝑥𝑥Ψ_𝑥+𝐶(𝑥,𝑡)Ψ, (3.11) for (𝑥,𝑡)∈(1−𝑇+𝜏,1)×(𝜏,𝑇). On the parabolic boundary, since 𝑥=1 is the only quenching point, if 𝜀 is small enough, then both Ψ(1−𝑇+𝜏,𝑡) and Ψ(𝑥,𝜏) are negative. At 𝑥=1, in view of (𝐻₄), we have Ψ_𝑥(1,𝑡)=−[1−𝜀(2−𝜎)]𝑔^′(𝑢(Ψ()(𝑔1,𝑡))1,𝑡−𝜀[1−𝜀2−𝜎)]^′(𝑢(1,𝑡))𝑑(𝑢(1,𝑡))−𝑑^′(𝑢(1,𝑡))𝑔(𝑢(1,𝑡))𝑔^{(𝑝−1)(2−𝜎)}(𝑢([(𝑔1,𝑡))≤−1−𝜀2−𝜎)]^′(𝑢(1,𝑡))Ψ(1,𝑡), (3.12) provided 𝜀 is sufficiently small. Hence, by the maximum principle, we have Ψ(𝑥,𝑡)≤0 on [1−𝑇+𝜏,1]×[𝜏,𝑇). In particular, Ψ(1,𝑡)≤0, that is, 𝑢_𝑡(1,𝑡)≤𝜀𝑑(𝑢(1,𝑡))𝑔^{(𝑝−1)(2−𝜎)}(𝑢(1,𝑡))=𝜀𝑎(𝑢(1,𝑡))𝑔^𝑝−1(𝑢(1,𝑡))𝑔^′(𝑢(1,𝑡)). (3.13) Integration of (3.13) over (𝑡,𝑇), then leads to ₀^𝑢(1,𝑡)𝜓^′(𝑠)𝑑𝑠−𝑔^𝑝−1(𝑠)𝑔^′(𝑠)≥𝐶₂(𝑇−𝑡). (3.14)

4. Results for Certain Nonlinearities

In this section, we give the concrete quenching rate of solutions for (1.8).

The Proof of Theorem 1.4
We first present the upper bound. Consider two cases.

Case 1. 𝑚+𝑞(𝑝−1)≥2. We only need to verify the hypothesis (𝐻₁). Since 𝜓^′′(𝑢)𝑔(𝑢)−2𝑝−1𝜓^′(𝑢)𝑔^′(𝑢)=𝑚𝑚+2𝑞𝑝−1−1𝑢^{𝑚−𝑞−2}𝜓≥0,_′(𝑢)𝑝−2𝑔^′2(𝑢)+𝑔(𝑢)𝑔^′′(𝑢)−𝜓^′′(𝑢)𝑔(𝑢)𝑔^′(𝑢)=𝑚𝑞𝑝𝑞−𝑞+𝑚−2𝑢^{𝑚−2𝑞−3}≥0, (4.1) then we have ₀^𝑢(1,𝑡)𝜓^′(𝑠)𝑑𝑠−𝑔^𝑝−1(𝑠)𝑔^′(𝑠)=₀^𝑢(1,𝑡)𝑚𝑠^𝑚−1−𝑠^{−𝑞(𝑝−1)}−𝑞𝑠^−𝑞−1𝑚𝑑𝑠=𝑞₀^𝑢(1,𝑡)𝑠^{𝑞𝑝+𝑚}=𝑚𝑑𝑠𝑞𝑢𝑚+𝑞𝑝+1^{𝑚+𝑞𝑝+1}(1,𝑡)≤𝐶₁(𝑇−𝑡). (4.2) Therefore, we get the upper bound as 𝑢(1,𝑡)(𝑇−𝑡)^{−(1/(𝑚+𝑞𝑝+1))}≤𝐶₁𝑚𝑞𝑚+𝑞𝑝+1^{1/(𝑚+𝑞𝑝+1)}∶=𝐶₃. (4.3)

Case 2. 𝑚+𝑞(𝑝−1)<2. We use a modification of an argument from [16]. For 𝑡∈[𝜂,𝑇) with some 𝜂 such that 𝑢(1,𝜂)<1, set 𝑦(𝑡)=𝑢^𝛾(1,𝑡)¹_{(𝑡)1−𝜉}𝑢^𝑚(𝑥,𝑡)𝑑𝑥, (4.4) with𝜉(𝑡)=𝑢^𝑞+1(1,𝑡), (4.5) where −(𝑚+𝑞+1)<𝛾≤−(𝑚+1)(𝑞+1).

A routine calculation shows 𝑦^′(𝑡)=𝛾𝑢^𝛾−1(1,𝑡)𝑢_𝑡(1,𝑡)¹_{(𝑡)1−𝜉}𝑢^𝑚(𝑥,𝑡)𝑑𝑥+𝑞+1𝑢^𝑞+𝛾(1,𝑡)𝑢^𝑚(1−𝜉(𝑡),𝑡)𝑢_𝑡(1,𝑡)+𝑢^𝛾(1,𝑡)𝑢_𝑥^𝑝−2𝑢_𝑥(1,𝑡)−𝑢_𝑥^𝑝−2𝑢_𝑥(1−𝜉(𝑡),𝑡)≥𝑢^𝛾−1(1,𝑡)𝑢_𝑡(1,𝑡)𝐼(𝑡)−𝑢^{𝛾−𝑞(𝑝−1)}(1,𝑡), (4.6) where 𝐼(𝑡)=𝛾∫¹_{1−𝜉(𝑡)}𝑢^𝑚(𝑥,𝑡)𝑑𝑥+(𝑞+1)𝑢^𝑞+1(1,𝑡)𝑢^𝑚(1−𝜉(𝑡),𝑡). Since 𝑢_𝑥≤0 and 𝑢_𝑥𝑥≤0 in [0,1]×[𝜂,𝑇), we find 𝑢(1,𝑡)≤𝑢(𝑥,𝑡)≤𝑢(1−𝜉(𝑡),𝑡)≤2𝑢(1,𝑡), (4.7) for any 𝑥∈[1−𝜉(𝑡),1] and 𝑡∈[𝜂,𝑇). By (4.4), (4.5) and (4.7), we have 𝑢^{𝛾+𝑚+𝑞+1}(1,𝑡)≤𝑦(𝑡)≤2^𝑚𝑢^{𝛾+𝑚+𝑞+1}(1,𝑡), (4.8) or equivalently, 𝑢(1,𝑡)≤𝑦^{1/(𝛾+𝑚+𝑞+1)}(𝑡)≤𝑐₄𝑢(1,𝑡)for𝑡∈𝜂,𝑇. (4.9) We now claim that 𝐼(𝑡)≤0 on [𝜂,𝑇). In fact 𝐼(𝑡)=𝛾+𝑞+1¹_{(𝑡)1−𝜉}𝑢^𝑚(𝑥,𝑡)𝑑𝑥−𝑞+1¹_{(𝑡)1−𝜉}𝑢^𝑚(𝑥,𝑡)−𝑢^𝑚((𝑡))=1−𝜉,𝑡𝑑𝑥𝛾+𝑞+1¹_{(𝑡)1−𝜉}𝑢^𝑚()−𝑥,𝑡𝑑𝑥𝑞+1¹_{(𝑡)1−𝜉}𝑚𝑢^𝑚−1(𝜁(𝑡),𝑡)𝑢_𝑥(𝜁(𝑡),𝑡)(𝑥−1+𝜉(𝑡))𝑑𝑥, (4.10) where 1−𝜉(𝑡)<𝜁(𝑡)<1. When 0<𝑚<1, 𝛾≤−(((𝑚+2)(𝑞+1))/2), because 𝑢_𝑥,𝑢_𝑥𝑥≤0 and ∫¹_{1−𝜉(𝑡)}(1−𝜉(𝑡),𝑡))𝑑𝑥=(1/2)𝑢^2𝑞+2(1,𝑡), we have 𝐼(𝑡)≤𝛾+𝑞+1¹_{(𝑡)1−𝜉}𝑢^𝑚(𝑥,𝑡)𝑑𝑥+𝑚𝑞+1¹_{(𝑡)1−𝜉}𝑢^𝑚−1(1,𝑡)−𝑢_𝑥()(𝑡=𝑚1,𝑡(𝑥−1+𝜉))𝑑𝑥𝛾+𝑞+1+𝑞+12𝑢^𝑚+𝑞+1(1,𝑡)≤0; (4.11) when 1≤𝑚<2, 𝛾≤−(𝑞+1)(𝑚+1), we have 𝐼(𝑡)≤𝛾+𝑞+1¹_{(𝑡)1−𝜉}𝑢^𝑚(𝑥,𝑡)𝑑𝑥+𝑚𝑞+1¹_{(𝑡)1−𝜉}(2𝑢(1,𝑡))^𝑚−1−𝑢_𝑥()(𝑡=1,𝑡(𝑥−1+𝜉))𝑑𝑥𝛾+𝑞+1+2^𝑚−2𝑚𝑞+1𝑢^𝑚+𝑞+1()≤𝑢1,𝑡𝛾+𝑞+1+𝑚𝑞+1^𝑚+𝑞+1(1,𝑡)≤0. (4.12) In conclusion, when 0<𝑚<2, 𝛾≤−(𝑞+1)(𝑚+1), we have 𝐼(𝑡)≤0. (4.13) From (4.6), (4.9), and (4.13), it then follows that 𝑦′(𝑡)≥−𝑢^{𝛾−𝑞(𝑝−1)}(1,𝑡)≥−𝑐₄^{−𝛾+𝑞(𝑝−1)}𝑦^{(𝛾−𝑞(𝑝−1))/(𝛾+𝑚+𝑞+1)}(1,𝑡). (4.14) Integrating the above inequality from 𝑡 to 𝑇, we obtain 𝑦^{(𝑚+𝑞𝑝+1)/(𝛾+𝑚+𝑞+1)}(𝑡)≤𝑚+𝑞𝑝+1𝑐𝛾+𝑚+𝑞+1^{4−𝛾+𝑞(𝑝−1)}(𝑇−𝑡), (4.15) that is, 𝑦^{1/(𝛾+𝑚+𝑞+1)}(1,𝑡)≤𝐶₃(𝑇−𝑡)^{1/(𝛾+𝑚+𝑞+1)}, (4.16) which in conjunction with (4.9) yields the desired upper bound.

We then give the lower bound. We examine the validity of hypotheses(𝐻₂)–(𝐻₄).

Firstly, for (𝐻₂), we find 𝑔^{(𝑝−1)(𝜎−1)}(𝑢)𝑔^′(𝑢)𝜓^′(𝑢)^′′𝑞=−𝑚𝑞⋅𝑞𝑢𝑝𝜎−𝜎−𝑝+2+𝑚𝑝𝜎−𝜎−𝑝+2+𝑚+1^{−𝑞(𝑝𝜎−𝜎−𝑝+2)−𝑚−2}<0, (4.17) provided 𝜎<𝜎₁=−((𝑚+1)/(𝑞(𝑝−1)))+((𝑝−2)/(𝑝−1)) or 𝜎>𝜎₂=−(𝑚/(𝑞(𝑝−1)))+((𝑝−2)/(𝑝−1)).

Secondly, for (𝐻₃), we have 5𝑝−2𝜎𝑝+2𝜎−6𝑔^{(𝑝−1)(𝜎−1)}(𝑢)𝑔^′(𝑢)^′𝜓^′(𝑢)−𝑝−1(3−𝜎)𝑔^{(𝑝−1)(𝜎−1)}(𝑢)𝑔^′(𝑢)𝜓^′′(𝑢)=𝑞𝑚−2𝑞(𝑝−1)²𝜎²+𝜎+𝑝−17𝑝𝑞−10𝑞−𝑚−1−5𝑝²𝑞+16𝑝𝑞−12𝑞+3𝑝𝑚+2𝑝−3𝑚−3𝑢^{−𝑞(𝑝−1)(𝜎−1)−𝑞+𝑚−3}≥0, (4.18) provided 𝜎₃≤𝜎≤𝜎₄ with 𝜎₃=𝑝−17𝑝𝑞−10𝑞−𝑚−1−√𝑏²−4𝑎𝑐4𝑞(𝑝−1)²,𝜎₄=𝑝−17𝑝𝑞−10𝑞−𝑚−1+√𝑏²−4𝑎𝑐4𝑞(𝑝−1)², (4.19) where 𝑎=−2𝑞(𝑝−1)², 𝑏=(𝑝−1)(7𝑝𝑞−10𝑞−𝑚−1), and 𝑐=−5𝑝²𝑞+16𝑝𝑞−12𝑞+3𝑝𝑚+2𝑝−3𝑚−3.

Thirdly, for (𝐻₄), we obtain 𝑔^{(𝑝−1)(𝜎−1)−1}(𝑢)𝑔^′(𝑢)𝜓^′(𝑢)^′𝑞=−𝑚−𝑞𝑝−1(𝜎−1)−𝑚𝑢^{−𝑞(𝑝−1)(𝜎−1)−𝑚−1}<0, (4.20) provided 𝜎<𝜎₅=1−(𝑞/(𝑚(𝑝−1))). Since 𝜎₂>𝜎₅, we choose a 𝜎 such that 𝜎₃<𝜎<min{𝜎₁,𝜎₄,𝜎₅} and hypotheses (𝐻₂)–(𝐻₄) hold. Thus the proof is completed.

Acknowledgments

This work is supported in part by NSF of PR China (11071266) and in part by Natural Science Foundation Project od CQ CSTC (2010BB9218).