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

and Applied Analysis 3 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 ∫u 1,t 0 ψ ′ s ds −gp−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 ∫u 1,t 0 ψ ′ s ds −gp−1 s g ′ s ≥ C2 T − t . 1.6 Furthermore, if H1 holds, then the quenching rates are C1 T − t ≥ ∫u 1,t 0 ψ ′ s ds −gp−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 t |ux|ux x, 0 < x < 1, t > 0, ux 0, t 0, ux 1, t −u−q 1, t , t > 0, u 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 u0 x ≤ 0 and u′′ 0 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. 4 Abstract and Applied Analysis 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: ut a u |ux|ux x, 0 < x < 1, t > 0, ux 0, t 0, ux 1, t −g u 1, t , t > 0, u 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 u0 x ≤ 0, u′′ 0 x ≤ 0 for 0 ≤ x ≤ 1. Then ux x, t < 0 and ut x, t < 0 in 0, 1 × 0, T0 . Proof. Let v x, t ux x, t . Then v x, t satisfies vt 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 ux 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 ux x, t is a classical solution of 2.2 . Similarly, letting w x, t ut x, t , we have wt a′ u |ux|ux xw ( p − 1)a u |ux|wx x, 0 < x < 1, 0 < t < T0, wx 0, t 0, wx 1, t −g ′ u 1, t w 1, t , 0 < t < T0, w x, 0 ( p − 1)a u0 x ∣∣u′0 x ∣∣p−2u′′0 x , 0 ≤ x ≤ 1. 2.3 Making use of the maximum principle, we obtain ut x, t < 0 in 0, 1 × 0, T0 . Hence, the solutions of problem 2.1 u ∈ C2,1 0, 1 × 0, T0 with ux x, t < 0 and ut 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, whereM max0≤x≤1u0 x . Define F t ∫1 0 ψ u x, t dx. Then F t satisfies F ′ t ∫1


Introduction
In this paper, we consider the following problem: 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 u 0 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 problem where q > 0. In 14 , it was shown that u quenches in finite time for all u 0 , 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:

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 , 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−2 u x x , see 11 , to a more general form ψ u t |u x | p−2 u 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 u 0 x ≤ 0 for 0 ≤ x ≤ 1, and one of the following conditions holds: Then every solution of 1.1 quenches in finite time, and the only quenching point is x 1.

Abstract and Applied Analysis 3
Next, we deal with the quenching rate.Before we establish upper bounds for the quenching rate, we introduce the following hypothesis: Theorem 1.2.Suppose that the conditions of Theorem 1.1 and the hypothesis H 1 hold.Then there exists a positive constant C 1 such that 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 Theorem 1.3.Suppose that the hypotheses of Theorem 1.1 hold.Furthermore, suppose that the hypotheses H 2 -H 4 hold.Then there exists a positive constant C 2 such that Furthermore, if H 1 holds, then the quenching rates are Next, as an application of the main results of this paper, we study the following concrete example: where q > 0, p > 1, and m > 0. We will verify that 1.8 satisfies the hypotheses H 1 -H 4 , and we give the following theorem.
Theorem 1.4.Suppose that u 0 x ≤ 0 and u 0 x ≤ 0 for 0 ≤ x ≤ 1.Then the solution of 1.8 satisfies where C 3 and C 4 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.

Quenching on the Boundary
In this section, we prove finite time quenching.We rewrite problem 1.1 into the following form: 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, T 0 for some T 0 > 0, and

2.2
The maximum principle leads to v x, t < 0, and thus u x x, t < 0 in 0, 1 × 0, T 0 .Then it is easy to see that the problem 2.2 is nondegenerate in 0, 1 × 0, T 0 .So u x x, t is a classical solution of 2.2 .Similarly, letting w x, t u t x, t , we have

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≤1 u 0 x .Define F t 1 0 ψ u x, t dx.Then F t satisfies

2.4
Abstract and Applied Analysis 5 Thus F t ≤ F 0 − g p−1 M t, which means that F t 0 0 for some t 0 > 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 ≤ T 0 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.
Thus by the maximum principle, we have h x, t ≤ 0 in 1/4 , 1 × η, T , which leads to So we have Integrating 2.7 from x to 1, we obtain

2.11
Integrating 2.11 from t to T , we obtain

2.12
Define G u u 0 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 where c 1 and c 2 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 x, η ≥ 0 if c 1 and c 2 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 ≥ c 1 1 − x 2 > 0 if x < 1.

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−2 u x ϕ p−1 x g p−1 u x, t in 0, 1 × η, T , where ϕ x is given as follows: with some x 0 < 1 and l ≥ max{3, 1/ p − 1 } is chosen so large that ϕ x ≤ − u 0 x / g u 0 x for x 0 < 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

3.3
By H 1 and the definition of ϕ x , it follows that 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,

3.7
Integrating 3.7 from t to T , we get 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 H 2 -H 4 are equivalent to , where ε is a positive constant.Through a fairly complicated calculation, we find that where

3.10
Abstract and Applied Analysis 9 Since the hypotheses H 2 -H 3 hold, and d u < 0, a u > 0, we see that R 2 x, t < 0. Thus, we have 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 H 4 , we have 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,

3.13
Integration of 3.13 over t, T , then leads to 3.14

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 H 1 .Since

4.14
Integrating the above inequality from t to T , we obtain y m qp 1 / γ m q 1 t ≤ m qp 1 γ m q 1 c −γ q p−1 4 T − t , 4.15 that is, y 1/ γ m q 1 1, t ≤ C 3 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 H 2 -H 4 .