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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 539161, 13 pages
http://dx.doi.org/10.1155/2012/539161
Research Article

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

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received 2 December 2011; Accepted 10 January 2012

Academic Editor: Kanishka Perera

Copyright © 2012 Xiliu Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper deals with a nonlinear -Laplacian equation with singular boundary conditions. Under proper conditions, the solution of this equation quenches in finite time and the only quenching point thatis 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: where is a monotone increasing function with , , , for , and . The initial value is positive and satisfying some compatibility conditions.

If with , (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 [35]. There are many results on the quenching phenomenon, see, for instance [612]. By the quenching phenomenon we mean that the solution approaches a constant but its derivative with respect to time variable tends to infinity as 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: (see [13]).

As an example of the type of results, we wish to obtain, let us recall results for a closely related problem where . In [14], it was shown that quenches in finite time for all , and the only quenching point is . Furthermore, the behavior of near quenching was described there. It is easily seen that (1.2) is a special case of (1.1).

If , then (1.1) reduces to the following equation: 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 , 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 .

In this paper, we extend the equation , see [11], to a more general form . We prove that quenching occurs only at . 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 and for , and one of the following conditions holds:(i) for ,(ii) for , .Then every solution of (1.1) quenches in finite time, and the only quenching point is .

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 hold. Then there exists a positive constant 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 such that(), (), ().

Theorem 1.3. Suppose that the hypotheses of Theorem 1.1 hold. Furthermore, suppose that the hypotheses hold. Then there exists a positive constant such that Furthermore, if holds, then the quenching rates are

Next, as an application of the main results of this paper, we study the following concrete example: where , , and . We will verify that (1.8) satisfies the hypotheses , and we give the following theorem.

Theorem 1.4. Suppose that and for . Then the solution of (1.8) satisfies where and are positive constants.

The plan of this paper is as follows. In Section 2, we prove that quenching occurs only at , 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 and , 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: where . Clearly, for .

Lemma 2.1. Assume the solution of problem (2.1) exists in for some , and , for . Then and in .

Proof. Let . Then satisfies The maximum principle leads to , and thus in . Then it is easy to see that the problem (2.2) is nondegenerate in . So is a classical solution of (2.2). Similarly, letting , we have Making use of the maximum principle, we obtain in . Hence, the solutions of problem (2.1) with and in .

The Proof of Theorem 1.1
By the maximum principle, we know that for all in the existence interval, where . Define . Then satisfies Thus , which means that for some . From the fact that for and for , we find that there exists a such that . By virtue of the singular nonlinearity in the boundary condition, must quench at . In what follows, we only need to prove that quenching cannot occur in for some . Consider two cases.

Case 1. for . Let in , where is a positive constant. Then satisfies for , since . On the parabolic boundary, for ; if is sufficiently small, for , and for . Thus by the maximum principle, we have in , which leads to So we have Integrating (2.7) from to 1, we obtain It then follows that if .

Case 2. for . Let in . Then satisfies for . On the boundary, for . Since and , if is close to and is small enough, for . Thus It is easily seen that for . Hence, the maximum principle yields that in . In particular, Integrating (2.11) from to , we obtain Define . Since for , the inverse exists. In view of (2.12), we can see Let , where and are positive constants. Since , , and by (2.13), we have that in provided , which is true since , where . On the other hand, ; if and if and are small enough. Thus by the maximum principle, we find that in , which implies that if .

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 in , where is given as follows: with some and is chosen so large that for . It is easy to see that , and . On the other hand, in , satisfies where By and the definition of , it follows that Thus, the maximum principle yields , that is Moreover, by the definition of the limit, we see that since . In fact, which means Integrating (3.7) from to , we get
We then give the lower bound.

The Proof of Theorem 1.3
Let . Notice that the hypotheses are equivalent to(), (), (), respectively. Letting be close to , we consider in , where is a positive constant. Through a fairly complicated calculation, we find that where Since the hypotheses hold, and , , we see that . Thus, we have for . On the parabolic boundary, since is the only quenching point, if is small enough, then both and are negative. At , in view of , we have provided is sufficiently small. Hence, by the maximum principle, we have on . In particular, , that is, Integration of (3.13) over , then leads to

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. . We only need to verify the hypothesis . Since then we have Therefore, we get the upper bound as

Case 2. . We use a modification of an argument from [16]. For with some such that , set with where .

A routine calculation shows where . Since and in , we find for any and . By (4.4), (4.5) and (4.7), we have or equivalently, We now claim that on . In fact where . When , , because and , we have when , , we have In conclusion, when , , we have From (4.6), (4.9), and (4.13), it then follows that Integrating the above inequality from to , we obtain that is, which in conjunction with (4.9) yields the desired upper bound.

We then give the lower bound. We examine the validity of hypotheses.

Firstly, for , we find provided or .

Secondly, for , we have provided with where , , and .

Thirdly, for , we obtain provided . Since , we choose a such that and hypotheses 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).

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