Quenching phenomenon for a nonNewtonian filtration equation with singular boundary flux
 Ying Yang^{1}Email author
Received: 1 August 2015
Accepted: 29 November 2015
Published: 10 December 2015
Abstract
This paper is concerned with the quenching phenomenon for the onedimensional nonNewtonian filtration equation with both source term and Neumann boundary condition. With two different kinds of initial data, we prove that the solution must quench in a finite time and the time derivative blows up at a quenching point. The corresponding quenching rate and a lower bound for the quenching time are also obtained.
Keywords
nonNewtonian filtration equation singular boundary flux finite time quenching quenching rateMSC
34B15 35K55 35K651 Introduction
However, as far as we know, there are very few papers concerned with the quenching phenomenon for singular or degenerate parabolic problem with two nonlinear heat sources, even if the linear diffusion equation holds. Obviously, in the model (1.1), the internal source \((1u)^{h}\) and the boundary flux \(u^{q}\) exist, both of which may become singular in some finite time, if the solution reach level \(u=1\) or \(u=0\), respectively. In the present paper, we will discuss these two cases by imposing conditions (A_{1})(A_{4}) upon the initial datum, which are give below. First of all, motivated by the work of [8], in Section 2, we will study the quenching phenomenon for the solution reaching the level \(u=1\). We will prove that quenching occurs in finite time under condition (A_{1}) and the only quenching point is \(x = 0\) under conditions (A_{1}) and (A_{2}). Furthermore, \(u_{t}\) blows up at quenching time is discussed. Then the bounds for the quenching rate and the lower bound for the quenching time are estimated. Second, in Section 3, we will do research on the quenching phenomenon for the solution reaching the level \(u=0\) under conditions (A_{2}) and (A_{4}). It will be shown that the solution quenches in finite time and \(u_{t}\) blows up at quenching time at the only quenching point \(x=1\). Finally, we will give bounds on the quenching rate.
 (A_{1}):

\((u_{0}'(x)^{p2}u_{0}'(x))'+(1u_{0}(x))^{h}\geq0\);
 (A_{2}):

\(u_{0}'(x)\le0\);
 (A_{3}):

\(u_{0}'(x)\lexu_{0}^{q}(x)\);
 (A_{4}):

\((u_{0}'(x)^{p2}u_{0}'(x))'+(1u_{0}(x))^{h}\le0\).
2 Quenching phenomenon for the solution reaching the level \(u = 1\)
In this section, we study the quenching phenomenon for the problem (1.1) under the conditions (A_{1}) and (A_{2}). Due to the degeneracy of the equation, the classical solutions might not exist and the weak solution should be considered. However, for simplicity of our arguments, we assume that the solution is appropriately smooth, since we may consider some approximate boundary and initial value conditions.
2.1 Quenching on the boundary and blowup of \(u_{t}\)
In this section, we prove the solution quenches in finite time and blowing up of \(u_{t}\) at the only quenching point \(x=0\).
Remark 2.1
The assumptions (A_{1}) and (A_{2}) on \(u_{0}(x)\) are proper. For example, for \(p=2\), \(h=9\), and \(q=\log_{30/7}3\), we can choose \(u_{0}(x)=0.9\frac{3}{2}x^{4.5}\), which satisfies (A_{1}), (A_{2}), and compatibility conditions.
In the following, we discuss the properties of the solution to the problem (1.1).
Lemma 2.1
Assume that (A_{1}), (A_{2}) hold and the solution u of the problem (1.1) exists in \((0,T_{0})\) for some \(T_{0}>0\). Then \(u\in C^{2,1}((0,1]\times(0,T_{0}))\) with \(u_{x}(x,t)<0\) and \(u_{t}(x,t)\geq0\) in \((0,1]\times(0,T_{0})\).
Proof
Now, we are in a position to show the quenching result.
Theorem 2.1
Assume that (A_{1}) and (A_{2}) hold. Then there exists a finite time T, such that every solution of (1.1) quenches in this time, and the only quenching point is \(x=0\).
Proof
Theorem 2.2
Assume that \(h\geq1\). Then \(u_{t}\) blows up at the quenching point \(x=0\).
Proof
2.2 Quenching rate and lower bound for the quenching time
In this section, a bound on the quenching rate is given and a lower bound for the quenching time is obtained. We present the quenching rate in the following:
Theorem 2.3
Proof
3 Quenching phenomenon for the solution reaching the level \(u = 0\)
In this section, we investigate the quenching phenomenon for the problem (1.1) under the conditions (A_{2}) and (A_{4}).
3.1 Quenching on the boundary and blowup of \(u_{t}\)
In this section, we prove the solution quenches in finite time and blowing up of \(u_{t}\) at the only quenching point \(x = 1\). First of all, we have the following:
Lemma 3.1
Assume that (A_{2}) and (A_{4}) hold and the solution u of the problem (1.1) exists in \((0,\tilde{T_{0}})\) for some \(\tilde{T_{0}}>0\). Then \(u_{x}(x,t)<0\) and \(u_{t}(x,t)<0\) in \((0,1]\times(0,\tilde{T_{0}})\).
The proof is similar to Lemma 2.1, so we omit it.
Theorem 3.1
Assume that (A_{2}) and (A_{4}) hold. Then there exists a finite time T, such that every solution of (1.1) quenches in this time, and the only quenching point is \(x=1\).
Proof
Theorem 3.2
\(u_{t}\) blows up at the quenching point \(x=1\).
Proof
3.2 Quenching rate
Now, we are in a position to investigate the bounds on the quenching rate. First of all, we will show the lower bound of the quenching rate.
Theorem 3.3
Proof
To end this section, we present the upper bound on the quenching rate.
Theorem 3.4
Proof
From Theorem 3.3 and Theorem 3.4, we have the following exact quenching rate.
Corollary 3.1
Declarations
Acknowledgements
This work is partially supported by the National Science Foundation of China (11201311, 11301345), Guangdong Natural Science Foundation (2014A030310074) and Natural Science Foundation of SZU (201425, 201545). The author would like to thank the referees for revising the manuscript and for many valuable comments, which helped to clarify this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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