Open Access

Quenching for a Reaction-Diffusion System with Coupled Inner Singular Absorption Terms

Boundary Value Problems20102010:797182

DOI: 10.1155/2010/797182

Received: 13 May 2010

Accepted: 5 July 2010

Published: 20 July 2010

Abstract

we devote to investigate the quenching phenomenon for a reaction-diffusion system with coupled singular absorption terms, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq2_HTML.gif . The solutions of the system quenches in finite time for any initial data are obtained, and the blow-up of time derivatives at the quenching point is verified. Moreover, under appropriate hypotheses, the criteria to identify the simultaneous and nonsimultaneous quenching are found, and the four kinds of quenching rates for different nonlinear exponent regions are given. Finally, some numerical experiments are performed, which illustrate our results.

1. Introduction

This paper deals with the following nonlinear parabolic equations with null Neumann boundary conditions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq3_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq5_HTML.gif is a bounded domain with smooth boundary, the initial data https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq6_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq7_HTML.gif are positive, smooth, and compatible with the boundary data.

Because of the singular nonlinearity inner absorption terms of (1.1), the so-called finite-time quenching may occur for the model. We say that the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq8_HTML.gif of the problem (1.1) quenches, if there exists a time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq9_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq10_HTML.gif denotes the quenching time, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq11_HTML.gif denotes quenching point), such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ2_HTML.gif
(1.2)

For a quenching solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq12_HTML.gif of (1.1), the inf norm of one of the components must tend to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq13_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq14_HTML.gif tends to the quenching time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq15_HTML.gif . The case when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq16_HTML.gif quenches and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq17_HTML.gif remains bounded from zero is called non-simultaneous quenching. We will call the case, when both components https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq18_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq19_HTML.gif quench at the same time, as simultaneous quenching. The purpose of this paper is to find a criteria to identify simultaneous and non-simultaneous quenching for (1.1) and then establish quenching rates for the different cases.

In order to motivate the main results for system (1.1), we recall some classical results for the related system. de Pablo et al., firstly distinguished non-simultaneous quenching from simultaneous one in [1]. They considered a heat system coupled via inner absorptions as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ3_HTML.gif
(1.3)

Recently, Zheng and Wang deduced problem (1.3) to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq20_HTML.gif -dimensional with positive Dirichlet boundary condition in [2]. Then, Zhou et al. have given a natural continuation for problem (1.3) beyond quenching time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq21_HTML.gif for the case of non-simultaneous quenching in [3].

Replacing the coupled inner absorptions in (1.1) by the coupled boundary fluxes, one gets
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ4_HTML.gif
(1.4)

Recently, the simultaneous and non-simultaneous quenching for problem (1.4), and what is related to it, was studied by many authors (see [47] and references therein).

In order to investigate the problem (1.1), it is necessary to recall the blow-up problem of the following reaction-diffusion system:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ5_HTML.gif
(1.5)

with positive powers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq22_HTML.gif has been extensively studied by many authors for various problems such as global existence and finite time blow-up, Fujita exponents, non-simultaneous and simultaneous blow-up, and blow-up rates, (see [810] and references therein). However, unlike the blow-up problem, there are less papers consider the weakly coupled quenching problem like (1.1), differently from the generally considered, there are two additional singular factors, namely, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq23_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq24_HTML.gif for the inner absorptions of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq25_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq26_HTML.gif , respectively. In this paper, we will show real contributions of the two additional singular factors to the quenching behavior of solutions. Our main results are stated as follows.

Theorem 1.1.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq27_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq28_HTML.gif , then the solution of the system (1.1) quenches in finite time for every initial data.

On the other hand, some authors understand quenching as blow-up of time derivatives while the solution itself remains bounded (see [1113]). In present paper, we assume that the initial data satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ6_HTML.gif
(1.6)

Theorem 1.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq29_HTML.gif and the radial initial function satisfies (1.6), then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq30_HTML.gif blows up in finite time.

Next, we characterize the ranges of parameters to distinguish simultaneous and non-simultaneous quenching. In order to simplify our work, we deal with the radial solutions of (1.1) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq31_HTML.gif , and the radial increasing initial data satisfies (1.6). Thus we, see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq32_HTML.gif is the only quenching point (see [2, 14]). Without loss of generality, we only consider the non-simultaneous quenching with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq33_HTML.gif remaining strictly positive, and our main results are stated as follows.

Theorem 1.3.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq34_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq35_HTML.gif , then any quenching in (1.1) must be simultaneous.

Theorem 1.4.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq36_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq37_HTML.gif , then any quenching in (1.1) is non-simultaneous with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq38_HTML.gif being strictly positive.

Theorem 1.5.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq39_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq40_HTML.gif , then both simultaneous and non-simultaneous quenching may occur in (1.1) depending on the initial data.

Remark 1.6.

In particular, if we choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq41_HTML.gif , then we obtain that the ranges of parameters to distinguish simultaneous and non-simultaneous quenching coincide with the problem (1.3) (see [1, 2]). Moreover, this criteria to identify the simultaneous and non-simultaneous quenching is the same with the problem (1.4) which coupled boundary fluxes (see [6]). This situation also happens for the blow-up problem (see [8, 10, 15]).

Next, we deal with quenching rates. To state our results more conveniently, we introduce the notation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq42_HTML.gif which means that there exist two finite positive constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq43_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq44_HTML.gif , and the two parameters https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq45_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq46_HTML.gif verifying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ7_HTML.gif
(1.7)
or equivalently,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ8_HTML.gif
(1.8)

In terms of parameters https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq47_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq48_HTML.gif , the quenching rates of problem (1.1) can be shown as follow.

Theorem 1.7.

If quenching is non-simultaneous and, for instance, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq49_HTML.gif is the quenching variable, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq50_HTML.gif .

Theorem 1.8.

If quenching is simultaneous, then for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq51_HTML.gif close to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq52_HTML.gif , we have

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq53_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq54_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq55_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq56_HTML.gif ;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq57_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq58_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq59_HTML.gif ;

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq60_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq61_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq62_HTML.gif .

The plan of this paper is organized as follows. In Section 2, we distinguish non-simultaneous quenching from simultaneous one. The four kinds of non-simultaneous and simultaneous quenching rates for different nonlinear exponent regions are given in Section 3. In the Section 4, we perform some numerical experiments which illustrate our results.

2. Simultaneous and Non-Simultaneous Quenching

Proof of Theorem 1.1.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq63_HTML.gif is the classical solution of (1.1) with the maximal existence time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq64_HTML.gif . The maximum principle implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq65_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq66_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq67_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq68_HTML.gif . Hence, integrating (1.1) in space and using Green's formula, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ9_HTML.gif
(2.1)
Consequently,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ10_HTML.gif
(2.2)

Thus, the solution of the problem (1.1) quenches in finite time. The prove of Theorem 1.1 is complete.

In order to prove Theorem 1.2, we need the following Lemma.

Lemma 2.1.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq69_HTML.gif and the radial nondecreasing initial data satisfy (1.6), then there exists a small https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq70_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ11_HTML.gif
(2.3)

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq71_HTML.gif . Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ12_HTML.gif
(2.4)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq72_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq73_HTML.gif are radial and nondecreasing in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq74_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq75_HTML.gif . A similar computation holds for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq76_HTML.gif , and we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ13_HTML.gif
(2.5)
with boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ14_HTML.gif
(2.6)
From (1.6), it is easy to deduce https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq77_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq78_HTML.gif (see [13, 14]). Choosing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq79_HTML.gif small enough, we have that the initial data verifying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ15_HTML.gif
(2.7)
Hence, by the comparison result, we derive that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ16_HTML.gif
(2.8)

This proves Lemma 2.1.

Proof of Theorem 1.2.

This theorem is the direct result of Theorem 1.1 and Lemma 2.1.

Next, we characterize the ranges of parameters to distinguish simultaneous and non-simultaneous quenching. By the hypothesis on the initial data, we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq80_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq81_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq82_HTML.gif (see [2, 14]). We collect the estimates of the time derivatives obtained before. Clearly, the only quenching point is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq83_HTML.gif (see [2]), we only care for the original point,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ17_HTML.gif
(2.9)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ18_HTML.gif
(2.10)

Proof of Theorem 1.3.

We argue by contradiction. Assume that there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq84_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq85_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq86_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq87_HTML.gif quenching at the time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq88_HTML.gif . Through (2.10), we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq89_HTML.gif , integrating from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq90_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq91_HTML.gif we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq92_HTML.gif . Together with (2.9) we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq93_HTML.gif . Integrating in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq94_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ19_HTML.gif
(2.11)

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq95_HTML.gif , we have the left hand of the above inequality diverged. So, we get a contradiction. The proof of Theorem 1.3 is finished.

Proof of Theorem 1.4.

First, assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq96_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq97_HTML.gif . Combining (2.9) with (2.10), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ20_HTML.gif
(2.12)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq98_HTML.gif , integrating the first inequality in the (2.12) from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq99_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq100_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ21_HTML.gif
(2.13)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq101_HTML.gif are positive constants, the above inequality requires that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq102_HTML.gif remains positive up to the quenching time. The case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq103_HTML.gif can be treated in an analogous way. The proof of Theorem 1.4 is complete.

Proof of Theorem 1.5.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq104_HTML.gif and the initial data https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq105_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq106_HTML.gif , thus, it is easy to see that for problem (1.1) simultaneous quenching occurs.

On the other hand, we want to choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq107_HTML.gif small in order that the quenching time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq108_HTML.gif (through Theorem 1.1, we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq109_HTML.gif ) be so small that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq110_HTML.gif does not have time to vanish.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq111_HTML.gif be fixed. From https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq112_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq113_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ22_HTML.gif
(2.14)
Together with the estimate (2.12), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ23_HTML.gif
(2.15)
Integrating in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq114_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ24_HTML.gif
(2.16)

It is easy to see that the last term of the above inequality is strictly positive, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq115_HTML.gif is small enough and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq116_HTML.gif , therefore, we prove that under the condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq117_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq118_HTML.gif , for the solution of (1.1) non-simultaneous quenching may occur. The proof of Theorem 1.5 is complete.

3. Quenching Rates

In this section, we deal with the all possible quenching rates in model (1.1).

Proof of Theorem 1.7.

Under the condition of Theorem 1.7, it holds that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq119_HTML.gif . By (2.10), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ25_HTML.gif
(3.1)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ26_HTML.gif
(3.2)

The proof of Theorem 1.7 is complete.

Proof of Theorem 1.8.
  1. (i)
    Assume that the quenching of problem (1.1) is simultaneous with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq120_HTML.gif , integrating (2.12) yields
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ27_HTML.gif
    (3.3)
     

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq121_HTML.gif . Since we assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq122_HTML.gif quench at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq123_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq124_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq125_HTML.gif .

On the other hand, from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq126_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq127_HTML.gif , we get, a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq128_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ28_HTML.gif
(3.4)
Similarly, we can show that there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq129_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ29_HTML.gif
(3.5)
Consequently,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ30_HTML.gif
(3.6)
Recalling the estimates (2.9) and (2.10), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ31_HTML.gif
(3.7)
Integrating from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq130_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq131_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ32_HTML.gif
(3.8)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq132_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq133_HTML.gif , we deduce the quenching rate by a bootstrap argument. First, by (2.9), we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq134_HTML.gif , it follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq135_HTML.gif . Employing (2.10), we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq136_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq137_HTML.gif . Repeating this procedure, we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq138_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq139_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq140_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ33_HTML.gif
(3.9)
One can check that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq141_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq142_HTML.gif define by (1.8)), and the all positive constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq143_HTML.gif are bounded. Therefore, passing to the limit, we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq144_HTML.gif . The reverse inequalities can be obtained in the same way.
  1. (ii)
    If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq145_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq146_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq147_HTML.gif . It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq148_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq149_HTML.gif , from (2.9) and (2.10), we obtain
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ34_HTML.gif
    (3.10)
     
 (iii) If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq150_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq151_HTML.gif , from (2.9), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ35_HTML.gif
(3.11)
Recalling the estimate (2.10), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ36_HTML.gif
(3.12)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ37_HTML.gif
(3.13)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq152_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ38_HTML.gif
(3.14)
It is known that the incomplete Gamma function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq153_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq154_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq155_HTML.gif . With https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq156_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ39_HTML.gif
(3.15)
and hence,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ40_HTML.gif
(3.16)
Next, we deduce the behaviour for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq157_HTML.gif . Combining with (2.9) and (3.16), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ41_HTML.gif
(3.17)
Integrating from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq158_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq159_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ42_HTML.gif
(3.18)
Setting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq160_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ43_HTML.gif
(3.19)
For the incomplete Gamma function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq161_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq162_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ44_HTML.gif
(3.20)

The proof of Theorem 1.8 is complete.

4. Numerical Experiments

In this section, we perform some numerical experiments, which illustrate our results. Now we introduce the numerical scheme for the space discretization, we discretize applying linear finite elements with mass lumping in a uniform mesh for the space variable and keeping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq163_HTML.gif continuous, it is well known that this discretization in space coincides with the classic central finite difference second-order scheme, (see [16]), Mass lumping is widely used in parabolic problems with blow-up and quenching, (see, e.g., [17, 18]).

Let us consider the uniform partition of size https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq164_HTML.gif of the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq165_HTML.gif , ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq166_HTML.gif ), and its associated standard piecewise linear finite element space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq167_HTML.gif . The semidiscrete approximation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq168_HTML.gif obtained by the finite element method with mass lumping is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ45_HTML.gif
(4.1)

where the superindex https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq169_HTML.gif denotes the Lagrange interpolation.

We denote with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq170_HTML.gif the values of the numerical approximation at the nodes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq171_HTML.gif and the time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq172_HTML.gif . Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ46_HTML.gif
(4.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq173_HTML.gif is the standard base of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq174_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq175_HTML.gif satisfies the following ODE system:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ47_HTML.gif
(4.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq176_HTML.gif is the mass matrix obtained with lumping, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq177_HTML.gif is the stiffness matrix, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq178_HTML.gif is the Lagrange interpolation of the initial datum https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq179_HTML.gif .

We take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq180_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq181_HTML.gif . Writing the system (4.3) explicitly, we get the following ODE system:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ48_HTML.gif
(4.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq182_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq183_HTML.gif . In order to show the evolution in time of a numerical solution, we chose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq184_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq185_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq186_HTML.gif which will be choose later.

First, we consider the case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq187_HTML.gif , and the initial data https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq188_HTML.gif , We observe that the solutions of (1.1) quenching only at the origin, if the symmetric initial data with a unique minimum at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq189_HTML.gif (see Figure 1), and the quenching is simultaneous (see Figure 2); If we take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq190_HTML.gif , and the same initial data (see Figures 3 and 4), then we obtain the results which accords with Theorem 1.3.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Fig1_HTML.jpg
Figure 1

The value of the solution at the quenching time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq191_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Fig2_HTML.jpg
Figure 2

Evolution at the point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq192_HTML.gif of the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq193_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Fig3_HTML.jpg
Figure 3

The value of the solution at the quenching time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq194_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Fig4_HTML.jpg
Figure 4

Evolution at the point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq195_HTML.gif of the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq196_HTML.gif .

Next, we take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq197_HTML.gif with the same initial data https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq198_HTML.gif . In this case the quenching in (1.1) is non-simultaneous with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq199_HTML.gif being strictly positive (see Figure 5); If we choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq200_HTML.gif with the initial data https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq201_HTML.gif (see Figure 6), then we can see that our results coincide with Theorem 1.4.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Fig5_HTML.jpg
Figure 5

Evolution at the point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq202_HTML.gif of the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq203_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Fig6_HTML.jpg
Figure 6

Evolution at the point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq204_HTML.gif of the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq205_HTML.gif .

Finally, we choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq206_HTML.gif In Figure 7, we take the initial data https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq207_HTML.gif , and in Figure 8 we take the different initial data both equal to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq208_HTML.gif , we can see that both non-simultaneous quenching and simultaneous quenching may occur in (1.1), depending on the initial data.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Fig7_HTML.jpg
Figure 7

Evolution at the point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq209_HTML.gif of the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq210_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Fig8_HTML.jpg
Figure 8

Evolution at the point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq211_HTML.gif of the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq212_HTML.gif .

Declarations

Acknowledgments

This work is supported in part by NSF of China (10771226) and in part by Innovative Talent Training Project, the Third Stage of "211 Project", Chongqing University, Project no.: S-09110.

Authors’ Affiliations

(1)
College of Mathematics and Statistics, Chongqing University

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© The Author(s) Shouming Zhou and Chunlai Mu. 2010

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