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

  • Shouming Zhou1Email author and

    Affiliated with

    • Chunlai Mu1

      Affiliated with

      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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq1_HTML.gif , http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ1_HTML.gif
      (1.1)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq3_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq4_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq6_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq9_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq10_HTML.gif denotes the quenching time, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq11_HTML.gif denotes quenching point), such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ2_HTML.gif
      (1.2)

      For a quenching solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq13_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq14_HTML.gif tends to the quenching time http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq15_HTML.gif . The case when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq16_HTML.gif quenches and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq18_HTML.gif and http://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:
      http://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 http://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 http://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
      http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ5_HTML.gif
      (1.5)

      with positive powers http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq23_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq24_HTML.gif for the inner absorptions of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq25_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq27_HTML.gif and http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ6_HTML.gif
      (1.6)

      Theorem 1.2.

      Let http://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 http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq34_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq36_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq39_HTML.gif and http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq43_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq44_HTML.gif , and the two parameters http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq45_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq46_HTML.gif verifying
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ7_HTML.gif
      (1.7)
      or equivalently,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ8_HTML.gif
      (1.8)

      In terms of parameters http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq47_HTML.gif and http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq49_HTML.gif is the quenching variable, then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq51_HTML.gif close to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq52_HTML.gif , we have

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

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

      (iii) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq60_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq61_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq64_HTML.gif . The maximum principle implies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq66_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq67_HTML.gif . Let http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ9_HTML.gif
      (2.1)
      Consequently,
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq70_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ11_HTML.gif
      (2.3)

      Proof.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq71_HTML.gif . Thus,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ12_HTML.gif
      (2.4)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq72_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq73_HTML.gif are radial and nondecreasing in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq74_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq75_HTML.gif . A similar computation holds for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq76_HTML.gif , and we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ13_HTML.gif
      (2.5)
      with boundary conditions
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq77_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq78_HTML.gif (see [13, 14]). Choosing http://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
      http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq80_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq81_HTML.gif for http://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 http://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,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ17_HTML.gif
      (2.9)
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq84_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq85_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq86_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq87_HTML.gif quenching at the time http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq88_HTML.gif . Through (2.10), we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq89_HTML.gif , integrating from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq90_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq91_HTML.gif we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq92_HTML.gif . Together with (2.9) we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq93_HTML.gif . Integrating in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq94_HTML.gif , we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ19_HTML.gif
      (2.11)

      If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq96_HTML.gif and http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ20_HTML.gif
      (2.12)
      Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq99_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq100_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ21_HTML.gif
      (2.13)

      where http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq104_HTML.gif and the initial data http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq105_HTML.gif on http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq108_HTML.gif (through Theorem 1.1, we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq109_HTML.gif ) be so small that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq111_HTML.gif be fixed. From http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq112_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq113_HTML.gif , we obtain
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ23_HTML.gif
      (2.15)
      Integrating in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq114_HTML.gif , we obtain
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq115_HTML.gif is small enough and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq117_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq119_HTML.gif . By (2.10), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ25_HTML.gif
      (3.1)
      Thus,
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq120_HTML.gif , integrating (2.12) yields
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ27_HTML.gif
        (3.3)
         

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

      On the other hand, from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq126_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq127_HTML.gif , we get, a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq128_HTML.gif such that
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq129_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ29_HTML.gif
      (3.5)
      Consequently,
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ31_HTML.gif
      (3.7)
      Integrating from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq130_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq131_HTML.gif , we get
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ32_HTML.gif
      (3.8)
      If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq132_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq134_HTML.gif , it follows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq135_HTML.gif . Employing (2.10), we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq136_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq137_HTML.gif . Repeating this procedure, we obtain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq138_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq139_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq140_HTML.gif satisfy
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ33_HTML.gif
      (3.9)
      One can check that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq141_HTML.gif ( http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq145_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq146_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq147_HTML.gif . It is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq148_HTML.gif as http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ34_HTML.gif
        (3.10)
         
       (iii) If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq150_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq151_HTML.gif , from (2.9), we get
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ36_HTML.gif
      (3.12)
      that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ37_HTML.gif
      (3.13)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq152_HTML.gif , we have
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq153_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq154_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq155_HTML.gif . With http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq156_HTML.gif , we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ39_HTML.gif
      (3.15)
      and hence,
      http://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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ41_HTML.gif
      (3.17)
      Integrating from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq158_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq159_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ42_HTML.gif
      (3.18)
      Setting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq160_HTML.gif , we get
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq161_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq162_HTML.gif , we obtain
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq164_HTML.gif of the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq165_HTML.gif , ( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq167_HTML.gif . The semidiscrete approximation http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ45_HTML.gif
      (4.1)

      where the superindex http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq171_HTML.gif and the time http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq172_HTML.gif . Thus,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ46_HTML.gif
      (4.2)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq173_HTML.gif is the standard base of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq174_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq175_HTML.gif satisfies the following ODE system:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ47_HTML.gif
      (4.3)

      where http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq177_HTML.gif is the stiffness matrix, and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq179_HTML.gif .

      We take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq180_HTML.gif and http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Equ48_HTML.gif
      (4.4)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq182_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq184_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq185_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq187_HTML.gif , and the initial data http://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 http://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 http://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.
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq191_HTML.gif .

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

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

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq194_HTML.gif .

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

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

      Next, we take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq197_HTML.gif with the same initial data http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq200_HTML.gif with the initial data http://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.
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Fig5_HTML.jpg
      Figure 5

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

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

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

      Finally, we choose http://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 http://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 http://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.
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_Fig7_HTML.jpg
      Figure 7

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

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

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

      References

      1. de Pablo A, Quirós F, Rossi JD: Nonsimultaneous quenching. Applied Mathematics Letters 2002,15(3):265-269. 10.1016/S0893-9659(01)00128-8MathSciNetView Article
      2. Zheng S, Wang W: Non-simultaneous versus simultaneous quenching in a coupled nonlinear parabolic system. Nonlinear Analysis: Theory, Methods & Applications 2008,69(7):2274-2285. 10.1016/j.na.2007.08.007MathSciNetView Article
      3. Zhou J, He Y, Mu C: Incomplete quenching of heat equations with absorption. Applicable Analysis 2008,87(5):523-529. 10.1080/00036810802001289MathSciNetView Article
      4. Ferreira R, de Pablo A, Quirós F, Rossi JD: Non-simultaneous quenching in a system of heat equations coupled at the boundary. Zeitschrift für Angewandte Mathematik und Physik 2006,57(4):586-594. 10.1007/s00033-005-0003-zView Article
      5. Ferreira R, de Pablo A, Pérez-Llanos M, Rossi JD: Incomplete quenching in a system of heat equations coupled at the boundary. Journal of Mathematical Analysis and Applications 2008,346(1):145-154. 10.1016/j.jmaa.2008.05.037MathSciNetView Article
      6. Ji R, Zheng S: Quenching behavior of solutions to heat equations with coupled boundary singularities. Applied Mathematics and Computation 2008,206(1):403-412. 10.1016/j.amc.2008.09.018MathSciNetView Article
      7. Zheng S, Song XF: Quenching rates for heat equations with coupled singular nonlinear boundary flux. Science in China. Series A 2008,51(9):1631-1643. 10.1007/s11425-007-0178-1MathSciNetView Article
      8. Escobedo M, Levine HA: Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations. Archive for Rational Mechanics and Analysis 1995,129(1):47-100. 10.1007/BF00375126MathSciNetView Article
      9. Guo J-S, Sasayama S, Wang C-J: Blowup rate estimate for a system of semilinear parabolic equations. Communications on Pure and Applied Analysis 2009,8(2):711-718.MathSciNetView Article
      10. Wang M: Blow-up rate estimates for semilinear parabolic systems. Journal of Differential Equations 2001,170(2):317-324. 10.1006/jdeq.2000.3823MathSciNetView Article
      11. Chan CY: Recent advances in quenching phenomena. Proceedings of Dynamic Systems and Applications, 1996, Atlanta, Ga, USA 2: 107-113.
      12. Kawarada H:On solutions of initial-boundary problem for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F797182/MediaObjects/13661_2010_Article_957_IEq213_HTML.gif . Publications of the Research Institute for Mathematical Sciences 1975,10(3):729-736.MathSciNetView Article
      13. Salin T: On quenching with logarithmic singularity. Nonlinear Analysis: Theory, Methods & Applications 2003,52(1):261-289. 10.1016/S0362-546X(02)00110-4MathSciNetView Article
      14. Mu C, Zhou S, Liu D: Quenching for a reaction-diffusion system with logarithmic singularity. Nonlinear Analysis: Theory, Methods & Applications 2009,71(11):5599-5605. 10.1016/j.na.2009.04.055MathSciNetView Article
      15. Pinasco JP, Rossi JD: Simultaneous versus non-simultaneous blow-up. New Zealand Journal of Mathematics 2000,29(1):55-59.MathSciNet
      16. Ciarlet PG: The Finite Element Method for Elliptic Problems, Studies in Mathematics and Its Applications. North-Holland, Amsterdam, The Netherlands; 1978:xix+530.
      17. Ferreira R: Numerical quenching for the semilinear heat equation with a singular absorption. Journal of Computational and Applied Mathematics 2009,228(1):92-103. 10.1016/j.cam.2008.08.041MathSciNetView Article
      18. Groisman P, Quirós F, Rossi JD: Non-simultaneous blow-up in a numerical approximation of a parabolic system. Computational & Applied Mathematics 2002,21(3):813-831.MathSciNet

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

      This article is published under license to BioMed Central Ltd. 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.