- Research Article
- Open Access
Quenching for a Reaction-Diffusion System with Coupled Inner Singular Absorption Terms
© The Author(s) Shouming Zhou and Chunlai Mu. 2010
- Received: 13 May 2010
- Accepted: 5 July 2010
- Published: 20 July 2010
we devote to investigate the quenching phenomenon for a reaction-diffusion system with coupled singular absorption terms, , . 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.
- Initial Data
- Incomplete Gamma Function
- Linear Finite Element
- Maximal Existence Time
- Mass Lump
where for , is a bounded domain with smooth boundary, the initial data and are positive, smooth, and compatible with the boundary data.
For a quenching solution of (1.1), the inf norm of one of the components must tend to as tends to the quenching time . The case when quenches and remains bounded from zero is called non-simultaneous quenching. We will call the case, when both components and 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.
Recently, Zheng and Wang deduced problem (1.3) to -dimensional with positive Dirichlet boundary condition in . Then, Zhou et al. have given a natural continuation for problem (1.3) beyond quenching time for the case of non-simultaneous quenching in .
with positive powers 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 [8–10] 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, and for the inner absorptions of and , 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.
If and , then the solution of the system (1.1) quenches in finite time for every initial data.
Let and the radial initial function satisfies (1.6), then 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 , and the radial increasing initial data satisfies (1.6). Thus we, see that is the only quenching point (see [2, 14]). Without loss of generality, we only consider the non-simultaneous quenching with remaining strictly positive, and our main results are stated as follows.
If and , then any quenching in (1.1) must be simultaneous.
If and , then any quenching in (1.1) is non-simultaneous with being strictly positive.
If and , then both simultaneous and non-simultaneous quenching may occur in (1.1) depending on the initial data.
In particular, if we choose , 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 ). This situation also happens for the blow-up problem (see [8, 10, 15]).
In terms of parameters and , the quenching rates of problem (1.1) can be shown as follow.
If quenching is non-simultaneous and, for instance, is the quenching variable, then .
If quenching is simultaneous, then for close to , we have
(i) for , or ;
(ii) for and ;
(iii) for and .
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.
Proof of Theorem 1.1.
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.
This proves Lemma 2.1.
Proof of Theorem 1.2.
This theorem is the direct result of Theorem 1.1 and Lemma 2.1.
Proof of Theorem 1.3.
If , 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.
where are positive constants, the above inequality requires that remains positive up to the quenching time. The case can be treated in an analogous way. The proof of Theorem 1.4 is complete.
Proof of Theorem 1.5.
If and the initial data on , thus, it is easy to see that for problem (1.1) simultaneous quenching occurs.
On the other hand, we want to choose small in order that the quenching time (through Theorem 1.1, we get ) be so small that does not have time to vanish.
It is easy to see that the last term of the above inequality is strictly positive, if is small enough and , therefore, we prove that under the condition and , for the solution of (1.1) non-simultaneous quenching may occur. The proof of Theorem 1.5 is complete.
In this section, we deal with the all possible quenching rates in model (1.1).
Proof of Theorem 1.7.
The proof of Theorem 1.7 is complete.
where . Since we assume that quench at , we have as .
The proof of Theorem 1.8 is complete.
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 continuous, it is well known that this discretization in space coincides with the classic central finite difference second-order scheme, (see ), Mass lumping is widely used in parabolic problems with blow-up and quenching, (see, e.g., [17, 18]).
where the superindex denotes the Lagrange interpolation.
where is the mass matrix obtained with lumping, is the stiffness matrix, and is the Lagrange interpolation of the initial datum .
where and . In order to show the evolution in time of a numerical solution, we chose , , and which will be choose later.
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.
- de Pablo A, Quirós F, Rossi JD: Nonsimultaneous quenching. Applied Mathematics Letters 2002,15(3):265-269. 10.1016/S0893-9659(01)00128-8MathSciNetView ArticleMATHGoogle Scholar
- 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 ArticleMATHGoogle Scholar
- Zhou J, He Y, Mu C: Incomplete quenching of heat equations with absorption. Applicable Analysis 2008,87(5):523-529. 10.1080/00036810802001289MathSciNetView ArticleMATHGoogle Scholar
- 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 ArticleMathSciNetMATHGoogle Scholar
- 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 ArticleMATHGoogle Scholar
- 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 ArticleMATHGoogle Scholar
- 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 ArticleMATHGoogle Scholar
- 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 ArticleMATHGoogle Scholar
- 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 ArticleMATHGoogle Scholar
- Wang M: Blow-up rate estimates for semilinear parabolic systems. Journal of Differential Equations 2001,170(2):317-324. 10.1006/jdeq.2000.3823MathSciNetView ArticleMATHGoogle Scholar
- Chan CY: Recent advances in quenching phenomena. Proceedings of Dynamic Systems and Applications, 1996, Atlanta, Ga, USA 2: 107-113.Google Scholar
- Kawarada H:On solutions of initial-boundary problem for . Publications of the Research Institute for Mathematical Sciences 1975,10(3):729-736.MathSciNetView ArticleMATHGoogle Scholar
- Salin T: On quenching with logarithmic singularity. Nonlinear Analysis: Theory, Methods & Applications 2003,52(1):261-289. 10.1016/S0362-546X(02)00110-4MathSciNetView ArticleMATHGoogle Scholar
- 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 ArticleMATHGoogle Scholar
- Pinasco JP, Rossi JD: Simultaneous versus non-simultaneous blow-up. New Zealand Journal of Mathematics 2000,29(1):55-59.MathSciNetMATHGoogle Scholar
- Ciarlet PG: The Finite Element Method for Elliptic Problems, Studies in Mathematics and Its Applications. North-Holland, Amsterdam, The Netherlands; 1978:xix+530.Google Scholar
- 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 ArticleMATHGoogle Scholar
- 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.MathSciNetMATHGoogle Scholar
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