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.
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.
2. Simultaneous and Non-Simultaneous Quenching
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.
3. Quenching Rates
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.
- (i)Assume that the quenching of problem (1.1) is simultaneous with , integrating (2.12) yields(3.3)
where . Since we assume that quench at , we have as .
- (ii)If and , we have . It is easy to see that as , from (2.9) and (2.10), we obtain(3.10)
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 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.
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