This paper deals with the following nonlinear parabolic equations with null Neumann boundary conditions:
where for , is a bounded domain with smooth boundary, the initial data and 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 of the problem (1.1) quenches, if there exists a time ( denotes the quenching time, denotes quenching point), such that
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.
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 . They considered a heat system coupled via inner absorptions as follows:
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 .
Replacing the coupled inner absorptions in (1.1) by the coupled boundary fluxes, one gets
Recently, the simultaneous and non-simultaneous quenching for problem (1.4), and what is related to it, was studied by many authors (see [4–7] 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:
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.
On the other hand, some authors understand quenching as blow-up of time derivatives while the solution itself remains bounded (see [11–13]). In present paper, we assume that the initial data satisfy
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]).
Next, we deal with quenching rates. To state our results more conveniently, we introduce the notation which means that there exist two finite positive constants such that , and the two parameters and verifying
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.