Critical curves for fast diffusion equations coupled via nonlinear boundary flux
- Zhengqiu Ling^{1}Email author
https://doi.org/10.1186/s13660-015-0695-3
© Ling 2015
Received: 21 July 2014
Accepted: 17 May 2015
Published: 2 June 2015
Abstract
Keywords
MSC
1 Introduction
The particular feature of equations (1.1) is their gradient-dependent diffusivity. Such equations can be used to provide a model for nonlinear heat propagation, they also appear in several branches of applied mathematics such as plasma physics, population dynamics, chemical reactions, and so on. At the same time, these equations are also called the Newtonian filtration equations, which have been intensively studied since the last century (see [1, 2] and the references therein). In addition, for the case \(s\geq2\), system (1.1)-(1.3) suggests that equations (1.1) are linked by the influx of energy input at the boundary \(x=0\). For instance, in the heat transfer process, \(-(u_{i}^{m_{i}})_{x}\) represents the heat flux, and hence the boundary conditions represent a nonlinear radiation law at the boundary. These kinds of boundary conditions appear also in combustion problem when the reaction happens only at the boundary of the container, for example, because of the presence of a solid catalyzer, see [3] for justification.
In general, system (1.1)-(1.3) does not possess classical solutions. This is due to the fact that the equations in (1.1) are parabolic only where \(u_{ix}>0\), but degenerate where \(u_{ix}=0\). However, local in time existence of weak solution \((u_{1}, u_{2}, \ldots, u_{s})\) to problem (1.1)-(1.3), defined in the usual integral way, as well as a comparison principle can be easily established as, for instance, in [2, 4, 5]. Let T be the maximal existence time of a solution \((u_{1}, u_{2}, \ldots, u_{s})\), which may be finite or infinite. If \(T<\infty\), then \(\| u_{1}\|_{\infty}+ \|u_{2}\|_{\infty}+ \cdots+ \|u_{s}\|_{\infty}\) becomes unbounded in finite time and we say that the solution blows up; while if \(T=\infty\), we say that the solution is global. In particular, the problem of determining critical Fujita exponents is very interesting for various nonlinear parabolic equations of mathematical physics. See the book [6] and the surveys [7, 8], where a full list of references can be found. Here, we recall some known results on system (1.1)-(1.3)
To our knowledge, however, there are few works in the literature dealing with the heat conduction systems such as (1.1)-(1.3). Motivated by the above mentioned works, in this paper we have a purpose to extend the results of the slow diffusion case [10] to the fast diffusion case and s components, and the aim is twofold. Firstly, we construct the self-similar super-solution and sub-solution to obtain the critical global existence curve of system (1.1)-(1.3). Secondly, the critical curve of Fujita type is conjectured with the aid of some new results. A very interesting feature of our results is that the critical curves are determined by a matrix and a linear algebraic system. The fact that we are dealing with a general system instead of a single equation and with nonlinear diffusion forces us to develop some new techniques.
Lemma 1
and \(2 q_{i} \alpha_{i+1} + \varepsilon_{0} = (1 + m_{i} - 2p_{i})\alpha_{i}\), \(\alpha_{s+1}:= \alpha_{1}\), \(i=1, 2, \ldots, s\).
Now we state the main results of this paper.
Theorem 1
Assume \(2p_{i}<1+m_{i}\) (\(i = 1, 2,\ldots,s\)). If \(\prod_{i=1}^{s} (1 + m_{i} - 2p_{i}) \geq2^{s} q_{1} q_{2} \cdots q_{s} \), i.e., \(\operatorname{det}A \geq0\), then every nonnegative solution of system (1.1)-(1.3) is global in time.
Theorem 2
Assume \(2p_{i} \leq1+m_{i}\) (\(i = 1, 2,\ldots ,s\)). If \(\prod_{i=1}^{s} (1 + m_{i} - 2p_{i}) < 2^{s} q_{1} q_{2} \cdots q_{s} \), i.e., \(\operatorname{det}A < 0\), then system (1.1)-(1.3) has a nonnegative solution blowing up in a finite time.
Remark 1
From Theorems 1 and 2, we see that the critical global existence curve for system (1.1)-(1.3) is \(\prod_{i=1}^{s} (1 + m_{i} - 2p_{i}) = 2^{s} q_{1} q_{2} \cdots q_{s} \), i.e., \(\operatorname{det}A = 0\).
Theorem 3
Remark 2
From Theorem 3, we conjecture that the critical curve of Fujita type is \(\min_{i}\{l_{i} - k_{i} \} =0\) if \(2p_{i} \leq 1+m_{i}\) (\(i = 1, 2,\ldots,s\)).
For the case \(2p_{1} > 1+m_{1}\) or … or \(2p_{s} > 1+ m_{s}\), we have the following.
Theorem 4
If there exists \(i = 1, 2,\ldots,s \) such that \(2p_{i} > 1+m_{i}\), then every nonnegative solution of (1.1)-(1.3) will blow up in a finite time.
Remark 3
By Theorem 4, it is seen that the critical global existence curve for system (1.1)-(1.3) is \(2p_{i} = 1 + m_{i}\) (\(i=1,2,\ldots,s\)) if \(2^{s} q_{1} q_{2} \cdots q_{s} = \prod_{i=1}^{s} (1+m_{i} -2p_{i})\).
The rest of this paper is organized as follows. In Section 2, we consider a critical global existence curve and prove Theorems 1 and 2. The proofs of Theorems 3 and 4 are given in Section 3.
2 Critical global existence curve
In this section, we characterize when all solutions to problem (1.1)-(1.3) are global in time or they blow up. Motivated by [10, 11], we base our methods on the construction of self-similar solutions and on the comparison arguments.
Throughout this paper, we always assume \((u_{0i}^{m_{i}})^{\prime\prime }(x) \geq0\), \(i=1,2,\ldots, s\). Now, let us prove Theorem 1 first.
Proof of Theorem 1
Next, we divide the proof into two cases.
Combining Cases (i) and (ii), the solutions of system (1.1)-(1.3) exist globally by comparison principle. The proof is complete. □
Proof of Theorem 2
3 Critical curve of Fujita type
Now we turn out attention to the results of Fujita type. This is, we shall show when all solutions of system (1.1)-(1.3) blow up in a finite time or both global and non-global solutions exist.
Proof of Theorem 3
Therefore, it follows from the comparison principle that \((\bar {u}_{1},\bar{u}_{2},\ldots,\bar{u}_{s})\) given by (3.1) is a super-solution of system (1.1)-(1.3) with \(\bar{u}_{1}(x,0)\geq u_{01}(x), \ldots, \bar{u}_{s}(x,0)\geq u_{0s}(x)\), which means that the solutions of (1.1)-(1.3) with small initial data have global existence.
Proof of Theorem 4
Declarations
Acknowledgements
The author would like to express many thanks to the editor and reviewers for their constructive suggestions, which helped to improve the previous version of this paper. This work was supported by the National Natural Science Foundation of China (11461076) and the University and College research foundation of Guangxi (ZD2014106).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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