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Critical curves for fast diffusion equations coupled via nonlinear boundary flux
Journal of Inequalities and Applications volume 2015, Article number: 175 (2015)
Abstract
This paper is concerned with fast diffusion equations for coupling via nonlinear boundary flux. By means of the theory of linear equations and constructing selfsimilar supersolutions and subsolutions, we obtain a critical global existence curve. The critical curve of Fujita type is conjectured with the aid of some new results. In addition, we show that the constant \(\varepsilon_{0}\) of the linear system
plays an important role in our discussion.
Introduction
In this paper, we investigate the existence and nonexistence of global weak solutions to the following porous medium equations:
coupled via nonlinear boundary flux
with continuous, nonnegative initial data
compactly supported in \(\mathbb{R}_{+}\), where \(s \geq2\), \(0 < m_{i} < 1\), \(p_{i} \geq0\), \(q_{i} >0\) (\(i=1,2,\ldots,s\)) are parameters.
The particular feature of equations (1.1) is their gradientdependent 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)
In 2001, Quirós and Rossi [9] considered the following degenerate equations coupled via variational nonlinear boundary flux (\(s = 2\)):
with \(m , n > 1\) and notations
They obtained that the critical global existence curve of (1.4) is \(pq = (\frac{1+m}{2})(\frac{1+n}{2})\) and the critical Fujita type curve is \(\min\{ \alpha_{1}+\beta_{1}, \alpha_{2}+\beta _{2} \} = 0\). Besides, it was Zheng et al. who dealt with the general system (1.1)(1.3) for \(s = 2\) with \(m_{1}, m_{2} >1\) in [10], in which the authors proved that for \(p_{1} < \frac{1+m_{1}}{2}\), \(p_{2} < \frac{1+m_{2}}{2}\), the critical global existence curve is \(q_{1} q_{2} = (\frac{1+m_{1}}{2}p_{1})(\frac{1+m_{2}}{2}p_{2})\) and the critical Fujita curve is \(\min\{ l_{1}k_{1}, l_{2} k_{2}\}=0\), while if \(p_{1} > \frac{1+m_{1}}{2}\) or \(p_{2} > \frac{1+m_{2}}{2}\), then the solutions may blow up in a finite time.
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 selfsimilar supersolution and subsolution 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.
In order to state our results, we introduce some useful symbols and a lemma. Denote by
and let
A series of standard computations yields
Next, we shall see that \(\operatorname{det}A=0\) is the critical global existence curve. In addition, we give the following lemma which comes from linear algebra, and its proof is obtained by using the Cramer theorem.
Lemma 1
For the matrix A which is defined by (1.5) and any constant \(\varepsilon_{0}\), according to the Cramer principle, if \(\operatorname{det}A\neq0\), then the following linear system
has a unique solution \((\alpha_{1}, \alpha_{2}, \ldots, \alpha_{s1}, \alpha_{s})^{\mathrm{T}}\) which is given by
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., \(\operatorname{det}A = 0\).
Let \((k_{1}, k_{2}, \ldots, k_{s})\) denote the unique positive solution of linear system (1.6) with constant \(\varepsilon_{0} = 1\), that is,
and define
Then we have the following.
Theorem 3
Assume \(2p_{i} \leq1+m_{i}\) (\(i = 1, 2,\ldots ,s\)) and \(\prod_{i=1}^{s} (1 + m_{i}  2p_{i}) < 2^{s} q_{1} q_{2} \cdots q_{s} \), i.e., \(\operatorname{det}A < 0\).

(1)
If \(\min_{i}\{l_{i}  k_{i} \}>0 \), there exist nonnegative solutions with blowup and nonnegative solutions that are global.

(2)
If \(\max_{i}\{ l_{i}  k_{i} \} < 0\), then every nonnegative, nontrivial solution of system (1.1)(1.3) blows up in a finite time.
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.
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 selfsimilar 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
In order to prove that the solution \((u_{1}, u_{2}, \ldots, u_{s})\) of (1.1)(1.3) is global, we look for a globally defined in time strict supersolution of selfsimilar form
where \(M \geq\max_{i}\{\u_{0i}\_{\infty}+1 \}\) and \(K_{i}, L_{i}>0\) are to be determined. It is easy to see that
By a direct computation, we obtain
Note that the function \(Z_{1}(x) = x e^{L_{1} x e^{\frac {K_{1}(1m_{1})t}{2}}}\) reaches its maximum \(Z(x_{0})= \frac{1}{e L_{1}}e^{\frac{K_{1}(1m_{1})t}{2}}\) at the point \(x_{0} = \frac {1}{L_{1}}e^{\frac{K_{1}(1m_{1})t}{2}}\). Then we have
At the same time,
Thus \(\bar{u}_{1}\) is a supersolution of Eq. (1.1) if
Similarly,
Therefore, we can first take M large enough so that
On the other hand, it remains to verify the boundary conditions (1.2), a simple computation yields
Then we have \((\bar{u}_{1}^{m_{1}})_{x}(0,t) \geq\bar{u}_{1}^{p_{1}}(0,t) \bar {u}_{2}^{q_{1}}(0,t)\), if we impose
Similarly,
Let
Hence, from (2.7) and (2.8) we get
Next, we divide the proof into two cases.
Case (i). If \(\prod_{i=1}^{s} (1+m_{i}  2p_{i}) > 2^{s} q_{1} q_{2} \cdots q_{s}\), then we choose that \((K_{1}, K_{2}, \ldots, K_{s})\) denotes the unique solution \((\alpha_{1}, \alpha_{2}, \ldots, \alpha_{s})\) of the linear system (1.6), i.e., \((K_{1}, K_{2}, \ldots, K_{s}) = (\alpha_{1}, \alpha_{2}, \ldots, \alpha_{s}) \) with
where
Thus, these constants \(K_{i}\) and \(\varepsilon_{0}\) can ensure that inequalities (2.4), (2.5) and (2.9) hold. Therefore, we have proved that \((\bar{u}_{1}, \bar{u}_{2}, \ldots, \bar {u}_{s})\) is a global supersolution of system (1.1)(1.3).
Case (ii). If \(\prod_{i=1}^{s} (1+m_{i}  2p_{i}) = 2^{s} q_{1} q_{2} \cdots q_{s}\), the linear system (1.6) with \(\varepsilon_{0} =0\) has nonzero positive solutions. Let \((K_{1}, K_{2}, \ldots, K_{s}) \) be such a positive solution of (1.6) and satisfy
which imply that (2.4), (2.5) and (2.9) hold. Thus, we have proved that \((\bar{u}_{1}, \bar{u}_{2}, \ldots, \bar{u}_{s})\) is a global supersolution of system (1.1)(1.3).
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
To prove the nonexistence of global solutions, we construct a blowup selfsimilar subsolution of system (1.1)(1.3). Let \(k_{i}\) and \(l_{i}\) satisfy (1.8) and (1.9), respectively. Then we have
Consider the functions
with a positive constant T and compactly supported functions
where \(A_{i}\), \(B_{i}\) are positive constants to be determined. By a direct calculation, we have
Thus, \((\tilde{u}_{1}, \tilde{u}_{2},\ldots, \tilde{u}_{s})\) is a subsolution of (1.1) and (1.2) provided that
and
Clearly,
Hence, inequalities (2.13) are satisfied if we choose the constants \(B_{1}, \ldots, B_{s}\) such that
On the other hand, the boundary conditions in (2.14) are satisfied provided that
Substituting (2.15) into (2.16), we obtain
For \(i=1,2, \ldots, s\), \(\prod_{i=1}^{s} (1 + m_{i}  2p_{i} ) < 2^{s} q_{1} q_{2} \cdots q_{s}\) implies that
At the same time, by (2.17) we get
where
Therefore, there exists \(A_{i+1}\) small enough such that inequality (2.18) is valid. Thus, if the initial data \(u_{01}(x), u_{02}(x), \ldots, u_{0s}(x)\) are large enough so that
then \((\tilde{u}_{1}, \tilde{u}_{2}, \ldots, \tilde{u}_{s})\) is a subsolution of system (1.1)(1.3) by the comparison principle, which implies that the solutions of system (1.1)(1.3) with large initial data blow up in a finite time. The proof is complete. □
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 nonglobal solutions exist.
Proof of Theorem 3
(1) In order to prove the conclusion, we only need to show that the solutions of system (1.1)(1.3) with small enough initial data have global existence, which will be proved by constructing kinds of selfsimilar global supersolutions
where \(\tau>0\) is a positive constant, \(k_{i}\) and \(l_{i}\) are defined by (1.8) and (1.9), respectively.
A direct computation together with (1.9) and (2.10), the function \((\bar {u}_{1},\ldots,\bar{u}_{s})\) is a supersolution of system (1.1)(1.3) provided that the nonnegative functions \(g_{1}(\xi_{1}),\ldots,g_{s}(\xi_{s})\) satisfy
We claim that (3.2) and (3.3) admit a solution of the form
Next, we will show that there exist suitable positive constants \(D_{i}\), \(d_{i}\), \(a_{i}\) (\(i=1,2,\ldots,s\)) such that inequalities (3.2) and (3.3) are satisfied. In fact, for \(i=1,2,\ldots ,s\), substituting \(g_{i}\) and \(g_{i}^{\prime}\) into (3.2), we have
That is,
Therefore, according to that \(k_{i} < l_{i}\), we may first take the constant \(D_{i}\) such that
Secondly, setting \(y_{i} = \xi_{i} + a_{i}\), then the inequality in (3.5) can be written as
For simplicity, we define the function \(h_{i}\) as
And then \(h_{i}(y_{i})\) reaches its maximum at the point
Hence, we only need that \(h_{i}(y_{i}^{*})\leq0\), it follows from the following:
On the other hand, for the above constants \(D_{i}\), \(d_{i}\) (\(i=1,2,\ldots,s\)), the boundary conditions (3.3) are satisfied if we have
where
Similar to the analysis of the proof in Theorem 2, for \(i=1,2,\ldots ,s\), \(\prod_{i=1}^{s} (1+m_{i}2p_{i}) < 2^{s} q_{1} q_{2} \cdots q_{s}\) implies that
In addition, by (3.7), there exists a positive constant \(\bar{C}_{i}\) as
such that
Thus, we can choose \(a_{1},a_{2},\ldots,a_{s}\) large enough for the above inequalities (3.8) to hold.
Therefore, it follows from the comparison principle that \((\bar {u}_{1},\bar{u}_{2},\ldots,\bar{u}_{s})\) given by (3.1) is a supersolution 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.
(2) We construct the selfsimilar subsolution of Eq. (1.1) in the following form:
with
where positive constants τ and c are to be determined. If we take
then it is easy to check that \(g_{i}\) satisfies
which implies that
Since \(u_{i}(x,t)\) (\(i=1,2,\ldots,s\)) are nonnegative, nontrivial functions, we see that \(u_{i}(0,t_{0})>0\) for some \(t_{0}>0\). It is well known that \(u_{i}(x,t_{0})>0\) (\(i=1,2,\ldots,s\)) are continuous (see [2, 4]). Then we can choose τ and c large enough so that
Thus, the selfsimilar solution \((\hat{u}_{1},\hat{u}_{2}, \ldots,\hat {u}_{s})\) is a subsolution of (1.1)(1.3) in \((0,\infty)\times(t_{0},T)\). The comparison principle follows
Recalling that \(\max_{i}\{l_{i}  k_{i}\}<0\), we see for large T that \(T^{l_{i}} \ll T^{k_{i}}\) (\(i=1,2,\ldots,s\)). So there exists \(t^{*} > t_{0}\) such that
Let \(\tilde{u}_{i}(x,t)\) be defined by (2.11) and (2.12) in the proof of Theorem 2, for any \(x>0\), the inequalities (3.11) imply that
By the comparison principle again, every nonnegative and nontrivial solution \((u_{1},u_{2}, \ldots,u_{s})\) of system (1.1)(1.3) blows up in a finite time. The proof is complete. □
Proof of Theorem 4
Without loss of generality, we may assume that \(2p_{1}>1+m_{1}\) and \((u_{0i}^{m_{i}})^{\prime\prime} \geq0\) (\(i=1,2,\ldots,s\)). Then \(u_{1t},u_{2t},\ldots,u_{st} \geq0 \) for \(x>0\), \(t>0\) (see [9, 10]). Furthermore, we have \(u_{1}^{p_{1}}(0,t)u_{2}^{q_{1}}(0,t) \geq u_{1}^{p_{1}}(0,t)u_{02}^{q_{1}}(0)\). Consider the following singleequation problem:
It is easy to verify that \((w,u_{02},\ldots,u_{0s})\) is a subsolution of system (1.1)(1.3). According to the results of [11], we know that the solutions of (3.12) with large initial data blow up in a finite time, and so the solutions of (1.1)(1.3) do too. The proof is complete. □
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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).
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Ling, Z. Critical curves for fast diffusion equations coupled via nonlinear boundary flux. J Inequal Appl 2015, 175 (2015). https://doi.org/10.1186/s1366001506953
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DOI: https://doi.org/10.1186/s1366001506953
MSC
 35B33
 35K50
 36K65
Keywords
 diffusion equation
 critical global existence curve
 critical Fujita curve
 Newtonian filtration equation
 nonlinear boundary flux