Open Access

Critical curves for fast diffusion equations coupled via nonlinear boundary flux

Journal of Inequalities and Applications20152015:175

https://doi.org/10.1186/s13660-015-0695-3

Received: 21 July 2014

Accepted: 17 May 2015

Published: 2 June 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 self-similar super-solutions and sub-solutions, 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
$$A(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{s})^{\mathrm{T}} = (\varepsilon_{0}, \varepsilon_{0}, \ldots, \varepsilon_{0})^{\mathrm{T}} $$
plays an important role in our discussion.

Keywords

diffusion equation critical global existence curve critical Fujita curve Newtonian filtration equation nonlinear boundary flux

MSC

35B33 35K50 36K65

1 Introduction

In this paper, we investigate the existence and non-existence of global weak solutions to the following porous medium equations:
$$ (u_{i} )_{t} = \bigl(u_{i}^{m_{i}} \bigr)_{xx}, \quad i=1, 2, \ldots, s, x>0, 0< t< T $$
(1.1)
coupled via nonlinear boundary flux
$$ - \bigl( u_{i}^{m_{i}} \bigr)_{x}(0,t)= u_{i}^{p_{i}}(0,t) u_{i+1}^{q_{i}}(0,t),\quad i=1, 2, \ldots, s, \qquad u_{s+1}:= u_{1},\quad 0< t< T, $$
(1.2)
with continuous, nonnegative initial data
$$ u_{i}(x,0) = u_{0 i}(x), \quad i=1, 2, \ldots, s, x>0 $$
(1.3)
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 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)

In 2001, Quirós and Rossi [9] considered the following degenerate equations coupled via variational nonlinear boundary flux (\(s = 2\)):
$$ \left \{ \begin{array}{l@{\quad}l} u_{t} = (u^{m})_{xx}, \qquad v_{t} = (v^{n})_{xx}, & x >0, t>0, \\ -(u^{m})_{x}(0,t) = v^{p}(0,t), \qquad -(v^{n})_{x}(0,t) = u^{q}(0,t),& t>0, \\ u(x,0)=u_{0}(x), \qquad v(x,0) = v_{0}(x), & x>0 \end{array} \right . $$
(1.4)
with \(m , n > 1\) and notations
$$\begin{aligned}& \alpha_{1} = \frac{1+n+2p}{(1+m)(1+n)-4pq},\qquad \alpha_{2} = \frac {1+m+2q}{(1+m)(1+n)-4pq}, \\& \beta_{1} = \frac{p(m-1-2q)+(1+n)m}{(1+m)(1+n)-4pq},\qquad \beta_{2} = \frac {q(n-1-2p)+(1+m)n}{(1+m)(1+n)-4pq}. \end{aligned}$$
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 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.

In order to state our results, we introduce some useful symbols and a lemma. Denote by
$$ A= \left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} 1+m_{1}-2p_{1} & -2q_{1} & 0 & \cdots& 0 & 0 \\ 0 & 1+m_{2}-2p_{2} & -2q_{2} & \cdots& 0 & 0 \\ \cdots& \cdots& \cdots & \cdots& \cdots& \cdots\\ 0 & 0 & 0 & \cdots& 1+m_{s-1}-2p_{s-1} & -2 q_{s-1} \\ -2 q_{s} & 0 & 0 & \cdots& 0 & 1+m_{s}-2 p_{s} \end{array} \right ) $$
(1.5)
and let
$$m_{s+i}:=m_{i},\qquad p_{s+i}:=p_{i}, \qquad q_{s+i}:=q_{i},\qquad k_{s+i}:=k_{i}, \qquad l_{s+i}:=l_{i}. $$
A series of standard computations yields
$$\operatorname{det}A = \prod_{i=1}^{s}(1+m_{i}-2p_{i}) - 2^{s} q_{1} q_{2} \cdots q_{s}. $$
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
$$ A(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{s-1}, \alpha_{s})^{\mathrm{T}} = ( \varepsilon_{0}, \varepsilon_{0}, \ldots, \varepsilon_{0}, \varepsilon_{0} )^{\mathrm{T}} $$
(1.6)
has a unique solution \((\alpha_{1}, \alpha_{2}, \ldots, \alpha_{s-1}, \alpha_{s})^{\mathrm{T}}\) which is given by
$$\begin{aligned} \alpha_{i} =& \frac{\varepsilon_{0}}{\prod_{j=1}^{s} (1+m_{j}-2p_{j}) - 2^{s} q_{1} q_{2} \cdots q_{s}} \Biggl( \prod _{j=1}^{s-1} (1+m_{i+j}-2p_{i+j}) \\ & {}+\sum_{j=1}^{s-2} \Biggl( \prod _{j_{1}=1}^{j}(2q_{i+j_{1}-1}) \prod _{j_{2} = j+1}^{s-1}(1 + m_{i+j_{2}}- 2p_{i+j_{2}}) \Biggr) + \prod_{j=1}^{s-1}(2 q_{i+j-1}) \Biggr) \end{aligned}$$
(1.7)

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,
$$ A(k_{1}, k_{2}, \ldots, k_{s-1}, k_{s})^{\mathrm{T}} = (-1, -1, \ldots, -1, -1 )^{\mathrm{T}} $$
(1.8)
and define
$$ l_{i} = \frac{1 + k_{i}(1-m_{i})}{2}, \quad i=1, 2, \ldots, s. $$
(1.9)
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. (1)

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

     
  2. (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.

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

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 super-solution of self-similar form
$$ \bar{u}_{i}(x,t) = e^{K_{i} t} \bigl( M + e^{-L_{i} x e^{\frac{K_{i}(1-m_{i}) t}{2}} } \bigr)^{\frac{1}{m_{i}}}, \quad x \geq0, t \geq0, i=1, 2, \ldots, s, $$
(2.1)
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
$$\bar{u}_{i}(x,0) \geq u_{0i}(x), \quad x \geq0, i=1, 2, \ldots, s. $$
By a direct computation, we obtain
$$\begin{aligned} (\bar{u}_{1})_{t} =& K_{1} e^{K_{1} t} \bigl( M + e^{-L_{1} x e^{\frac {K_{1}(1-m_{1}) t}{2}}} \bigr)^{\frac{1}{m_{1}}} \\ &{}- \frac{K_{1} L_{1} (1-m_{1})}{2 m_{1}}e^{K_{1} t} \bigl( M + e^{-L_{1} x e^{\frac{K_{1}(1-m_{1}) t}{2}}} \bigr)^{\frac{1-m_{1}}{m_{1}}} e^{\frac{K_{1}(1-m_{1})t}{2}} x e^{-L_{1} x e^{\frac {K_{1}(1-m_{1})t}{2}}}. \end{aligned}$$
Note that the function \(Z_{1}(x) = x e^{-L_{1} x e^{\frac {K_{1}(1-m_{1})t}{2}}}\) reaches its maximum \(Z(x_{0})= \frac{1}{e L_{1}}e^{-\frac{K_{1}(1-m_{1})t}{2}}\) at the point \(x_{0} = \frac {1}{L_{1}}e^{-\frac{K_{1}(1-m_{1})t}{2}}\). Then we have
$$ (\bar{u}_{1})_{t} \geq K_{1} e^{K_{1} t} M^{\frac{1}{m_{1}}} - \frac {K_{1}(1-m_{1})}{2 e m_{1}}e^{K_{1} t}(1+M)^{\frac{1-m_{1}}{m_{1}}}. $$
(2.2)
At the same time,
$$ \bigl(\bar{u}_{1}^{m_{1}}\bigr)_{xx} = L_{1}^{2} e^{K_{1} t} e^{-L_{1} x e \frac {K_{1}(1-m_{1})t}{2}} \leq L_{1}^{2} e^{K_{1} t}. $$
(2.3)
Thus \(\bar{u}_{1}\) is a super-solution of Eq. (1.1) if
$$ K_{1} \biggl( M^{\frac{1}{m_{1}}} - \frac{(1-m_{1})}{2 e m_{1}}(1+M)^{\frac {1-m_{1}}{m_{1}}} \biggr) \geq L_{1}^{2}. $$
(2.4)
Similarly,
$$ K_{i} \biggl( M^{\frac{1}{m_{i}}} - \frac{(1-m_{i})}{2 e m_{i}}(1+M)^{\frac {1-m_{i}}{m_{i}}} \biggr) \geq L_{i}^{2},\quad i=2, \ldots, s. $$
(2.5)
Therefore, we can first take M large enough so that
$$ M^{\frac{1}{m_{i}}} - \frac{(1-m_{i})}{2 e m_{i}}(1+M)^{\frac{1-m_{i}}{m_{i}}}>0,\quad i=1,2,\ldots,s. $$
(2.6)
On the other hand, it remains to verify the boundary conditions (1.2), a simple computation yields
$$-\bigl(\bar{u}_{1}^{m_{1}}\bigr)_{x}(0,t) = L_{1} e^{\frac{K_{1}(1+m_{1})t}{1}}, \qquad \bar {u}_{1}^{p_{1}}(0,t) \bar{u}_{2}^{q_{1}}(0,t) = e^{(p_{1} K_{1}+q_{1} K_{2})t}(1+M)^{\frac{p_{1}}{m_{1}}+\frac{q_{1}}{m_{2}}}. $$
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
$$ L_{1} e^{\frac{K_{1}(1+m_{1})t}{2}} \geq e^{(p_{1} K_{1}+q_{1} K_{2})t}(1+M)^{\frac {p_{1}}{m_{1}}+\frac{q_{1}}{m_{2}}}. $$
(2.7)
Similarly,
$$ L_{i} e^{\frac{K_{i}(1 + m_{i})t}{2}} \geq e^{(p_{i} K_{i}+q_{i} K_{i+1})t}(1+M)^{\frac{p_{i}}{m_{i}}+\frac{q_{i}}{m_{i+1}}}, \quad i=2, \ldots, s. $$
(2.8)
Let
$$L_{i} = (1 + M)^{\frac{p_{i}}{m_{i}}+\frac{q_{i}}{m_{i+1}}}, \quad i=1, 2, \ldots, s. $$
Hence, from (2.7) and (2.8) we get
$$ e^{\frac{K_{i}(1 + m_{i})t}{2}} \geq e^{(p_{i} K_{i}+q_{i} K_{i+1})t}, \quad i=1, 2, \ldots, s. $$
(2.9)

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
$$\varepsilon_{0} = \max_{i} \biggl\{ \frac{L_{i}^{2} ( \prod_{j=1}^{s} (1 + m_{j} - 2p_{j}) - 2^{s} q_{1} q_{2} \cdots q_{s} )}{\beta_{i} ( M^{\frac{1}{m_{i}}} - \frac{1-m_{i}}{2 e m_{i}}(1+M)^{\frac{1-m_{i}}{m_{i}}} ) } \biggr\} , $$
where
$$\beta_{i} = \prod_{j=1}^{s-1} (1 + m_{i+j} - 2 p_{i+j}) + \sum_{j=1}^{s-2} \Biggl( \prod_{j_{1}=1}^{j}(2 q_{i+j_{1}-1}) \prod_{j_{2}=j+1}^{s-1}(1 + m_{i+j_{2}} - 2 p_{i+j_{2}}) \Biggr) + \prod _{j=1}^{s-1}(2 q_{i+j-1}). $$
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 super-solution 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 non-zero positive solutions. Let \((K_{1}, K_{2}, \ldots, K_{s}) \) be such a positive solution of (1.6) and satisfy
$$K_{i} \geq\frac{L_{i}^{2}}{M^{\frac{1}{m_{i}}} - \frac{1-m_{i}}{2 e m_{i}}(1+M)^{\frac{1-m_{i}}{m_{i}}} }, \quad i=1, 2, \ldots, s, $$
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 super-solution 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 non-existence of global solutions, we construct a blow-up self-similar sub-solution of system (1.1)-(1.3). Let \(k_{i}\) and \(l_{i}\) satisfy (1.8) and (1.9), respectively. Then we have
$$ m_{i} k_{i} + l_{i} = p_{i} k_{i} + q_{i} k_{i+1}, \quad i=1, 2, \ldots, s. $$
(2.10)
Consider the functions
$$ \tilde{u}_{i}(x,t) = (T-t)^{-k_{i}}f_{i}( \xi_{i}),\quad \xi_{i} = x (T-t)^{-l_{i}}, i=1, 2, \ldots, s, $$
(2.11)
with a positive constant T and compactly supported functions
$$ f_{i}(\xi_{i}) = ( A_{i} + B_{i} \xi_{i} )^{\frac{2}{m_{i}-1}}, \quad i=1, 2, \ldots, s, $$
(2.12)
where \(A_{i}\), \(B_{i}\) are positive constants to be determined. By a direct calculation, we have
$$\left \{ \begin{array}{l} (\tilde{u}_{i})_{t} = (T-t)^{-k_{i}-1} ( k_{i} f_{i}(\xi_{i}) + l_{i} \xi_{i} f_{i}^{\prime}(\xi_{i}) ), \\ (\tilde{u}_{i}^{m_{i}})_{xx} = (T-t)^{-m_{i} k_{i} - 2 l_{i}} (f_{i}^{m_{i}})^{\prime\prime}, \\ (\tilde{u}_{i}^{m_{i}})_{x}(0,t) = (T-t)^{-m_{i} k_{i} -l_{i}}(f_{i}^{m_{i}})^{\prime}(0), \\ \tilde{u}_{i}^{p_{i}}(0,t) \tilde{u}_{i+1}^{q_{i}}(0,t) = (T-t)^{-p_{i} k_{i} -q_{i} k_{i+1}} f_{i}^{p_{i}}(0) f_{i+1}^{q_{i}}(0). \end{array} \right . $$
Thus, \((\tilde{u}_{1}, \tilde{u}_{2},\ldots, \tilde{u}_{s})\) is a sub-solution of (1.1) and (1.2) provided that
$$ k_{i} f_{i}(\xi_{i}) + l_{i} \xi_{i} f_{i}^{\prime}( \xi_{i}) \leq\bigl(f_{i}^{m_{i}}\bigr)^{\prime \prime}( \xi_{i}), \quad i=1, 2, \ldots, s $$
(2.13)
and
$$ -\bigl(f_{i}^{m_{i}}\bigr)^{\prime}(0) \leq f_{i}^{p_{i}}(0) f_{i+1}^{q_{i}}(0), \quad i=1, 2, \ldots, s, \qquad f_{s+1}:=f_{1}. $$
(2.14)
Clearly,
$$\begin{aligned}& \bigl(f_{i}^{m_{i}}\bigr)^{\prime\prime}(\xi_{i}) = \frac{2 m_{i} (m_{i} + 1 )B_{i}^{2} }{(m_{i} - 1)^{2}} ( A_{i} + B_{i} \xi_{i} )^{\frac{2}{m_{i} - 1}}, \\& k_{i} f_{i}(\xi_{i}) + l_{i} \xi_{i} f_{i}^{\prime}(\xi_{i}) = k_{i} ( A_{i} + B_{i} \xi _{i} )^{\frac{2}{m_{i} - 1}} + \frac{2 l_{i} B_{i} }{m_{i} -1 } \xi_{i} ( A_{i} + B_{i} \xi_{i} )^{\frac{3 - m_{i}}{m_{i} - 1}} \leq k_{i} ( A_{i} + B_{i} \xi_{i} )^{\frac{2}{m_{i} - 1}}. \end{aligned}$$
Hence, inequalities (2.13) are satisfied if we choose the constants \(B_{1}, \ldots, B_{s}\) such that
$$ B_{i}^{2} = \frac{k_{i} (1-m_{i})^{2}}{2 m_{i} (1+m_{i})},\quad i=1, 2, \ldots, s. $$
(2.15)
On the other hand, the boundary conditions in (2.14) are satisfied provided that
$$ \frac{2 m_{i}}{1-m_{i}} B_{i} \leq A_{i}^{\frac{1+m_{i} - 2 p_{i}}{1-m_{i}}} A_{i+1}^{\frac{2 q_{i}}{m_{i+1}-1}},\qquad A_{s+1}:= A_{1},\quad i=1, 2, \ldots, s. $$
(2.16)
Substituting (2.15) into (2.16), we obtain
$$ \biggl( \frac{2 k_{i} m_{i}}{1 + m_{i}} \biggr)^{\frac{1 - m_{i+1}}{2(1 + m_{i} - 2p_{i})}} A_{i+1}^{\frac{2 q_{i}}{1 + m_{i} - 2 p_{i}}} \leq A_{i}^{\frac{1-m_{i+1}}{1-m_{i}}}, \quad i=1,2, \ldots, s. $$
(2.17)
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
$$\frac{2 q_{i} }{1+m_{i} -2p_{i}} > \prod_{j=1}^{s-1} \frac{1 + m_{i+j} - 2p_{i+j}}{2 q_{i+j}}. $$
At the same time, by (2.17) we get
$$ A_{i+1}^{\frac{2 q_{i}}{1+m_{i} - 2 p_{i}}} \leq C_{i} A_{i+1}^{\prod _{j=1}^{s-1}\frac{1 + m_{i+j} - 2p_{i+j}}{2 q_{i+j}}}, $$
(2.18)
where
$$C_{i} = \delta_{i}^{-1}\prod _{j=1}^{s-1} \bigl( \delta_{i+j}^{-\frac{1 - m_{i+1}}{1 - m_{i+j+1}} \prod_{j_{1} = j}^{s-1}\frac{1+m_{i+j_{1}}-2 p_{i+j_{1}}}{2 q_{i+j_{1}}} } \bigr),\qquad \delta_{i} = \biggl( \frac{2 k_{i} m_{i}}{1 + m_{i}} \biggr)^{\frac{1 - m_{i+1}}{2(1 + m_{i} - 2p_{i})}},\qquad \delta _{s+i}:=\delta_{i}. $$
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
$$\tilde{u}_{i}(x,0) \leq u_{0i}(x), \quad i=1, 2, \ldots, s, $$
then \((\tilde{u}_{1}, \tilde{u}_{2}, \ldots, \tilde{u}_{s})\) is a sub-solution 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. □

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

(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 self-similar global super-solutions
$$ \bar{u}_{i}(x,t) = (\tau+ t)^{-k_{i}}g_{i}( \xi_{i}), \quad \xi_{i} = x (\tau+ t)^{-l_{i}}, i=1,2,\ldots,s, $$
(3.1)
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 super-solution of system (1.1)-(1.3) provided that the nonnegative functions \(g_{1}(\xi_{1}),\ldots,g_{s}(\xi_{s})\) satisfy
$$\begin{aligned}& \bigl(g_{i}^{m_{i}}\bigr)^{\prime\prime}( \xi_{i}) + l_{i} \xi_{i} g_{i}^{\prime}( \xi_{i}) + k_{i} g_{i}(\xi_{i}) \leq0, \quad i=1,2,\ldots,s, \end{aligned}$$
(3.2)
$$\begin{aligned}& -\bigl(g_{i}^{m_{i}}\bigr)^{\prime}(0) \geq g_{i}^{p_{i}}(0)g_{i+1}^{q_{i}}(0),\quad i=1,2, \ldots,s, \qquad g_{s+1}:=g_{1}. \end{aligned}$$
(3.3)
We claim that (3.2) and (3.3) admit a solution of the form
$$ g_{i}(\xi_{i}) = D_{i} \bigl( d_{i}^{2} a_{i}^{2} + ( \xi_{i} + a_{i})^{2} \bigr)^{\frac {1}{m_{i} - 1}}, \quad i=1,2,\ldots,s. $$
(3.4)
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
$$\biggl(\frac{2 m_{i} D_{i}^{m_{i}}}{m_{i} -1}+k_{i} D_{i} \biggr) \bigl(d_{i}^{2} a_{i}^{2} + (\xi _{i} + a_{i})^{2} \bigr)+\frac{4 m_{i} D_{i}^{m_{i}}}{(m_{i}-1)^{2}}( \xi_{i}+a_{i})^{2} + \frac{2 l_{i} D_{i}}{m_{i}-1} \xi_{i} (\xi _{i}+a_{i}) \leq0. $$
That is,
$$\begin{aligned}& - \biggl( \frac{2 m_{i} D_{i}^{m_{i} -1}}{1-m_{i}} - k_{i} \biggr) \bigl(d_{i}^{2} a_{i}^{2} + ( \xi_{i}+a_{i})^{2} \bigr) \\& \quad {}- \frac{2 (\xi_{i}+a_{i})}{1-m_{i}} \biggl( \biggl(l_{i} - \frac{2 m_{i} D_{i}^{m_{i}-1}}{1-m_{i}} \biggr) ( \xi_{i}+a_{i}) -l_{i} a_{i} \biggr) \leq0. \end{aligned}$$
(3.5)
Therefore, according to that \(k_{i} < l_{i}\), we may first take the constant \(D_{i}\) such that
$$k_{i} < \frac{2 m_{i} D_{i}^{m_{i}-1}}{1-m_{i}}< l_{i}. $$
Secondly, setting \(y_{i} = \xi_{i} + a_{i}\), then the inequality in (3.5) can be written as
$$ \biggl( \frac{2 m_{i} (1+m_{i})}{1-m_{i}}D_{i}^{m_{i}-1}-1 \biggr)y_{i}^{2} + 2 l_{i} a_{i} y_{i} - (1-m_{i}) \biggl( \frac{2 m_{i} D_{i}^{m_{i}-1}}{1-m_{i}} - k_{i} \biggr) d_{i}^{2} a_{i}^{2} \leq0. $$
(3.6)
For simplicity, we define the function \(h_{i}\) as
$$h_{i}(y_{i}) = \biggl( \frac{2 m_{i} (1+m_{i})}{1-m_{i}}D_{i}^{m_{i}-1}-1 \biggr)y_{i}^{2} + 2 l_{i} a_{i} y_{i} - (1-m_{i}) \biggl( \frac{2 m_{i} D_{i}^{m_{i}-1}}{1-m_{i}} - k_{i} \biggr) d_{i}^{2} a_{i}^{2}, \quad y_{i} \geq a_{i}. $$
And then \(h_{i}(y_{i})\) reaches its maximum at the point
$$y_{i}^{*} = \frac{l_{i} a_{i}}{1 - \gamma_{i} (1+m_{i})}, \quad \gamma_{i} = \frac{2 m_{i} D_{i}^{m_{i}-1}}{1-m_{i}}. $$
Hence, we only need that \(h_{i}(y_{i}^{*})\leq0\), it follows from the following:
$$d_{i} \geq l_{i} \bigl( (1-m_{i}) ( \gamma_{i} - k_{i}) \bigl( 1 - (1+m_{i}) \gamma_{i} \bigr) \bigr)^{-\frac{1}{2}}. $$
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
$$ a_{i}^{\frac{1+m_{i} - 2 p_{i}}{2q_{i}}} \leq R_{i} a_{i+1}^{\frac {1-m_{i}}{1-m_{i+1}}}, \quad i=1,2,\ldots,s, $$
(3.7)
where
$$\left \{ \begin{array}{l} R_{i} = D_{i+1}^{\frac{m_{i} -1}{2}} ( \frac{2 m_{i} D_{i}^{m_{i}-p_{i}}}{1-m_{i}} )^{\frac{1-m_{i}}{2q_{i}}} ( d_{i+1}^{2}+1 )^{\frac{1-m_{i}}{2(1-m_{i+1})}} ( d_{i}^{2} + 1 )^{\frac {p_{i}-1}{2q_{i}}}, \\ D_{s+1}:=D_{1},\qquad a_{s+1}:=a_{1}, \qquad d_{s+1}:=d_{1}. \end{array} \right . $$
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
$$\frac{1+m_{i}-2p_{i}}{2q_{i}} < \prod_{j=1}^{s-1} \frac{2q_{i+j}}{1+m_{i+j}-2p_{i+j}}. $$
In addition, by (3.7), there exists a positive constant \(\bar{C}_{i}\) as
$$\bar{C}_{i} = R_{i} \prod_{j=1}^{s-1} \bigl( R_{i+j}^{ \frac {1-m_{i}}{1-m_{i+j}}\prod_{j_{1}=1}^{j} \frac{2 q_{i+j_{1}}}{1+m_{i+j_{1}}-2p_{i+j_{1}}} } \bigr), \qquad R_{s+i}:=R_{i} $$
such that
$$ a_{i}^{\frac{1+m_{i} -2p_{i}}{2q_{i}}} \leq\bar{C}_{i} a_{i}^{\prod _{j=1}^{s-1}\frac{2q_{i+j}}{1+m_{i+j}-2p_{i+j}} },\quad i=1,2,\ldots,s. $$
(3.8)
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 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.

(2) We construct the self-similar sub-solution of Eq. (1.1) in the following form:
$$ \hat{u}_{i} (x,t) = (\tau+ t)^{-\frac{1}{m_{i}+1}}g_{i}( \xi_{i}),\quad \xi_{i} = x (\tau+ t)^{-\frac{1}{m_{i}+1}}, i=1,2, \ldots,s $$
(3.9)
with
$$g_{i}(\xi_{i}) = D_{i} \bigl(c^{2} + \xi_{i}^{2}\bigr)^{\frac{1}{m_{i}-1}}, $$
where positive constants τ and c are to be determined. If we take
$$D_{i} = \biggl( \frac{1-m_{i}}{2 m_{i} (m_{i}+1)} \biggr)^{\frac{1}{m_{i}-1}},\quad i=1, 2, \ldots, s, $$
then it is easy to check that \(g_{i}\) satisfies
$$ \bigl(g_{i}^{m_{i}}(\xi_{i}) \bigr)^{\prime\prime} + \frac{1}{m_{i}+1} \xi_{i} g_{i}^{\prime}( \xi_{i}) + \frac{1}{m_{i}+1}g_{i}(\xi_{i})=0, \qquad g_{i}^{\prime}(0) =0, \quad i=1,2,\ldots,s, $$
(3.10)
which implies that
$$\left \{ \begin{array}{l@{\quad}l} (\hat{u}_{i})_{t} = (\hat{u}_{i}^{m_{i}})_{xx}, & x>0, t>0, \\ -(\hat{u}_{i}^{m_{i}})_{x} (0,t)=0, & t>0, \end{array} \right . \quad i=1,2,\ldots,s. $$
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
$$u_{i}(x,t_{0})\geq\hat{u}_{i}(x,t_{0}), \quad x>0, i=1,2,\ldots,s. $$
Thus, the self-similar solution \((\hat{u}_{1},\hat{u}_{2}, \ldots,\hat {u}_{s})\) is a sub-solution of (1.1)-(1.3) in \((0,\infty)\times(t_{0},T)\). The comparison principle follows
$$u_{i}(x,t)\geq\hat{u}_{i}(x,t), \quad x>0, t\geq t_{0}, i=1,2,\ldots,s. $$
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
$$ T^{l_{i}} \ll\bigl(\tau+ t^{*}\bigr)^{\frac{1}{m_{i}+1}} \ll T^{k_{i}},\quad i=1,2,\ldots,s. $$
(3.11)
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
$$\tilde{u}_{i}(x,0) \leq\hat{u}_{i}\bigl(x,t^{*}\bigr) \leq u_{i}\bigl(x,t^{*}\bigr). $$
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 single-equation problem:
$$ \left \{ \begin{array}{l@{\quad}l} w_{t} = (w^{m_{1}})_{xx}, & x>0, 0< t< T, \\ -(w^{m_{1}})_{x}(0,t) = w^{p_{1}}(0,t)u_{02}^{q_{1}}(0), & 0< t< T, \\ w(x,0) = u_{01}(x), & x>0. \end{array} \right . $$
(3.12)
It is easy to verify that \((w,u_{02},\ldots,u_{0s})\) is a sub-solution 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. □

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

(1)
Institute of Mathematics and Information Science, Yulin Normal University

References

  1. Glansdorff, P, Prigogine, I: Thermodynamic Theory of Structure, Stability and Fluctuation. Wiley-Interscience, London (1971) MATHGoogle Scholar
  2. Kalashnikov, AS: Some problems of the qualitative theory of nonlinear degenerate parabolic equations of second order. Russ. Math. Surv. 42, 169-222 (1987) View ArticleMATHMathSciNetGoogle Scholar
  3. Mancebo, FJ, Vega, JM: A model of porous catalyst accounting for incipiently non-isothermal effects. J. Differ. Equ. 151, 79-110 (1999) View ArticleMATHMathSciNetGoogle Scholar
  4. DiBenedetto, E: Degenerate Parabolic Equations. Springer, New York (1993) View ArticleMATHGoogle Scholar
  5. Wu, ZQ, Zhao, JN, Yin, JX, Li, HL: Nonlinear Diffusion Equations. World Scientific, Singapore (2001) View ArticleMATHGoogle Scholar
  6. Samarskii, AA, Galaktionov, VA, Kurdyumov, SP, Mikhailov, AP: Blow-up in Quasilinear Parabolic Equations. de Gruyter, Berlin (1995) View ArticleMATHGoogle Scholar
  7. Deng, K, Levine, HA: The role of critical exponents in blow-up theorems: the sequel. J. Math. Anal. Appl. 243, 85-126 (2000) View ArticleMATHMathSciNetGoogle Scholar
  8. Levine, HA: The role of critical exponents in blow up theorems. SIAM Rev. 32, 262-288 (1990) View ArticleMATHMathSciNetGoogle Scholar
  9. Quirós, F, Rossi, JD: Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions. Indiana Univ. Math. J. 50, 629-654 (2001) View ArticleMATHMathSciNetGoogle Scholar
  10. Zheng, SN, Song, XF, Jiang, ZX: Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux. J. Math. Anal. Appl. 298, 308-324 (2004) View ArticleMATHMathSciNetGoogle Scholar
  11. Ferreira, R, de Pablo, A, Quirós, F, Rossi, JD: The blow-up profile for a fast diffusion equation with a nonlinear boundary condition. Rocky Mt. J. Math. 33, 123-146 (2003) View ArticleMATHMathSciNetGoogle Scholar

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