In this paper, we deal with Newtonian filtration equations on the exterior domain of the unit ball in $\mathbb R^N$, i.e.,

$ \frac{\partial {{u}_{i}}}{\partial t}=\Delta u_{i}^{{{m}_{i}}}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in {{\mathbb{R}}^{N}}\backslash {{B}_{1}}(0), t > 0, i=1, 2, \cdots, n, $

(1.1)

$ \nabla u_{i}^{{{m}_{i}}}\cdot \vec{\nu }=u_{i+1}^{{{p}_{i}}}(x, t), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in \partial {{B}_{1}}(0), t > 0, i=1, 2, \cdots, n, $

(1.2)

$ {{u}_{i}}(x, 0)={{u}_{0, i}}(x), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in {{\mathbb{R}}^{N}}\backslash {{B}_{1}}(0), i=1, 2, \cdots, n, $

(1.3)

where $m_i > 1$, $p_i\ge0$, $u_{n+1}=u_1$, $B_1(0)$ is the unit ball in $\mathbb R^N$ with boundary $\partial B_1(0)$, $\vec \nu$ is the inward normal vector on $\partial B_1(0)$, and $u_{0, i}(x)$ are nonnegative, suitably smooth and bounded functions with compact support that satisfy the appropriate compatibility conditions.

The equation in (1.1) has been intensively used in the models of chemical reactions, population dynamics, heat transfer, and so on. For the problem (1.1)-(1.3), the local existence and the comparison principle of the weak solutions can be established, see [1, 2]. In this paper, we investigate the large time behavior of solutions to the system (1.1)-(1.3), such as global existence and blow-up in a finite time.

Since the beginning work on critical exponent done by Fujita [3] in 1966, lots of Fujita type results are established for various of problems, see the survey papers [4-6] and the references therein. It was Glalaktionov and Levine who first discussed the critical exponents for the one-dimensional nonlinear diffusion equations with boundary sources in [7]:

$ \frac{\partial u}{\partial t}=\frac{{{\partial }^{2}}{{u}^{m}}}{\partial {{x}^{2}}}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x > 0, t > 0, $

(1.4)

$ \frac{\partial {{u}^{m}}}{\partial x}(0, t)={{u}^{\alpha }}(0, t), \ \ \ \ \ \ \ \ \ \ \ \ t > 0, $

(1.5)

$ u(x, 0)={{u}_{0}}(x), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in (0, +\infty ), $

(1.6)

here $m > 1, \alpha\ge0$. For the problems (1.4)-(1.6), they proved that $\alpha_0={(m+1)}/{2}, \alpha_c=m+1$. Here, we call $\alpha_0$ as the critical global existence exponent and $\alpha_c$ as the critical Fujita exponent respectively,

(ⅰ) if $0 < \alpha < \alpha_0$, then every nontrivial nonnegative solution is global in time;

(ⅱ) if $\alpha_0 < \alpha < \alpha_c$, then the nontrivial nonnegative solutions blow up in a finite time;

(ⅲ) if $\alpha > \alpha_c$, then the solutions exist globally for the small initial data and blow up in a finite time for the large initial data.

In fact, instead of critical exponents there exist the critical global existence curve and the critical Fujita curve for the coupled system of diffusion equations, see [8]. For the one-dimensional nonlinear diffusion equations, Quirós and Rossi [9] considered the coupled Newtonian filtration equations as follows

$ \frac{\partial u}{\partial t}=\frac{{{\partial }^{2}}{{u}^{{{m}_{1}}}}}{\partial {{x}^{2}}}, \ \ \quad \frac{\partial v}{\partial t}=\frac{{{\partial }^{2}}{{v}^{{{m}_{2}}}}}{\partial {{x}^{2}}}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x > 0, t > 0, \ $

$ -\frac{\partial {{u}^{{{m}_{1}}}}}{\partial x}(0, t)={{v}^{\alpha }}(0, t), \ \ \ -\frac{\partial {{v}^{{{m}_{2}}}}}{\partial x}(0, t)={{u}^{\beta }}(0, t), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t > 0, $

$ u(x, 0)={{u}_{0}}(x), \ \ \ \ v(x, 0)={{v}_{0}}(x), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x > 0. $

They showed that the critical global existence curve is given by $\alpha\beta=({{m_1}+1})({{m_2}+1})/{4}$ and the critical Fujita curve is given by $\min\{\alpha_1+\beta_1, \alpha_2+\beta_2\}=0$, where

$ {{\alpha }_{1}}=\frac{2\alpha +{{m}_{2}}+1}{({{m}_{1}}+1)({{m}_{2}}+1)-4\alpha \beta }, \ \ \ \ {{\beta }_{1}}=\frac{\alpha ({{m}_{1}}-1-2\beta )+({{m}_{2}}+1){{m}_{1}}}{({{m}_{1}}+1)({{m}_{2}}+1)-4\alpha \beta }, $

$ {\alpha _2} = \frac{{2\beta + {m_1} + 1}}{{({m_1} + 1)({m_2} + 1) - 4\alpha \beta }},\;\;\;\;{\beta _2} = \frac{{\beta ({m_2} - 1 - 2\alpha ) + ({m_1} + 1){m_2}}}{{({m_1} + 1)({m_2} + 1) - 4\alpha \beta }}. $

The similar results were established in [10-12].

In the present paper, we consider the critical curves for the multi-dimensional system (1.1)-(1.3), the case of single equation is studied in [13] and proved that both the critical global existence exponent and the critical Fujita exponent are given by $p=m. $ We extend the results in [13] to the problem on multiple equations. Furthermore, by virtue of the radial symmetry of the exterior domain of the unit ball, we can extend our results to the following more general equations

$ \frac{\partial }{\partial t}(|x{{|}^{{{{\tilde{\lambda }}}_{i}}}}{{u}_{i}})=\text{div}(|x{{|}^{{{{\tilde{\lambda }}}_{i}}}}\nabla u_{i}^{{{m}_{i}}}), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in {{\mathbb{R}}^{N}}\backslash {{B}_{1}}(0), t > 0 $

(1.7)

with $\tilde\lambda_i > 2-N$, $N\ge 1$.

The rest of this paper is organized as follows. Section 2 is devoted to the large time behavior of solutions to the nonlinear boundary problem for the Newtonian filtration equations, namely (1.1)-(1.3) and (1.7), (1.2), (1.3).

In this section, we first introduce our results on the system of Newtonian filtration equations coupled with boundary conditions, then we give the proofs.

Theorem 2.1  The critical global existence curve and the critical Fujita curve for the system (1.1)-(1.3) are given by

$ \prod\limits_{i=1}^{n}{{{p}_{i}}}=\prod\limits_{i=1}^{n}{{{m}_{i}}}. $

Namely, if $\prod\limits_{i=1}^n p_i\le{\prod\limits_{i=1}^n m_i}$, then all nonnegative nontrivial solutions to the system (1.1)-(1.3) exist globally in time; while if $ \prod\limits_{i=1}^n p_i > {\prod\limits_{i=1}^n m_i}$, then the nonnegative solutions to the system (1.1)-(1.3) blow up in finite time for large initial data and exist globally for small initial data.

Theorem 2.2  Assume $\tilde\lambda_i>2-N$, $N\ge1$. For the equation (1.7) with the initial and boundary conditions (1.2), (1.3), the critical global existence curve and the critical Fujita curve both are given by $ \prod\limits_{i=1}^n p_i={\prod\limits_{i=1}^n m_i}$.

Before we give the proof of Theorem 2.1 and Theorem 2.2, we consider the problem

$ \frac{\partial {{u}_{i}}}{\partial t}=\frac{{{\partial }^{2}}u_{i}^{{{m}_{i}}}}{\partial {{r}^{2}}}+\frac{{{\lambda }_{i}}}{r}\frac{\partial u_{i}^{{{m}_{i}}}}{\partial r}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ r > 1, t > 0, i=1, 2, \cdots, n, $

(2.1)

$ -\frac{\partial u_{i}^{{{m}_{i}}}}{\partial r}(1, t)=u_{i+1}^{{{p}_{i}}}(1, t), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t > 0, i=1, 2, \cdots, n, $

(2.2)

$ {{u}_{i}}(r, 0)={{u}_{0, i}}(r), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ r > 1, i=1, 2, \cdots, n, $

(2.3)

where $r=|x|$, $m_i > 1$, $p_i\ge0$, $N\ge1$, $\lambda_i > 1$, and $u_{0, 1}(r), u_{0, 2}(r), \cdots, u_{0, n}(r)$ are nonnegative, nontrivial functions with compact supports. Clearly, the solution $(u_1, u_2, \cdots, u_n)$ of the system (2.1)-(2.3) with $\lambda_i= N-1$ is also the solution of the system (1.1)-(1.3) if $u_{0, 1}(x), u_{0, 2}(x), \cdots, u_{0, n}(x)$ are radially symmetrical. If $\lambda_i=\tilde\lambda_i+N-1$, the same facts also hold valid between the system (2.1)-(2.3) with the system (1.7), (1.2), (1.3). In order to obtain Theorem 2.1, Theorem 2.2, we firstly show the following results on the system (2.1)-(2.3).

Proposition 2.1  If $\prod\limits_{i=1}^n p_i < {\prod\limits_{i=1}^n m_i}$, then all nonnegative nontrivial solutions to the system (2.1)-(2.3) exist globally in time.

Proposition 2.2  If $ \prod\limits_{i=1}^n p_i > {\prod\limits_{i=1}^n m_i}$, then the nonnegative solutions with large initial data to the system (2.1)-(2.3) blow up in a finite time.

Remark 1  It can be seen from Proposition 2.1 and Proposition 2.2 that the critical global existence curve for the systems (2.1)-(2.3) is $ \prod\limits_{i=1}^n p_i={\prod\limits_{i=1}^n m_i}$.

Proposition 2.3  If $\prod\limits_{i=1}^n p_i\neq{\prod\limits_{i=1}^n m_i}$, then every nonnegative nontrivial solution with small initial data to the problems (2.1)-(2.3) exists globally.

Remark 2  From Propositions 2.1-Proposition 2.3, we have the critical Fujita curve for the system (2.1)-(2.3) is given by $ \prod\limits_{i=1}^n p_i=\prod\limits_{i=1}^n m_i$.

Now, we prove Proposition 2.1-Proposition 2.3.

Proof of Proposition 2.1.  We prove this proposition by constructing a kind of global supersolution in the following form

$ {\tilde u_i}(r, t) = {(T + t)^{{k_i}}}{h_i}({\xi _i}), \;\;\;\;{\xi _i} = \frac{{r - 1}}{{{{(T + t)}^{{l_i}}}}}, \;\;\;\;r > 1, t > 0, i = 1, 2, \cdots, n, $

(2.4)

where $T > 0$, and $m_{n+1}=m_1, k_{i+1}=k_i \gamma_i=k_1\prod\limits_{j=1}^i \gamma_j, \gamma_i < \frac{m_i}{p_i}, i=1, 2, \cdots, n-1$ with $k_1$ being the positive constants to be determined and

$ {l_i} = \frac{{1 + {k_i}({m_i} - 1)}}{2}, \;\;\;\;i = 1, 2, \cdots, n. $

Fix $\xi_i > 0$, for any $i\in \{1, 2, \cdots, n\}$ we take

$ {h_i}({\xi _i}) = {(\frac{{{m_i} - 1}}{{{m_i}}}{(1 - {\xi _i})_ + })^{1/({m_i} - 1)}}, \;\;\;\;i = 1, 2, \cdots, n. $

Denote

$ {{\bar u}_i}(r, t) = {r^{ - {\alpha _i}/{m_i}}}{{\tilde u}_i}(r, t), \;\;\;\;r > 1, t > 0, i = 1, 2, \cdots, n, $

(2.5)

where $\alpha_i$ are given by the following

$ {\alpha _i} = \left\{ \begin{array}{l} {\lambda _i} - 1, {\rm{ }}1 < {\lambda _i} < 2, \;\;\\ \frac{1}{2}{\lambda _i}, \;\;\;\;\;\;\;\;{\lambda _i} \ge 2. \end{array} \right. $

We claim that $(\bar{u}_1, \bar{u}_2, \cdots, \bar{u}_n)$ is a upper solution to the problems (2.1)-(2.3). First, it needs to verify the boundary conditions

$ - \frac{{\partial \bar u_i^{{m_i}}}}{{\partial r}}(1, t) \ge \bar u_{i + 1}^{{p_i}}(1, t), \;\;\;\;i = 1, 2, \cdots, n - 1, $

(2.6)

$ - \frac{{\partial \bar u_n^{{m_n}}}}{{\partial r}}(1, t) \ge \bar u_1^{{p_n}}(1, t). $

(2.7)

Hence, due to that

$ \frac{{\partial \tilde u_i^{{m_i}}}}{{\partial r}} = {(T + t)^{{m_i}{k_i} - {l_i}}}{(h_i^{{m_i}})^\prime }({\xi _i}) \le 0, \;\;\;\;i = 1, 2, \cdots, n, $

we have

$ \begin{array}{l} - \frac{{\partial \bar u_i^{{m_i}}}}{{\partial r}}(1, t) = {\alpha _i}\tilde u_i^{{m_i}}(1, t) - \frac{{\partial \tilde u_i^{{m_i}}}}{{\partial r}}(1, t)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ge {\alpha _i}\tilde u_i^{{m_i}}(1, t) = {\alpha _i}{(T + t)^{{k_i}{m_i}}}{(\frac{{{m_i} - 1}}{{{m_i}}})^{{m_i}/({m_i} - 1)}}. \end{array} $

Similarly, it is clear that

$ \bar u_{i + 1}^{{p_i}}(1, t) = {(T + t)^{{k_{i + 1}}{p_i}}}{(\frac{{{m_{i + 1}} - 1}}{{{m_{i + 1}}}})^{{p_i}/({m_{i + 1}} - 1)}}. $

Thus, the inequalities in (2.6) and (2.7) are valid if

$ {\alpha _i}{(\frac{{{m_i} - 1}}{{{m_i}}})^{{m_i}/({m_i} - 1)}}{(T + t)^{{k_i}{m_i}}} \ge {(\frac{{{m_{i + 1}} - 1}}{{{m_{i + 1}}}})^{{p_i}/({m_{i + 1}} - 1)}}{(T + t)^{{k_{i + 1}}{p_i}}}, $

(2.8)

$ {\alpha _n}{(\frac{{{m_n} - 1}}{{{m_n}}})^{{m_n}/({m_n} - 1)}}{(T + t)^{{k_n}{m_n}}} \ge {(\frac{{{m_1} - 1}}{{{m_1}}})^{{p_n}/({m_1} - 1)}}{(T + t)^{{k_1}{p_n}}}. $

(2.9)

Noticing that $ \prod\limits_{i=1}^n p_i < {\prod\limits_{i=1}^n m_i} $, there exists constant $\gamma_i > 0, i=1, 2, \cdots, n-1$, such that $ \prod\limits_{i=1}^{n-1} \gamma_i > \frac {p_n}{m_n}. $ For any fixed $ k_1 > 0 $ and $ k_{i+1}=k_i \gamma_i=k_1\prod\limits_{j=1}^i \gamma_j, \gamma_i < \frac{m_i}{p_i}, i=1, 2, \cdots, n-1, $ (2.8) and (2.9) hold for the large enough $T$.

Second, we verify that $ (\bar{u}_1, \bar{u}_2, \cdots, \bar{u}_n )$ is the upper solutions to the equations in (2.1). A simple computation yields

$ {(h_i^{{m_i}})^{\prime \prime }}({\xi _i}) + {\xi _i}{h_{i'}}({\xi _i}) - \frac{1}{{{m_i} - 1}}{h_i}({\xi _i}) = 0, \;\;\;\;\;\;\;\;\;\;{\xi _i} > 0, i = 1, 2, \cdots, n. $

Then

$ \begin{align} & \frac{\partial {{{\tilde{u}}}_{i}}t}{\partial t}-\frac{{{\partial }^{2}}\tilde{u}_{i}^{{{m}_{i}}}}{\partial {{r}^{2}}}={{(T+t)}^{{{k}_{i}}-1}}({{k}_{i}}{{h}_{i}}({{\xi }_{i}})-{{l}_{i}}{{\xi }_{i}}h_{i}^{'}({{\xi }_{i}})-(h_{i}^{{{m}_{i}}}{)}''({{\xi }_{i}})) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ={{(T+t)}^{{{k}_{i}}-1}}[({{k}_{i}}-\frac{1}{{{m}_{i}}-1}){{h}_{i}}({{\xi }_{i}})-({{l}_{i}}-1){{\xi }_{i}}h_{i}^{'}({{\xi }_{i}})]. \\ \end{align} $

Recall that $k_{i+1}=k_i \gamma_i=k_1\prod\limits_{j=1}^i \gamma_j, i=1, 2, \cdots, n-1$, we choose $k_1 > 0$ such that

$ {{k}_{1}}>\max \{\frac{1}{{{m}_{1}}-1}, \frac{1}{{{\gamma }_{1}}({{m}_{2}}-1)}, \frac{1}{{{\gamma }_{1}}{{\gamma }_{2}}({{m}_{3}}-1)}, \cdots, \prod\limits_{i=1}^{n-1}{\frac{1}{{{\gamma }_{i}}({{m}_{n}}-1)}}\} $

implies $l_i > 1, k_i > \frac1{m_i-1}$. Combing with that $ h_i'(\xi_i)\le0 $, we get

$ \frac{\partial {{{\tilde{u}}}_{i}}}{\partial t}\ge \frac{{{\partial }^{2}}\tilde{u}_{i}^{{{m}_{i}}}}{\partial {{r}^{2}}}\ge \frac{{{\partial }^{2}}\tilde{u}_{i}^{{{m}_{i}}}}{\partial {{r}^{2}}}+\frac{{{{\tilde{\alpha }}}_{i}}}{r}\frac{\partial \tilde{u}_{i}^{{{m}_{i}}}}{\partial r} $

(2.10)

with

$ {{{\tilde{\alpha }}}_{i}}=\left\{ \begin{align} &2-{{\lambda }_{i}}, \ \ \ \ 1 < {{\lambda }_{i}} < 2, \\ &0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\lambda }_{i}}\ge 2. \\ \end{align} \right. $

Note that

$ \begin{array}{l} \frac{{\partial {{\tilde u}_i}}}{{\partial t}} = {r^{{\alpha _i}/{m_i}}}\frac{{\partial {{\bar u}_i}}}{{\partial t}},\\ \frac{{\partial \tilde u_i^{{m_i}}}}{{\partial r}} = {\alpha _i}{r^{{\alpha _i} - 1}}\bar u_i^{{m_i}} + {r^{{\alpha _i}}}\frac{{\partial \bar u_i^{{m_i}}}}{{\partial r}},\\ \frac{{{\partial ^2}{{\tilde u}^{{m_i}}}}}{{\partial {r^2}}} = {r^{{\alpha _i}}}\frac{{{\partial ^2}\bar u_i^{{m_i}}}}{{\partial {r^2}}} + 2{\alpha _i}{r^{{\alpha _i} - 1}}\frac{{\partial \bar u_i^{{m_i}}}}{{\partial r}} + {\alpha _i}({\alpha _i} - 1){r^{{\alpha _i} - 2}}\bar u_i^{{m_i}}. \end{array} $

Substituting the above equalities into (2.10), we obtain that

$ \begin{array}{l} {r^{{\alpha _i}/{m_i}}}\frac{{\partial {{\bar u}_i}}}{{\partial t}} \ge {r^{{\alpha _i}}}\frac{{{\partial ^2}\bar u_i^{{m_i}}}}{{\partial {r^2}}} + (2{\alpha _i} + {{\tilde \alpha }_i}){r^{{\alpha _i} - 1}}\frac{{\partial \bar u_i^{{m_i}}}}{{\partial r}} + {\alpha _i}({\alpha _i} - 1 + {{\tilde \alpha }_i}){r^{{\alpha _i} - 2}}\bar u_i^{{m_i}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ge {r^{{\alpha _i}}}(\frac{{{\partial ^2}\bar u_i^{{m_i}}}}{{\partial {r^2}}} + \frac{{{\lambda _i}}}{r}\frac{{\partial \bar u_i^{{m_i}}}}{{\partial r}}), \end{array} $

due to that $\alpha_i-1+\tilde\alpha_i\ge0$ and $2\alpha_i+\tilde\alpha_i=\lambda_i$. The above inequality implies that for $r > 1$,

$ \frac{\partial {{{\bar{u}}}_{i}}}{\partial t}\ge {{r}^{{{\alpha }_{i}}(1-1/{{m}_{i}})}}(\frac{{{\partial }^{2}}\bar{u}_{i}^{{{m}_{i}}}}{\partial {{r}^{2}}}+\frac{{{\lambda }_{i}}}{r}\frac{\partial \bar{u}_{i}^{{{m}_{i}}}}{\partial r})\ge \frac{{{\partial }^{2}}\bar{u}_{i}^{{{m}_{i}}}}{\partial {{r}^{2}}}+\frac{{{\lambda }_{i}}}{r}\frac{\partial \bar{u}_{i}^{{{m}_{i}}}}{\partial r}. $

Finally, we verify the initial conditions that

$ {{u}_{i}}(r, 0)\ge {{u}_{0, i}}(r), ~~~~i=1, 2, \cdots, n. $

(2.11)

Denote

$ {M_i} = \mathop {\max }\limits_{r > 1} {u_{0,i}}(r),\;\;\;\;{\rm{supp}}{u_{0,i}} = [1,{R_i}],\;\;\;\;i = 1,2, \cdots ,n. $

Then (2.11) holds provided with

$ {{\bar{u}}_{i}}({{R}_{i}}, 0)\ge {{M}_{i}}, \ \ \ \ i=1, 2, \cdots, n, $

(2.12)

since that $\bar{u}_i(r, 0)$ are decreasing with respect to $r$. In fact, we can choose $T$ to be large enough to satisfy that

$ R_{i}^{-{{\alpha }_{i}}/{{m}_{i}}}{{T}^{{{k}_{i}}}}{{(\frac{{{m}_{i}}-1}{{{m}_{i}}}{{(1-\frac{{{R}_{i}}-1}{{{T}^{{{l}_{i}}}}})}_{+}})}^{1/({{m}_{i}}-1)}}\ge {{M}_{i}}, \ \ \ \ i=1, 2, \cdots, n. $

From the above, for large $T$ satisfying (2.8) and (2.12), it is seen that $(\bar{u}_1, \bar{u}_2, \cdots, \bar{u}_n)$ is a supersolution of the systems (2.1)-(2.3). Therefore, we have the solution of the problems (2.1)-(2.3) exists globally by the comparison principle. The proof is completed.

Proof of Proposition 2.2  The proposition is proved by constructing a kind of lower blow-up lower solution of the systems (2.1)-(2.3). For $r > 1, 0 < t < T$, set

$ {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u}}_{i}}(r, t)={{(T-t)}^{-{{\mu }_{i}}}}{{f}_{i}}(\eta ), \ \ \ \ \eta =(r-1)(T+t), \ \ \ \ i=1, 2, \cdots, n, $

where $T > 0, \mu_1 > 0$ is a given constant,

$ \begin{align} &{{\mu }_{1}} > \max \{\frac{1}{{{m}_{1}}-1}, \prod\limits_{j=1}^{i}{\frac{{{p}_{j}}}{{{m}_{j}}}}\frac{1}{({{m}_{i+1}}-1)}\}, \ \ \ \ i=1, 2, \cdots, n-1, \\ &-{{\mu }_{i}}{{m}_{i}}+{{\mu }_{i+1}}{{p}_{i}}=0, \ \ \ \ i=1, 2, \cdots, n-1. \\ \end{align} $

Assume that $f_i$ satisfy $f_i\ge0, f_i'\le0, (f_i^{m_i})''\ge0, i=1, 2, \cdots, n$, by a direct calculation, we can see that $(\underline{u}_1, \underline{u}_2, \cdots, \underline{u}_n)$ with $\underline{u}_i(r, 0)\le u_{0, i}(r)$ is a subsolution to the system (2.1)-(2.3), if the following inequalities hold

$ {T^2}{(f_i^{{m_i}})^{\prime \prime }} + 2T\frac{{{\lambda _i}}}{r}{(f_i^{{m_i}})^\prime } - {\mu _i}{(T - t)^{{\mu _i}{m_i} - {\mu _i} - 1}}{f_i}(\eta ) \ge 0,\;\;\;\;i = 1,2, \cdots ,n,{\rm{ }} $

(2.13)

$ - 2T{(T - t)^{ - {\mu _i}{m_i} + {\mu _{i + 1}}{p_i}}}{(f_i^{{m_i}})^\prime }(0) \le f_{i + 1}^{{p_i}}(0),\;\;\;\;i = 1,2, \cdots ,n. $

(2.14)

Note that $r > 1$ and

$ \begin{align} &{{\mu }_{i}}{{m}_{i}}-{{\mu }_{i}}-1 > 0, \ \ \ \ i=1, 2, \cdots, n, \\ &-{{\mu }_{i}}{{m}_{i}}+{{\mu }_{i+1}}{{p}_{i}}=0, ~\ \ \ \ i=1, 2, \cdots, n-1. \\ \end{align} $

Thus (2.13) and (2.14) hold if

$ {{T}^{2}}{{(f_{i}^{{{m}_{i}}})}^{\prime \prime }}+2T{{\lambda }_{i}}{{(f_{i}^{{{m}_{i}}})}^{\prime }}-{{\mu }_{i}}{{T}^{{{\mu }_{i}}{{m}_{i}}-{{\mu }_{i}}-1}}{{f}_{i}}(\eta )\ge 0, ~~~~i=1, 2, \cdots, n, $

(2.15)

$ -2T{{(f_{i}^{{{m}_{i}}})}^{\prime }}(0)\le f_{i+1}^{{{p}_{i}}}(0), ~~~~i=1, 2, \cdots, n $

(2.16)

with $f_{n+1}=f_1$. For any $i\in\{1, 2, \cdots, n\}, $ namely,

$ {{f}_{i}}(\eta )={{A}_{i}}({{B}_{i}}-\eta )_{+}^{1/({{m}_{i}}-1)}, \ \ \ \ \eta > 0, $

(2.17)

where $A_i$ are positive constants to be determined, and

$ {B_i} = \min \{ \frac{T}{{4{\lambda _i}({m_i}-1)}}, \frac{{{m_i}{T^{3 + {\mu _i}-{\mu _i}{m_i}}}}}{{2{\mu _i}{{({m_i}-1)}^2}}}\} . $

In the following, we verify that the above $f_i$ satisfy (2.15) and (2.16). Substituting (2.17) into (2.15) yields that

$ \begin{array}{l} A_i^{{m_i}}\frac{{{m_i}}}{{{{({m_i}-1)}^2}}}{T^2}-A_i^{{m_i}}{\lambda _i}\frac{{{m_i}}}{{{m_i}-1}}(2T){({B_i} - \eta )_ + } - {A_i}{\mu _i}{T^{{\mu _i}{m_i} - {\mu _i} - 1}}{({B_i} - \eta )_ + } \ge 0, \\ i = 1, 2, \cdots, n, \end{array} $

which can be obtained by the following

$ \begin{array}{l} A_i^{{m_i}}\frac{{{m_i}}}{{{{({m_i} - 1)}^2}}}{T^2} \ge \frac{1}{2}A_i^{{m_i}}{\lambda _i}\frac{{{m_i}}}{{{m_i} - 1}}(2T){B_i}, \\ A_i^{{m_i}}\frac{{{m_i}}}{{{{({m_i} - 1)}^2}}}{T^2} \ge \frac{1}{2}{A_i}{\mu _i}{T^{{\mu _i}{m_i} - {\mu _i} - 1}}{B_i}. \end{array} $

The choice of $B_i$ makes us to conclude that the above inequalities hold for $A_i > 1$. This indicated that $f_i$ satisfy (2.15).

Next, substitute (2.17) into (2.16), we have

$ 2T\frac{{{m_i}}}{{{m_i} - 1}}B_i^{1/({m_i} - 1)}A_i^{{m_i}} \le B_{i + 1}^{{p_i}/({m_{i + 1}} - 1)}A_{i + 1}^{{p_i}}, \;\;\;\;i = 1, 2, \cdots, n $

(2.18)

with $A_{n+1}=A_1$. Since $ \prod\limits_{i=1}^n p_i > {\prod\limits_{i=1}^n m_i}$, there exists a constant $k > 1$, such that

$ {k^{n - 1}} < \frac{{\prod\limits_{i = 1}^n {{p_i}} }}{{\prod\limits_{i = 1}^n {{m_i}} }}. $

(2.19)

For $ i\in \{1, 2, \cdots, n-1\}, $ set $A_{i+1}=A_i^{k\sigma_i}$ with $\sigma_i=\frac{m_i}{p_i}$. Then （2.18） can also be written as

$ 2T\frac{{{m_i}}}{{{m_i} - 1}}B_i^{1/({m_i} - 1)}A_i^{{m_i}} \le B_{i + 1}^{{p_i}/({m_{i + 1}} - 1)}A_i^{k{m_i}}, \;\;\;\;i = 1, 2, \cdots, n. $

(2.20)

Choose $A_i$ large enough to satisfy (2.20), then we get (2.16).

Therefore, the solution $(u_1, u_2, \cdots, u_n)$ with $u_{0, i}(r)\ge\underline{u}_i(r, 0)~~ (i=1, 2, \cdots, n)$ of the problems (2.1)-(2.3) blows up in a finite time. The proof is completed.

Proof of Proposition 2.3  The proof is completed by constructing the following global upper solution

$ {\bar u_i}(r, t) = {({B_i}{r^{1-{\lambda _i}}})^{1/{m_i}}}, \;\;\;\;r > 1, t > 0, \;\;\;\;i = 1, 2, \cdots, n, $

where

$ {B_1} = \prod\limits_{i = 1}^n {{{({\lambda _i}-1)}^{{\mu _i}}}}, \;\;\;\;{B_{i + 1}} = {({B_i}({\lambda _i}-1))^{{m_{i + 1}}/{p_i}}}, \;\;\;\;i = 1, 2, \cdots, n-1 $

with $\mu_i=\frac{m_1\prod\limits_{j=1}^{i-1} p_j \prod\limits_{j=i+1}^n m_j}{\prod\limits_{j=1}^n p_j- \prod\limits_{j=1}^n m_j}$. It can easily be checked that

$ \begin{array}{l} \frac{{\partial {{\bar u}_i}}}{{\partial t}} - \frac{{{\partial ^2}\bar u_i^{{m_i}}}}{{\partial {r^2}}} - \frac{{{\lambda _i}}}{r}\frac{{\partial \bar u_i^{{m_i}}}}{{\partial r}} = 0,\;\;\;\;r > 1,t > 0,i = 1,2, \cdots ,n,\\ - \frac{{\partial \bar u_i^{{m_i}}}}{{\partial r}}(1,t) = \bar u_{i + 1}^{{p_i}}(1,t),\;\;\;\;t > 0,i = 1,2, \cdots ,n \end{array} $

with $\bar{u}_{n+1}=\bar{u}_1.$ Furthermore, by the comparison principle, if the initial data

$ ({u_{0, 1}}, {u_{0, 2}} \ldots, {u_{0, n}}) $

is small enough that

$ {u_{0,i}}(r) \le {\bar u_{0,i}}(r,0),\;\;\;\;i = 1,2, \ldots ,n, $

the solutions of the problems(2.1)-(2.3) exist globally in time.

Now, we prove the main result for the systems (2.1)-(2.3), i.e., Theorem 2.1.

Proof of Theorem 2.1  Noticing that the functions $u_{0, 1}(x), u_{0, 2}(x), \cdots, u_{0, n}(x)$ have compact supports, we can choose $n$ bounded, radially symmetrical functions, denoted by $u_i(x)=u_i(|x|)\ge u_{0, i}(x), i=1, 2, \cdots, n $. By using Proposition 2.1 and the comparison principle, we can obtain the global existence of solutions for the systems (1.1)-(1.3). However, for the large and radially symmetric functions $\underline u_1(|x|, 0), \underline u_2(|x|, 0), \cdots, \underline u_n(|x|, 0)$ defined in the proof of Proposition 2.2, if $(u_{0, 1}, u_{0, 2}, \cdots, u_{0, n})$ is large enough such that $u_{0, i}(x)\ge \underline u_i(|x|, 0) $, $ i=1, 2, \cdots, n, $ then the solutions of the systems (1.1)-(1.3) with $ \prod\limits_{i=1}^n p_i>{\prod\limits_{i=1}^n m_i}$ blow up by the comparison principle and Proposition 2.2. This clarifies that the critical global existence curve is $ \prod\limits_{i=1}^n p_i={\prod\limits_{i=1}^n m_i}$ for the system (1.1)-(1.3).

On the other hand, using the comparison principle again, we conclude that the solution $(u_1, u_2, \cdots, u_n)$ of (1.1)-(1.3) with

$ {{u}_{0, i}}(x)\le {{({{B}_{i}}|x{{|}^{2-N}})}^{1/{{m}_{i}}}}, \ \ \ \ x\in {{\mathbb{R}}^{\mathbb{N}}}\backslash {{B}_{1}}(0), i=1, 2, \cdots, n, $

(2.21)

where

$ {{B}_{1}}=\prod\limits_{i=1}^{n}{{{(N-2)}^{{{\mu }_{i}}}}}, \quad {{B}_{i+1}}={{({{B}_{i}}(N-2))}^{{{m}_{i+1}}/{{p}_{i}}}}, ~~~~i=1, 2, \cdots, n-1, $

with $\mu_i=\frac{m_1\prod\limits_{j=1}^{i-1} p_j \prod\limits_{j=i+1}^n m_j}{\prod\limits_{j=1}^n p_j- \prod\limits_{j=1}^n m_j}$, exists globally for $\prod\limits_{i=1}^n p_i > {\prod\limits_{i=1}^n m_i}$ by Proposition 2.3. This combined with Proposition 2.2} indicates that the critical Fujita curve $\prod\limits_{i=1}^n p_i={\prod\limits_{i=1}^n m_i}$ for the systems (1.1)-(1.3).

Proof of Theorem 2.2  By virtue of the same discussion in the proof of Theorem 2.1, if we prove this theorem by taking $\lambda_i=\tilde\lambda_i+N-1$ in the systems (2.1)-(2.3), and replacing (2.21) with

$ {{u}_{0, i}}(x)\le {{({{B}_{i}}|x{{|}^{-{{{\tilde{\lambda }}}_{i}}+2-N}})}^{1/{{m}_{i}}}}, ~~~~i=1, 2, \cdots, n, $

where

$ {{B}_{1}}=\prod\limits_{i=1}^{n}{{{({{{\tilde{\lambda }}}_{i}}+N-2)}^{{{\mu }_{i}}}}}, \quad {{B}_{i+1}}={{({{B}_{i}}({{\tilde{\lambda }}_{i}}+N-2))}^{{{m}_{i+1}}/{{p}_{i}}}}, ~~~~i=1, 2, \cdots, n-1, $

with $\mu_i=\frac{m_1\prod\limits_{j=1}^{i-1} p_j \prod\limits_{j=i+1}^n m_j}{\prod\limits_{j=1}^n p_j- \prod\limits_{j=1}^n m_j}$. The proof is completed.