本文研究如下具有范数型源的反应扩散方程组解的爆破性质:

$\begin{aligned} &{u_t} = \Delta {u^m} + a\left\| {{u^{{p_1}}}{v^{{q_1}}}} \right\|_\alpha ^{{r_1}}, x \in \Omega, t > 0, \\ &{v_t} = \Delta {v^n} + b\left\| {{v^{{p_2}}}{w^{{q_2}}}} \right\|_\beta ^{{r_2}}, x \in \Omega, t > 0, \\ &{w_t} = \Delta {w^h} + c\left\| {{w^{{p_3}}}{u^{{q_3}}}} \right\|_\gamma ^{{r_3}}, x \in \Omega, t > 0, \\ &u(x, t) = v(x, t)= w(x, t) = 0, x \in \partial \Omega, t > 0, \\ &u(x, 0) = {u_0}(x), v(x, 0) = {v_0}(x), w(x, 0)= {w_0}(x), x \in \Omega, \end{aligned}$

(1.1)

其中 $\Omega $是${R^N}(N \ge 1)$中一个具有光滑边界$\partial \Omega$的有界区域, 常数$m, n, h > 1, a, b, c > 0, $ $\alpha, \beta, \gamma \ge1$, ${r_i} > 0$及${p_i} \ge 0, {q_i} > 0$ $(i = 1, 2, 3)$, 初值函数${u_0}(x), {v_0}(x), {w_0}(x)$ 是$\Omega$上的非负有界连续函数, 这里的范数

$\left\| . \right\|_\alpha ^\alpha = \displaystyle\int_\Omega {{{\left| . \right|}^\alpha }dx} .$

问题(1.1)是很值得研究的, 因为它能够描述不同的物理现象, 例如生物种群的研究中, 可以用$u$、$v$及$w$来刻画同一种环境下, 三种不同的物种在生长过程的密度(见文[1-4]及相应的文献);它也能刻画混合物燃烧后的热传播过程和化学反应过程.

在过去的几十年里, 对于带有局部源和没有局部源的反应扩散方程组得到了很多数学工作者的研究.例如, 在文[5] 中, Deng研究了下列带齐次Dirichlet边界条件的初边值问题:

$\begin{aligned} {u_t} = \Delta {u^m} + {u^\alpha }{v^p}, {v_t} = \Delta {v^n} + {u^q}{v^\beta }, {\kern 1pt} {\kern 1pt} {\kern 1pt} x \in \Omega , t > 0{\kern 1pt} {\kern 1pt} {\kern 1pt}, \end{aligned}$

(1.2)

他们证明了如果$m > \alpha, n > \beta $ 及$pq ＜ (m - \alpha )(n -\beta )$, 则问题(1.2)的每一个非负解整体存在, 而如果$m ＜ \alpha $或$n ＜ \beta $或 $pq > (m - \alpha )(n - \beta)$, 则该问题既存在整体非负解, 也有爆破的非负解.

在相同的初边值条件下, Li等人在文[6] 中考虑了下列问题:

$\begin{aligned} {u_t} = \Delta u + {u^\alpha }({x_0}, t){v^p}({x_0}, t), {\kern 1pt} {\kern 1pt} {v_t} = \Delta v + {u^q}({x_0}, t){v^\beta }({x_0}, t), {\kern 1pt} {\kern 1pt} x \in \Omega, t > 0{\kern 1pt} , {\kern 1pt} {\kern 1pt} {\kern 1pt} \end{aligned}$

(1.3)

作者给出了上述问题解$(u, v)$同时爆破的充分必要条件, 并得到了一致爆破模式和爆破率的精确估计.

对于具有非局部源项的问题, 其解的整体存在和有限时刻爆破行为也得到了相应的研究. 如文[7]中, Deng 等人研究了

$\begin{aligned} {u_t} = \Delta {u^m} + a\left\| v \right\|_\alpha ^p, {\kern 1pt} {\kern 1pt} {\kern 1pt} {v_t} = \Delta {v^n} + b\left\| u \right\|_\beta ^q, x \in \Omega, t > 0, {\kern 1pt} {\kern 1pt} {\kern 1pt} \end{aligned}$

(1.4)

在齐次Dirichlet边界条件下, 他们证明了当$pq ＜ mn$ 时, 每一个非负解是整体存在的; 如果$pq > mn$, 整体解与爆破解同时存在;他们的结果表明${p_c} = pq - mn$是问题(1.4)的临界指标.

近年来, Kong等人在文[8]中研究了如下的方程组

$\begin{aligned} {u_t} = \Delta u + \int_\Omega {{u^m}(x, t){v^n}(x, t)dx}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {v_t} = \Delta v + \int_\Omega {{v^p}(x, t){u^q}(x, t)dx}, \end{aligned}$

(1.5)

作者建立了相应问题的解是整体存在和有限时刻爆破的条件.

关于非局部退化抛物方程的研究, 读者还可以查阅其他的相关文献(如文[9-12])．

受上述文献的启发, 以构造的技巧, 用更加简单的方法推广了前述文献的研究结果, 讨论问题(1.1)解的整体存在与爆破准则, 并得到如下结果.

定理 1.1  若下列条件之一成立, 那么问题(1.1)的解整体存在.

(1) $ m > {p_1}{r_1}, n > {p_2}{r_2}, h > {p_3}{r_3}$, 且$({r_1}{q_1})({r_2}{q_2})({r_3}{q_3}) ＜ (m - {r_1}{p_1})(n - {r_2}{p_2})(h - {r_3}{p_3})$;

(2) $ m \le {p_1}{r_1}$或$n \le {p_2}{r_2}$或$h \le {p_3}{r_3}$或

$m > {p_1}{r_1}, n > {p_2}{r_2}, h > {p_3}{r_3}, ({r_1}{q_1})({r_2}{q_2})({r_3}{q_3}) > (m - {r_1}{p_1})(n -{r_2}{p_2})(h - {r_3}{p_3}), $ 且初值${u_0}(x), {v_0}(x), {w_0}(x)$充分小;

(3) $ m > {p_1}{r_1}, n > {p_2}{r_2}, h > {p_3}{r_3}$, $({r_1}{q_1})({r_2}{q_2})({r_3}{q_3}) = (m - {r_1}{p_1})(n -{r_2}{p_2})(h - {r_3}{p_3})$, 且$a, b, c$充分小.

定理 1.2  若下列条件之一成立, 那么问题(1.1)的解在有限时刻爆破.

(1) $ m > {p_1}{r_1}, n > {p_2}{r_2}, h >{p_3}{r_3}$,

$({r_1}{q_1})({r_2}{q_2})({r_3}{q_3}) > (m - {r_1}{p_1})(n - {r_2}{p_2})(h - {r_3}{p_3}), $

且初值${u_0}(x), {v_0}(x), {w_0}(x)$充分大;

(2) $ m \le {p_1}{r_1}$或$n \le {p_2}{r_2}$或$h \le {p_3}{r_3}$, 且初值 ${u_0}(x), {v_0}(x), {w_0}(x)$充分大;

(3) $ m > {p_1}{r_1}, n > {p_2}{r_2}, h >{p_3}{r_3}$,

$({r_1}{q_1})({r_2}{q_2})({r_3}{q_3}) = (m - {r_1}{p_1})(n - {r_2}{p_2})(h - {r_3}{p_3}), $

并且区域$\Omega$包含一个充分大的球, 而初值${u_0}(x), {v_0}(x), {w_0}(x)$ 在$\Omega$ 上为正的连续函数.

在这里给出问题(1.1)的弱解的定义．对任意的$0 ＜ T ＜ \infty$, 设${Q_T} = \Omega \times (0, T)$, ${S_T} = \partial \Omega\times (0, T)$.

定义 2.1  称函数组$(u(x, t), v(x, t), w(x, t))$是问题(1.1)在${Q_T}$上的弱下解, 如果下列条件都成立:

(1) $(u(x, t), v(x, t), w(x, t)) \in {L^\infty }({Q_T})$;

(2) $u(x, t), v(x, t), w(x, t) \le 0, (x, t) \in {S_T}$, $u(x, 0) \le{u_0}(x), v(x, 0) \le {v_0}(x), w(x, 0) \le {w_0}(x), x \in \Omega $;

(3)

$\begin{array}{l} \displaystyle\int_\Omega {(u(x, t){\psi _1}(x, t) - {u_0}(x){\psi _1}(x, 0))dx} \le \int_0^t {\int_\Omega {(u{\psi _{1s}} + {u^m}\Delta {\psi _1} + a\left\| {{u^{{p_1}}}{v^{{q_1}}}} \right\|_\alpha ^{{r_1}}{\psi _1})dx} } ds, \\ \displaystyle\int_\Omega {(v(x, t){\psi _2}(x, t) - {v_0}(x){\psi _2}(x, 0))dx} \le \int_0^t {\int_\Omega {(v{\psi _{2s}} + {v^n}\Delta {\psi _2} + b\left\| {{v^{{p_2}}}{w^{{q_2}}}} \right\|_\beta ^{{r_2}}{\psi _2})dx} } ds, \\ \displaystyle\int_\Omega {(w(x, t){\psi _3}(x, t) - {w_0}(x){\psi _3}(x, 0))dx} \le \int_0^t {\int_\Omega {(w{\psi _{3s}} + {w^n}\Delta {\psi _3} + c\left\| {{w^{{p_3}}}{u^{{q_3}}}} \right\|_\gamma ^{{r_3}}{\psi _3})dx} } ds, \\ \end{array}$

其中$t \in [0, T]$, 且${\psi _1}, {\psi _2}, {\psi _3}$属于函数族

$\Psi \equiv \{ \psi \in C(\overline {{Q_T}} );{\psi _t}, \Delta \psi \in C({Q_T}) \cap {L^2}({Q_T});\psi \ge 0, \psi (x, t)\left| {_{\partial \Omega \times (0, T)}} \right. = 0\}.$

类似的, 改变上式中所有不等号的方向可以得到弱上解的定义.称$(u(x, t), v(x, t), w(x, t))$ 是问题(1.1)的弱解, 如果它既是(1.1)的弱下解, 又是弱上解. 显然, 问题(1.1)的每个非负古典解都是定义 2.1意义下的弱解.

类似于文[8]那样, 能论证问题(1.1)弱解的局部存在性和相应的强极值原理, 这里省略了, 而下列的比较原理成立.

引理 2.2 (比较原理)  设$(\overline u, \overline v, \overline w )$与 $(\underline u, \underline v, \underline w)$分别是问题(1.1)在 ${\overline Q _T}$上的非负上、下解.如果$({\underline u _0}, {\underline v _0}, {\underline w _0}) \le(\overline {{u_0}}, \overline {{v_0}}, \overline {{w_0}} )$或者存在常数$\delta $, 使得

$\begin{aligned} \underline u, \underline v, \underline w \ge \delta > 0 \mbox{或} \overline u, \overline v, \overline w \ge \delta > 0 \end{aligned}$

(2.1)

成立. 那么 在${Q_T}$ 上, $(\underline u, \underline v, \underline w ) \le (\overline u, \overline v, \overline w )$ .

由比较原理, 只需要对任意 $T > 0$, 构造出有界的正弱上解即可.

设$\varphi (x)$ 是下列椭圆问题的解

$\begin{aligned} - \Delta \varphi (x) = 1, x \in \Omega; \varphi (x) = 1, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x \in \partial \Omega. \end{aligned}$

(3.1)

记 $C = \mathop {\max }\limits_{x \in \Omega } \varphi (x)$, 则 $1\le \varphi (x) \le C$. 定义$\overline u, \overline v, \overline w$ 为如下形式

$\begin{aligned} \overline u = {K^{{l_1}}}{\varphi ^{\frac{1}{m}}}(x), {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \overline v = {K^{{l_2}}}{\varphi ^{\frac{1}{n}}}(x), {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \overline \omega = {K^{{l_3}}}{\varphi ^{\frac{1}{h}}}(x), \end{aligned}$

(3.2)

其中${l_1}, {l_2}, {l_3} > 0$, $K > 0$ 将在后面被确定. 显然, 对于任意$T > 0$, $(\overline u, \overline v, \overline w )$是有界的, 且 $\overline u \ge {K^{{l_1}}} > 0, \overline v \ge{K^{{l_2}}} > 0, \overline w \ge {K^{{l_3}}} > 0$.通过一系列的计算, 可得

$\begin{aligned} {\overline u _t} - \Delta {\overline u ^m} - a\left\| {{{\overline u }^{{p_1}}}{{\overline v }^{{q_1}}}} \right\|_\alpha ^{{r_1}} &= {K^{m{l_1}}} - a\left\| {{K^{{l_1}}}^{{p_1}}{\varphi ^{\frac{{{p_1}}}{m}}}{K^{{l_2}{q_1}}}{\varphi ^{\frac{{{q_1}}}{n}}}} \right\|_\alpha ^{{r_1}} \\ &\ge {K^{m{l_1}}} - a{K^{{r_1}({l_1}{p_1} + {l_2}{q_1})}}{C^{{r_1}(\frac{{{p_1}}}{m} + \frac{{{q_1}}}{n})}}{\left| \Omega \right|^{\frac{{{r_1}}}{\alpha }}}, \\ {\overline v _t} - \Delta {\overline v ^n} - b\left\| {{{\overline v }^{{p_2}}}{{\overline w }^{{q_2}}}} \right\|_\beta ^{{r_2}} &= {K^{n{l_2}}} - b\left\| {{K^{{l_2}{p_2}}}{\varphi ^{\frac{{{p_2}}}{n}}}{K^{{l_3}{q_2}}}{\varphi ^{\frac{{{q_2}}}{h}}}} \right\|_\beta ^{{r_2}} \\ &\ge {K^{n{l_2}}} - b{K^{{r_2}({l_2}{p_2} + {l_3}{q_2})}}{C^{{r_2}(\frac{{{p_2}}}{n} + \frac{{{q_2}}}{h})}}{\left| \Omega \right|^{\frac{{{r_2}}}{\beta }}}, \\ {\overline w _t} - \Delta {\overline w ^h} - c\left\| {{{\overline w }^{{p_3}}}{{\overline u }^{{q_3}}}} \right\|_\gamma ^{{r_3}} &= {K^{h{l_3}}} - c\left\| {{K^{{l_3}}}^{{p_3}}{\varphi ^{\frac{{{p_3}}}{h}}}{K^{{l_1}{q_3}}}{\varphi ^{\frac{{{q_3}}}{m}}}} \right\|_\gamma ^{{r_3}} \\ &\ge {K^{h{l_3}}} - c{K^{{r_3}({l_3}{p_3} + {l_1}{q_3})}}{C^{{r_3}(\frac{{{p_3}}}{h} + \frac{{{q_3}}}{m})}}{\left| \Omega \right|^{\frac{{{r_3}}}{\gamma }}}. \\ \end{aligned}$

(3.3)

记

$\begin{array}{l} {C_1} = {(a{C^{{r_1}(\frac{{{p_1}}}{m} + \frac{{{q_1}}}{n})}}{\left| \Omega \right|^{\frac{{{r_1}}}{\alpha }}})^{1/(m{l_1} - {l_1}{r_1}{p_1} - {l_2}{r_1}{q_1})}},\\ {C_2} = {(b{C^{{r_2}(\frac{{{p_2}}}{n} + \frac{{{q_2}}}{h})}}{\left| \Omega \right|^{\frac{{{r_2}}}{\beta }}})^{1/(n{l_2} - {l_2}{r_2}{p_2} - {l_3}{r_2}{q_2})}},\\ {C_3} = {(c{C^{{r_3}(\frac{{{p_3}}}{h} + \frac{{{q_3}}}{m})}}{\left| \Omega \right|^{\frac{{{r_3}}}{\gamma }}})^{1/(h{l_3} - {l_3}{r_3}{p_3} - {l_1}{r_3}{q_3})}}. \end{array}$

(3.4)

(1) 若$m > {p_1}{r_1}, n > {p_2}{r_2}, h > {p_3}{r_3}$, $({r_1}{q_1})({r_2}{q_2})({r_3}{q_3}) ＜ (m - {r_1}{p_1})(n -{r_2}{p_2})(h - {r_3}{p_3}), $ 那么存在正常数${l_1}, {l_2}$ 使得

$\begin{aligned} \frac{{{r_1}{q_1}}}{{m - {r_1}{p_1}}} ＜ \frac{{{l_1}}}{{{l_2}}} ＜ \frac{{(h - {r_3}{p_3})(n - {r_2}{p_2})}}{{({r_2}{q_2})({r_3}{q_3})}}. \end{aligned}$

(3.5)

同时, 也存在正数${l_3}$ 使得

$\begin{aligned} \frac{{{l_3}}}{{{l_2}}} ＜ \frac{{n - {r_2}{p_2}}}{{{r_2}{q_2}}}, \frac{{{l_1}}}{{{l_3}}} ＜ \frac{{h - {r_3}{p_3}}}{{{r_3}{q_3}}}. \end{aligned}$

(3.6)

从而

$\begin{aligned} {r_1}{q_1}{l_2} ＜ (m - {r_1}{p_1}){l_1}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {r_2}{q_2}{l_3} ＜ (n - {r_2}{p_2}){l_2}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {r_3}{q_3}{l_1} ＜ (h - {r_3}{p_3}){l_3}. \end{aligned}$

(3.7)

根据 (3.7)式, 能选择充分大的$K$ 使得 $K \ge \max \{{C_1}, {C_2}, {C_3}\} $, 且

$\begin{aligned} {K^{{l_1}}} \ge {u_0}(x), {\kern 1pt} {\kern 1pt} {\kern 1pt} {K^{{l_2}}} \ge {v_0}(x), {\kern 1pt} {\kern 1pt} {\kern 1pt} {K^{{l_3}}} \ge {w_0}(x). \end{aligned}$

(3.8)

根据 (3.3)--(3.8)式, 那么由(3.2)式定义的 $(\overline u, \overline v, \overline w )$ 是问题(1.1)的正弱上解, 因此由比较原理知$(u, v, w) \le (\overline u, \overline v, \overline w )$, 进而意味着$(u, v, w)$ 整体存在.

(2) 若 $m \le {p_1}{r_1}$或 $n \le {p_2}{r_2}$ 或$h \le {p_3}{r_3}$或

$m > {p_1}{r_1}, n > {p_2}{r_2}, h >{p_3}{r_3}, ({r_1}{q_1})({r_2}{q_2})({r_{}}{q_3}) > (m -{r_1}{p_1})(n - {r_2}{p_2})(h -{r_3}{p_3}), $那么存在三个正常数${l_1}, {l_2}, {l_3}$ 使得

$\begin{aligned} {r_1}{q_1}{l_2} > (m - {r_1}{p_1}){l_1}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {r_2}{q_2}{l_3} > (n - {r_2}{p_2}){l_2}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {r_3}{q_3}{l_1} > (h - {r_3}{p_3}){l_3}. \end{aligned}$

(3.9)

那么能选择充分小的 $K$, 使得$K \le \min \{{C_1}, {C_2}, {C_3}\}$.进一步, 假设 ${u_0}(x), {v_0}(x), {w_0}(x)$也是充分小, 并满足式(3.8). 同样的, 由(3.2)式定义的 $(\overline u, \overline v, \overline w )$ 也是问题(1.1)的弱上解, 因此$(u, v, w)$ 整体存在.

(3) 若$m > {p_1}{r_1}, n > {p_2}{r_2}, h > {p_3}{r_3}, $

$({r_1}{q_1})({r_2}{q_2})({r_3}{q_3}) = (m - {r_1}{p_1})(n - {r_2}{p_2})(h - {r_3}{p_3}), $

那么存在正常数 ${l_1}, {l_2}, {l_3}$使得

$\begin{aligned} \frac{{{r_1}{q_1}}}{{m - {r_1}{p_1}}} = \frac{{{l_1}}}{{{l_2}}}, \frac{{n - {r_2}{p_2}}}{{{r_2}{q_2}}} = \frac{{{l_3}}}{{{l_2}}}, \frac{{h - {r_3}{p_3}}}{{{r_3}{q_3}}} = {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{{l_1}}}{{{l_3}}}. \end{aligned}$

(3.10)

从而

$\begin{aligned} {r_1}{q_1}{l_2} = (m - {r_1}{p_1}){l_1}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {r_2}{q_2}{l_3} = (n - {r_2}{p_2}){l_2}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {r_3}{q_3}{l_1} = (h - {r_3}{p_3}){l_3}. \end{aligned}$

(3.11)

假定$a, b, c$ 充分小, 且满足

$\begin{eqnarray} a &\le& {({C^{{r_1}(\frac{{{p_1}}}{m} + \frac{{{q_1}}}{n})}}{\left| \Omega \right|^{\frac{{{r_1}}}{\alpha }}})^{ - 1}}, \nonumber\\ b &\le& {({C^{{r_2}(\frac{{{p_2}}}{n} + \frac{{{q_2}}}{h})}}{\left| \Omega \right|^{\frac{{{r_2}}}{\beta }}})^{ - 1}}, \nonumber\\ c &\le& {({C^{{r_3}(\frac{{{p_3}}}{h} + \frac{{{q_3}}}{m})}}{\left| \Omega \right|^{\frac{{{r_3}}}{\gamma }}})^{ - 1}}. \end{eqnarray}$

(3.12)

另外, 可以选择充分大的 $K$ 使得(3.8)式成立. 那么根据 (3.3), (3.8), (3.10)--(3.12)式, 由(3.2)式定义的 $(\overline u, \overline v, \overline w )$ 也是问题(1.1)的弱上解. 根据比较原理 $(u, v, w) \le(\overline u, \overline v, \overline w )$. 因此可以得到 $(u, v, w)$是整体存在的. 定理 1证毕.

在本节将通过构造问题(1.1)的爆破弱下解来证明定理 2.

设 $\lambda $和 $\phi(x)$是下列特征问题的第一特征值和相应的特征函数

$\begin{aligned} - \Delta \phi (x) = \lambda \phi (x), {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x \in \Omega ;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \phi (x) = 0, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x \in \partial \Omega . \end{aligned}$

(4.1)

显然$ \phi (x)$在 $ \Omega $上可正则化成 $ \phi (x) > 0$, 且 $\mathop {\max }\limits_{\overline \Omega } \phi (x) = 1$.在$\partial \Omega $上, $\partial \phi {\rm{ }}/\partial n{\rm{ ＜ }}0$, 其中 $n$为 $\partial \Omega $的外法线方向. 记 $d = \mathop {\min }\limits_{x \in \overline\Omega } \phi (x).$

定义$\underline u (x, t), \underline v (x, t), \underline w (x, t)$如下

$\begin{aligned} \underline u (x, t) = {s^{{l_1}}}(t){\phi ^{\frac{1}{m}}}(x), {\kern 1pt} {\kern 1pt} {\kern 1pt} \underline v (x, t) = {s^{{l_2}}}(t){\phi ^{\frac{1}{n}}}(x), {\kern 1pt} {\kern 1pt} {\kern 1pt} \underline w (x, t) = {s^{{l_3}}}(t){\phi ^{\frac{1}{h}}}(x), \end{aligned}$

(4.2)

其中$s(t)$为初值问题

$\begin{aligned} s'(t) = k{s^\theta }, s(0) = {s_0} > 0 \end{aligned}$

(4.3)

的解, 这里$k, {s_0} > 0, \theta > 1$ 待定. 显然$s(t) \ge {s_0}$, 且在有限时刻无界, 而且

$\begin{aligned} \underline u \ge {d^{{l_1}}}s_0^{^{\frac{1}{m}}} > 0, \underline v \ge {d^{{l_2}}}s_0^{^{\frac{1}{n}}} > 0, \underline w \ge {d^{{l_3}}}s_0^{^{\frac{1}{h}}} > 0. \end{aligned}$

(4.4)

直接计算知

$\begin{eqnarray} \Delta {\underline u ^m} + a\left\| {{{\underline u }^{{p_1}}}{{\underline v }^{{q_1}}}} \right\|_\alpha ^{{r_1}} &=& - \lambda {s^{m{l_1}}}\phi + a{s^{({l_1}{p_1} + {l_2}{q_1}){r_1}}}\left\| {{\phi ^{\frac{{{p_1}}}{m} + \frac{{{q_1}}}{n}}}} \right\|_\alpha ^{{r_1}} \nonumber\\ &=& {l_1}{s^{{l_1} - 1}}{\phi ^{\frac{1}{m}}}( - \frac{{\lambda {s^{m{l_1} - {l_1} + 1}}{\phi ^{1 - \frac{1}{m}}}}}{{{l_1}}} + \frac{{a{c_1}{\phi ^{ - \frac{1}{m}}}}}{{{l_1}}}{s^{({l_1}{p_1} + {l_2}{q_1}){r_1} - {l_1} + 1}}) \nonumber\\ &\ge& {l_1}{s^{{l_1} - 1}}{\phi ^{\frac{1}{m}}}\frac{{a{c_1}}}{{{l_1}}}{s^{m{l_1} - {l_1} + 1}}( - \frac{\lambda }{{a{c_1}}} + {s^{({l_1}{p_1} + {l_2}{q_1}){r_1} - m{l_1}}}), \nonumber\\ {\underline u _t} &=& {l_1}{s^{{l_1} - 1}}{\phi ^{\frac{1}{m}}}s'(t). \end{eqnarray}$

(4.5)

类似有

$\begin{eqnarray} \Delta {\underline v ^n} + b\left\| {{{\underline v }^{{p_2}}}{{\underline w }^{{q_2}}}} \right\|_\beta ^{{r_2}}{\kern 1pt} &=& - \lambda {s^{n{l_2}}}\phi + b{s^{({l_2}{p_2} + {l_3}{q_2}){r_2}}}\left\| {{\phi ^{\frac{{{p_2}}}{n} + \frac{{{q_2}}}{h}}}} \right\|_\beta ^{{r_2}} \nonumber\\ &\ge& {l_2}{s^{{l_2} - 1}}{\phi ^{\frac{1}{n}}}\frac{{b{c_2}}}{{{l_2}}}{s^{n{l_2} - {l_2} + 1}}( - \frac{\lambda }{{b{c_2}}} + {s^{({l_2}{p_2} + {l_3}{q_2}){r_2} - n{l_2}}}), \\ {\underline v _t} &=& {l_2}{s^{{l_2} - 1}}{\phi ^{\frac{1}{n}}}s'(t), \nonumber\\ \end{eqnarray}$

(4.6)

$\begin{eqnarray} \Delta {\underline w ^h} + c\left\| {{{\underline w }^{{p_3}}}{{\underline u }^{{q_3}}}} \right\|_\gamma ^{{r_3}}{\kern 1pt} &=& - \lambda {s^{h{l_3}}}\phi + c{s^{({l_3}{p_3} + {l_1}{q_3}){r_3}}}\left\| {{\phi ^{\frac{{{p_3}}}{h} + \frac{{{q_3}}}{m}}}} \right\|_\gamma ^{{r_3}} \nonumber\\ &\ge& {l_3}{s^{{l_3} - 1}}{\phi ^{\frac{1}{h}}}\frac{{c{c_3}}}{{{l_3}}}{s^{h{l_3} - {l_3} + 1}}( - \frac{\lambda }{{c{c_3}}} + {s^{({l_3}{p_3} + {l_1}{q_3}){r_3} - h{l_3}}}), \\ {\underline w _t} &=& {l_3}{s^{{l_3} - 1}}{\phi ^{\frac{1}{h}}}s'(t), \nonumber \end{eqnarray}$

(4.7)

其中

$\begin{aligned} {c_1} = \left\| {{\phi ^{\frac{{{p_1}}}{m} + \frac{{{q_1}}}{n}}}} \right\|_\alpha ^{{r_1}}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {c_2} = \left\| {{\phi ^{\frac{{{p_2}}}{n} + \frac{{{q_2}}}{h}}}} \right\|_\beta ^{{r_2}}, {c_3} = \left\| {{\phi ^{\frac{{{p_3}}}{h} + \frac{{{q_3}}}{m}}}} \right\|_\gamma ^{{r_3}}. \end{aligned}$

(4.8)

(1) 如果 $m > {p_1}{r_1}, n > {p_2}{r_2}, h >{p_3}{r_3}, ({r_1}{q_1})({r_2}{q_2})({r_3}{q_3}) > (m -{r_1}{p_1})(n - {r_2}{p_2})(h - {r_3}{p_3}), $那么存在正数${l_1}, {l_2}, {l_3}$ 使得

$\begin{aligned} {r_1}{q_1}{l_2} > (m - {r_1}{p_1}){l_1}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {r_2}{q_2}{l_3} > (n - {r_2}{p_2}){l_2}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {r_3}{q_3}{l_1} > (h - {r_3}{p_3}){l_3}. \end{aligned}$

(4.9)

选取

$\begin{array}{l} k\; = \min \{ \frac{{a{c_1}}}{{{l_1}}},\frac{{b{c_2}}}{{{l_2}}},\frac{{c{c_3}}}{{{l_3}}}\} ,\\ \theta \; = \min \{ m{l_1} - {l_1} + 1,n{l_2} - {l_2} + 1,h{l_3} - {l_3} + 1\} > 1,\\ {s_0} = \max \{ {(\frac{\lambda }{{a{c_1}}} + 1)^{1/(({l_1}{p_1} + {l_2}{q_1}){r_1} - m{l_1})}},\\ \;\;\;\;\;\;\;\;{(\frac{\lambda }{{b{c_2}}} + 1)^{1/(({l_2}{p_2} + {l_3}{q_2}){r_2} - n{l_2})}},{(\frac{\lambda }{{c{c_3}}} + 1)^{1/(({l_3}{p_3} + {l_1}{q_3}){r_3} - h{l_3})}}\} > 1. \end{array}$

(4.10)

假设 ${u_0}(x), {v_0}(x), {w_0}(x)$充分大, 并满足

$\begin{aligned} {u_0}(x) \ge {s_0}^{{l_1}}, {v_0}(x) \ge {s_0}^{{l_2}}, {w_0}(x) \ge {s_0}^{{l_3}}. \end{aligned}$

(4.11)

那么 根据 (4.3)--(4.11)式, 由(4.2)式定义的$(\underline u, \underline v, \underline w )$ 是问题(1.1)的弱下解. 由于$s(t)$在有限时刻无界, 因此 $(\underline u, \underline v, \underline w )$在有限时刻爆破. 由比较原理知, $(\underline u, \underline v, \underline w ) \le (u, v, w)$ ; 这意味着$(u, v, w)$在有限时刻爆破.

(2)若$m \le {p_1}{r_1}$ 或 $n \le {p_2}{r_2}$ 或 $h \le{p_3}{r_3}$, 那么存在正常数${l_1}, {l_2}, {l_3}, $ 使得 (4.9)式仍然成立, 因此可以用类似的方法证明结果, 在这儿省略掉.

(3) 下面考虑 $m > {p_1}{r_1}, n > {p_2}{r_2}, h > {p_3}{r_3}, $

$({r_1}{q_1})({r_2}{q_2})({r_3}{q_3}) = (m - {r_1}{p_1})(n - {r_2}{p_2})(h - {r_3}{p_3}).$

显然, 存在正常数${l_1}, {l_2}, {l_3}$使得

$\begin{aligned} {r_1}{q_1}{l_2} = (m - {r_1}{p_1}){l_1}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {r_2}{q_2}{l_3} = (n - {r_2}{p_2}){l_2}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {r_3}{q_3}{l_1} = (h - {r_3}{p_3}){l_3}. \end{aligned}$

(4.12)

不失一般性, 假设$0 \in \Omega $, 并设 ${B_R}(0) \subset \Omega $ 是一个半径为 $R$ 的开球.

记${\lambda _{{B_R}}} > 0$ 与${\phi _R}(r)$是下列问题的第一特征值和相应的特征函数

$\begin{aligned} - \phi ''(r) - \frac{{N - 1}}{r}\phi '(r) = \lambda \phi (r), {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} r \in (0, R);{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \phi '(0) = 0, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \phi (R) = 0. \end{aligned}$

(4.13)

在${B_R}$中, 将 ${\phi _R}(r)$正则化成${\phi _R}(r) > 0$, 且${\phi _R}(0) = \mathop {\max }\limits_{{B_R}} {\phi _R}(r) = 1$ .根据特征值及特征函数的性质 (设$\tau = r/R$)可知

${\lambda _{{B_R}}} = {R^{ - 2}}{\lambda _{{B_1}}},{\phi _R}(r) = {\phi _1}(r/R{\rm{ }}) = {\phi _1}(\tau ),$

这里${\lambda _{{B_1}}}$ 与 ${\phi _1}(\tau )$是单位球${B_1}(0)$ 内的第一特征值问题的特征值和相应的标准特征函数, 且

$\mathop {\max }\limits_{{B_1}} {\phi _1} = {\phi _1}(0) = {\phi _R}(0) = \mathop {\max }\limits_{{B_R}} {\phi _R} = 1.$

现在 定义$\underline u (x, t), \underline v (x, t), \underline w (x, t)$ 如下:

$\begin{aligned} \underline u (x, t) = {s^{{l_1}}}(t){\phi _R}^{\frac{1}{m}}(\left| x \right|), \underline v (x, t) = {s^{{l_2}}}(t){\phi _R}^{\frac{1}{n}}(\left| x \right|), \underline w (x, t) = {s^{{l_3}}}(t){\phi _R}^{\frac{1}{h}}(\left| x \right|), \end{aligned}$

(4.14)

其中$s(t)$ 如式(4.3)的定义, $k, {s_0} > 0, \theta > 1$待定. 那么, 类似于 (4.5)--(4.7)式 的计算可得

$\begin{aligned} {\underline u _t} - \Delta {\underline u ^m} - a\left\| {{{\underline u }^{{p_1}}}{{\underline v }^{{q_1}}}} \right\|_\alpha ^{{r_1}}{\kern 1pt} {\kern 1pt} \le {l_1}{s^{{l_1} - 1}}{\phi _R}^{\frac{1}{m}}(s'(t) -\frac{{a{c_1} - {\lambda _{{B_R}}}}}{{{l_1}}}{s^{m{l_1} - {l_1} + 1}}), \end{aligned}$

(4.15)

$\begin{aligned} {\underline v _t} - \Delta {\underline v ^n} + b\left\| {{{\underline v }^{{p_2}}}{{\underline w }^{{q_2}}}} \right\|_\beta ^{{r_2}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \le {l_2}{s^{{l_2} - 1}}{\phi_R}^{\frac{1}{n}}(s'(t) - \frac{{b{c_2} - {\lambda _{{B_R}}}}}{{{l_2}}}{s^{n{l_2} - {l_2} + 1}}), \end{aligned}$

(4.16)

$\begin{aligned} {\underline w _t} - \Delta {\underline w ^h} - c\left\| {{{\underline w }^{{p_3}}}{{\underline u }^{{q_3}}}} \right\|_\gamma ^{{r_3}}{\kern 1pt} \le {l_3}{s^{{l_3} - 1}}{\phi _R}^{\frac{1}{h}}(s'(t) - \frac{{c{c_3} - {\lambda _{{B_R}}}}}{{{l_3}}}{s^{h{l_3} - {l_3} + 1}}), \end{aligned}$

(4.17)

其中

$\begin{eqnarray} {c_1} &=& \left\| {{\phi _R}^{\frac{{{p_1}}}{m} + \frac{{{q_1}}}{n}}} \right\|_\alpha ^{{r_1}} = {(\displaystyle\int_{{B_R}} {{\phi _R}^{\alpha (\frac{{{p_1}}}{m} + \frac{{{q_1}}}{n})}(\left| x \right|)} dx)^{\frac{{{r_1}}}{\alpha }}} \nonumber\\ &=& {R^{\frac{{N{r_1}}}{\alpha }}}\displaystyle\int_{{B_1}} {{\phi _1}^{\alpha (\frac{{{p_1}}}{m} + \frac{{{q_1}}}{n})}\left| y \right|)} dy{)^{\frac{{{r_1}}}{\alpha }}}{\kern 1pt} {\kern 1pt}\le {K_1}{R^{\frac{{N{r_1}}}{\alpha }}}, \nonumber\\ {c_2} &=& \left\| {{\phi _R}^{\frac{{{p_2}}}{n} + \frac{{{q_2}}}{h}}} \right\|_\beta ^{{r_2}} \le {K_2}{R^{\frac{{N{r_2}}}{\beta }}}, \nonumber\\ {c_3} &=& \left\| {{\phi _R}^{\frac{{{p_3}}}{h} + \frac{{{q_3}}}{m}}} \right\|_\gamma ^{{r_3}} \le {K_3}{R^{\frac{{N{r_3}}}{\gamma }}}, \end{eqnarray}$

(4.18)

这里${K_1}, {K_2}, {K_3}$ 是与$R$无关的常数. 由于${\lambda _{{B_R}}}= {R^{ - 2}}{\lambda _{{B_1}}}$, 可以假设球${B_R}$ 的半径$R$充分大, 并使得

$\begin{aligned} {\lambda _{{B_R}}} ＜ \min \{ a{c_1}, b{c_2}, c{c_3}\}, \end{aligned}$

(4.19)

因此

$\begin{aligned} {\sigma _1} \equiv \frac{{a{c_1} - {\lambda _{{B_R}}}}}{{{l_1}}} > 0, {\sigma _2} \equiv \frac{{b{c_2}}}{{{l_2}}} - {\lambda _{{B_R}}} > 0, {\sigma _3} \equiv \frac{{c{c_3}}}{{{l_3}}} - {\lambda _{{B_R}}} > 0. \end{aligned}$

(4.20)

记

${\theta _1} = m{l_1} - {l_1} + 1, {\theta _2} = n{l_2} - {l_2} + 1, {\theta _3} = h{l_3} - {l_3} + 1.$

根据 $m, n, h >1$得${\theta _1}, {\theta _2}, {\theta _3} > 1$.因为${u_0}, {v_0}, {w_0}$ 在$\Omega $ 内是正的连续函数, 所以可以选取充分小的${s_0} > 0$, 在球${B_R}(0)$内满足

$\begin{aligned} {u_0} \ge {s_0}^{{l_1}}{\phi _R}^{\frac{1}{m}}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {v_0} \ge {s_0}^{{l_2}}{\phi _R}^{\frac{1}{n}}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {w_0} \ge {s_0}^{{l_3}}{\phi _R}^{\frac{1}{h}}{\kern 1pt} {\kern 1pt}. \end{aligned}$

(4.21)

最后, 选择 $\theta, k$ 使得

$\begin{aligned} 1 ＜ \theta ＜ \min \{ {\theta _1}, {\theta _2}, {\theta _3}\}, 0 ＜ k ＜ \min \{ {\sigma _1}{s_0}^{{\theta _1} - \theta }, {\sigma _2}{s_0}^{{\theta _2} - \theta }, {\sigma _3}{s_0}^{{\theta _3} - \theta }\}. \end{aligned}$

(4.22)

从而根据(4.3)式, 得到

$\begin{array}{l} s(t) \ge s(0) = {s_0} > \max \{ {(\frac{k}{{{\sigma _1}}})^{1/({\theta _1} - \theta )}},{(\frac{k}{{{\sigma _2}}})^{1/({\theta _2} - \theta )}},{(\frac{k}{{{\sigma _3}}})^{1/({\theta _3} - \theta )}}\} .\\ \end{array}$

(4.23)

即

$\begin{aligned} k{s^\theta }(t) \le {\sigma _1}{s^{{\theta _1}}}(t), k{s^\theta }(t) \le {\sigma _2}{s^{{\theta _2}}}(t), k{s^\theta }(t) \le {\sigma _3}{s^{{\theta _3}}}(t). \end{aligned}$

(4.24)

根据 (4.15)--(4.24)式 知, 由(4.14)式定义的$(\underline u, \underline v, \underline w )$ 是问题(1.1)在 ${B_R}(0)$内正的弱下解, 它在有限时刻爆破. 从而 $(u, v, w)$ 在充分大的$\Omega $上爆破. 定理2 证毕.