0 引言

本文讨论了下列一类交叉耦合抛物型方程组解的整体存在及有限时刻爆破问题

$ \left\{ \begin{array}{l} {u_t} = \Delta {u^{{m_1}}} + {v^{{p_1}}}\int_\mathit{\Omega } {{{\left( {\ln w} \right)}^{{q_1}}}{\rm{d}}x} ,\;\;\;\;x \in \mathit{\Omega ,t > }0,\\ {v_t} = \Delta {v^{{m_2}}} + {w^{{p_2}}}\int_\mathit{\Omega } {{{\left( {\ln u} \right)}^{{q_2}}}{\rm{d}}x} ,\;\;\;\;x \in \mathit{\Omega ,t > }0,\\ {w_t} = \Delta {w^{{m_3}}} + {u^{{p_3}}}\int_\mathit{\Omega } {{{\left( {\ln v} \right)}^{{q_3}}}{\rm{d}}x} ,\;\;\;\;x \in \mathit{\Omega ,t > }0. \end{array} \right. $

(1)

具有非局部边界流

$ \left\{ \begin{array}{l} u\left( {x,t} \right) = \int_\mathit{\Omega } {{\phi _1}\left( {x,y} \right)u\left( {y,t} \right){\rm{d}}y} ,\;\;\;\;x \in \partial \mathit{\Omega ,t > }0,\\ v\left( {x,t} \right) = \int_\mathit{\Omega } {{\phi _2}\left( {x,y} \right)v\left( {y,t} \right){\rm{d}}y} ,\;\;\;\;x \in \partial \mathit{\Omega ,t > }0,\\ w\left( {x,t} \right) = \int_\mathit{\Omega } {{\phi _3}\left( {x,y} \right)w\left( {y,t} \right){\rm{d}}y} ,\;\;\;\;x \in \partial \mathit{\Omega ,t > }0 \end{array} \right. $

(2)

及连续有界初值

$ \left\{ \begin{array}{l} u\left( {x,0} \right) = {u_0}\left( x \right),\;\;\;\;\;x \in \mathit{\Omega },\\ v\left( {x,0} \right) = {v_0}\left( x \right),\;\;\;\;\;x \in \mathit{\Omega },\\ w\left( {x,0} \right) = {w_0}\left( x \right),\;\;\;\;\;x \in \mathit{\Omega }. \end{array} \right. $

(3)

式中, Ω⊂R^N是有界光滑区域, m_i>1, p_j, q_k>0, 且满足当i≠j, j≠k, k≠i时, m_ip_jq_k=1, i, j, k=1, 2, 3, ϕ_i(x, y)(i=1, 2, 3) 是∂Ω×Ω上的非负连续函数, 初值u₀(x), v₀(x), w₀(x)∈C^2+α(Ω) (0 < α < 1) 非负, 且在边界上满足相容条件.

方程组(1)~(3) 有明显的实际背景, 可用来描述3种混合物质燃烧的热传导过程, 或者3种化学反应中反应物的反应情况, u, v, w分别表示3种介质的温度或反应物的浓度.许多学者对此类方程组做了大量研究.文献[1]研究了由两个幂函数交叉耦合的抛物型方程组

$ \left\{ \begin{array}{l} {u_t} = \Delta {u^{{m_1}}} - \int_\mathit{\Omega } {{u^\alpha }{v^p}{\rm{d}}x,{v_t}} = \Delta {v^{{m_2}}} - \int_\mathit{\Omega } {{u^q}{v^\beta }{\rm{d}}x,x \in \mathit{\Omega ,t > }0} ,\\ u\left( {x,t} \right) = \int_\mathit{\Omega } {\phi \left( {x,y} \right)u\left( {y,t} \right){\rm{d}}y,v\left( {x,t} \right)} = \int_\mathit{\Omega } {\psi \left( {x,y} \right)v\left( {y,t} \right){\rm{d}}y,x \in \partial \mathit{\Omega ,t > }0} ,\\ u\left( {x,0} \right) = {u_0}\left( x \right),v\left( {x,0} \right) = {v_0}\left( x \right),x \in \mathit{\Omega }, \end{array} \right. $

得到解的整体存在及解在有限时刻爆破的充分条件.文献[2]研究了如下由幂函数和指数函数交叉耦合的抛物型方程组

$ \left\{ \begin{array}{l} {u_t} = \Delta {u^{{m_1}}} - \int_\mathit{\Omega } {{v^\beta }{\rm{exp}}\left( {\alpha u} \right){\rm{d}}x} ,\;\;\;\;\;\;\;\;\;\;x \in \mathit{\Omega ,t > }0,\\ {v_t} = \Delta {v^{{m_2}}} - \int_\mathit{\Omega } {{u^p}{\rm{exp}}\left( {qv} \right){\rm{d}}x} ,\;\;\;\;\;\;\;\;\;\;\;x \in \mathit{\Omega ,t > }0,\\ u\left( {x,t} \right) = v\left( {x,t} \right) = 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \partial \mathit{\Omega ,t > }0,\\ u\left( {x,0} \right) = {u_0}\left( x \right),v\left( {x,0} \right) = {v_0}\left( x \right),\;\;\;\;\;x \in \mathit{\Omega }\mathit{.} \end{array} \right. $

得到了解在有限时刻爆破的充分条件及同时爆破的一个必要条件.其他相关结果见文献[3-19].

综上所述, 已有的结果主要研究了由幂函数和指数函数两个变量交叉耦合的抛物型方程组解的整体存在及有限时刻爆破问题, 本文将变量个数推广到3个变量, 且引入了对数函数为耦合变量的抛物型方程组解的整体存在及爆破问题, 使得此类方程组更能广泛地描述热传导和化学反应问题.

1 预备知识

定义1  令T>0, 正函数u(x, t), v(x, t), w(x, t)∈C^{1, 0}(×0, T))∩C^{2, 1}(Ω×[0, T), 且满足

$ \left\{ \begin{array}{l} {{\bar u}_t} = \Delta {{\bar u}^{{m_1}}} + {{\bar v}^{{p_1}}}\int_\mathit{\Omega } {{{\left( {\ln \bar w} \right)}^{{q_1}}}{\rm{d}}x} ,\;\;\;\;x \in \mathit{\Omega ,t > }0,\\ {{\bar v}_t} = \Delta {{\bar v}^{{m_2}}} + {{\bar w}^{{p_2}}}\int_\mathit{\Omega } {{{\left( {\ln \bar u} \right)}^{{q_2}}}{\rm{d}}x} ,\;\;\;\;x \in \mathit{\Omega ,t > }0,\\ {{\bar w}_t} = \Delta {{\bar w}^{{m_3}}} + {{\bar u}^{{p_3}}}\int_\mathit{\Omega } {{{\left( {\ln \bar v} \right)}^{{q_3}}}{\rm{d}}x} ,\;\;\;\;x \in \mathit{\Omega ,t > }0. \end{array} \right. $

(4)

具有非局部边界流

$ \left\{ \begin{array}{l} \bar u\left( {x,t} \right) = \int_\mathit{\Omega } {{\phi _1}\left( {x,y} \right)\bar u\left( {y,t} \right){\rm{d}}y} ,\;\;\;\;x \in \partial \mathit{\Omega ,t > }0,\\ \bar v\left( {x,t} \right) = \int_\mathit{\Omega } {{\phi _2}\left( {x,y} \right)\bar v\left( {y,t} \right){\rm{d}}y} ,\;\;\;\;x \in \partial \mathit{\Omega ,t > }0,\\ \bar w\left( {x,t} \right) = \int_\mathit{\Omega } {{\phi _3}\left( {x,y} \right)\bar w\left( {y,t} \right){\rm{d}}y} ,\;\;\;\;x \in \partial \mathit{\Omega ,t > }0. \end{array} \right. $

(5)

及连续有界初值

$ \left\{ \begin{array}{l} \bar u\left( {x,0} \right) = {{\bar u}_0}\left( x \right),\;\;\;\;\;x \in \mathit{\Omega },\\ \bar v\left( {x,0} \right) = {{\bar v}_0}\left( x \right),\;\;\;\;\;x \in \mathit{\Omega },\\ \bar w\left( {x,0} \right) = {{\bar w}_0}\left( x \right),\;\;\;\;\;x \in \mathit{\Omega }. \end{array} \right. $

(6)

其中, m_i>1, p_j, q_k>0, (i=1, 2, 3) 则称((x, t), (x, t), (x, t))为方程组(1)~(3) 的上解.改变不等号的方向, 类似可以定义下解.

由文献[8, 20], 有如下的比较定理.

引理1  设(u(x, t), v(x, t), w(x, t))和(u(x, t), v(x, t), w(x, t))是方程组(1)~(3) 在×[0, T)上的一对有序上、下解, 则方程组(1)~(3) 存在唯一古典解(u(x, t), v(x, t), w(x, t))在E×[0, T)上有定义且满足

$ \left\{ \begin{array}{l} \underline u \left( {x,t} \right) \le u\left( {x,t} \right) \le \bar u\left( {x,t} \right),\left( {x,t} \right) \in \bar E \times \left[ {0,T} \right),\\ \underline v \left( {x,t} \right) \le v\left( {x,t} \right) \le \bar v\left( {x,t} \right),\left( {x,t} \right) \in \bar E \times \left[ {0,T} \right),\\ \underline w \left( {x,t} \right) \le w\left( {x,t} \right) \le \bar w\left( {x,t} \right),\left( {x,t} \right) \in \bar E \times \left[ {0,T} \right). \end{array} \right. $

引理2  设(u(x, t), v(x, t), w(x, t))>(0, 0, 0) 是(1) 的下解, 如果(u, v, w)在有限时刻爆破, 则方程组(1)~(3) 的解(u(x, t), v(x, t), w(x, t))在有限时刻爆破.

2 解的整体存在

定理1  如果m₁m₂m₃>p₁p₂p₃+q₁q₂q₃+3时, 对于小初值u₀(x), v₀(x), w₀(x), 方程组(1)~(3) 的解整体存在.

证明  设ϕ(x)满足

$ \left\{ \begin{array}{l} - \Delta \phi = {\varepsilon _0},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \mathit{\Omega ,}\\ \phi \left( x \right) = \int_\mathit{\Omega } {\tau \left( {x,y} \right){\rm{d}}y} ,\;\;\;\;x \in \partial \mathit{\Omega }\mathit{.} \end{array} \right. $

(7)

式中:τ(x, y)=max{ϕ₁(x, y), ϕ₂(x, y), ϕ₃(x, y), ϕ_i(x, y)≠0, i=1, 2, 3}是定义在∂Ω×Ω上的非负连续函数, ε₀>0使得0 < ϕ(x)≤1, 记${K_1} = \mathop {\max }\limits_{x \in \mathit{\Omega }} \phi \left( x \right)$, ${K_2} = \mathop {\max }\limits_{x \in \mathit{\Omega }} \phi \left( x \right)$.则对于任意正实数η, 利用拉格朗日中值定理证明可知ln(1+η) < η, 同理,

$ \ln \left( {k{\phi ^{1/{m_i}}}\left( x \right)} \right) = \ln \left[ {1 + \left( {k{\phi ^{1/{m_i}}}\left( x \right) - 1} \right)} \right] < \left( {k{\phi ^{1/{m_i}}}\left( x \right) - 1} \right) < k{\phi ^{1/{m_i}}}\left( x \right). $

令u=aϕ^1/m₁(x), v=bϕ^1/m₂(x), w=cϕ^1/m₃(x), 结合式(4)~(7), 则有

$ {{\bar u}_t} - \Delta {{\bar u}^{{m_1}}} - {{\bar v}^{{p_1}}}\int_\mathit{\Omega } {{{\left[ {\ln \bar w} \right]}^{{q_1}}}{\rm{d}}x} = $

$ {a^{{m_1}}}{\varepsilon _0} - {b^{{p_1}}}{\phi ^{{p_1}/{m_2}}}\left( x \right)\int_\mathit{\Omega } {{{\left[ {\ln \left( {c{\phi ^{1/{m_3}}}\left( x \right)} \right)} \right]}^{{q_1}}}{\rm{d}}x} \ge $

$ {a^{{m_1}}}{\varepsilon _0} - {b^{{p_1}}}{\phi ^{{p_1}/{m_2}}}\left( x \right)\int_\mathit{\Omega } {{{\left[ {c{\phi ^{1/{m_3}}}\left( x \right) - 1} \right]}^{{q_1}}}{\rm{d}}x} \ge $

$ {a^{{m_1}}}{\varepsilon _0} - {b^{{p_1}}}{\phi ^{{p_1}/{m_2}}}\left( x \right)\int_\mathit{\Omega } {{{\left[ {c{\phi ^{1/{m_3}}}\left( x \right)} \right]}^{{q_1}}}{\rm{d}}x} = $

$ {a^{{m_1}}}{\varepsilon _0} - {b^{{p_1}}}{c^{{q_1}}}{\phi ^{{p_1}/{m_2}}}\left( x \right)\int_\mathit{\Omega } {{\phi ^{{q_1}/{m_3}}}\left( x \right){\rm{d}}x} \ge $

$ {a^{{m_1}}}{\varepsilon _0} - {b^{{p_1}}}{c^{{q_1}}}K_1^{\left( {{p_1}{m_3} + {q_1}{m_2}} \right)/{m_2}{m_3}}\mathit{\Omega }\mathit{.} $

同理,

$ \begin{array}{l} {{\bar v}_t} - \Delta {{\bar v}^{{m_2}}} - {{\bar w}^{{p_2}}}\int_\mathit{\Omega } {{{\left[ {\ln \bar u} \right]}^{{q_2}}}{\rm{d}}x} \ge {b^{{m_2}}}{\varepsilon _0} - {c^{{p_2}}}{a^{{q_2}}}K_1^{\left( {{p_2}{m_1} + {q_2}{m_3}} \right)/{m_1}{m_3}}\mathit{\Omega },\\ {{\bar w}_t} - \Delta {{\bar w}^{{m_3}}} - {{\bar u}^{{p_3}}}\int_\mathit{\Omega } {{{\left[ {\ln \bar v} \right]}^{{q_3}}}dx} \ge {c^{{m_3}}}{\varepsilon _0} - {a^{{p_3}}}{b^{{q_3}}}K_1^{\left( {{p_3}{m_2} + {q_3}{m_1}} \right)/{m_1}{m_2}}\mathit{\Omega }\mathit{.} \end{array} $

边界:

$ \begin{array}{*{20}{c}} {{{\bar u}_{\partial \mathit{\Omega }}} = a{{\left( {\int_\mathit{\Omega } {\tau \left( {x,y} \right){\rm{d}}y} } \right)}^{1/{m_1}}} \ge a\int_\mathit{\Omega } {\tau \left( {x,y} \right){\rm{d}}y} \ge a\int_\mathit{\Omega } {{\phi _1}\left( {x,y} \right){\rm{d}}y} \ge }\\ {a\int_\mathit{\Omega } {{\phi _1}\left( {x,y} \right){\phi ^{1/{m_1}}}\left( x \right){\rm{d}}y} = \int_\mathit{\Omega } {{\phi _1}\left( {x,y} \right)\bar u\left( {x,y} \right){\rm{d}}y} .} \end{array} $

同理, ${\bar v_{\partial \mathit{\Omega }}} \ge \int_\mathit{\Omega } {{\phi _2}\left( {x, y} \right)} \bar v\left( {x, y} \right){\rm{d}}y, {\bar w_{\partial \mathit{\Omega }}} \ge \int_\mathit{\Omega } {{\phi _3}\left( {x, y} \right)\bar w\left( {x, y} \right){\rm{d}}y} $, ${\bar w_{\partial \mathit{\Omega }}} \ge \int_\mathit{\Omega } {{\phi _3}\left( {x, y} \right)\bar w\left( {x, y} \right){\rm{d}}y} $.

初值:

$ \bar u\left( {x,0} \right) = a{\phi ^{1/{m_1}}}\left( x \right) \ge aK_2^{1/{m_1}},\bar v\left( {x,0} \right) = bK_2^{1/{m_1}},\bar w\left( {x,0} \right) = c{\phi ^{1/{m_3}}}\left( x \right) \ge cK_2^{1/{m_3}}. $

综上可知, 只要存在a, b, c, 使得

$ \begin{array}{l} {a^{{m_1}}}{\varepsilon _0} - {b^{{p_1}}}{c^{{q_1}}}K_1^{\left( {{p_1}{m_3} + {q_1}{m_2}} \right)/{m_2}{m_3}}\mathit{\Omega } \ge 0,\\ {b^{{m_1}}}{\varepsilon _0} - {c^{{p_2}}}{a^{{q_2}}}K_1^{\left( {{p_2}{m_1} + {q_2}{m_3}} \right)/{m_1}{m_3}}\mathit{\Omega } \ge 0,\\ {c^{{m_3}}}{\varepsilon _0} - {a^{{p_3}}}{b^{{q_3}}}K_1^{\left( {{p_3}{m_2} + {q_3}{m_1}} \right)/{m_1}{m_2}}\mathit{\Omega } \ge 0,\\ aK_2^{1/{m_1}} \ge {u_0}\left( x \right),bK_2^{1/{m_1}} \ge {v_0}\left( x \right),cK_2^{1/{m_3}} \ge {w_0}\left( x \right) \end{array} $

(8)

成立, 则(u, v, w)是方程组(1)~(3) 的上解, 由引理1知, 方程组(1)~(3) 的解整体存在.

下证这样的a, b, c存在.令b^p₁=a^m₁c^-q₁K₁^{-(p₁m₃+q₁m₂)/m₂m₃}Ω^-1ε₀, 代入式(8) 可得关于a的不等式，即

$ a\frac{{\left( {{m_1}{m_2} - {p_1}{q_2}} \right)\left( {{m_3}{p_1} - {q_1}{q_3}} \right)}}{{\left( {{p_1}{q_2} + {m_2}{q_1}} \right)\left( {{p_1}{p_3} + {m_1}{q_3}} \right)}} \ge K_1^{\frac{{{p_1}{q_2} + {m_2}{q_1}}}{{{m_3}{p_1} + {q_1}{q_3}}}}\left[ {\frac{{{p_3}{m_2} + {q_3}{m_1}}}{{{m_1}{m_2}}} - \frac{{{q_3}\left( {{p_1}{m_3} + {q_1}{m_2}} \right)}}{{{p_1}{m_2}{m_3}}}} \right]\mathit{\Omega }\frac{{\left( {{p_1}{q_2} + {m_2}{q_1}} \right)\left( {{p_1} - {q_3}} \right)}}{{{p_1}\left( {{m_3}{p_1} + {q_1}{q_3}} \right)}}\varepsilon _0^{\frac{{\left( {{p_1}{q_2} + {m_2}{q_1}} \right)\left( {{p_1} - 1} \right)}}{{{p_1}\left( {{m_3}{p_1} + {q_1}{q_3}} \right)}}}. $

(9)

由定理1条件m₁m₂m₃>p₁p₂p₃+q₁q₂q₃+3知, (m₁m₂-p₁q₂)(m₃p₁+q₁q₃)>(p₁q₂+m₂q₁)·(p₁p₃+m₁q₃), 只要取a充分大时, 可使得式(9) 成立.另只要a, b, c充分大, 又对于小初值u₀(x), v₀(x), w₀(x), 就可以保证式(8) 的后3个式子成立.定理1证毕.

3 解的有限时刻爆破

讨论解的整体存在问题时, 引入以下两个引理.

引理3  设θ>λ>1, k, l>0, h(t)是问题

$ \left\{ \begin{array}{l} h'\left( t \right) = - h{h^\lambda }\left( t \right) + l{h^\theta }\left( t \right),t > 0,\\ h\left( 0 \right) = {h_0} > 0 \end{array} \right. $

(10)

的正解, 则当h₀充分大时, h(t)在有限时刻爆破.

引理4  设λ₂>λ₁>1, θ₂>θ₁>1, 则存在如引理3的h(t)是满足

$ \left\{ \begin{array}{l} h'\left( t \right) \le - k{h^{{\lambda _1}}}\left( t \right) + l{h^{{\lambda _2}}}\left( t \right),\\ h'\left( t \right) \le - k{h^{{\theta _1}}}\left( t \right) + l{h^{{\theta _2}}}\left( t \right). \end{array} \right. $

引理3及引理4的证明见文献[1, 16].

定理2  如果2m₁m₂m₃ < p₁p₂p₃-q₁q₂q₃-3, 则当初值u₀(x), v₀(x), w₀(x)充分大时, 方程组(1)~(3) 的解在有限时刻爆破.

证明  设ϕ(x)是满足方程

$ \left\{ \begin{array}{l} - \Delta \phi {\rm{ = }}1,\;\;\;\;\;\;x \in \mathit{\Omega ,}\\ \phi \left( x \right) = 0,\;\;\;\;x \in \partial \mathit{\Omega } \end{array} \right. $

(11)

的解, 则∃C>0, 使得0≤ϕ(x)≤C.令

$ \underline u \left( {x,t} \right) = {h^{{l_1}}}\left( t \right){\phi ^{{l_1}}}\left( x \right),\underline v \left( {x,t} \right) = {h^{{l_2}}}\left( t \right){\phi ^{{l_2}}}\left( x \right),\underline w \left( {x,t} \right) = {h^{{l_3}}}\left( t \right){\phi ^{{l_3}}}\left( x \right). $

其中, l₁, l₂, l₃>1, h(t)待定, 由式(10) 及(11) 可知, 取适当的h(t), 使得h^l_i(t)ϕ^l_i(x)>2(i=1, 2, 3), 则对于任意正实数α, 利用拉格朗日中值定理证明可知

$ {\ln ^\alpha }\left[ {{h^{li}}\left( t \right){\phi ^{li}}\left( x \right)} \right] > {h^{ - \alpha li}}\left( t \right){\phi ^{ - \alpha li}}\left( x \right). $

记

$ \begin{array}{l} k = \max \left\{ {{m_1}\left( {{l_1}{m_1} - 1} \right){C^{{l_1}\left( {{m_1} - 1} \right) - 2}},{m_2}\left( {{l_2}{m_2} - 1} \right){C^{{l_2}\left( {{m_2} - 1} \right) - 2}},{m_3}\left( {{l_3}{m_3} - 1} \right){C^{{l_3}\left( {{m_3} - 1} \right) - 2}}} \right\},\\ l = \min \left\{ {\frac{1}{{{l_1}}}{C^{{l_2}{p_1} - {l_1}}}\int_\mathit{\Omega } {{\phi ^{ - {l_3}{q_1}}}\left( x \right){\rm{d}}x} ,\frac{1}{{{l_2}}}{C^{{l_3}{p_2} - {l_2}}}\int_\mathit{\Omega } {{\phi ^{ - {l_1}{q_2}}}\left( x \right){\rm{d}}x} ,\frac{1}{{{l_3}}}{C^{{l_1}{p_3} - {l_3}}}\int_\mathit{\Omega } {{\phi ^{ - {l_2}{q_3}}}\left( x \right){\rm{d}}x} } \right\}. \end{array} $

则

$ \begin{array}{l} \underline {{u_t}} - \Delta {\underline u ^{{m_1}}} - {\underline v ^{{p_1}}}\int_\mathit{\Omega } {{{\left[ {\ln \underline w } \right]}^{{q_1}}}{\rm{d}}x} = \\ {l_1}{h^{{l_1} - 1}}\left( t \right){\phi ^{{l_1}}}\left( x \right)h'\left( t \right) - {l_1}{m_1}\left( {{l_1}{m_1} - 1} \right){h^{{l_1}{m_1}}}\left( t \right){\phi ^{{l_1}{m_1} - 2}}\left( x \right)\Delta \phi \left( x \right) - \\ {h^{{l_2}{p_1}}}\left( t \right){\phi ^{{l_2}{p_1}}}\left( x \right)\int_\mathit{\Omega } {{{\ln }^{{q_1}}}\left( {{h^{{l_3}}}\left( t \right){\phi ^{{l_3}}}\left( x \right)} \right){\rm{d}}x} \le \\ {l_1}{h^{{l_1} - 1}}\left( t \right){\phi ^{{l_1}}}\left( x \right)h'\left( t \right) - {l_1}{m_1}\left( {{l_1}{m_1} - 1} \right){h^{{l_1}{m_1}}}\left( t \right){\phi ^{{l_1}{m_1} - 2}}\left( x \right)\Delta \phi \left( x \right) - \\ {h^{{l_2}{p_1}}}\left( t \right){\phi ^{{l_2}{p_1}}}\left( x \right)\int_\mathit{\Omega } {{h^{ - {l_3}{q_1}}}\left( t \right){\phi ^{ - {l_3}{q_1}}}\left( x \right){\rm{d}}x} = \\ {l_1}{h^{{l_1} - 1}}\left( t \right){\phi ^{{l_1}}}\left( x \right)\left[ {h'\left( t \right) + {m_1}\left( {{l_1}{m_1} - 1} \right){h^{{l_1}{m_1} - {l_1} + 1}}\left( t \right){\phi ^{{l_1}{m_1} - {l_1} - 2}}\left( x \right) - } \right.\\ \left. {\frac{{{h^{{l_2}{p_1} - {l_3}{q_1} - {l_1} + 1}}\left( t \right)}}{{{l_1}}}{\phi ^{{l_2}{p_1} - {l_1}}}\left( x \right)\int_\mathit{\Omega } {{\phi ^{ - {l_3}{q_1}}}\left( x \right){\rm{d}}x} } \right] \le \\ {l_1}{h^{{l_1} - 1}}\left( t \right){\phi ^{{l_1}}}\left( x \right)\left[ {h'\left( t \right) + {m_1}\left( {{l_1}{m_1} - 1} \right){h^{{l_1}{m_1} - {l_1} + 1}}\left( t \right){C^{{l_1}{m_1} - {l_1} - 2}} - } \right.\\ \left. {\frac{{{h^{{l_2}{p_1} - {l_3}{q_1} - {l_1} + 1}}\left( t \right)}}{{{l_1}}}{C^{{l_2}{p_1} - {l_1}}}\int_\mathit{\Omega } {{\phi ^{ - {l_3}{q_1}}}\left( x \right){\rm{d}}x} } \right] \le \\ {l_1}{h^{{l_1} - 1}}\left( t \right){\phi ^{{l_1}}}\left( x \right)\left[ {h'\left( t \right) + k{h^{{l_1}{m_1} - {l_1} + 1}}\left( t \right) - l{h^{{l_2}{p_1} - {l_3}{q_1} - {l_1} + 1}}\left( t \right)} \right]. \end{array} $

同理:v_t-Δv^m₂-w^p₂∫_Ω[lnu]^q₂dx≤l₂h^l₂-1(t)ϕ^l₂(x)[h′(t)+kh^{l₂m₂-l₂+1}(t)-lh^{l₃p₂-l₁q₂-l₂+1}(t)], w_t-Δw^m₃-u^p₃∫_Ω[lnv]^q₃dx≤l₃h^l₃-1(t)ϕ^l₃(x)[h′(t)+kh^{l₃m₃-l₃+1}(t)-lh^{l₁p₃-l₂q₃-l₃+1}(t)].

综上由引理3的条件可知, 只要存在l₁, l₂, l₃, 使得

$ \left\{ \begin{array}{l} {l_2}{p_1} - {l_3}{q_1} - {l_1} + 1 > {l_1}\left( {{m_1} - 1} \right) + 1\\ {l_3}{p_2} - {l_1}{q_2} - {l_2} + 1 > {l_2}\left( {{m_2} - 1} \right) + 1\\ {l_1}{p_3} - {l_2}{q_3} - {l_3} + 1 > {l_3}\left( {{m_3} - 1} \right) + 1 \end{array} \right. \Rightarrow \left\{ \begin{array}{l} {l_2}{p_1} - {l_3}{q_1} > {l_1}{m_1}\\ {l_3}{p_2} - {l_1}{q_2} > {l_2}{m_2}\\ {l_1}{p_3} - {l_2}{q_3} > {l_3}{m_3} \end{array} \right. $

(12)

成立, 则由引理4知, 存在满足引理3的h(t)使得

$ \begin{array}{l} h'\left( t \right) \le - k{h^{{l_1}\left( {{m_1} - 1} \right) + 1}}\left( t \right) + l{h^{{l_2}{p_1} - {l_3}{q_1} - {l_1} + 1}}\left( t \right),\\ h'\left( t \right) \le - k{h^{{l_2}\left( {{m_2} - 1} \right) + 1}}\left( t \right) + l{h^{{l_3}{p_2} - {l_1}{q_2} - {l_2} + 1}}\left( t \right),\\ h'\left( t \right) \le - k{h^{{l_3}\left( {{m_3} - 1} \right) + 1}}\left( t \right) + l{h^{{l_1}{p_3} - {l_2}{q_3} - {l_3} + 1}}\left( t \right). \end{array} $

所以, u_t≤Δu^m₁+∫_Ωv^p₁[lnw]^q₁dx, v_t≤Δv^m₂+∫_Ωw^p₂[lnu]^q₂dx, w_t≤Δw^m₃+∫_Ωu^p₃[lnv]^q₃dx.边界:由式(11) 知, ϕ(x)=0, x∈∂Ω, 有

$ \begin{array}{*{20}{c}} {\underline u \left( {x,t} \right) = 0 \le \int_\mathit{\Omega } {{\phi _1}\left( {x,y} \right)u\left( {y,t} \right){\rm{d}}y,v\left( {x,t} \right)} = 0 \le \int_\mathit{\Omega } {{\phi _2}\left( {x,y} \right)v\left( {y,t} \right){\rm{d}}y,} }\\ {\underline w \left( {x,t} \right) = 0 \le \int_\mathit{\Omega } {{\phi _3}\left( {x,y} \right)w\left( {y,t} \right){\rm{d}}y} .} \end{array} $

初值:当初值u₀(x), v₀(x), w₀(x)充分大时, 有

$ \begin{array}{l} \underline u \left( {x,0} \right) = {h^{{l_1}}}\left( 0 \right){\phi ^{{l_1}}}\left( x \right) \le {u_0}\left( x \right),\\ \underline v \left( {x,0} \right) = {h^{{l_2}}}\left( 0 \right){\phi ^{{l_2}}}\left( x \right) \le {v_0}\left( x \right),\\ \underline w \left( {x,0} \right) = {h^{{l_3}}}\left( 0 \right){\phi ^{{l_3}}}\left( x \right) \le {w_0}\left( x \right). \end{array} $

故(u, v, w)为方程组(1)~(3) 的下解, 而且(u, v, w)在有限时刻爆破.由引理2知, 方程组(1)~(3) 的解在有限时刻爆破.

下证满足式(12) 的l₁, l₂, l₃存在, 取${l_1} = \frac{{{l_2}{p_1}-{l_3}{q_1}}}{{2{m_1}}}$, 代入式(12) 的第3式, 得

$ {l_2} > \frac{{2{m_1}{m_3} + {p_3}{q_1}}}{{{p_1}{p_3} - 2{m_1}{q_3}}}{l_3}. $

再将其代入式(12) 式的第2式, 得

$ \left( {{q_1}{q_2} + 2{m_1}{p_3}} \right){l_3} > \frac{{\left( {{p_1}{q_2} + 2{m_1}{m_2}} \right)\left( {2{m_1}{m_3} + {p_3}{q_1}} \right){l_3}}}{{\left( {{p_1}{p_3} - 2{m_1}{q_3}} \right)}}, $

由定理2条件2m₁m₂m₃ < p₁p₂p₃-q₁q₂q₃-3知, (q₁q₂+2m₁p₂)(p₁p₃-2m₁q₃)>(p₁q₂+2m₁m₂)(2m₁m₃+p₃q₁), 即上式成立, 定理2证毕.