非线性系统在自然科学学术界中是一个热门的研究对象^{[1, 2]}.非线性奇异摄动问题近来已有很多近似解法,包括边界层校正法、匹配法和多重尺度法.许多科学研究者,如Tian等^[3], Samusenko^[4], Kellogg等^[5], Skrynnikov^[6]和Martinez等^[7]已做了很多工作.利用非线性奇摄动和微分不等式等理论和方法,莫嘉琪,韩祥临等也研究了一类非线性奇异摄动微分方程和数学物理等问题^[8-18].本文是利用广义解的微分不等式等理论和方法研究一类带有广义反应扩散系统的奇异摄动问题.

考虑如下一类非线性微分-积分时滞奇摄动广义反应扩散系统初始-边值问题

(1.1) $\varepsilon(\psi(x), \frac{\partial u_{i}(t, x)}{\partialt})-\mu^{2}(\psi(x), L_{i}u_{i}(t, x))+(\psi(x), T_{i}u_{i}(t-\varepsilon, x))=(\psi(x), f_{i}(x, u_{i}(t, x))), \\(t, x)\in(0, \infty)\times\Omega, \ \ \forall\psi(x)\inC^{\infty}_{0}(\Omega), \ \ i=1, 2, \cdots, N, $

(1.2) $(\psi(x), u_{i}(t, x))=(\psi(x), g_{i}(t, x)), \ \ x\in\partial\Omega, \\ \forall\psi(x)\in C^{\infty}_{0}(\Omega) , \ \ i=1, 2, \cdots, N, $

(1.3) $(\psi(x), u_{i}(t, x))=(\psi(x), h_{i}(x)), \ \ t\in[-\varepsilon, 0], \ \ \forall\psi(x)\in C^{\infty}_{0}(\Omega) , \ \i=1, 2, \cdots, N, $

其中$\mu$为正的小参数, $\varepsilon$为小滞时数, $x=(x_{1}, x_{2}, \cdots, x_{n}), u=(u_{1}, u_{2}, \cdots, u_{N})$, $\Omega$为有界凸域; $\partial\Omega$为$\Omega$的$C^{1+\alpha}$类的边界($\alpha$为Hölder系数), $C^{\infty}_{0}(\Omega)$为$C^{\infty}(\Omega)$在$\Omega$中的紧致子集, $K_{i}, f_{i}, g_{i}$和$h_{i}$为关于它们的变量在$C^{\infty}_{0}(\Omega)$内的函数,积分算子$Tu_{i}(t, x)=\int_{\Omega}K_{i}(x)u_{i}(t, x){\rm d}x$, $L$是在$C^{\infty}_{0}(\Omega)$上定义的有界函数$a^{jk}$的一致椭圆型的算子

$L_{i}=\sum\limits^{n}_{j, k=1}(D^{j}a^{jk}_{i}D^{k}), \ \ i=1, 2, \cdots, N, $

$D_{j}=\frac{\partial}{\partial x}, \ \D^{\alpha}=D^{\alpha_{1}}D^{\alpha_{2}}\cdots D^{\alpha_{n}}, \ \\alpha=\sum\limits^{n}_{j=1}\alpha_{j}, $

在Sobolev空间上的函数的有界模为

$\|\phi\|=\bigg\{\sum\limits^{n}_{\alpha_{j}=1}\int_{\Omega}|D^{\alpha}\phi(x)|^{2}{\rm d}x\bigg\}^{1/2}, \\ \forall\phi\in C^{\infty}_{0}(\Omega), $

而$(v, u)$为在$H^{\infty}_{0}(\Omega)$上的Hilbert内积.

首先假设

(H1) $0 <\mu/\varepsilon\ll 1$;

(H2) $\alpha^{jk}_{i}, \ f_{i}, \ g_{i}, \ h_{i}, \ K_{i}$,为关于其变量在相应区域内为Hölder连续的函数;

(H3)对于函数$u, v, $存在常数$C_{1}, C_{2}$,使得

$|(v, L_{i}[u])|\leq C_{1}\|v\|\|u\|, \ |(u, L_{i}[u])|\geqC_{2}\|u_{i}\|, \ \ \forall v, u\in H^{\infty}_{0}(\Omega), \ \i=1, 2, \cdots, N;$

(H4)系数$a^{jk}_{i}$在$\Omega$上有界,且有正常数$C_{3}$,使得$|a^{jk}_{i}(x)-a^{jk}_{i}(\overline{x})|\leqC_{3}(|x-\overline{x}|), $$ \forall x, \overline{x}\in\Omega, $$j, k=1, 2, \cdots, n, \ i=1, 2, \cdots, N$;

(H5)存在正常数$\delta_{i1}, \delta_{i2}$,使得

$\delta_{i1}\leq(\psi, f_{iu})\leq\delta_{i2}, \ \ \forallx\in\overline{\Omega}, \ \ \forall\psi\in H^{\infty}_{0}(\Omega), \\ i=1, 2, \cdots, N;$

(H6)积分方程

$(\psi(x), T_{i}u_{i})=(\psi(x), f_{i}(x, u_{i})), \ \ x\in\Omega, \ \\forall\psi(x)\in H^{\infty}_{0}(\Omega), \ \ i=1, 2, \cdots, N$

存在一组广义解$U_{i}\in H^{\infty}_{0}(\Omega), \ i=1, 2, \cdots, N$.

非线性微分-积分时滞奇摄动广义反应扩散系统的退化问题为

(2.1) $(\psi(x), T_{i}u_{i})=(\psi(x), f_{i}(x, u_{i})), \ \ \forall\psi(x)\inC^{\infty}_{0}(\Omega), \ \ i=1, 2, \cdots, N.$

由假设(H4), Fredholm积分方程(2.1)有一组广义解$U_{i00}\inH^{\infty}_{0}(\Omega)\ \ i=1, 2, \cdots, N$.

再将$T_{i}u_{i}(t-\varepsilon, x)$按时滞参数的幂展开

(2.2) $T_{i}u_{i}(t-\varepsilon, x)=T_{i0}u_{i}(t, x)+\sum\limits^{\infty}_{j=1}\bigg[\frac{1}{j!}\frac{\partial^{j}}{\partial\varepsilon^{j}}T_{i}u_{i}(t-\varepsilon, x)\bigg]_{\varepsilon=0}\varepsilon^{j}, \ \ i=1, 2, \cdots, N, $

并设广义初始-边值问题(1.1)-(1.3)的外部解$U_{i}(t, x), \ \i=1, 2, \cdots, N$为

(2.3) $U_{i}=\sum\limits^{\infty}_{j, k=1}U_{ijk}\varepsilon^{j}\mu^{k}, \ \i=1, 2, \cdots, N.$

将(2.1)和(2.2)式代入系统(1.1),按照$\varepsilon, \mu$的幂展开非线性项,合并对应$\varepsilon^{j}\mu^{k}$项的系数,令同次幂$\varepsilon^{j}\mu^{k}$$\ (i, k=0, 1, \cdots)$项的系数为零.取$\varepsilon^{0}\mu^{0}$项的系数为零时,得到的系统就是问题(1.1)-(1.3)的退化系统,故其解为$U_{i00}, \i=1, 2, \cdots, N$.取$\varepsilon^{j}, \mu^{k}, \ j, k=0, 1, \cdots, j+k\neq 0 $项的系数为零时,得到

(2.4) $(\psi(x), TU_{ijk})=(\psi(x), L_{i}U_{ij(k-2)})-(\psi(x), L_{i}\frac{\partial U_{(j-1)k}}{\partialt})+(\psi(x), F_{ijk}), \\\ \ \forall\psi(x)\in C^{\infty}_{0}(\Omega), \ \ j, k=0, 1, \cdots, \j+k\neq 0, \ \ i=1, 2, \cdots, N.$

上述和下面带有负下标的项均设为零,其中$F_{ijk}$为

$F_{ijk}=\sum\limits^{\infty}_{j, k=1}\bigg[\frac{1}{j!k!}\frac{\partial^{j+k}}{\partial\varepsilon^{j}\partial\mu^{k}}f_{i}(x, \sum\limits^{\infty}_{p, q=0}U_{i}\varepsilon^{p}\mu^{q})\bigg]_{\varepsilon=\mu=0}, \ \ j, k=0, 1, \cdots, j+k\neq 0$

逐次已知的函数.由假设, Fredholm积分方程(2.4)可依次得到解$U_{ijk}(t, x)$.于是由(2.1)式可得到一组广义外部解$U=(U_{1}, U_{2}, \cdots, U_{N})$.但是它未必满足边界条件(1.2)和初始条件(1.3),所以尚需构造在$t=0$附近的初始层校正项$Y$和在$\partial\Omega$附近的边界层校正项$W$.

现构造初始层校正项$Y$.引入伸长变量$\tau=t/\varepsilon$,并设

(3.1) $\begin{equation}u_{i}=\sum\limits^{\infty}_{j, k=1}U_{ijk}\varepsilon^{j}\mu^{k}+Y_{i}, \\ i=1, 2, \cdots, N, \end{equation}$

这里

(3.2) $\begin{equation}Y_{i}=\sum\limits^{\infty}_{j, k=1}y_{ijk}(\tau, x)\varepsilon^{j}\sigma^{k}, \ \ i=1, 2, \cdots, N, \end{equation}$

其中$\sigma=\mu/\varepsilon$.

将(3.1), (3.2)式代入(1.1), (1.3)式,按照$\varepsilon, \sigma$的幂展开非线性项,合并对应的$\varepsilon^{j}\sigma^{k}$项的系数,并令同次幂$\varepsilon^{j}\sigma^{k}\ (i, j=0, 1, \cdots)$项的系数为零有

(3.3) $\begin{eqnarray}& (\psi(x), \frac{\partial y_{i00}(\tau, x)}{\partial\tau})+(\psi(x), T_{i}y_{i00}(\tau, x))=(\psi(x), f_{i}(x, y_{i00}(\tau, x)+U_{i00}(0, x))), \\[2mm]&\forall\psi(x)\in C^{\infty}_{0}(\Omega), \ \ i=1, 2, \cdots, N, \end{eqnarray}$

(3.4) $\begin{equation}(\psi(x), y_{i00}(0, x))=(\psi(x), h_{i}(x)), \ \ x\in\partial\Omega, \\ \forall\psi(x)\in C^{\infty}_{0}(\Omega), \ \ i=1, 2, \cdots, N, \end{equation}$

(3.5) $\begin{eqnarray}&&(\psi(x), \frac{\partial y_{ijk}(\tau, x)}{\partial\tau})+(\psi(x), T_{i}y_{ijk}(\tau, x))-(\psi(x), f_{iy}(x, y_{i00}(0, x)))y_{ijk}(\tau, x)\\&=&(\psi(x), L_{i}y_{ij(k-2)}(\tau, x))+\widetilde{F}_{ijk}, \ \\forall\psi(x)\in C^{\infty}_{0}(\Omega), \ \ i=1, 2, \cdots, N, \end{eqnarray}$

(3.6) $\begin{equation}(\psi(x), y_{ijk}(0, x))=0, \ \ x\in\partial\Omega, \ \ \forall\psi(x)\inC^{\infty}_{0}(\Omega), \ \ i=1, 2, \cdots, N, \end{equation}$

其中$\widetilde{F}_{ijk}\ (j, k=0, 1, \cdots, \ j+k\neq 0, \i=1, 2, \cdots, N)$为逐次已知的函数,其结构从略.

由初值问题(3.3), (3.4)和(3.5), (3.6),可依次得到$y_{ijk}, \j, k=0, 1, \cdots, \ j+k\neq 0, \ i=1, 2, \cdots, N$.因此由(3.2)式,可得到在$t=0$邻域内的一组初始层校正项$Y=(Y_{1}, Y_{2}, \cdots, Y_{N})$,且不难看出

(3.7) $y_{ijk}=O(\exp(-\overline{k}_{ijk}\frac{t}{\varepsilon})), \ \j, k=0, 1, \cdots, \ \ i=1, 2, \cdots, N, \ \ 0 <\varepsilon\ll 1, $

其中$\overline{k}_{ijk}\ j, k=0, 1, \cdots, \ i=1, 2, \cdots, N$为正常数.

在$\Omega$的边界$\partial\Omega$的邻域内构造一个局部坐标系$(\rho, \phi)$:在$\Omega$的边界$\partial\Omega$的邻域内每一点$P$的$\rho(0\leq\rho_{0})$为点$P$到$\partial\Omega$的距离,这里$\rho_{0}$为适当小的正常数,使得在边界$\partial\Omega$上各点的内法线在$\partial\Omega$的邻域$0\leq\rho\leq\rho_{0}$内相互不相交.坐标$\phi=(\phi_{1}, \phi_{2}, \cdots, \phi_{n-1})$是边界$\partial\Omega$上的一个$n-1$维非奇异坐标系,设点$P$的坐标$\phi$是通过点$P$的内法线和边界$\partial\Omega$的交点$Q$的坐标$\phi$相同.

在$\partial\Omega$的邻域$0\leq\rho\leq\rho_{0}$中,设

$L=\overline{D}^{n}_{i}\overline{a}^{nn}_{i}(\rho, \phi)\overline{D}^{n}_{i}+\sum\limits^{n-1}_{j, k=1}\overline{D}^{j}_{i}\overline{a}^{jk}_{i}(\rho, \phi)\overline{D}^{k}_{i}+\overline{D}^{n}_{i}\overline{b}^{n}_{i}(\rho, \phi)+\sum\limits^{n-1}_{j=1}\overline{D}^{j}_{i}\overline{b}^{j}_{i}(\rho, \phi), \ \ i=1, 2, \cdots, N, $

$\overline{D}^{n}_{i}=\frac{\partial}{\partial \rho}, \ \\overline{D}^{j}_{i}=\frac{\partial}{\partial \phi_{j}}, \ \i=1, 2, \cdots, N, \ j=1, 2, \cdots, n-1, $

其中$\overline{a}^{jk}_{i}, \overline{a}^{j}_{i}, \overline{b}^{j}_{i}$为已知函数,它们的结构从略.

再在$0\leq\rho\leq\rho_{0}$上引入多重尺度变量^{[1, 2]}

(4.1) $\zeta_{i}=\frac{h_{i}(\rho, \phi)}{\mu}, \ \overline{\rho}=\rho, \\phi=\phi, \ \ i=1, 2, \cdots, N, $

这里$h_{i}(\rho, \phi)$为待定函数.不失一般性,以下仍用$\rho$来代替$\overline{\rho}$.由(4.1)式,我们有

(4.2) $L=\frac{1}{\mu^{2}}K_{i0}+\frac{1}{\mu}K_{i1}+K_{i2}, $

其中$K_{i0}=\overline{a}^{nn}_{i}h^{2}_{i}\frac{\partial^{2}}{\partial\zeta^{2}}$,而$K_{i1}$和$K_{i2}$的构造从略.设积分-微分方程初-边值问题(1.1)-(1.3)的广义解$(u_{1}, u_{2}, \cdots, u_{N})$为

(4.3) $u_{i}=U_{i}+W_{i}, \ \ i=1, 2, \cdots, N, $

其中

(4.4) $W_{i}=\sum\limits^{\infty}_{0}w_{ijk}(t, \zeta_{i}, \phi)\varepsilon^{j}\mu^{k}, \\ i=1, 2, \cdots, N.$

选取$h_{i}(\rho, \phi)=\int^{\rho}_{0}1/\sqrt{a^{nn}_{i}(\rho)}$,将(2.3), (3.2), (4.3)和(4.4)式代入(1.1), (1.2)式,按$\varepsilon, \mu$的幂级数展开非线性项,合并$\varepsilon^{j}\mu^{k}$同次幂项,并令其的系数为零,得

(4.5) $(\psi(\rho, \phi), K_{i0}w_{i00})-(\psi(\rho, \phi), T_{i}w_{i00})=(\psi(\rho, \phi), f_{i}(\rho, \phi, U_{i00}+w_{i00})-f_{i}(\rho, \phi, U_{i00})), \\\forall\psi(\rho, \phi)\in C^{\infty}_{0}(\Omega)\ \i=1, 2, \cdots, N, $

(4.6) $(\psi(\rho, \phi), w_{i00})=(\psi(\rho, \phi), g_{i}(t, 0, \phi)), ~~x\in\partial\Omega, \ \forall\psi(\rho, \phi)\in C^{\infty}_{0}(\Omega) , \ i=1, 2, \cdots, N, $

(4.7) $(\psi(\rho, \phi), K_{i0}w_{ijk})-(\psi(\rho, \phi), T_{i}w_{ijk})=(\psi(\rho, \phi), f_{iy_{i}}(\rho, \phi, U_{i00}+w_{i00})w_{ijk}+G_{ijk}), \\\forall\psi(\rho, \phi)\in C^{\infty}_{0}(\Omega), \ \j, k=0, 1, \cdots, \ j+k\neq 0, \ \ i=1, 2, \cdots, N, \ \$

(4.8) $(\psi(\rho, \phi), w_{ijk})=(\psi(\rho, \phi), U_{ijk}(t, 0, \phi)), \\x\in\partial\Omega, \ \forall\psi(\rho, \phi)\in C^{\infty}_{0}(\Omega), \ j, k=0, 1, \cdots, \ j+k\neq 0, \ \i=1, 2, \cdots, N, $

其中$G_{ijk}\ (j, k=0, 1, \cdots, \ j+k\neq 0)$为逐次已知函数,其结构也从略.

由(4.5), (4.6)式以及(4.7), (4.8)式和假设,可依次得到具有衰减性质的解$w_{ijk}(t, \rho, \phi)$为

(4.9) $w_{ijk}=O(\exp(-\widetilde{k}_{ijk}\zeta_{i})), \ \ j, k=0, 1, \cdots, \ \ i=1, 2, \cdots, N, $

这里$\widetilde{k}_{ijk}\ (j, k=0, 1, \cdots, i=1, 2, \cdots, N)$为适当小的正常数.

再引入一个充分光滑的分割函数$\overline{\psi}(\rho)$为

$\overline{\psi}(\rho)=1, \ \ 0\leq\rho\leq (2/3)\rho_{0}; \ \ \ \\overline{\psi}(\rho)=0, \ \ (2/3)\leq\rho.$

令$\overline{w}_{ijk}=\overline{\psi}(\rho)w_{ijk}\(j, k=0, 1, \cdots, \ i=1, 2, \cdots, N )$.为了方便,以下仍将$w_{ijk}$来代替$\overline{w}_{ijk}$.将$w_{ijk}$代入(4.4)式,便得到具有边界层校正性质的一组广义解$W=(W_{1}, W_{2}, \cdots, W_{N})$.

由上面的讨论,可得非线性微分-积分时滞奇摄动广义反应扩散系统(1.1)-(1.3)的一组广义解$u(t, x)=(u_{1}(t, x), u_{2}(t, x), \cdots, u_{N}(t, x))$,并有如下渐近展开式

(5.1) $\begin{eqnarray}u_{i}(t, x)&=&\sum^{\infty}_{j, k=0}U_{ijk}(t, x)\varepsilon^{j}\mu^{k}+\sum^{\infty}_{j, k=0}y_{ijk}(\tau, x)\varepsilon^{j}\sigma^{k}+\sum^{\infty}_{j, k=0}w_{ijk}(t, \zeta)\varepsilon^{j}\mu^{k}, \\&&~~~~ i=1, 2, \cdots, N, \ \ 0 <\varepsilon, \sigma, \mu\ll 1.\end{eqnarray}$

下面来讨论广义解存在性及其渐近展开式(5.1)的一致有效性.

首先建立如下非线性微分-积分时滞奇摄动广义反应扩散系统的微分不等式理论.

定义  设有两组函数$\overline{u}_{i}, \i=1, 2, \cdots, N$和$\underline{u}_{i}, \ i=1, 2, \cdots, N, \\overline{u}_{i}\geq\underline{u}_{i}$,并分别满足不等式

$\begin{eqnarray*}&&\varepsilon(\psi(x), \frac{\partial \overline{u}_{i}(t, x)}{\partialt})-\mu^{2}(\psi(x), L_{i}\overline{u}_{i}(t, x))+(\psi(x), T_{i}\overline{u}_{i}(t-\varepsilon, x))\\&\geq&(\psi(x), f_{i}(x, \overline{u}_{i}(t, x))), \ \(t, x)\in(0, \infty)\times\Omega, \ \ \forall\psi(x)\inC^{\infty}_{0}(\Omega), \ \ i=1, 2, \cdots, N, \end{eqnarray*}$

$(\psi(x), \overline{u}_{i}(t, x))\geq(\psi(x), A_{i}(t, x)), \ \x\in\partial\Omega, \ \ \forall\psi(x)\in C^{\infty}_{0}(\Omega) , \\ i=1, 2, \cdots, N, $

$(\psi(x), \overline{u}_{i}(t, x))\geq(\psi(x), B_{i}(t, x)), \ \(t, x)\in [-\varepsilon, 0]\times\Omega, \ \ \forall\psi(x)\inC^{\infty}_{0}(\Omega) , \ \ i=1, 2, \cdots, N$

和

$\varepsilon(\psi(x), \frac{\partial\underline{u}_{i}(t, x)}{\partialt})-\mu^{2}(\psi(x), L_{i}\underline{u}_{i}(t, x))+(\psi(x), T_{i}\underline{u}_{i}(t-\varepsilon, x))\leq(\psi(x), f_{i}(x, \underline{u}_{i}(t, x))), $

$(t, x)\in(0, \infty)\times\Omega, \ \ \forall\psi(x)\inC^{\infty}_{0}(\Omega), \ \$

$(\psi(x), \underline{u}_{i}(t, x))\leq(\psi(x), A_{i}(t, x)), \ \x\in\partial\Omega, \ \ \forall\psi(x)\in C^{\infty}_{0}(\Omega) , \ \i=1, 2, \cdots, N, $

$(\psi(x), \underline{u}_{i}(t, x))\leq(\psi(x), B_{i}(t, x)), \ \(t, x)\in [-\varepsilon, 0]\times\Omega, \ \ \forall\psi(x)\inC^{\infty}_{0}(\Omega) , \ \ i=1, 2, \cdots, N, $

则分别称$\overline{u}_{i}(t, x)$和$\underline{u}_{i}(t, x)$为初始-边值问题(1.1)-(1.3)的上解和下解.

现有如下定理.

定理1  在假设(H1)-(H5)下, $\underline{u}_{i}(t, x), i=1, 2, \cdots, N$和$\overline{u}_{i}(t, x), i=1, 2, \cdots, N$,分别为非线性微分-积分时滞奇摄动广义反应扩散系统初始-边值问题(1.1)-(1.3)的下解和上解,则问题(1.1)-(1.3)存在一组广义解$u_{i}(t, x), i=1, 2, \cdots, N$,并成立$\underline{u}_{i}(t, x)\leq u_{i}(t, x)\leq \overline{u}_{i}(t, x), i=1, 2, \cdots, N$.

证  首先按以下关系式构造迭代序列

(5.2) $\begin{eqnarray}&&\varepsilon(\psi(x), \frac{\partial u_{i(k+1)}(t, x)}{\partialt})-\mu^{2}(\psi(x), L_{i}u_{i(k+1)}(t, x))+(\psi(x), T_{i}u_{i(k+1)}(t-\varepsilon, x))\\&=&(\psi(x), f_{i}(x, u_{ik}(t, x))), \ \(t, x)\in(0, \infty)\times\Omega, \ \forall\psi(x)\inC^{\infty}_{0}(\Omega), \ \\end{eqnarray}$

(5.3) $(\psi(x), u_{i(k+1)}(t, x))=(\psi(x), A_{i}(t, x)), \ \\forall\psi(x)\in C^{\infty}_{0}(\Omega) , \ \ i=1, 2, \cdots, N, $

(5.4) $(\psi(x), u_{i(k+1)}(t, x))=(\psi(x), B_{i}(t, x)), \\(t, x)\in [-\varepsilon, 0]\times\Omega, \ \ \forall\psi(x)\inC^{\infty}_{0}(\Omega) , \ \ i=1, 2, \cdots, N.$

由(5.2)-(5.4)式,设$\overline{u}_{i0}=\overline{u}_{i}$和$\underline{u}_{i0}=\overline{u}_{i}$为两组的零次函数,我们可依次地可得$\overline{u}_{in}, n=1, 2\cdots$和$\underline{u}_{in}, n=1, 2\cdots$.因此能得到两组函数列$\{\overline{u}_{in}\i=1, 2, \cdots, N\}$和$\{\underline{u}_{in}$\ $i=1, 2, \cdots, N\}$.现讨论其收敛性态.

设$\overline{v}_{i0}=\overline{u}_{i0}-\overline{u}_{i1}$,由假设和(5.2)-(5.4)式有

$\begin{eqnarray*}&&\varepsilon(\psi(x), \frac{\partial\overline{v}_{i0}(t, x)}{\partialt})-\mu^{2}(\psi(x), L_{i}\overline{v}_{i0}(t, x))+(\psi(x), T_{i}\overline{v}_{i0}(t-\varepsilon, x))\\&=&\varepsilon(\psi(x), \frac{\partial\overline{u}_{i0}(t, x)}{\partialt})-\mu^{2}(\psi(x), L_{i}\overline{u}_{i0}(t, x))+(\psi(x), T_{i}\overline{u}_{i0}(t-\varepsilon, x))\\&&-\varepsilon(\psi(x), \frac{\partial\overline{u}_{i1}(t, x)}{\partialt})-\mu^{2}(\psi(x), L_{i}\overline{u}_{i1}(t, x))+(\psi(x), T_{i}\overline{u}_{i1}(t-\varepsilon, x))\\&\geq& (\psi(x), f_{i}(x, \overline{u}_{i0}(t, x)))-(\psi(x), f_{i}(x, \overline{u}_{i0}(t, x)))=0, \\&&~~(t, x)\in(0, \infty)\times\Omega, \ \ \forall\psi(x)\inC^{\infty}_{0}(\Omega) , \ \ i=1, 2, \cdots, N, \end{eqnarray*}$

$(\psi(x), \overline{v}_{i0}(t, x))=(\psi(x), \overline{u}_{i0}(t, x))-(\psi(x), \overline{u}_{i1}(t, x))=0, $

$x\in\partial\Omega, \ \ \forall\psi(x)\in C^{\infty}_{0}(\Omega) , \\ i=1, 2, \cdots, N, $

$(\psi(x), \overline{v}_{i0}(0, x))=(\psi(x), \overline{u}_{i0}(0, x))-(\psi(x), \overline{u}_{i1}(0, x))=0, $

$(t, x)\in [-\varepsilon, 0]\times\Omega, \forall\psi(x)\inC^{\infty}_{0}(\Omega) , \ \ i=1, 2, \cdots, N.$

于是由线性问题的极值原理得到$\overline{v}_{i0}\geq 0, i=1, 2, \cdots, N$.即

$\overline{u}_{i0}\geq\overline{u}_{i1}, ~ i=1, 2, \cdots, N.$

若$\overline{u}_{in}\geq\overline{u}_{i(n+1)}$,设$\overline{v}_{in}=\overline{u}_{in}-\overline{u}_{i(n+1)}$,这时有

$\begin{eqnarray*}&&\varepsilon(\psi(x), \frac{\partial\overline{v}_{in}(t, x)}{\partialt})-\mu^{2}(\psi(x), L_{i}\overline{v}_{in}(t, x))+(\psi(x), T_{i}\overline{v}_{in}(t-\varepsilon, x))\\&=&\varepsilon(\psi(x), \frac{\partial\overline{u}_{in}(t, x)}{\partialt})-\mu^{2}(\psi(x), L_{i}\overline{u}_{in}(t, x))+(\psi(x), T_{i}\overline{u}_{in}(t-\varepsilon, x))\\&&-\varepsilon(\psi(x), \frac{\partial\overline{u}_{i(n+1)}(t, x)}{\partial{t}})-\mu^{2}(\psi(x), L_{i}\overline{u}_{i(n+1)(t, x)})+(\psi(x), T_{i}\overline{u}_{i(n+1)}(t-\varepsilon, x))\\&\geq& (\psi(x), f_{i}(x, \overline{u}_{in}(t, x)))-(\psi(x), f_{i}(x, \overline{u}_{i(n+1)}(t, x)))\geq 0, \\&&~~(t, x)\in(0, \infty)\times\Omega, \ \ \forall\psi(x)\inC^{\infty}_{0}(\Omega) , \ \ i=1, 2, \cdots, N, \end{eqnarray*}$

$(\psi(x), \overline{v}_{in}(t, x))=(\psi(x), \overline{u}_{in}(t, x))-(\psi(x), \overline{u}_{i(n+1)}(t, x))=0, $

$x\in\partial\Omega, \ \ \forall\psi(x)\in C^{\infty}_{0}(\Omega) , \\ i=1, 2, \cdots, N, $

$(\psi(x), \overline{v}_{in}(0, x))=(\psi(x), \overline{u}_{in}(0, x))-(\psi(x), \overline{u}_{i(n+1)}(0, x))=0, $

$(t, x)\in [-\varepsilon, 0]\times\Omega, \forall\psi(x)\inC^{\infty}_{0}(\Omega) , \ \ i=1, 2, \cdots, N.$

于是由线性问题的极值原理得到$\overline{v}_{in}\geq 0, i=1, 2, \cdots, N$.即

$\overline{u}_{in}\geq\overline{u}_{i(n+1)}, \ \ i=1, 2, \cdots, N.$

由此我们得到

$\overline{u}_{i}=\overline{u}_{i0}\geq\overline{u}_{i1}\geq\cdots\geq\overline{u}_{in}\geq\overline{u}_{i(n+1)}\geq\cdots, \ \i=1, 2, \cdots, N.$

类似有

$\underline{u}_{i}=\underline{u}_{i0}\leq\underline{u}_{i1}\leq\cdots\leq\underline{u}_{in}\leq\underline{u}_{i(n+1)}\leq\cdots, \\ i=1, 2, \cdots, N.$

同样可得

$\overline{u}_{in}\geq\underline{u}_{in}, \ \ n=0, 1, \cdots, \ \i=1, 2, \cdots, N.$

由上述的结果和Arzela定理,非线性微分-积分时滞奇摄动广义反应扩散系统初始-边值问题(1.1)-(1.3)存在一组广义解$u(t, x)=(u_{1}(t, x), u_{2}(t, x), \cdots, u_{N}(t, x))$,且成立不等式$\underline{u}_{i}(t, x)\leq u_{i}(t, x)\leq\overline{u}_{i}(t, x), \ i=1, 2, \cdots, N$.定理1证毕.

定理2  在假设(H1)-(H6)下,非线性微分-积分时滞奇摄动广义反应扩散系统初始-边值问题(1.1)-(1.3)存在一组广义解$u(t, x)=(u_{1}(t, x), u_{t, 2}(t, x), \cdots, u_{N}(t, x))$,并具有形如(5.1)式的一致有效的渐近展开式.

证  构造两组辅助函数$\alpha_{i}, \beta_{i}, \ i=1, 2, \cdots, N$

(5.5) $\alpha_{i}=Z_{iM}-r_{i}\lambda, \ \ i=1, 2, \cdots, N, $

(5.6) $\beta_{i}=Z_{iM}+r_{i}\lambda, \ \ i=1, 2, \cdots, N, $

其中$r_{i}$为适当大的待定正常数, $\lambda=\max(\varepsilon^{M}\mu^{M+1}, \varepsilon^{M}\mu^{M+1})$,而

$Z_{iM}(t, x)=\sum\limits^{M}_{j, k=0}U_{ijk}(t, x)\varepsilon^{j}\mu^{k}+\sum\limits^{M}_{j, k=0}y_{ijk}(\tau, x)\varepsilon^{j}\sigma^{k}+\sum\limits^{M}_{j, k=0}w_{ijk}(t, \zeta)\varepsilon^{j}\mu^{k}.$

显然,选取足够大的$r_{i}, i=1, 2, \cdots, N$,当$r_{i}\geqr_{i0}$时,恒有

(5.7) $(\psi(x), \alpha_{i}(t, x))\leq(\psi(x), \beta_{i}(t, x)), \\(t, x)\in [-\varepsilon, \infty)\times\Omega\ \ \forall\psi(x)\inC^{\infty}_{0}(\Omega) , \ \ i=1, 2, \cdots, N, $

(5.8) $(\psi(x), \alpha_{i}(t, x))\leq(\psi(x), g_{i}(t, x))\leq(\psi(x), \beta_{i}(t, x)), \\x\in\partial\Omega, \ \ \forall\psi(x)\in C^{\infty}_{0}(\Omega), \\ i=1, 2, \cdots, N.$

由假设,存在正常数$D_{i1}$有

$\begin{eqnarray*}(\psi(x), \alpha_{i}(t, x))&=&(\psi(x), Z_{iM}-r_{i}\lambda)\\&=&(\psi(x), \sum^{M}_{j, k=0}U_{ijk}(t, x)\varepsilon^{j}\mu^{k}+\sum^{M}_{j, k=0}y_{ijk}(\tau, x)\varepsilon^{j}\sigma^{k}+\sum^{M}_{j, k=0}w_{ijk}(t, \zeta)\varepsilon^{j}\mu^{k}-r_{i}\lambda)\\&\leq& (\psi(x), h_{i}(x))+(D_{i1}-r_{i})\lambda), \\&&(t, x)\in[-\varepsilon, 0]\times\Omega, \ \ \forall\psi(x)\inC^{\infty}_{0}(\Omega), \ \ i=1, 2, \cdots, N.\end{eqnarray*}$

选取$r_{i}\geq D_{i1}$,可得

$(\psi(x), \alpha_{i}(t, x))\leq(\psi(x), h_{i}(x)), $

$(t, x)\in[-\varepsilon, 0]\times\Omega, \ \ \forall\psi(x)\inC^{\infty}_{0}(\Omega), \ \ i=1, 2, \cdots, N.$

同理可得

$(\psi(x), \beta_{i}(t, x))\geq(\psi(x), h_{i}(x)), $

$(t, x)\in[-\varepsilon, 0]\times\Omega, \ \ \forall\psi(x)\inC^{\infty}_{0}(\Omega), \ \ i=1, 2, \cdots, N.$

即成立

(5.9) $(\psi(x), \alpha_{i}(t, x))\leq(\psi(x), h_{i}(x))\leq(\psi(x), \beta_{i}(x)), \\(t, x)\in[-\varepsilon, 0]\times\Omega, \ \ \forall\psi(x)\inC^{\infty}_{0}(\Omega), \ \ i=1, 2, \cdots, N.$

下面来证明

(5.10) $\varepsilon(\psi(x), \frac{\partial \alpha_{i}(t, x)}{\partialt})-\mu^{2}(\psi(x), L_{i}\alpha_{i}(t, x))+(\psi(x), T_{i}\alpha_{i}(t-\varepsilon, x))-(\psi(x), f_{i}(x, \alpha_{i}(t, x)))\leq 0, \\(t, x)\in(0, \infty)\times\Omega, \ \ \forall\psi(x)\inC^{\infty}_{0}(\Omega), \ \ i=1, 2, \cdots, N, $

(5.11) $\varepsilon(\psi(x), \frac{\partial \beta_{i}(t, x)}{\partialt})-\mu^{2}(\psi(x), L_{i}\beta_{i}(t, x))+(\psi(x), T_{i}\beta_{i}(t-\varepsilon, x))-(\psi(x), f_{i}(x, \beta_{i}(t, x)))\geq 0, \\(t, x)\in (0, \infty)\times\Omega, \ \ \forall\psi(x)\inC^{\infty}_{0}(\Omega), \ \ i=1, 2, \cdots, N.$

由假设和(3.7), (4.9)式,对足够小的$\varepsilon, \mu$,存在正常数$D_{i2}$有

$\begin{eqnarray*}&&\varepsilon(\psi(x), \frac{\partial \alpha_{i}(t, x)}{\partialt})-\mu^{2}(\psi(x), L_{i}\alpha_{i}(t, x))+(\psi(x), T_{i}\alpha_{i}(t-\varepsilon, x))-(\psi(x), f_{i}(x, \alpha_{i}(t, x)))\\&=&\varepsilon(\psi(x), \frac{\partial(Z_{iM}(t, x)-r_{i}\lambda)}{\partialt})-\mu^{2}(\psi(x), L_{i}(Z_{iM}(t, x)-r_{i}\lambda))\\&&+(\psi(x), T_{i}(Z_{iM}(t-\varepsilon, x)-r_{i}\lambda))-(\psi(x), f_{i}(x, (Z_{iM}(t, x))-r_{i}\lambda))\\&=&(\psi(x), T_{i}U_{i00})-(\psi(x), f_{i}(x, U_{i00}))+\sum^{M}_{j, k=0, j+k\neq0}[(\psi(x), TU_{ijk})-(\psi(x), L_{i}[U_{ij(k-2)}])\\&&+(\psi(x), L_{i}[\frac{\partial U_{(j-1)k}}{\partialt}])-(\psi(x), F_{ijk})]\varepsilon^{j}\mu^{k}+(\psi(x), \frac{\partial y_{i00}(\tau, x)}{\partial\tau})\\&&+(\psi(x), T_{i}y_{i00}(\tau, x))-(\psi(x), f_{i}(x, y_{i00}(\tau, x))+U_{i00}(0, x))\\&&+\sum^{M}_{j, k=1, j+k\neq 0}[(\psi(x), \frac{\partialy_{ijk}(\tau, x)}{\partial \tau})+(\psi(x), T_{i}y_{ijk}(\tau, x))-(\psi(x), f_{iy}(x, y_{i00}(0, x))y_{ijk}(\tau, x))\\&&-(\psi(x), L_{i}y_{ij(k-2)}(\tau, x))-\widetilde{F}_{ijk}]\varepsilon^{j}\sigma^{k}+(\psi(\rho, \phi), K_{i0}w_{i00})-(\psi(\rho, \phi), T_{i}w_{i00})\\&&-(\psi(\rho, \phi), f_{i}(\rho, \phi, U_{i00}+w_{i00})+f_{i}(\rho, \phi, U_{i00}))\\&&+\sum^{M}_{j, k=1, j+k\neq0}[(\psi(\rho, \phi), K_{i0}w_{ijk})-(\psi(\rho, \phi), T_{i}w_{ijk})\\&&-(\psi(\rho, \phi), -f_{iy_{i}}(\rho, \phi, U_{i00}+w_{i00})w_{ijk}-G_{ijk})]\varepsilon^{j}\mu^{k}+(\psi(x), D_{i2}\lambda)-(\psi(x), \sum^{N}_{i=1}\delta_{i2}r_{i}\lambda)\\&\leq&(\psi(x), D_{i2}\lambda-\sum^{N}_{i=1}\delta_{i2}r_{i}\lambda)\leq(\psi(x), (D_{i2}\lambda))-(\psi(x), \sum^{N}_{i=1}\delta_{i2}r_{i}\lambda)\\&=&(\psi(x), (D_{i2}-\sum^{N}_{i=1}\delta_{i2}r_{i})\lambda).\end{eqnarray*}$

因此选取$r_{i}\geq D_{i2}/\sum\limits^{N}_{i=1}\delta_{i2}$,不等式(5.10)成立.同理可证不等式(5.11)也成立.

综上所证,选择足够大的$r_{i}$,并对足够小的$\varepsilon, \mu$,关系式(5.7)-(5.11)成立.于是由定理1,初始-边值问题(1.1)-(1.3)存在一组广义解$u=(u_{1}, u_{1}, \cdots, u_{N})$,且成立

$\alpha_{i}(t, x)\leq u_{i}(t, x)\leq \beta_{i}(t, x), \ \(t, x)\in[0, \infty)\times(\Omega+\partial\Omega).$

再由(5.5), (5.6)式知,对于$i=1, 2, \cdots, N$,有

$u_{i}(t, x)=\sum\limits^{M}_{j, k=0}U_{ijk}(t, x)\varepsilon^{j}\mu^{k}+\sum\limits^{M}_{j, k=0}y_{ijk}(\tau, x)\varepsilon^{j}\sigma^{k}+\sum\limits^{M}_{j, k=0}w_{ijk}(t, \zeta_{i})\varepsilon^{j}\mu^{k}+O(\lambda), $

$\lambda=\max(\varepsilon^{M+1}\mu^{M}, \varepsilon^{M}\mu^{M+1}), \ \i=1, 2, \cdots, N, \ \ 0 <\varepsilon, \sigma, \mu\ll 1.$

即非线性微分-积分时滞奇摄动广义反应扩散系统初始-边值问题(1.1)-(1.3)的解

$u(t, x)=(u_{1}(t, x), u_{2}(t, x), \cdots, u_{N}(t, x))$

的分量$u_{i}(t, x), \ i=1, 2, \cdots, N$具有形如(5.1)式的一致有效的渐近展开式.定理证毕.