具常重特征的拟线性双曲组的经典解的奇性形成

Formation of Singularities for Quasilinear Hyperbolic Systems with Characteristic with Constant Multiplicity

摘要: 文章研究的是带有非齐次项的且具常重特征的拟线性双曲组，其中源项满足相应的匹配条件。不加重特征是线性退化的限制，考虑具有一定衰减性的初值Cauchy问题的C¹解的整体存在性及破裂现象并估计出解的生命跨度。

Abstract: The problem which we research on is inhomogeneous quasilinear hyperbolic systems with char-acteristics with constant multiplicity, where the source term satisfied the corresponding matching condition. And we don’t restrict the condition that the characteristics with multiplicity are weakly linearly degenerate. Next, we shall investigate the global existence or the blow-up phenomenon of C¹ solutions to the Cauchy problem with weaker decaying initial data and estimate the life-span of C¹ solutions.

1. 引言及主要结果

考虑下面的一阶拟线性双曲组

$\frac{\partial u}{\partial t}+A\left(u\right)\frac{\partial u}{\partial x}=F\left(u\right),$ (1.1)

其中 $u={\left(u_1,\cdots ,u_n\right)}^{\text{T}}$ 是变量 $\left(t,x\right)$ 的未知向量函数，而 $A\left(u\right)=\left(a_{ij}\left(u\right)\right)\left(i,j=1,\cdots ,n\right)$ 是元素适当光滑的 $n\times n$ 矩阵，非齐次项 $F\left(u\right)={\left(F_1\left(u\right),\cdots ,F_n\left(u\right)\right)}^{\text{T}}$ 是元素适当光滑的向量函数。由双曲性可知，对所考虑范围上任一给定的u值， $A\left(u\right)$ 有n个实的特征值 ${\lambda }_1\left(u\right),\cdots {\lambda }_n\left(u\right)$ 和一个完备的左(右)特征向量组。对于 $i=1,\cdots ,n$，令 $l_i\left(u\right)=\left(l_{i1}\left(u\right),\cdots ,l_{in}\left(u\right)\right)$ (相应地， $r_i\left(u\right)={\left(r_{i1}\left(u\right),\cdots ,r_{in}\left(u\right)\right)}^{\text{T}}$ )是对应于 ${\lambda }_i\left(u\right)$ 的左(相应地，右)特征向量：

$l_i\left(u\right)A\left(u\right)={\lambda }_i\left(u\right)l_i\left(u\right)(相应地,A\left(u\right)r_i\left(u\right)={\lambda }_i\left(u\right)r_i(u))$

有

$\det \left|l_{ij}\left(u\right)\right|\ne 0\left(等价地,\det \left|r_{ij}\left(u\right)\right|\ne 0\right).$

不失一般性，假设在所考虑的范围上

$l_i\left(u\right)r_j\left(u\right)\equiv {\delta }_{ij},\left(i,j=1,\cdots ,n\right),$ (1.2)

$r_i^{\text{T}}\left(u\right)r_i\left(u\right)\equiv 1,\left(i=1,\cdots ,n\right),$

其中 ${\delta }_{ij}$ 是Kronecker符号。

在方程组是严格双曲的假设下，早期利用弱线性退化的概念，李大潜、周忆和孔德兴在 [1] [2] 中，对方程组(1.1)具一定衰减性的小C¹初值的Cauchy问题给出了C¹解的整体存在性和破裂现象的结果。之后，这些结果又被推广到具有常重特征的非严格双曲组的情况( [3] 及 [4] )，这里所有的常重特征均为或均假设为线性退化的。其后，王利彬在不加重特征是线性退化的限制下，对具有一定衰减性的小C¹初值的Cauchy问题解决了C¹解的破裂问题( [5] )。本文在王利彬的基础上，将齐次项改为非齐次项 $F\left(u\right)$，考虑当 $F\left(u\right)\in C^3$ 且满足匹配条件时，对上面的Cauchy问题，其C¹解在有限时间内的奇性形成问题。

在本文中，对于具常重特征的双曲组(1.1)恒假设所有 ${\lambda }_i\left(u\right),l_{ij}\left(u\right)$ 及 $r_{ij}\left(u\right)\left(i,j=1,\cdots ,n\right)$ 均与 $a_{ij}\left(u\right)\left(i,j=1,\cdots ,n\right)$ 有同样的正规性。

不失一般性，设在 $u=0$ 的某个邻域内成立

$\lambda \left(u\right)={\lambda }_1\left(u\right)\equiv \cdots \equiv {\lambda }_p\left(u\right)<{\lambda }_{p+1}\left(u\right)<\cdots <{\lambda }_n\left(u\right),$ (1.3)

其中 $1\le p\le n$。当 $p=1$ 时，方程组(1.1)是严格双曲的；当 $p>1$ 时，方程组(1.1)是具常重特征 $p\left(>1\right)$ 的非严格双曲组。

为了简练，本文不再赘述弱线性退化，标准坐标和匹配条件的概念(看 [6] )。

考虑方程组(1.1)具如下初值

$u\left(0,x\right)=\epsilon \psi \left(x\right)$ (1.4)

的Cauchy问题，其中 $\epsilon >0$ 是一个小参数。本文的主要结果是：

定理1.1：设方程组(1.1)是非严格双曲组，(1.3)式成立，且在 $u=0$ 的某个邻域中，A是适当光滑的，进一步假设 $F\in C^3$ 满足匹配条件且系统(1.1)不是弱线性退化的，且

$\alpha =\min \left\{{\alpha }_i|i\in J\right\}<+\infty ,$

其中 ${\alpha }_i$ 的定义见 [2] 。最后假设 $\psi \in C^1$ 且满足

$\underset{x\in R}{\sup }\left\{{\left(1+\left|x\right|\right)}^{\mu }\left(\left|\psi \left(x\right)\right|+\left|{\psi }^{\prime }\left(x\right)\right|\right)\right\}<+\infty ,$ (1.5)

其中 $\frac{{\alpha }^2+\left(2+\rho \right)\alpha +\rho +1}{{\alpha }^2+\left(2+\rho \right)\alpha +\rho +2\rho }<\mu <1$。 $\rho \in \left(1,2\right)$ 是一个常数，令

$J_1=\left\{i|i\in J,{\alpha }_i=\alpha \right\},$

如果存在 $m\in J_1$，使得

$l_m\left(0\right)\psi \left(x\right)\cancel{\equiv }0,$

那么一定存在适当小的 ${\epsilon }_0>0$，使得对任意给定的 $\epsilon \in \left(0,{\epsilon }_0\right]$，Cauchy问题(1.1)和(1.4)的C¹解 $u=u\left(t,x\right)$ 的一阶偏导数 $u_x$ 必在有限时间内破裂且生命跨度 $\tilde{T}\left(\epsilon \right)$ 满足

$\underset{\epsilon \to 0^{+}}{\lim }{\epsilon }^{1+\alpha }\tilde{T}\left(\epsilon \right)=M_0,$

其中

$M_0={\left\{\underset{i\in J_1}{\max }\underset{x\in R}{\sup }\left(-\frac{1}{\alpha !}{\frac{{\text{d}}^{\alpha +1}{\lambda }_i\left(u^{\left(i\right)}\left(s\right)\right)}{\text{d}{s}^{\alpha +1}}|}_{s=0}{\left(l_i\left(0\right)\psi \left(x\right)\right)}^{\alpha }{l}_i\left(0\right){\psi }^{\prime }\left(x\right)\right)\right\}}^{-1}.$ (1.6)

$u=u^{\left(i\right)}\left(s\right)$ 的定义见 [2] 。

2. 预备知识

令

$w_i=l_i\left(u\right)u_x,\left(i=1,\cdots ,n\right).$ (2.1)

由(1.2)易知

$u_x={\displaystyle \underset{k=1}{\overset{n}{\sum }}{w}_k{r}_k\left(u\right)}.$ (2.2)

令

$\frac{\text{d}}{{\text{d}}_{i}t}=\frac{\partial }{\partial t}+{\lambda }_i\left(u\right)\frac{\partial }{\partial x}$

表示沿第i特征关于t的方向导数，有(参见 [6] )

$\frac{\text{d}u}{{\text{d}}_{i}t}={\displaystyle \underset{k=1}{\overset{n}{\sum }}\left({\lambda }_i\left(u\right)-{\lambda }_k\left(u\right)\right)w_k{r}_k+F\left(u\right)}\left(i=1,\cdots ,n\right),$

然后，在广义的标准坐标下，易得

$\frac{\text{d}{u}_i}{{\text{d}}_{i}t}={\displaystyle \underset{j,k=1}{\overset{n}{\sum }}\left({\rho }_{ijk}\left(u\right)u_j{w}_k+f_{ijk}\left(u\right)u_j{u}_k\right)}\left(i=1,\cdots ,n\right),$ (2.3)

其中

${\rho }_{ijk}\left(u\right)=\left({\lambda }_k\left(u\right)-{\lambda }_i\left(u\right)\right)\left(1-{\delta }_{jk}\right){\displaystyle {\int }_0^1\frac{\partial r_{ki}}{\partial u_j}}\left(\tau u_1,\cdots ,\tau u_{k-1},u_k,\tau u_{k+1},\cdots ,\tau u_n\right)\text{d}\tau ,$

$f_{ijk}\left(u\right)=\left(1-{\delta }_{ij}\right){\displaystyle {\int }_0^1{\theta }_k\left(\tau \right)}{\displaystyle {\int }_0^1\frac{{\partial }^2{F}_i\left(\sigma \tau u_1,\cdots ,\sigma \tau u_{i-1},\sigma u_i,\sigma \tau u_{i+1},\cdots ,\sigma \tau u_n\right)}{\partial u_j\partial u_k}\text{d}\sigma \text{d}\tau ,}$

${\theta }_k\left(\tau \right)=\{\begin{array}{l}\tau ,k\ne i,\\ 1,k=i.\end{array}$

显然

${\rho }_{ijj}\left(u\right)\equiv 0,\forall i,j,$ (2.4)

${\rho }_{ijk}\left(u\right)\equiv 0,\forall i,k\in \left\{1,\cdots ,p\right\},\forall j,$ (2.5)

$f_{iik}\left(u\right)\equiv 0,\forall i,k.$ (2.6)

而且，由(2.2)和(2.3)，有

$\text{d}\left[u_i\left(\text{d}x-{\lambda }_i\left(u\right)\text{d}t\right)\right]={\displaystyle \underset{j,k=1}{\overset{n}{\sum }}\left(F_{ijk}\left(u\right)u_j{w}_k+f_{ijk}\left(u\right)u_j{u}_k\right)\text{d}t\wedge \text{d}x,}$

其中

$F_{ijk}\left(u\right)={\rho }_{ijk}\left(u\right)+\nabla {\lambda }_j\left(u\right)r_k\left(u\right){\delta }_{ij}.$ (2.7)

另一方面，有

$\frac{\text{d}{w}_i}{{\text{d}}_{i}t}={\displaystyle \underset{j,k=1}{\overset{n}{\sum }}\left(Q_{ijk}\left(u\right)u_k{w}_j+{\gamma }_{ijk}\left(u\right)w_j{w}_k\right)}\left(i=1,\cdots ,n\right),$ (2.8)

其中

$Q_{ijk}\left(u\right)=\left(1-{\delta }_{jk}\right){\displaystyle {\int }_0^1\frac{\partial {\tilde{v}}_{ij}}{\partial u_k}\left(\tau u_1,\cdots ,\tau u_{j-1},u_j,\tau u_{j+1},\cdots ,\tau u_n\right)\text{d}\tau },$

${\gamma }_{ijk}\left(u\right)=\frac{1}{2}\left\{\left({\lambda }_j\left(u\right)-{\lambda }_k\left(u\right)\right)l_i\left(u\right)\nabla r_k\left(u\right)r_j\left(u\right)-\nabla {\lambda }_k\left(u\right)r_j\left(u\right){\delta }_{ik}+\left(j|k\right)\right\},$

其中 $\left(j|k\right)$ 表示在前面各项中交换j与k后所得的项。

${\tilde{v}}_{ij}\left(u\right)={\displaystyle \underset{k=1}{\overset{n}{\sum }}\left(-l_i\left(u\right)\nabla r_j\left(u\right)F\left(u\right)\right)+l_i\left(u\right)\nabla F\left(u\right)r_j\left(u\right).}$ (2.9)

因此，

$Q_{ijj}\left(u\right)\equiv 0,\forall i,j,$ (2.10)

${\gamma }_{ijj}\left(u\right)\equiv 0,\forall j\ne i\left(i,j=1,\cdots ,n\right),$ (2.11)

${\gamma }_{ijk}\left(u\right)\equiv 0,\forall j,k\in \left\{1,\cdots ,p\right\},i\in \left\{p+1,\cdots ,n\right\},$ (2.12)

${\gamma }_{iii}\left(u\right)=-\nabla {\lambda }_i\left(u\right)r_i\left(u\right)\left(i=1,\cdots ,n\right).$ (2.13)

而且，由(2.2)和(2.8)，有

$\text{d}\left[w_i\left(\text{d}x-{\lambda }_i\left(u\right)\text{d}t\right)\right]={\displaystyle \underset{j,k=1}{\overset{n}{\sum }}\left(Q_{ijk}\left(u\right)u_k{w}_j+{\Gamma }_{ijk}\left(u\right)w_j{w}_k\right)\text{d}t\wedge \text{d}x},\left(i=1,\cdots ,n\right),$

其中

${\Gamma }_{ijk}\left(u\right)={\gamma }_{ijk}\left(u\right)+\nabla {\lambda }_i\left(u\right)r_k\left(u\right){\delta }_{ij}.$

再由(1.3)有

${\Gamma }_{ijk}\left(u\right)\equiv 0,\forall j,k\in \left\{1,\cdots ,p\right\},i\in \left\{p+1,\cdots ,n\right\}.$

在标准坐标下，初始条件(1.4)可以写作(参见 [2] )

$u\left(0,x\right)=\epsilon \Psi \left(x,\epsilon \right),$ (2.14)

其中

$\Psi \left(x,\epsilon \right)=L\left(0\right)\psi \left(x\right)+O\left(\epsilon \right),$ (2.15)

$\frac{\partial \Psi \left(x,\epsilon \right)}{\partial x}=L\left(0\right){\psi }^{\prime }\left(x\right)+O\left(\epsilon \right).$ (2.16)

而且，根据 [7] 中的注2.4.1可以选取一个适当的标准坐标，使得

$r_i\left(u_i{e}_i\right)\equiv e_i,\forall \left|u_i\right|小\left(i=1,\cdots ,n\right).$

因此，有

$R\left(0\right)=L\left(0\right)=I.$

接下来，将给出本文所要用到的两个重要引理(证明见 [8] 或 [9] )。

引理2.1：假设 $w=w\left(t\right)$ 是常微分方程

$\frac{\text{d}w}{\text{d}t}=a_0\left(t\right)w^2+a_1\left(t\right)w+a_2\left(t\right),$ (2.17)

在区间 $\left[0,T\right]$ 上的 $C^1$ 解，其中T是给定的正数， $a_i\left(i=0,1,2\right)$ 是 $\left[0,T\right]$ 上的连续函数，且

$a_0\left(t\right)\ge 0,\forall t\in \left[0,T\right],$

令

$K={\displaystyle {\int }_0^T\left|a_2\left(t\right)\right|}\exp \left(-{\displaystyle {\int }_0^t{a}_1\left(s\right)\text{d}s}\right)\text{d}t.$ (2.18)

如果

$w_0>K,$

那么

${\displaystyle {\int }_0^T{a}_0\left(t\right)\exp \left({\displaystyle {\int }_0^t{a}_1\left(s\right)\text{d}s}\right)\text{d}t}<{\left(w_0-K\right)}^{-1}.$

引理2.2：假设 $a_i\left(i=0,1,2\right)$ 是 $\left[0,T\right]$ 上的连续函数，令

$a_0^{\pm }\left(t\right)=\max \left\{\pm a_0\left(t\right),0\right\},$

且K如(2.18)所定义。如果

$w_0\ge 0,$

且有

${\displaystyle {\int }_0^T{a}_0^{+}\left(t\right)\exp \left({\displaystyle {\int }_0^t{a}_1\left(s\right)\text{d}s}\right)\text{d}t}<{\left(w_0+K\right)}^{-1},$

${\displaystyle {\int }_0^T{a}_0^{-}}\left(t\right)\exp \left({\displaystyle {\int }_0^t{a}_1\left(s\right)\text{d}s}\right)\text{d}t<K^{-1},$

那么(2.17)连同初值 $w\left(0\right)=w_0$ 在 $\left[0,T\right]$ 上有唯一的解 $w=w\left(t\right)$。

且如果 $w\left(T\right)>0$，则

${\left(w\left(T\right)\right)}^{-1}\ge {\left(w_0+K\right)}^{-1}-{\displaystyle {\int }_0^T{a}_0^{+}}\left(t\right)\exp \left({\displaystyle {\int }_0^t{a}_1\left(s\right)\text{d}s}\right)\text{d}t.$

如果 $w\left(T\right)<0$，则

${\left|w\left(T\right)\right|}^{-1}\ge K^{-1}-{\displaystyle {\int }_0^T{a}_0^{-}\left(t\right)\exp \left({\displaystyle {\int }_0^t{a}_1\left(s\right)\text{d}s}\right)\text{d}t.}$

3. 定理1.1的证明

不失一般性，定理1.1的证明将在标准化坐标下进行，且与 [2] 中类似，可以假设

$0<\lambda \left(0\right)={\lambda }_1\left(0\right)=\cdots ={\lambda }_p\left(0\right)<{\lambda }_{p+1}\left(0\right)<\cdots <{\lambda }_n\left(0\right).$ (3.1)

由(1.3)知，存在适当小的正常数 $\delta$ 及 ${\delta }_0$，使得

${\lambda }_{p+1}\left(u\right)-{\lambda }_i\left(v\right)\ge 4{\delta }_0,\forall \left|u\right|,\left|v\right|\le \delta ,\left(i=1,\cdots ,p\right),$ (3.2)

${\lambda }_{j+1}\left(u\right)-{\lambda }_j\left(v\right)\ge 4{\delta }_0,\forall \left|u\right|,\left|v\right|\le \delta ,\left(j=p+1,\cdots ,n-1\right),$ (3.3)

$\left|{\lambda }_i\left(u\right)-{\lambda }_i\left(v\right)\right|\le \frac{{\delta }_0}{2},\forall \left|u\right|,\left|v\right|\le \delta ,\left(i=1,\cdots ,n\right).$ (3.4)

先假设在 $C^1$ 解 $u=u\left(t,x\right)$ 的任一存在域上，恒成立

$\left|u\left(t,x\right)\right|\le \delta .$ (3.5)

在引理3.3的证明最后，将解释此假设的合理性。对任意给定的 $T>0$，令

$D_{+}^T=\left\{\left(t,x\right)|0\le t\le T,x\ge \left({\lambda }_n\left(0\right)+{\delta }_0\right)t\right\},$

$D_{-}^T=\left\{\left(t,x\right)|0\le t\le T,x\le -t\right\},$

$D_0^T=\left\{\left(t,x\right)|0\le t\le T,-t\le x\le \left(\lambda \left(0\right)-\delta \right)t\right\},$

$D^T=\left\{\left(t,x\right)|0\le t\le T,\left(\lambda \left(0\right)-{\delta }_0\right)t\le x\le \left({\lambda }_n\left(0\right)+{\delta }_0\right)t\right\}.$

对 $i=1,\cdots ,n$，令

$D_i^T=\left\{\left(t,x\right)|0\le t\le T,-\left[{\delta }_0+\eta \left({\lambda }_i\left(0\right)-\lambda \left(0\right)\right)\right]t\le x-{\lambda }_i\left(0\right)t\le \left[{\delta }_0+\eta \left({\lambda }_n\left(0\right)-{\lambda }_i\left(0\right)\right)\right]t\right\}.$

由(3.1)~(3.3)，当 $\eta >0$ 适当小时，有

$D_1^T=D_2^T=\cdots =D_p^T,$

$D_i^T\cap D_j^T=\varnothing ,\forall i,j\in \left\{1,p+1,\cdots ,n\right\},$

$D_1^T\cup {\displaystyle \underset{i=p+1}{\overset{n}{\cup }}{D}_i^T}\subset D^T.$

令

$U\left(D_{\pm }^T\right)=\underset{i=1,\cdots ,n}{\max }{\Vert {\left(1+\left|x\right|\right)}^{\mu }{u}_i\left(t,x\right)\Vert }_{L^{\infty }\left(D_{\pm }^T\right)},$

$U\left(D_0^T\right)=\underset{i=1,\cdots ,n}{\max }{\Vert {\left(1+\left|t\right|\right)}^{\mu }{u}_i\left(t,x\right)\Vert }_{L^{\infty }\left(D_0^T\right)},$

$W\left(D_{\pm }^T\right)=\underset{i=1,\cdots ,n}{\max }{\Vert {\left(1+\left|x\right|\right)}^{\mu }{w}_i\left(t,x\right)\Vert }_{L^{\infty }\left(D_{\pm }^T\right)},$

$W\left(D_0^T\right)=\underset{i=1,\cdots ,n}{\max }{\Vert {\left(1+\left|t\right|\right)}^{\mu }{w}_i\left(t,x\right)\Vert }_{L^{\infty }\left(D_0^T\right)},$

$U_{\infty }^c\left(T\right)=\underset{i=1,\cdots ,n}{\max }\underset{\left(t,x\right)\in D^T\backslash D_i^T}{\sup }\left\{{\left(1+\left|x-{\lambda }_i\left(0\right)t\right|\right)}^{\mu }\left|u_i\left(t,x\right)\right|\right\},$

$W_{\infty }^c\left(T\right)=\underset{i=1,\cdots ,n}{\max }\underset{\left(t,x\right)\in D^T\backslash D_i^T}{\sup }\left\{{\left(1+\left|x-{\lambda }_i\left(0\right)t\right|\right)}^{\mu }\left|w_i\left(t,x\right)\right|\right\},$

$U_1\left(T\right)=\underset{i=1,\cdots ,n}{\max }\underset{0\le t\le T}{\sup }{\displaystyle {\int }_{D_i^T\left(t\right)}\left|u_i\left(t,x\right)\right|\text{d}x},$

$W_1\left(T\right)=\underset{i=1,\cdots ,n}{\max }\underset{0\le t\le T}{\sup }{\displaystyle {\int }_{D_i^T\left(t\right)}\left|w_i\left(t,x\right)\right|\text{d}x},$

其中

$D_i^T\left(t\right)=\left\{\left(\tau ,x\right)|\tau =t,\left(\tau ,x\right)\in D_i^T\right\}.$

${\tilde{U}}_1\left(T\right)=\max \left\{\underset{i=1,\cdots ,p}{\max }\underset{j\in \left\{p+1,\cdots ,n\right\}}{\max }\underset{{\tilde{C}}_j}{\sup }{\displaystyle {\int }_{{\tilde{C}}_j}\left|u_i\left(t,x\right)\right|\text{d}t},\underset{i=p+1,\cdots ,n}{\max }\underset{j\ne i}{\max }\underset{{\tilde{C}}_j}{\sup }{\displaystyle {\int }_{{\tilde{C}}_j}\left|u_i\left(t,x\right)\right|\text{d}t}\right\},$

${\tilde{W}}_1\left(T\right)=\max \left\{\underset{i=1,\cdots ,p}{\max }\underset{j\in \left\{p+1,\cdots ,n\right\}}{\max }\underset{{\tilde{C}}_j}{\sup }{\displaystyle {\int }_{{\tilde{C}}_j}\left|w_i\left(t,x\right)\right|\text{d}t},\underset{i=p+1,\cdots ,n}{\max }\underset{j\ne i}{\max }\underset{{\tilde{C}}_j}{\sup }{\displaystyle {\int }_{{\tilde{C}}_j}\left|w_i\left(t,x\right)\right|\text{d}t}\right\},$

其中 ${\tilde{C}}_j$ 表示在 $D_i^T$ 上的任一第j特征。

$U_{\infty }\left(T\right)=\underset{i=1,\cdots ,n}{\max }\underset{\underset{x\in R}{0\le t\le T}}{\sup }\left|u_i\left(t,x\right)\right|,$

$W_{\infty }\left(T\right)=\underset{i=1,\cdots ,n}{\max }\underset{\underset{x\in R}{0\le t\le T}}{\sup }\left|w_i\left(t,x\right)\right|.$

由 $D_i^T$ 及 $D^T$ 的定义容易得到(参见 [2] )。

引理3.1：对于 $i=1,\cdots ,n$，在区域 $D^T\backslash D_i^T$ 上成立

$ct\le \left|x-{\lambda }_i\left(0\right)t\right|\le Ct,$

$cx\le \left|x-{\lambda }_i\left(0\right)t\right|\le Cx,$

其中c和C是不依赖于T的正常数。

引理3.2：假设(3.1)成立， $\psi \in C^1$，满足(1.5)，且在 $u=0$ 的一个邻域内， $A\in C^2$。那么一定存在适当小的 ${\epsilon }_0>0$，使得对于任意给定的 $\epsilon \in \left[0,{\epsilon }_0\right]$，对于Cauchy问题(1.1)及(1.4)的 $C^1$ 解 $u=u\left(t,x\right)$，在其任一给定的存在区域 $0\le t\le T$ (其中 $T{\epsilon }^{2+\alpha }\le 1$ )上，存在不依赖于 $\epsilon$ 及T的正常数 ${\kappa }_1$ 和 ${\kappa }_2$ 使得下面的一致先验估计式成立：

$U\left(D_{\pm }^T\right),W\left(D_{\pm }^T\right)\le {\kappa }_1\epsilon ,$

$U\left(D_0^T\right),W\left(D_0^T\right)\le {\kappa }_2\epsilon .$

引理3.3：在定理1.1的假设下，在标准坐标下，一定存在适当小的 ${\epsilon }_0>0$，使得对于任意给定的 $\epsilon \in \left[0,{\epsilon }_0\right]$，对Cauchy问题(1.1)和(1.4)的 $C^1$ 解 $u=u\left(t,x\right)$，在其任一给定的存在区域 $0\le t\le T$ 上，存在不依赖于 $\epsilon$ 及T的正常数 ${\kappa }_i\left(i=3,4,5,6,7,8\right)$ 使得下面的一致先验估计式成立：

$W_{\infty }^c\left(T\right)\le {\kappa }_3\epsilon ,$ (3.6)

${\tilde{W}}_1\left(T\right),W_1\left(T\right)\le {\kappa }_4{\epsilon }^{\gamma },$ (3.7)

$U_{\infty }^c\left(T\right)\le {\kappa }_5\epsilon ,$ (3.8)

$U_{\infty }\left(T\right)\le {\kappa }_6{\epsilon }^{\gamma },$ (3.9)

${\tilde{U}}_1\left(T\right),U_1\left(T\right)\le {\kappa }_7{\epsilon }^{\tau }+{\kappa }_8{\epsilon }^{\gamma \left(2+\alpha \right)}T,$ (3.10)

其中

$\gamma =\left(2+\alpha \right)\mu -\left(1+\alpha \right)>0,$

$\tau =1+\gamma \left(\rho +\alpha \right)\left(\mu -1\right),\rho \in \left(1,2\right),$

而且

$T{\epsilon }^{\gamma \left(1+\alpha \right)}\le 1,$ (3.11)

$W_{\infty }\left(T\right)\le {\kappa }_9\epsilon .$ (3.12)

证明：先估计 ${\tilde{W}}_1\left(T\right)$。在 $D_i^T$ 上任作一条第j特征 ${\tilde{C}}_j:x=x_j\left(t\right)$ 交 $D_i^T$ 的边界于点P₁与点P₂ (当 $i\in \left\{1,\cdots ,p\right\}$ 时， $j\in \left\{p+1,\cdots ,n\right\}$ ；当 $i\in \left\{p+1,\cdots ,n\right\}$ 时， $j\ne i$ )。过原点作第i特征交 ${\tilde{C}}_j$ 于点P₀。设点P₁，P₂及P₀的t坐标分别为 $t_1,t_2$ 及 $t_0\left(0\le t_1,t_0,t_2\le T\right)$。过P₁作第i特征交直线 $x=\left(\lambda \left(0\right)-{\delta }_0\right)t$ 于点 $A_1\left({\bar{t}}_1,{\bar{y}}_1\right)$，过P₂作第i特征交直线 $x=\left({\lambda }_n\left(0\right)+{\delta }_0\right)t$ 于点 $A_2\left({\bar{t}}_2,{\bar{y}}_2\right)$。有

${\displaystyle {\int }_{{\tilde{C}}_j}\left|w_i\left(t,x\right)\right|\text{d}t}={\displaystyle {\int }_{t_1}^{t_0}\left|w_i\left(t,x_j\left(t\right)\right)\right|\text{d}t}+{\displaystyle {\int }_{t_0}^{t_2}\left|w_i\left(t,x_j\left(t\right)\right)\right|\text{d}t}.$

为了估计 ${\displaystyle {\int }_{t_1}^{t_0}\left|w_i\left(t,x_j\left(t\right)\right)\right|\text{d}t}$，在区域P₁A₁OP₀上用Stokes公式，就得到

$\begin{array}{l}{\displaystyle {\int }_{t_1}^{t_0}\left|w_i\left(t,x_j\left(t\right)\right)\left({\lambda }_j\left(u\left(t,x_j\left(t\right)\right)\right)-{\lambda }_i\left(u\left(t,x_j\left(t\right)\right)\right)\right)\right|\text{d}t}\\ \le {\displaystyle {\int }_0^{{\bar{t}}_1}\left|w_i\left(t,\left(\lambda \left(0\right)-{\delta }_0\right)t\right)\left(\lambda \left(0\right)-{\delta }_0-{\lambda }_i\left(u\left(t,\left(\lambda \left(0\right)-{\delta }_0\right)t\right)\right)\right)\right|\text{d}t}\\ +{\displaystyle {\iint }_{P_1{A}_{1}O{P}_0}\left|{\displaystyle \underset{j,k=1}{\overset{n}{\sum }}{Q}_{ijk}\left(u\right)u_k{w}_j+{\Gamma }_{ijk}\left(u\right)w_j{w}_k}\right|\text{d}t\text{d}x},\end{array}$ (3.13)

由(3.2)~(3.3)及(3.5)，并注意到(2.9)~(2.12)及引理3.1~3.2，可得

$\begin{array}{c}{\displaystyle {\int }_{t_1}^{t_0}\left|w_i\left(t,x_j\left(t\right)\right)\right|\text{d}t}\le c\{W\left(D_0^T\right)+U_{\infty }^c\left(T\right)W_1\left(T\right)+W_{\infty }^c\left(T\right)U_1\left(T\right)+W_{\infty }^c\left(T\right)W_1\left(T\right)\stackrel{}{}\\ +\left[W_{\infty }^c\left(T\right)U_{\infty }^c\left(T\right)+{\left(W_{\infty }^c\left(T\right)\right)}^2\right]{\left(1+T\right)}^{1-\mu }\}{\left(1+T\right)}^{1-\mu },\end{array}$

今后c是不依赖于 $\epsilon$ 和T的正常数。同时，对 ${\displaystyle {\int }_{t_0}^{t_2}\left|w_i\left(t,x_j\left(t\right)\right)\right|\text{d}t}$ 有类似的估计式。这样得到

$\begin{array}{c}{\tilde{W}}_1\left(T\right)\le c\{\epsilon +U_{\infty }^c\left(T\right)W_1\left(T\right)+W_{\infty }^c\left(T\right)U_1\left(T\right)+W_{\infty }^c\left(T\right)W_1\left(T\right)\stackrel{}{}\\ +\left[W_{\infty }^c\left(T\right)U_{\infty }^c\left(T\right)+{\left(W_{\infty }^c\left(T\right)\right)}^2\right]{\left(1+T\right)}^{1-\mu }\}{\left(1+T\right)}^{1-\mu }.\end{array}$

类似的，有

$\begin{array}{c}{W}_1\left(T\right)\le c\{\epsilon +U_{\infty }^c\left(T\right)W_1\left(T\right)+W_{\infty }^c\left(T\right)U_1\left(T\right)+W_{\infty }^c\left(T\right)W_1\left(T\right)\stackrel{}{}\\ +\left[W_{\infty }^c\left(T\right)U_{\infty }^c\left(T\right)+{\left(W_{\infty }^c\left(T\right)\right)}^2\right]{\left(1+T\right)}^{1-\mu }\}{\left(1+T\right)}^{1-\mu }.\end{array}$

下面估计 $W_{\infty }^c\left(T\right)$。当 $i\in \left\{1,\cdots ,p\right\}$ 时，过任意给定的点 $\left(t,x\right)\in D^T\backslash D_1^T$ 作第i特征交直线 $x=\left({\lambda }_n\left(0\right)+{\delta }_0\right)t$ 于点 $D\left(t_0,y\right)$，沿该特征从 $t_0$ 到t积分(2.8)，得

$w_i\left(t,x\right)=w_i\left(t_0,y\right)+{\displaystyle {\int }_{t_0}^t{\displaystyle \underset{j,k=1}{\overset{n}{\sum }}\left[Q_{ijk}\left(u\right)u_k{w}_j+{\gamma }_{ijk}\left(u\right)w_j{w}_k\right]\text{d}s.}}$ (3.14)

由引理3.2，有

$\left|w_i\left(t_0,y\right)\right|\le {\left(1+y\right)}^{-\mu }W\left(D_{+}^T\right)\le c\epsilon {\left(1+t_0\right)}^{-\mu }.$ (3.15)

另一方面，由(2.10)和(3.5)，并利用引理3.1，有

$\begin{array}{l}{\displaystyle {\int }_{t_0}^t\left|{\displaystyle \underset{j,k=1}{\overset{n}{\sum }}\left[Q_{ijk}\left(u\right)u_k{w}_j+{\gamma }_{ijk}\left(u\right)w_j{w}_k\right]}\right|\text{d}s}\\ \le c\{W_{\infty }^c\left(T\right){\tilde{U}}_1\left(T\right)+U_{\infty }^c\left(T\right){\tilde{W}}_1\left(T\right)+W_{\infty }^c\left(T\right){\tilde{W}}_1\left(T\right)\stackrel{}{}\\ +\left[W_{\infty }^c\left(T\right)U_{\infty }^c\left(T\right)+{\left(W_{\infty }^c\left(T\right)\right)}^2\right]{\left(1+T\right)}^{1-\mu }\}{\left(1+t_0\right)}^{-\mu }.\end{array}$ (3.16)

由(3.4)，有

$x-\left(\lambda \left(0\right)+\frac{{\delta }_0}{2}\right)t\le y-\left(\lambda \left(0\right)+\frac{{\delta }_0}{2}\right)t_0,$

再由 $D_1^T$ 的定义并注意到 $y=\left({\lambda }_n\left(0\right)+{\delta }_0\right)t_0$，易得

$\eta t\le t_0\le t.$

于是从(3.14)~(3.16)，并应用引理3.1，就得到

$\begin{array}{l}{\left(1+\left|x-\lambda \left(0\right)t\right|\right)}^{\mu }\left|w_i\left(t,x\right)\right|\\ \le c\{\epsilon +W_{\infty }^c\left(T\right){\tilde{U}}_1\left(T\right)+U_{\infty }^c\left(T\right){\tilde{W}}_1\left(T\right)+W_{\infty }^c\left(T\right){\tilde{W}}_1\left(T\right)\stackrel{}{}\\ +\left[W_{\infty }^c\left(T\right)U_{\infty }^c\left(T\right)+{\left(W_{\infty }^c\left(T\right)\right)}^2\right]{\left(1+T\right)}^{1-\mu }\}.\end{array}$

当 $i\in \left\{p+1,\cdots ,n\right\}$ 时，由(2.10)，类似地可得

$\begin{array}{l}{\left(1+\left|x-{\lambda }_i\left(0\right)t\right|\right)}^{\mu }\left|w_i\left(t,x\right)\right|\\ \le c\{\epsilon +W_{\infty }^c\left(T\right){\tilde{U}}_1\left(T\right)+U_{\infty }^c\left(T\right){\tilde{W}}_1\left(T\right)+W_{\infty }^c\left(T\right){\tilde{W}}_1\left(T\right)\underset{}{}\\ +\left[W_{\infty }^c\left(T\right)U_{\infty }^c\left(T\right)+{\left(W_{\infty }^c\left(T\right)\right)}^2\right]{\left(1+T\right)}^{1-\mu }\},\end{array}$

然后得到

$\begin{array}{l}{W}_{\infty }^c\left(T\right)\le c\{\epsilon +W_{\infty }^c\left(T\right){\tilde{U}}_1\left(T\right)+U_{\infty }^c\left(T\right){\tilde{W}}_1\left(T\right)+W_{\infty }^c\left(T\right){\tilde{W}}_1\left(T\right)\underset{}{}\\ +\left[W_{\infty }^c\left(T\right)U_{\infty }^c\left(T\right)+{\left(W_{\infty }^c\left(T\right)\right)}^2\right]{\left(1+T\right)}^{1-\mu }\}.\end{array}$

类似地，有

$U_{\infty }^c\left(T\right)\le c\left\{\epsilon +U_{\infty }^c\left(T\right){\tilde{U}}_1\left(T\right)+W_{\infty }^c\left(T\right){\tilde{U}}_1\left(T\right)+\left[W_{\infty }^c\left(T\right)U_{\infty }^c\left(T\right)+{\left(U_{\infty }^c\left(T\right)\right)}^2\right]{\left(1+T\right)}^{1-\mu }\right\}.$

现在来估计 ${\tilde{U}}_1\left(T\right)$ 和 $U_1\left(T\right)$。类似于(3.13)，由(2.4)~(2.7)我们有

$\begin{array}{l}{\displaystyle {\int }_{t_1}^{t_0}\left|u_i\left(t,x_j\left(t\right)\right)\left({\lambda }_j\left(u\left(t,x_j\left(t\right)\right)\right)-{\lambda }_i\left(u\left(t,x_j\left(t\right)\right)\right)\right)\right|\text{d}t}\\ \le {\displaystyle {\int }_0^{{\bar{t}}_1}\left|u_i\left(t,\left(\lambda \left(0\right)-{\delta }_0\right)t\right)\left(\lambda \left(0\right)-{\delta }_0-{\lambda }_i\left(u\left(t,\left(\lambda \left(0\right)-{\delta }_0\right)t\right)\right)\right)\right|\text{d}t}\\ +{\displaystyle {\iint }_{P_1{A}_{1}O{P}_0}\left|{\displaystyle \underset{j,k=1}{\overset{n}{\sum }}{F}_{ijk}\left(u\right)u_j{w}_k+f_{ijk}\left(u\right)u_j{u}_k}\right|\text{d}t\text{d}x}\\ \le c\left\{\left[\epsilon +W_{\infty }^c\left(T\right)U_{\infty }^c\left(T\right){\left(1+T\right)}^{1-\mu }+{\left(U_{\infty }^c\left(T\right)\right)}^2{\left(1+T\right)}^{1-\mu }+U_{\infty }^c\left(T\right)U_1\left(T\right)\right]{\left(1+T\right)}^{1-\mu }\right\}.\end{array}$

类似地可以估计 ${\displaystyle {\int }_{t_0}^{t_2}\left|u_i\left(t,x_j\left(t\right)\right)\right|\text{d}t}$，因此得到

${\tilde{U}}_1\left(T\right)\le c\left\{\left[\epsilon +W_{\infty }^c\left(T\right)U_{\infty }^c\left(T\right){\left(1+T\right)}^{1-\mu }+{\left(U_{\infty }^c\left(T\right)\right)}^2{\left(1+T\right)}^{1-\mu }+U_{\infty }^c\left(T\right)U_1\left(T\right)\right]{\left(1+T\right)}^{1-\mu }\right\}.$

类似地，有

$U_1\left(T\right)\le c\left\{\left[\epsilon +W_{\infty }^c\left(T\right)U_{\infty }^c\left(T\right){\left(1+T\right)}^{1-\mu }+{\left(U_{\infty }^c\left(T\right)\right)}^2{\left(1+T\right)}^{1-\mu }+U_{\infty }^c\left(T\right)U_1\left(T\right)\right]{\left(1+T\right)}^{1-\mu }\right\}.$

而且，对于任意给定的点 $\left(t,x\right)\in D^T$，有

$u\left(t,x\right)=u\left(t,x_0\right)+{\displaystyle {\int }_{x_0}^x{u}_{\xi }\left(t,\xi \right)\text{d}\xi },$

其中 $\left(t,x_0\right)$ 位于直线 $x=\left(\lambda \left(0\right)-{\delta }_0\right)t$ 上。于是，由(3.5)和引理3.1，并利用引理3.2易得：当 $\left(t,x\right)\in D^T$ 时，成立

$\left|u\left(t,x\right)\right|\le c\left\{U\left(D_0^T\right)+W_{\infty }^c\left(T\right){\left(1+T\right)}^{1-\mu }+W_1\left(T\right)\right\}.$

因此，得到

$U_{\infty }\left(T\right)\le c\left\{U\left(D_0^T\right)+W_{\infty }^c\left(T\right){\left(1+T\right)}^{1-\mu }+W_1\left(T\right)\right\}.$

由 $U_{\infty }\left(T\right)\le {\kappa }_6{\epsilon }^{\gamma }$。对于适当小的 ${\delta }_0>0$，恒成立

$\left|u\left(t,x\right)\right|\le {\kappa }_6{\epsilon }_0^{\gamma }\le \frac{1}{2}\delta .$

这说明了假设(3.5)的合理性。利用连续归纳假设即证(3.6)~(3.10)。

最后，证明 $W_{\infty }\left(T\right)\le {\kappa }_9\epsilon$。过任意给定的点 $\left(t,x\right)\in D_i^T$，作第i特征 $\xi =x_i\left(s;t,x\right)$ 交 $D^T$ 的边界之一于点 $D\left(t_0,y\right)$。沿该特征从 $t_0$ 到t积分(2.8)，得到

$w_i\left(t,x\right)=w_i\left(t_0,y\right)+{\displaystyle {\int }_{t_0}^t{\displaystyle \underset{j,k=1}{\overset{n}{\sum }}\left[Q_{ijk}\left(u\right)u_k{w}_j+{\gamma }_{ijk}\left(u\right)w_j{w}_k\right]\left(s,x_i\left(s;t,x\right)\right)\text{d}s}}.$

注意到引理(3.1)~(3.2)，和

$\left|{\gamma }_{iii}\left(u_i{e}_i\right)\right|\le c{\left|u_i\right|}^{\alpha },$

对任意固定的点 $\left(t,x\right)\in D_i^T$，有

$\begin{array}{c}\left|w_i\left(t,x\right)\right|\le c\{W\left(D_0^T\right)+W\left(D_{+}^T\right)+U_{\infty }^c\left(T\right)W_{\infty }^c\left(T\right)+{\left(W_{\infty }^c\left(T\right)\right)}^2\\ +\left[W_{\infty }^c\left(T\right)\left(U_{\infty }\left(T\right)+W_{\infty }\left(T\right)\right)+U_{\infty }^c\left(T\right)\left(W_{\infty }\left(T\right)+{\left(W_{\infty }\left(T\right)\right)}^2\right)\right]{\left(1+T\right)}^{1-\mu }\\ +{\left(U_{\infty }\left(T\right)\right)}^{\alpha }{\left(W_{\infty }\left(T\right)\right)}^{2}T\},\end{array}$

由引理3.2和(3.11)，并应用连续归纳假设，就能立即得到(3.12)。引理3.3得证。

注3.1：由(3.9)和(3.12)，当 ${\epsilon }_0>0$ 适当小时，Cauchy问题(1.1)和(1.4)在 $\left[0,T\right]$ 上存在唯一的 $C^1$ 解 $u=u\left(t,x\right)$，其中T满足(3.11)。因此我们得到了 $C^1$ 解 $u=u\left(t,x\right)$ 的生命跨度的下界

$\tilde{T}\ge {\epsilon }^{-\gamma \left(1+\alpha \right)}.$

下面证明定理1.1。

显然，为了证明定理1.1，只要证明

$\overline{\underset{\epsilon \to 0^{+}}{\lim }}\left\{{\epsilon }^{1+\alpha }\tilde{T}\left(\epsilon \right)\right\}\le M_0,$ (3.17)

及

$\underset{\epsilon \to 0^{+}}{\underset{\_}{\lim }}\left\{{\epsilon }^{1+\alpha }\tilde{T}\left(\epsilon \right)\right\}\ge M_0.$ (3.18)

其中 $M_0$ 由(1.6)定义。

接下来，将在标准坐标下考虑问题，并且假设 ${\epsilon }_0>0$ 是适当小的。

先来证明(3.17)。由注3.1，对于任意固定的 $\epsilon \in \left(0,{\epsilon }_0\right]$，Cauchy问题(1.1)和(1.4)在区域 $\left[0,T_1\right]$ 上存在唯一的 $C^1$ 解 $u=u\left(t,x\right)$，其中

$T_1={\epsilon }^{-\left(\rho +\alpha \right)}\le \tilde{T}\left(\epsilon \right)-1=\bar{T},$ (3.19)

$\rho =\gamma \left(1+\alpha \right)-\alpha =\left({\alpha }^2+3\alpha +2\right)\mu -\left({\alpha }^2+3\alpha +1\right)>0.$

先假设成立

$\tilde{T}\left(\epsilon \right)\le {\epsilon }^{-\gamma \left(\rho +\alpha \right)},$ (3.20)

并将在证明的最后说明此假设的合理性。

由(1.5)和(1.6)知，存在 $i\in J_1$ 和 $x_0\in ℝ$ 使得

$M_0={\left\{-\frac{1}{\alpha !}{\frac{{\text{d}}^{\alpha +1}{\lambda }_i\left(u^{\left(i\right)}\left(s\right)\right)}{\text{d}{s}^{\alpha +1}}|}_{s=0}{\left(l_i\left(0\right)\psi \left(x_0\right)\right)}^{\alpha }{l}_i\left(0\right){\psi }^{\prime }\left(x_0\right)\right\}}^{-1},$

令

$a=-\frac{1}{\alpha !}{\frac{{\text{d}}^{\alpha +1}{\lambda }_i\left(u^{\left(i\right)}\left(s\right)\right)}{\text{d}{s}^{\alpha +1}}|}_{s=0}.$

不妨设 $i=m\in J_1$，且

$a{\left(l_m\left(0\right)\psi \left(x_0\right)\right)}^{\alpha }>0,l_m\left(0\right){\psi }^{\prime }\left(x_0\right)>0,x_0>0.$

过 $\left(0,x_0\right)$ 作第m特征 $x=x_m\left(t,x_0\right)$。由 $D_m^T$ 的定义知，该特征一定在有限时间 $t_0>0$ 进入 $D_m^T$ 并从此落在其中(参见 [2] )。在 $C^1$ 解的存在区域上，由(2.8)沿该特征有

$\frac{\text{d}{w}_m}{{\text{d}}_{m}t}=a_0\left(t\right)w_m^2+a_1\left(t\right)w_m+a_2\left(t\right),$ (3.21)

其中

$\{\begin{array}{l}{a}_0\left(t\right)={\gamma }_{mmm}\left(u\right),\\ a_1\left(t\right)={\displaystyle \underset{\begin{array}{c}j=1\\ j\ne m\end{array}}{\overset{n}{\sum }}\left[Q_{mmj}\left(u\right)u_j+2{\gamma }_{mmj}\left(u\right)w_j\right],}\\ a_2\left(t\right)={\displaystyle \underset{\begin{array}{l}j=1\\ j\ne m\end{array}}{\overset{n}{\sum }}{\displaystyle \underset{k=1}{\overset{n}{\sum }}{Q}_{mjk}\left(u\right)w_j{u}_k}}+{\displaystyle \underset{\begin{array}{l}j,k=1\\ {\lambda }_j\ne {\lambda }_k\\ j,k\ne m\end{array}}{\overset{n}{\sum }}{\gamma }_{mjk}\left(u\right)w_j{w}_k,}\end{array}$ (3.22)

由(3.19)，得到

$t_0<T_0={\epsilon }^{-\alpha }<T_1.$ (3.23)

由(2.4)~(2.6)，并应用引理3.1~3.3，沿 $x=x_m\left(t,x_0\right)$ 积分(2.3)

$\begin{array}{l}\left|u_m\left(t,x_m\left(t,x_0\right)\right)-u_m\left(0,x_0\right)\right|\\ \le \left|{\displaystyle {\int }_0^t{\displaystyle \underset{\begin{array}{l}j=1\\ j\ne m\end{array}}{\overset{n}{\sum }}{\displaystyle \underset{k=1}{\overset{n}{\sum }}\left[{\rho }_{mjk}\left(u\right)u_k{w}_j+f_{mjk}\left(u\right)u_j{u}_k\right]\left(s,x_m\left(s,x_0\right)\right)\text{d}s}}}\right|\\ \le c\left\{\left[U\left(D_{+}^t\right)+W\left(D_{+}^t\right)\right]U\left(D_{+}^t\right)+U_{\infty }^c\left(t\right)U_{\infty }\left(t\right){\left(1+t\right)}^{1-\mu }+\left[U_{\infty }^c\left(t\right)+W_{\infty }^c\left(t\right)\right]U_{\infty }^c\left(t\right)\right\}\\ \le c{\epsilon }^{1+\gamma -\gamma \left(\rho +\alpha \right)\left(1-\mu \right)},\forall t\in \left[0,\bar{T}\right]\end{array}$ (3.24)

由(2.14)~(2.15)，得到

$\left|u_m\left(t,x_m\left(t,x_0\right)\right)-\epsilon l_m\left(0\right)\psi \left(x_0\right)\right|\le c{\epsilon }^{1+\gamma -\gamma \left(\rho +\alpha \right)\left(1-\mu \right)},\forall t\in \left[0,\bar{T}\right]$ (3.25)

另一方面，在 $C^1$ 解 $u=u\left(t,x\right)$ 的存在域 $D^T\cap \left\{\left(t,x\right)|T_0\le t\le \bar{T}\right\}$ 上，由Hadamard公式和引理3.3，沿着 $x=x_m\left(t,x_0\right)$，有

$\left|{\gamma }_{mmm}\left(u\right)-{\gamma }_{mmm}\left(u_m{e}_m\right)\right|\le c{U}_{\infty }^c\left(\bar{T}\right){\left(1+t\right)}^{-\mu }\le c{\epsilon }^{1+\alpha \mu },\forall t\in \left[T_0,\bar{T}\right]$ (3.26)

而且，由(2.13)并由 $\alpha$ 及 $J_1$ 的定义，有

${\gamma }_{mmm}\left(u_m{e}_m\right)=a{\left(\epsilon l_m\left(0\right)\psi \left(x_0\right)\right)}^{\alpha }+a\left[{\left(u_m\left(t,x_m\left(t,x_0\right)\right)\right)}^{\alpha }-{\left(\epsilon l_m\left(0\right)\psi \left(x_0\right)\right)}^{\alpha }\right]+O\left({\left|u_m\right|}^{1+\alpha }\right).$ (3.27)

然后，应用(3.9)，并由(3.25)~(3.26)得到

$a_0\left(t\right)={\gamma }_{mmm}\left(u\right)=a{\left(\epsilon l_m\left(0\right)\psi \left(x_0\right)\right)}^{\alpha }+O\left({\epsilon }^{\varpi }\right),\forall t\in \left[T_0,\bar{T}\right]$ (3.28)

其中 $\varpi =\min \left\{\gamma \left(1+\alpha \right),1+\alpha \gamma -\gamma \left(\rho +\alpha \right)\left(1-\mu \right)\right\}$。

因此，对于适当小的 ${\epsilon }_0>0$，有

$a_0\left(t\right)\ge \frac{1}{2}a{\left(\epsilon l_m\left(0\right)\psi \left(x_0\right)\right)}^{\alpha }>0,\forall t\in \left[T_0,\bar{T}\right]$

应用Hadamard公式和引理3.3，得到

$\left|\frac{\partial u_m\left(t,x\right)}{\partial x}-w_m\left(t,x\right)\right|=\left|{\displaystyle \underset{k=1}{\overset{n}{\sum }}{w}_k{\left(r_k\left(u\right)-r_k\left(u_k{e}_k\right)\right)}^T{e}_m}\right|\le c{\epsilon }^{1+\gamma },\forall t\in \left[0,T_0\right]$ (3.29)

与(3.24)类似，应用引理3.2~3.3，并且由(3.23)，有

$\left|w_m\left(t,x_m\left(t,x_0\right)\right)-w_m\left(0,x_0\right)\right|\le c{\epsilon }^{\min \left\{1+\gamma -\alpha \left(1-\mu \right),2+\alpha \left(\gamma -1\right)\right\}},\forall t\in \left[0,T_0\right]$ (3.30)

注意到(2.16)，由(3.29)~(3.30)，得

$w_m\left(T_0,x_m\left(T_0,x_0\right)\right)=\epsilon l_m\left(0\right){\psi }^{\prime }\left(x_0\right)+O\left({\epsilon }^{\min \left\{1+\gamma -\alpha \left(1-\mu \right),2+\alpha \left(\gamma -1\right)\right\}}\right).$ (3.31)

对于适当小的 ${\epsilon }_0>0$，有

$w_m\left(T_0,x_m\left(T_0,x_0\right)\right)\ge \frac{1}{2}\epsilon l_m\left(0\right){\psi }^{\prime }\left(x_0\right).$

应用引理3.3，易得

${\displaystyle {\int }_{T_0}^{\bar{T}}\left|a_1\left(t\right)\right|\text{d}t}\le c\left(U_{\infty }^c\left(\bar{T}\right)+W_{\infty }^c\left(\bar{T}\right)\right){\left(1+\bar{T}\right)}^{1-\mu }\le c{\epsilon }^{1-\gamma \left(\rho +\alpha \right)\left(1-\mu \right)}.$ (3.32)

$\begin{array}{c}{\displaystyle {\int }_{T_0}^{\bar{T}}\left|a_2\left(t\right)\right|\text{d}t}\le c\left\{W_{\infty }^c\left(\bar{T}\right)U_{\infty }\left(\bar{T}\right){\left(1+\bar{T}\right)}^{1-\mu }+W_{\infty }^c\left(\bar{T}\right)U_{\infty }^c\left(\bar{T}\right)+{\left(W_{\infty }^c\left(\bar{T}\right)\right)}^2\right\}\\ \le c{\epsilon }^{1+\gamma -\gamma \left(\rho +\alpha \right)\left(1-\mu \right)}.\end{array}$

然后，有

$\begin{array}{c}K={\displaystyle {\int }_{T_0}^{\bar{T}}\left|a_2\left(t\right)\right|\exp \left(-{\displaystyle {\int }_{T_0}^t{a}_1\left(s\right)\text{d}s}\right)\text{d}t}\\ \le {\displaystyle {\int }_{T_0}^{\bar{T}}\left|a_2\left(t\right)\right|\text{d}t}\exp \left({\displaystyle {\int }_{T_0}^{\bar{T}}\left|a_1\left(s\right)\right|\text{d}s}\right)\le c{\epsilon }^{1+\gamma -\gamma \left(\rho +\alpha \right)\left(1-\mu \right)},\end{array}$ (3.33)

则

$w_m\left(T_0,x_m\left(T_0,x_0\right)\right)>\frac{1}{2}\epsilon l_m\left(0\right){\psi }^{\prime }\left(x_0\right)>K.$ (3.34)

对常微分方程(3.21)具初值

$t=T_0:w_m=w_m\left(T_0,x_m\left(T_0,x_0\right)\right)$

的Cauchy问题应用引理2.1，得到

$\begin{array}{c}{\displaystyle {\int }_{T_0}^{\bar{T}}{a}_0\left(t\right)\exp \left(-{\displaystyle {\int }_{T_0}^{\bar{T}}{a}_1\left(s\right)\text{d}s}\right)\text{d}t}\le {\displaystyle {\int }_{T_0}^{\bar{T}}{a}_0\left(t\right)\exp \left({\displaystyle {\int }_{T_0}^t{a}_1\left(s\right)\text{d}s}\right)\text{d}t}\\ \le {\left(w_m\left(T_0,x_m\left(T_0,x_0\right)\right)-K\right)}^{-1}.\end{array}$

因此，由(3.28)，(3.31)~(3.34)，易得，存在不依赖于 $\epsilon$ 和T的正常数 $\bar{C}$，使得

$\tilde{T}\left(\epsilon \right)\le \bar{C}{\epsilon }^{-\left(1+\alpha \right)},$

其中

$\bar{C}>{\left[a{\left(l_m\left(0\right)\psi \left(x_0\right)\right)}^{\alpha }{l}_m\left(0\right){\psi }^{\prime }\left(x_0\right)\right]}^{-1}.$

因此，当 ${\epsilon }_0>0$ 适当小时，成立

$\tilde{T}\left(\epsilon \right){\epsilon }^{\gamma \left(\rho +\alpha \right)}\le \bar{C}{\epsilon }^{\gamma \left(\rho +\alpha \right)-\left(1+\alpha \right)}\le 1,$

这就说明了假设(3.20)的合理性。

现在来证明(3.18)。只需证明对于任意给定的正常数 $M<M_0$，成立

$\tilde{T}\left(\epsilon \right)>M{\epsilon }^{-\left(1+\alpha \right)}.$ (3.35)

为了证明这个，只需在任意给定的区域 $D^T$ 上建立有关Cauchy问题(1.1)和(1.4)的解的 $C^1$ 范数的一致先验估计：

$0<T\le M{\epsilon }^{-\left(1+\alpha \right)}.$ (3.36)

由引理3.2~3.3，有

${\Vert u\left(t,x\right)\Vert }_{C^0\left(D^T\right)}\le c{\epsilon }^{\gamma },$

${\Vert w_i\left(t,x\right)\Vert }_{C^0\left(D^T\backslash D_i^T\right)}\le c\epsilon ,\left(i=1,\cdots ,n\right)$

因此，下面估计 ${\Vert w_i\left(t,x\right)\Vert }_{C^0\left(D_i^T\right)}\left(i=1,\cdots ,n\right)$，仅估计 ${\Vert w_m\left(t,x\right)\Vert }_{C^0\left(D_m^T\right)}$，其他的情况可类似地估计出来。

沿着穿过x-轴上任意固定的点 $\left(0,y\right)$ 的m阶特征 $x=x_m\left(t,y\right)$，仍然有(3.21)成立，其中 $u=u\left(t,x_m\left(t,y\right)\right)$。可假设

$w_m\left(0,y\right)\ge 0.$

(否则，用 $-w_1$ 代替 $w_1$ )。

下面首先估计 $w_m\left(0,y\right){\displaystyle {\int }_0^T{a}_0^{+}\left(t\right)\text{d}t}$，其中T满足(3.36)。应用(3.27)并注意到(3.9)和(3.25)，由(3.22)得

$a_0\left(t\right)=a{\left(\epsilon l_m\left(0\right)\psi \left(y\right)\right)}^{\alpha }+\left({\gamma }_{mmm}\left(u\right)-{\gamma }_{mmm}\left(u_m{e}_m\right)\right)+O\left({\epsilon }^{\varpi }\right).$

那么，由(2.1)，(2.14)，(2.16)，得到

$\begin{array}{c}{w}_m\left(0,y\right){\displaystyle {\int }_0^T{a}_0^{+}\left(t\right)\text{d}t}\le \left({\left(l_m\left(0\right){\psi }^{\prime }\left(y\right)\right)}^{+}\epsilon +c{\epsilon }^2\right){\displaystyle {\int }_0^T[{\left(a{\left(l_m\left(0\right)\psi \left(y\right)\right)}^{\alpha }\right)}^{+}{\epsilon }^{\alpha }}\\ \stackrel{}{}+\left|{\gamma }_{mmm}\left(u\right)-{\gamma }_{mmm}\left(u_m{e}_m\right)\right|+c{\epsilon }^{\varpi }]\text{d}t.\end{array}$ (3.37)

而且，正如上面指出的， $x=x_m\left(t,y\right)$ 一定在一个独立于 $\epsilon$ 的有限时间 $t_0>0$ 时进入 $D_m^T$ 并在这之后就在其中。那么，对于适当小的 $\epsilon >0$，由(3.36)并应用Hadamard公式和引理3.2~3.3，有

$\begin{array}{c}{w}_m\left(0,y\right){\displaystyle {\int }_0^T{a}_0^{+}\left(t\right)\text{d}t}\le {\left(a{\left(l_m\left(0\right)\psi \left(y\right)\right)}^{\alpha }{l}_m\left(0\right){\psi }^{\prime }\left(y\right)\right)}^{+}M+c{\epsilon }^{\varpi -\alpha }\\ +c\epsilon \left\{\left[U\left(D_{\pm }^T\right)+U\left(D_0^T\right)+U_{\infty }^c\left(T\right)\right]{\left(1+T\right)}^{1-\mu }+{\tilde{U}}_1\left(T\right)\right\}\\ \le {\left(a{\left(l_m\left(0\right)\psi \left(y\right)\right)}^{\alpha }{l}_m\left(0\right){\psi }^{\prime }\left(y\right)\right)}^{+}M+c{\epsilon }^{\varpi -\alpha }.\end{array}$

而且，有

$\begin{array}{c}{\displaystyle {\int }_0^T\left|a_1\left(t\right)\right|\text{d}t}\le c\{[U\left(D_{\pm }^T\right)+U\left(D_0^T\right)+W\left(D_{\pm }^T\right)+W\left(D_0^T\right)\\ +U_{\infty }^c\left(T\right)+W_{\infty }^c\left(T\right)]{\left(1+T\right)}^{1-\mu }+{\tilde{U}}_1\left(T\right)+{\tilde{W}}_1\left(T\right)\}\\ \le c{\epsilon }^{\gamma \left(2+\alpha \right)-\left(1+\alpha \right)},\end{array}$ (3.38)

$\begin{array}{c}{\displaystyle {\int }_0^T\left|a_2\left(t\right)\right|\text{d}t}\le c\{U\left(D_{\pm }^T\right)W\left(D_{\pm }^T\right)+U\left(D_0^T\right)W\left(D_0^T\right)+W{\left(D_{\pm }^T\right)}^2+W{\left(D_0^T\right)}^2\\ +W_{\infty }^c\left(T\right)U_{\infty }^c\left(T\right)+W_{\infty }^c\left(T\right)U_{\infty }\left(T\right){\left(1+T\right)}^{1-\mu }\\ +{\left(W_{\infty }^c\left(T\right)\right)}^2+U_{\infty }^c\left(T\right){\tilde{W}}_1\left(T\right)+{\tilde{U}}_1\left(T\right)W_{\infty }^c\left(T\right)\}\\ \le c{\epsilon }^{\gamma \left(2+\alpha \right)-\alpha },\end{array}$ (3.39)

$K={\displaystyle {\int }_0^T\left|a_2\left(t\right)\right|\exp \left(-{\displaystyle {\int }_0^t{a}_1\left(s\right)\text{d}s}\right)\text{d}t}\le c{\epsilon }^{\gamma \left(2+\alpha \right)-\alpha },$ (3.40)

及

${\displaystyle {\int }_0^T\left|a_0\left(t\right)\right|\text{d}t}\le c{\epsilon }^{-1}.$ (3.41)

因此，由 $M_0$ 的定义及(3.37)并应用(3.40)~(3.41)，对于适当小的 $\epsilon >0$，有

$\begin{array}{c}\left(w_m\left(0,y\right)+K\right){\displaystyle {\int }_0^T{a}_0^{+}\left(t\right)\text{d}t}\le \frac{M}{M_0}+c{\epsilon }^{\varpi -\alpha }+K{\displaystyle {\int }_0^T\left|a_0\left(t\right)\right|\text{d}t}\\ \le \frac{M}{M_0}+c{\epsilon }^{\min \left\{\varpi -\alpha ,\gamma \left(2+\alpha \right)-\left(1+\alpha \right)\right\}}\le 1.\end{array}$ (3.42)

因此，由(3.38)~(3.42)，应用引理2.2，可以得到

$\left|w_m\left(t,x_m\left(t,y\right)\right)\right|\le c\epsilon ,\forall t\in \left[0,T\right]$

因为y是任意的，于是有

$\left|w_m\left(t,x\right)\right|\le c\epsilon ,\forall \left(t,x\right)\in D^T$

那些 $i=1,\cdots ,m-1,m+1,\cdots ,n$ 的情形可以被类似的证明。

那么，注意到(2.2)，最终得到

${\Vert u_x\left(t,x\right)\Vert }_{c^0\left(D^T\right)}\le c\epsilon .$

因此，对于任意正常数 $M<M_0$，Cauchy问题(1.1)和(1.4)在区域 $D^t\left(0\le t\le M{\epsilon }^{-\left(1+\alpha \right)}\right)$ 上存在唯一解。这意味着当 $\epsilon >0$ 适当小时，(3.35)成立，这就证明了(3.18)。定理1.1证毕。

致谢

本文是在导师徐玉梅副教授的悉心指导下完成的，徐老师不仅传授知识，悉心指导论文需要注意的地方，对本篇论文的推导提供了很多的帮助，而且其对学术的不懈追求激励着我不断进取。

同时，作者衷心地感谢李傅山、王培合等各位老师的精心指导和帮助，在课上，两位老师传授了很多偏微分方向的知识，拓宽了自己的知识面，掌握了解决问题的一些基本方法。

最后，还要感谢同专业的同学对我的帮助，在我感到迷茫的时候，经过反复的讨论交流，给了我很多的启发。