具有梯度吸收项的非局部反应扩散方程解的性质

Properties of Solutions to Nonlocal Reaction-Diffusion Equations with Gradient Absorption Terms

摘要: 针对在非局部边界条件下具有梯度吸收项的反应扩散方程解的性质的问题，通过构造适当的辅助函数，利用散度定理、Hölder不等式、Young不等式、Sobolev不等式和微分不等式技巧，得到解的全局存在性和爆破时间的下界的估计。

Abstract: For the properties of solutions of reaction-diffusion equations with gradient absorption terms under non-local boundary conditions, by using auxiliary function, divergence theorem, Hölder inequalities, Young inequalities, Sobolev inequalities and a differential inequality technique, the global existence of the solutions and lower bound for blow-up time are derived.

1. 引言

本文考虑下列具有梯度吸收项的非局部反应扩散方程在非齐次Neumann边界条件下解的全局存在性和爆破时间的下界

$\{\begin{array}{l}{u}_t=\Delta u^m+k\left(t\right)u^p{\left({\displaystyle {\int }_{\Omega }{u}^{\alpha }}\text{d}x\right)}^s-{\left|\nabla u\right|}^q,\left(x,t\right)\in \Omega \times \left(0,t^{\ast }\right),\\ \frac{\partial u}{\partial \upsilon }={\displaystyle {\int }_{\Omega }g\left(u\right)\text{d}x},\left(x,t\right)\in \partial \Omega \times \left(0,t^{\ast }\right),\\ u\left(x,0\right)=u_0\left(x\right)\ge 0,x\in \bar{\Omega },\end{array}$ (1.1)

其中 $\Omega \subset {ℝ}^N\left(N\ge 3\right)$ 是一个具有光滑边界 $\partial \Omega$ 的有界域， $\frac{\partial u}{\partial \upsilon }$ 表示u在 $\partial \Omega$ 上的外法向量导数， $t^{\ast }$ 是爆破时间，初值 $u_0\in C^1\left(\bar{\Omega }\right)$ 是正函数且满足相容性条件。

近年来，许多学者对抛物方程和抛物系统解的全局存在性、解的爆破时间的上下界、爆破集、爆破速率和解的渐近行为等解的其他性态进行广泛研究，并取得丰硕的成果 [1] - [6] 。Payne等学者起初在低维空间中研究不同边值条件下解的性态。后来，学者们将其推广到高维空间，继续研究解的性态。目前，关于时变系数和空变系数的非局部反应扩散方程(组)解的性态研究较多 [5] [6] [7] ；其中，有很多学者对反应扩散方程(组)解的全局存在性与爆破进行研究。有关抛物方程和抛物系统下界的研究成果在物理学、生物学、天文学、化学等领域都有重要应用 [8] [9] 。影响解全局存在与爆破的主要因素有初边值条件、时变系数、空变系数、空间维数、吸收项、非线性项以及局部或者非局部等。问题(1.1)可以用于描述物理中气体流量、多孔介质力学、流体力学等两种介质的反应扩散问题 [10] [11] 。

本文的研究受下面文献的启发。

文献 [3] 研究了具有加权非局部源和吸收项的抛物方程

$\{\begin{array}{l}{u}_t=\Delta u^m+a\left(x\right)u^p{\displaystyle {\int }_{\Omega }{u}^q\text{d}x}-u^s,\left(x,t\right)\in \Omega \times \left(0,t^{\ast }\right),\\ u\left(x,t\right),\left(x,t\right)\in \partial \Omega \times \left(0,t^{\ast }\right),\\ u\left(x,0\right)=u_0\left(x\right)\ge 0,x\in \Omega \end{array}$

解的爆破问题，得到了全空间上解爆破的充分条件和爆破发生时爆破时间上下界的估计。

文献 [12] 研究了具有非局部和梯度项的反应扩散方程

$\{\begin{array}{l}{u}_t=\Delta u+a{u}^p{\left({\displaystyle {\int }_{\Omega }{u}^{\alpha }\text{d}x}\right)}^m-{\left|\nabla u\right|}^q,\left(x,t\right)\in \Omega \times \left(0,t^{\ast }\right),\\ \frac{\partial u}{\partial \upsilon }=h\left(u\right),\left(x,t\right)\in \partial \Omega \times \left(0,t^{\ast }\right),\\ u\left(x,0\right)=u_0\left(x\right)\ge 0,x\in \bar{\Omega },\end{array}$

解的爆破问题，运用Sobolev不等式和微分不等式技巧得到了高维空间中爆破时间的下界估计。

文献 [2] 研究了齐次Dirichlet和齐次Neumann边界条件下具有吸收项的非局部多孔介质方程

$u_t=\Delta u^m+u^p{\displaystyle {\int }_{\Omega }{u}^q\text{d}x}-u^s$ ， $\left(x,t\right)\in \Omega \times \left(0,t^{\ast }\right)$ ，

解的爆破问题，得到了在三维空间中爆破时间的下界估计。

目前为止，并未发现有学者研究问题(1.1)具有时变系数和梯度吸收项的非局部多孔介质解的全局存在性和爆破现象。本文的研究目标是非线性边界条件下 ${ℝ}^N\left(N\ge 3\right)$ 上解的全局存在性条件及爆破时间的下界估计。本文研究的难点是构造适当的辅助函数并且恰当处理高维空间、非线性边界条件、时变系数、非局部项、以及吸收项对解的爆破的影响。

2. 全局存在性

引理1 [6] 设 $\Omega$ 是 ${ℝ}^N\left(N\ge 3\right)$ 上的有界凸区域，则对于 $u\in C^1\left(\bar{\Omega }\right)$ ， $w>0$ ，有不等式

${\displaystyle {\int }_{\partial \Omega }{u}^w}\text{d}S\le \frac{N}{{\rho }_0}{\displaystyle {\int }_{\Omega }{u}^w}\text{d}x+\frac{wd}{{\rho }_0}{\displaystyle {\int }_{\Omega }{u}^{w-1}}\left|\nabla u\right|\text{d}x$ ， (2.1)

其中， ${\rho }_0=\underset{x\in \partial \Omega }{\min }\left(x\cdot \upsilon \right)$ ， $d=\underset{x\in \Omega }{\max }\left|x\right|$ 。

引理2 [13] 设 ${\lambda }_1$ 是固定膜问题

$\{\begin{array}{l}\Delta w+\lambda w=0,x\in \Omega ,\\ w=0,x\in \partial \Omega ,\\ w>0,x\in \Omega \end{array}$

的第一正特征值，则下列不等式成立

${\displaystyle {\int }_{\Omega }{u}^{ns-1}{\left|\nabla u\right|}^q\text{d}x}\ge {\left(\frac{2\sqrt{{\lambda }_1}}{ns+q-1}\right)}^q{\displaystyle {\int }_{\Omega }{u}^{ns+q-1}\text{d}x}$ 。 (2.2)

定理1设 $u\left(x,t\right)$ 是问题(1.1)的非负古典解， $0<g\left(\theta \right)\le {\theta }^r$ ， $0<m<1$ ， $1<r<\frac{3-m}{2}$ ， $1<s<\frac{1-p}{\alpha }$ ， $\xi =\max \{m+2r-2,p+\alpha s\}<q<1$ ， $0<\alpha +p<1$ ， $\frac{k^{\prime }\left(t\right)}{k\left(t\right)}<\beta$ ， $\beta >0$ ， $\theta >0$ ，则 $u\left(x,t\right)$ 全局存在。

证明 构造辅助函数

$\phi \left(t\right)=k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x$ ， $\sigma >1$ 。 (2.3)

对 $\phi \left(t\right)$ 进行求导，可得

$\begin{array}{c}{\phi }^{\prime }\left(t\right)=k^{\prime }\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x+\sigma k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma -1}}{u}_t\text{d}x\\ \le \beta \phi \left(t\right)+\sigma k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma -1}}\Delta u^m\text{d}x+\sigma k^2\left(t\right){\displaystyle {\int }_{\Omega }{u}^{p+\sigma -1}}\text{d}x{\left({\displaystyle {\int }_{\Omega }{u}^{\alpha }}\text{d}x\right)}^s\\ -\sigma k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma -1}}\text{d}x{\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^q\text{d}x}.\end{array}$ (2.4)

运用散度定理、(2.1)式和Young不等式，可得

$\begin{array}{c}{\displaystyle {\int }_{\Omega }{u}^{\sigma -1}}\Delta u^m\text{d}x={\displaystyle {\int }_{\partial \Omega }{u}^{\sigma -1}\frac{\partial u^m}{\partial \upsilon }\text{d}S}-m\left(\sigma -1\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +m-3}{\left|\nabla u\right|}^2\text{d}x}\\ \le mb{\displaystyle {\int }_{\partial \Omega }{u}^{\sigma +m-2}\frac{\partial u^m}{\partial \upsilon }\text{d}S{\displaystyle {\int }_{\Omega }{u}^r\text{d}x}}-m\left(\sigma -1\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +m-3}{\left|\nabla u\right|}^2\text{d}x}\\ \le \frac{Nmb\left|\Omega \right|}{{\rho }_0}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+r-2}\text{d}x}+\frac{\left(\sigma +m-2\right)dmb\left|\Omega \right|}{{\rho }_0}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+r-3}\left|\nabla u\right|\text{d}x}\\ -m\left(\sigma -1\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +m-3}{\left|\nabla u\right|}^2\text{d}x}\\ \le \frac{Nmb\left|\Omega \right|}{{\rho }_0}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+r-2}\text{d}x}+\frac{\left(\sigma +m-2\right)dmb{\epsilon }_1\left|\Omega \right|}{2{\rho }_0}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m-3}{\left|\nabla u\right|}^2\text{d}x}\\ +\frac{\left(\sigma +m-2\right)dmb\left|\Omega \right|}{2{\epsilon }_1{\rho }_0}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x-m\left(\sigma -1\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +m-3}{\left|\nabla u\right|}^2\text{d}x}.\end{array}$ (2.5)

由(2.5)可得

$\sigma k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma -1}}\Delta u^m\text{d}x\le r_1\sigma k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +m+r-2}\text{d}x+\frac{r_2\sigma }{2{\epsilon }_1}k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x}+r_3\sigma k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +m-3}{\left|\nabla u\right|}^2\text{d}x}$ ，

其中 $r_1=\frac{Nmb\left|\Omega \right|}{{\rho }_0}$ ， $r_2=\frac{\left(\sigma +m-2\right)dmb\left|\Omega \right|}{{\rho }_0}$ ， $r_3=\frac{\left(\sigma +m-2\right)dmb{\epsilon }_1\left|\Omega \right|}{2{\rho }_0}-m\left(\sigma -1\right)$ ，

选取适当的 ${\epsilon }_1$ 使得 $r_3\le 0$ ，可得

$\sigma k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma -1}}\Delta u^m\text{d}x\le r_1\sigma k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +m+r-2}\text{d}x}+\frac{r_2\sigma }{2{\epsilon }_1}k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x$ 。 (2.6)

在定理1的条件下，由Hölder不等式和Young不等式，可得

${\displaystyle {\int }_{\Omega }{u}^{\sigma +m+r-2}\text{d}x}\le \frac{m+r-2}{m+2r-3}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x+\frac{r-1}{m+2r-3}{\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x$ 。 (2.7)

将(2.7)式代入(2.6)式，可得

$\sigma k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma -1}}\Delta u^m\text{d}x\le r_{4}k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x+r_{5}k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x$ 。 (2.8)

其中 $r_4=\frac{\left(m+r-2\right)r_1\sigma }{m+2r-3}+\frac{r_2\sigma }{2{\epsilon }_1}$ ， $r_5=\frac{\left(r-1\right)r_1\sigma }{m+2r-3}$ 。

在定理1的条件下，对(2.4)式右端第三项运用Hölder不等式，可得

$\sigma k^2\left(t\right){\displaystyle {\int }_{\Omega }{u}^{p+\sigma -1}}\text{d}x{\left({\displaystyle {\int }_{\Omega }{u}^{\alpha }}\text{d}x\right)}^s\le \sigma k^2\left(t\right){\left|\Omega \right|}^s{\displaystyle {\int }_{\Omega }{u}^{p+\sigma +\alpha s-1}}\text{d}x$ 。 (2.9)

对(2.4)式右端第三项运用(2.2)式，可得

$\sigma k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma -1}}\text{d}x{\displaystyle {\int }_{\Omega }{\left|\nabla u\right|}^q\text{d}x}\ge \sigma k\left(t\right){\left(\frac{2\sqrt{{\lambda }_1}}{\sigma +q-1}\right)}^q{\displaystyle {\int }_{\Omega }{u}^{\sigma +q-1}\text{d}x}$ 。 (2.10)

将(2.8)~(2.10)式代入(2.4)式，可得

${\phi }^{\prime }\left(t\right)\le \left(r_5+\beta \right)\phi \left(t\right)+r_{4}k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x+r_{6}k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{p+\sigma +\alpha s-1}}\text{d}x-r_{7}k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +q-1}\text{d}x}$ ， (2.11)

其中 $r_6=\sigma k\left(t\right){\left|\Omega \right|}^s$ ， $r_7=\sigma {\left(\frac{2\sqrt{{\lambda }_1}}{\sigma +q-1}\right)}^q$ 。

由 $\xi =\max \{m+2r-2,p+\alpha s\}$ ，可得

${\phi }^{\prime }\left(t\right)\le \left(r_5+\beta \right)\phi \left(t\right)+\left(r_4+r_6\right)k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +\xi -1}}\text{d}x-r_{7}k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +q-1}\text{d}x}$ 。 (2.12)

在定理1的条件下，由Hölder不等式和Young不等式，可得

${\displaystyle {\int }_{\Omega }{u}^{\sigma +\xi -1}}\text{d}x\le {\left({\displaystyle {\int }_{\Omega }{u}^{\sigma +q-1}}\text{d}x\right)}^{\frac{\xi -1}{q-1}}{\left({\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x\right)}^{\frac{q-\xi }{q-1}}\le \frac{\xi -1}{q-1}{\epsilon }_2{\displaystyle {\int }_{\Omega }{u}^{\sigma +q-1}}\text{d}x+\frac{q-\xi }{q-1}{\epsilon }_2^{-\frac{\xi -1}{q-\xi }}{\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x$ 。 (2.13)

将(2.13)式代入(2.12)式，可得

${\phi }^{\prime }\left(t\right)\le r_8\phi \left(t\right)-r_{9}k\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +q-1}\text{d}x}$ ， (2.14)

其中 $r_8=r_5+\beta +\frac{q-\xi }{q-1}{\epsilon }_2^{-\frac{\xi -1}{q-\xi }}\left(r_4+r_6\right)$ ， $r_9=r_7-\frac{\xi -1}{q-1}{\epsilon }_2\left(r_4+r_6\right)$ 。

在定理1的条件下，由Hölder不等式，可得

${\displaystyle {\int }_{\Omega }{u}^{\sigma +q-1}\text{d}x}\ge {\left({\displaystyle {\int }_{\Omega }{u}^{\sigma }\text{d}x}\right)}^{\frac{\sigma +q-1}{\sigma }}{\left|\Omega \right|}^{\frac{1-q}{\sigma }}$ 。 (2.15)

将(2.15)式代入(2.14)式，可得

${\phi }^{\prime }\left(t\right)\le \phi \left(t\right)\left(r_8-r_9{\left|\Omega \right|}^{\frac{1-q}{\sigma }}{k}^{\frac{1-q}{\sigma }}\left(t\right){\phi }^{\frac{q-1}{\sigma }}\left(t\right)\right)$ 。 (2.16)

如果u在 $\phi \left(t\right)$ 度量下的某个 $t^{\ast }$ 爆破，即 $\underset{t\to t^{\ast -}}{\lim }\phi \left(t\right)=\infty$ ；由(2.16)式可知，对 $\forall t<t^{\ast }$ 有 ${\phi }^{\prime }\left(t\right)\le 0$ ，于是产生矛盾。

3. 爆破时间的下界

引理3 [14] 将 $W^{1,2}\left(\Omega \right)$ 嵌入 $L^{\frac{2N}{N-2}}\left(\Omega \right)$ ， $N\ge 3$ ，可知

${\left({\displaystyle {\int }_{\Omega }{\omega }^{2N/\left(N-2\right)}\text{d}x}\right)}^{\left(N-2\right)/2N}\le C{\left({\displaystyle {\int }_{\Omega }{\omega }^2\text{d}x}+{\displaystyle {\int }_{\Omega }{\left|\nabla \omega \right|}^2\text{d}x}\right)}^{1/2}$ ， (3.1)

其中 $\omega \in W^{1,2}\left(\Omega \right)$ ， $C=C\left(N,\Omega \right)$ 。

定理2 设 $u\left(x,t\right)$ 是问题(1.1)的非负古典解， $0<g\left(\theta \right)\le {\theta }^r$ ， $\theta >0$ ， $m>1$ ， $r>\max \{1,\frac{m-2}{q-2}\}$ ， $q>2$ ， $m+2r>\max \{3,p+\alpha s+2\}$ ， $\frac{k^{\prime }\left(t\right)}{k\left(t\right)}<\beta$ ， $\beta >0$ ，则 $u\left(x,t\right)$ 在有限时间 $t^{\ast }$ 发生爆破，其中 $t^{\ast }\ge {\displaystyle {\int }_{\varphi \left(0\right)}^{+\infty }\frac{\text{d}\eta }{\delta \beta \eta +M_5{\eta }^A+M_4}}$ ， $M_4$ 、 $M_5$ 和A分别在(3.20)式和(3.22)式给出。

证明 构造辅助函数

$\varphi \left(t\right)=k^{\delta }\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x$ ， (3.2)

其中 $\delta >1$ ， $\sigma >N\left(r-1\right)$ 。

对 $\varphi \left(t\right)$ 进行求导并运用散度定理，可得

$\begin{array}{c}{\varphi }^{\prime }\left(t\right)=\delta k^{\delta -1}\left(t\right)k^{\prime }\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x+\sigma k^{\delta }\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma -1}{u}_t}\text{d}x\\ \le \delta \beta \varphi \left(t\right)+\sigma k^{\delta }\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma -1}\Delta u^m}\text{d}x+\sigma k^{\delta +1}\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +p-1}}\text{d}x{\left({\displaystyle {\int }_{\Omega }{u}^{\alpha }}\text{d}x\right)}^s-\sigma k^{\delta }\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma -1}{\left|\nabla u\right|}^q}\text{d}x\\ \le \delta \beta \varphi \left(t\right)-\sigma k^{\delta }\left(t\right)m\left(\sigma -1\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +m-3}{\left|\nabla u\right|}^2}\text{d}x+\sigma k^{\delta }\left(t\right)m{\displaystyle {\int }_{\partial \Omega }{u}^{\sigma +m-2}}\text{d}S{\displaystyle {\int }_{\Omega }{u}^r}\text{d}x\\ +\sigma k^{\delta +1}\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +p-1}}\text{d}x{\left({\displaystyle {\int }_{\Omega }{u}^{\alpha }}\text{d}x\right)}^s-\sigma k^{\delta }\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma -1}{\left|\nabla u\right|}^q}\text{d}x.\end{array}$ (3.3)

在定理2的条件下，由Hölder不等式和Young不等式，可得

$\begin{array}{c}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m-3}{\left|\nabla u\right|}^2}\text{d}x\le {\left({\displaystyle {\int }_{\Omega }{u}^{\sigma -1}{\left|\nabla u\right|}^q}\text{d}x\right)}^{\frac{2}{q}}{\left({\displaystyle {\int }_{\Omega }{u}^{\sigma +m-3+2\left(m-2\right)/\left(q-2\right)}}\text{d}x\right)}^{\frac{q-2}{q}}\\ \le {\displaystyle {\int }_{\Omega }{u}^{\sigma -1}{\left|\nabla u\right|}^q}\text{d}x+\frac{q-2}{q}{\left(\frac{q}{2}\right)}^{2/\left(2-q\right)}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m-3+2\left(m-2\right)/\left(q-2\right)}}\text{d}x.\end{array}$ (3.4)

将(3.4)式代入(3.3)式，可得

$\begin{array}{c}{\varphi }^{\prime }\left(t\right)\le \delta \beta \varphi \left(t\right)-\frac{4\sigma \left(m\left(\sigma -1\right)+1\right)}{{\left(\sigma +m-1\right)}^2}{k}^{\delta }\left(t\right){\displaystyle {\int }_{\Omega }{\left|\nabla u^{\left(\sigma +m-1\right)/2}\right|}^2}\text{d}x\\ +\sigma k^{\delta }\left(t\right)\frac{q-2}{q}{\left(\frac{q}{2}\right)}^{2/\left(2-q\right)}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m-3+2\left(m-2\right)/\left(q-2\right)}}\text{d}x\\ +\sigma k^{\delta }\left(t\right)m{\displaystyle {\int }_{\partial \Omega }{u}^{\sigma +m-2}}\text{d}S{\displaystyle {\int }_{\Omega }{u}^r}\text{d}x+\sigma k^{\delta +1}\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +p-1}}\text{d}x{\left({\displaystyle {\int }_{\Omega }{u}^{\alpha }}\text{d}x\right)}^s.\end{array}$ (3.5)

在定理2的条件下，由(2.1)式、Hölder不等式和Young不等式，可得

$\begin{array}{c}{\displaystyle {\int }_{\partial \Omega }{u}^{\sigma +m-2}}\text{d}S{\displaystyle {\int }_{\Omega }{u}^r}\text{d}x\le \frac{N\left|\Omega \right|}{{\rho }_0}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+r-2}}\text{d}x+\frac{\left(\sigma +m-2\right)d\left|\Omega \right|}{{\rho }_0}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+r-3}\left|\nabla u\right|\text{d}x}\\ \le \frac{N\left|\Omega \right|}{{\rho }_0}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+r-2}}\text{d}x+\frac{\left(\sigma +m-2\right)2{\tau }_{1}d\left|\Omega \right|}{{\rho }_0{\left(\sigma +m-1\right)}^2}{\displaystyle {\int }_{\Omega }{\left|\nabla u^{\left(\sigma +m-1\right)/2}\right|}^2}\text{d}x\\ +\frac{\left(\sigma +m-2\right)d\left|\Omega \right|}{2{\rho }_0{\tau }_1}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x.\end{array}$ (3.6)

取 ${\tau }_1=\frac{{\rho }_0\left(\left(m\left(\sigma -1\right)+1\right)\right)}{\left(\sigma +m-2\right)\sigma md\left|\Omega \right|}$ 并将(3.6)式代入(3.5)式，可得

$\begin{array}{c}{\varphi }^{\prime }\left(t\right)\le \delta \beta \varphi \left(t\right)-\frac{2\sigma \left(m\left(\sigma -1\right)+1\right)}{{\left(\sigma +m-1\right)}^2}{k}^{\delta }\left(t\right){\displaystyle {\int }_{\Omega }{\left|\nabla u^{\left(\sigma +m-1\right)/2}\right|}^2}\text{d}x\\ +\frac{\left(\sigma +m-2\right)d\sigma m\left|\Omega \right|}{2{\rho }_0{\tau }_1}{k}^{\delta }\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x+\frac{N\sigma m\left|\Omega \right|}{{\rho }_0}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+r-2}}\text{d}x\\ +\sigma k^{\delta }\left(t\right)\frac{q-2}{q}{\left(\frac{q}{2}\right)}^{2/\left(2-q\right)}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m-3+2\left(m-2\right)/\left(q-2\right)}}\text{d}x+\sigma k^{\delta +1}\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +p-1}}\text{d}x{\left({\displaystyle {\int }_{\Omega }{u}^{\alpha }}\text{d}x\right)}^s.\end{array}$ (3.7)

在定理2的条件下，由Hölder不等式和Young不等式，可得

$\begin{array}{c}{\displaystyle {\int }_{\Omega }{u}^{\sigma +p-1}}\text{d}x{\left({\displaystyle {\int }_{\Omega }{u}^{\alpha }}\text{d}x\right)}^s\le \left[{\left({\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x\right)}^{\left(\sigma +p+\alpha s-1\right)/\left(\sigma +m+2r-3\right)}{\left|\Omega \right|}^{\frac{m+2r-p-\alpha s-2}{\sigma +m+2r-3}}\right]{\left|\Omega \right|}^s\\ \le \frac{\sigma +p+\alpha s-1}{\sigma +m+2r-3}{\left|\Omega \right|}^s{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x+\frac{m+2r-p-\alpha s-2}{\sigma +m+2r-3}{\left|\Omega \right|}^{1+s},\end{array}$ (3.8)

$\begin{array}{c}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+r-2}}\text{d}x\le {\left({\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x\right)}^{\left(\sigma +m+r-2\right)/\left(\sigma +m+2r-3\right)}{\left|\Omega \right|}^{\frac{r-1}{\sigma +m+2r-3}}\\ \le \frac{\sigma +m+r-2}{\sigma +m+2r-3}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x+\frac{r-1}{\sigma +m+2r-3}\left|\Omega \right|,\end{array}$ (3.9)

$\begin{array}{c}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m-3+2\left(m-2\right)/\left(q-2\right)}}\text{d}x\le {\left({\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x\right)}^{\left(\sigma +m-3+2\left(m-2\right)/\left(q-2\right)\right)/\left(\sigma +m+2r-3\right)}{\left|\Omega \right|}^{\frac{2r-2\left(m-2\right)/\left(q-2\right)}{\sigma +m+2r-3}}\\ \le \frac{\sigma +m-3+2\left(m-2\right)/\left(q-2\right)}{\sigma +m+2r-3}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x+\frac{2r-2\left(m-2\right)/\left(q-2\right)}{\sigma +m+2r-3}\left|\Omega \right|.\end{array}$ (3.10)

将(3.8)~(3.10)式代入(3.7)，可得

${\varphi }^{\prime }\left(t\right)\le \delta \beta \varphi \left(t\right)-\frac{2\sigma \left(m\left(\sigma -1\right)+1\right)}{{\left(\sigma +m-1\right)}^2}{k}^{\delta }\left(t\right){\displaystyle {\int }_{\Omega }{\left|\nabla u^{\left(\sigma +m-1\right)/2}\right|}^2}\text{d}x+M_1{k}^{\delta }\left(t\right){\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x+M_2$ ， (3.11)

其中

$\begin{array}{c}{M}_1=\frac{\left(\sigma +m-2\right)d\sigma m\left|\Omega \right|}{2{\rho }_0{\tau }_1}+\frac{N\sigma m\left|\Omega \right|\left(\sigma +m+r-2\right)}{{\rho }_0\left(\sigma +m+2r-3\right)}\\ +\frac{\sigma \left(q-2\right)\left(\sigma +m-3+2\left(m-2\right)/\left(q-2\right)\right)}{q\left(\sigma +m+2r-3\right)}{\left(\frac{q}{2}\right)}^{2/2-q}\\ +\frac{\left(\sigma +p+\alpha s-1\right)\sigma k\left(t\right){\left|\Omega \right|}^s}{\sigma +m+2r-3},\end{array}$

$\begin{array}{c}{M}_2=\frac{N\sigma m{\left|\Omega \right|}^2\left(r-1\right)}{{\rho }_0\left(\sigma +m+2r-3\right)}+\frac{\sigma \left(q-2\right)\left(2r-2\left(m-2\right)/\left(q-2\right)\right)\left|\Omega \right|}{q\left(\sigma +m+2r-3\right)}{\left(\frac{q}{2}\right)}^{2/\left(2-q\right)}\\ +\frac{\left(m+2r-p-\alpha s-2\right)\sigma k^{\delta +1}\left(t\right){\left|\Omega \right|}^{1+s}}{\sigma +m+2r-3}.\end{array}$

由 $\sigma >N\left(r-1\right)$ 、Hölder不等式和(3.1)式，可得

$\begin{array}{c}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x\le {\left({\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x\right)}^{\frac{2\sigma +N\left(m-1\right)-\left(m+2r-3\right)\left(N-2\right)}{2\sigma +N\left(m-1\right)}}{\left({\displaystyle {\int }_{\Omega }{\left(u^{\frac{\sigma +m-1}{2}}\right)}^{\frac{2N}{N-2}}}\text{d}x\right)}^{\frac{\left(m+2r-3\right)\left(N-2\right)}{2\sigma +N\left(m-1\right)}}\\ \le {\left({\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x\right)}^{\frac{2\sigma +N\left(m-1\right)-\left(m+2r-3\right)\left(N-2\right)}{2\sigma +N\left(m-1\right)}}{\left[C^{\frac{2N}{N-2}}{\left({\displaystyle {\int }_{\Omega }{u}^{\sigma +m-1}}\text{d}x+{\displaystyle {\int }_{\Omega }{\left|\nabla u^{\frac{\sigma +m-1}{2}}\right|}^2}\text{d}x\right)}^{\frac{N}{N-2}}\right]}^{\frac{\left(m+2r-3\right)\left(N-2\right)}{2\sigma +N\left(m-1\right)}}\\ ={\left({\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x\right)}^{\frac{2\sigma +N\left(m-1\right)-\left(m+2r-3\right)\left(N-2\right)}{2\sigma +N\left(m-1\right)}}{C}^{\frac{2N\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}}{\left({\displaystyle {\int }_{\Omega }{u}^{\sigma +m-1}}\text{d}x+{\displaystyle {\int }_{\Omega }{\left|\nabla u^{\frac{\sigma +m-1}{2}}\right|}^2}\text{d}x\right)}^{\frac{N\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}}.\end{array}$ (3.12)

通过基本不等式

${\left(b_1+b_2\right)}^i\le b_1^i+b_2^i$ ， $b_1,b_2>0,0\le i<1$ 。 (3.13)

将(3.13)式代入(3.12)，可得

$\begin{array}{c}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x\le {\left({\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x\right)}^{\frac{2\sigma +N\left(m-1\right)-\left(m+2r-3\right)\left(N-2\right)}{2\sigma +N\left(m-1\right)}}{C}^{\frac{2N\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}}{\left({\displaystyle {\int }_{\Omega }{u}^{\sigma +m-1}}\text{d}x\right)}^{\frac{N\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}}\\ +{\left({\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x\right)}^{\frac{2\sigma +N\left(m-1\right)-\left(m+2r-3\right)\left(N-2\right)}{2\sigma +N\left(m-1\right)}}{C}^{\frac{2N\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}}{\left({\displaystyle {\int }_{\Omega }{\left|\nabla u^{\frac{\sigma +m-1}{2}}\right|}^2}\text{d}x\right)}^{\frac{N\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}}.\end{array}$ (3.14)

由 $\sigma >N\left(r-1\right)$ 、Hölder不等式和Young不等式，可得

$\begin{array}{l}{\left({\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x\right)}^{\frac{2\sigma +N\left(m-1\right)-\left(m+2r-3\right)\left(N-2\right)}{2\sigma +N\left(m-1\right)}}{C}^{\frac{2N\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}}{\left({\displaystyle {\int }_{\Omega }{u}^{\sigma +m-1}}\text{d}x\right)}^{\frac{N\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}}\\ \le {\left({\left(\frac{N{C}^2\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}\right)}^{\frac{N\left(m+2r-3\right)}{2\sigma +2N\left(1-r\right)}}{\left({\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x\right)}^{\frac{2\sigma +N\left(m-1\right)-\left(m+2r-3\right)\left(N-2\right)}{2\sigma +2N\left(1-r\right)}}\right)}^{\frac{2\sigma +2N\left(1-r\right)}{2\sigma +N\left(m-1\right)}}{\left(\frac{2\sigma +N\left(m-1\right)}{N\left(m+2r-3\right)}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m-1}}\text{d}x\right)}^{\frac{N\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}}\\ \le \frac{2\sigma +2N\left(1-r\right)}{2\sigma +N\left(m-1\right)}{\left(\frac{2N{C}^2\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}\right)}^{\frac{N\left(m+2r-3\right)}{2\sigma +2N\left(1-r\right)}}{\left({\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x\right)}^{\frac{2\sigma +N\left(m-1\right)-\left(m+2r-3\right)\left(N-2\right)}{2\sigma +2N\left(1-r\right)}}+{\displaystyle {\int }_{\Omega }{u}^{\sigma +m-1}}\text{d}x,\end{array}$ (3.15)

$\begin{array}{c}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m-1}}\text{d}x\le {\left(\frac{\sigma +m+2r-3}{2\left(\sigma +m-1\right)}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x\right)}^{\frac{\sigma +m-1}{\sigma +m+2r-3}}{\left({\left(\frac{\sigma +m+2r-3}{2\left(\sigma +m-1\right)}\right)}^{-\frac{\left(\sigma +m-1\right)}{2\left(r-1\right)}}\left|\Omega \right|\right)}^{\frac{2\left(r-1\right)}{\sigma +m+2r-3}}\\ \le \frac{1}{2}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x+\frac{2\left(r-1\right)}{\sigma +m+2r-3}{\left(\frac{\sigma +m+2r-3}{2\left(\sigma +m-1\right)}\right)}^{-\frac{\left(\sigma +m-1\right)}{2\left(r-1\right)}}\left|\Omega \right|,\end{array}$ (3.16)

$\begin{array}{l}{\left({\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x\right)}^{\frac{2\sigma +N\left(m-1\right)-\left(m+2r-3\right)\left(N-2\right)}{2\sigma +N\left(m-1\right)}}{C}^{\frac{2N\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}}{\left({\displaystyle {\int }_{\Omega }{\left|\nabla u^{\frac{\sigma +m-1}{2}}\right|}^2}\text{d}x\right)}^{\frac{N\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}}\\ \le {\left({\left({\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x\right)}^{\frac{2\sigma +N\left(m-1\right)-\left(m+2r-3\right)\left(N-2\right)}{2\sigma +2N\left(m-1\right)}}{\left(C^2{\tau }_2^{-1}\right)}^{\frac{N\left(m+2r-3\right)}{2\sigma +2N\left(1-r\right)}}\right)}^{\frac{2\sigma +2N\left(1-r\right)}{2\sigma +N\left(m-1\right)}}{\left({\tau }_2{\displaystyle {\int }_{\Omega }{\left|\nabla u^{\frac{\sigma +m-1}{2}}\right|}^2}\text{d}x\right)}^{\frac{N\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}}\\ \le \frac{2\sigma +2N\left(1-r\right)}{2\sigma +N\left(m-1\right)}{\left({\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x\right)}^{\frac{2\sigma +N\left(m-1\right)-\left(m+2r-3\right)\left(N-2\right)}{2\sigma +2N\left(m-1\right)}}{\left(C^2{\tau }_2^{-1}\right)}^{\frac{N\left(m+2r-3\right)}{2\sigma +2N\left(1-r\right)}}+\frac{N{\tau }_2\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}{\displaystyle {\int }_{\Omega }{\left|\nabla u^{\frac{\sigma +m-1}{2}}\right|}^2}\text{d}x.\end{array}$ (3.17)

将(3.15)~(3.17)式代入(3.14)，可得

$\begin{array}{c}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x\le \frac{2\sigma +2N\left(1-r\right)}{2\sigma +N\left(m-1\right)}{\left(\frac{2N{C}^2\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}\right)}^{\frac{N\left(m+2r-3\right)}{2\sigma +2N\left(1-r\right)}}{\left({\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x\right)}^{\frac{2\sigma +N\left(m-1\right)-\left(m+2r-3\right)\left(N-2\right)}{2\sigma +2N\left(1-r\right)}}+\frac{1}{2}{\displaystyle {\int }_{\Omega }{u}^{\sigma +m+2r-3}}\text{d}x\\ +\frac{2\left(r-1\right)}{\sigma +m+2r-3}{\left(\frac{\sigma +m+2r-3}{2\left(\sigma +m-1\right)}\right)}^{-\frac{\sigma +m-1}{2\left(r-1\right)}}\left|\Omega \right|+\frac{N{\tau }_2\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}{\displaystyle {\int }_{\Omega }{\left|\nabla u^{\frac{\sigma +m-1}{2}}\right|}^2}\text{d}x\\ +\frac{2\sigma +2N\left(1-r\right)}{2\sigma +N\left(m-1\right)}{\left({\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x\right)}^{\frac{2\sigma +N\left(m-1\right)-\left(m+2r-3\right)\left(N-2\right)}{2\sigma +2N\left(m-1\right)}}{\left(C^2{\tau }_2^{-1}\right)}^{\frac{N\left(m+2r-3\right)}{2\sigma +2N\left(1-r\right)}}\\ \le 2\left(\frac{2\sigma +2N\left(1-r\right)}{2\sigma +N\left(m-1\right)}{\left(\frac{2N{C}^2\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}\right)}^{\frac{N\left(m+2r-3\right)}{2\sigma +2N\left(1-r\right)}}+\frac{2\sigma +2N\left(1-r\right)}{2\sigma +N\left(m-1\right)}{\left(C^2{\tau }_2^{-1}\right)}^{\frac{N\left(m+2r-3\right)}{2\sigma +2N\left(1-r\right)}}\right)\\ \times {\left({\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x\right)}^{\frac{2\sigma +N\left(m-1\right)-\left(m+2r-3\right)\left(N-2\right)}{2\sigma +2N\left(1-r\right)}}+\frac{4\left(r-1\right)}{\sigma +m+2r-3}{\left(\frac{\sigma +m+2r-3}{2\left(\sigma +m-1\right)}\right)}^{-\frac{\left(\sigma +m-1\right)}{2\left(r-1\right)}}\left|\Omega \right|\\ +\frac{2N{\tau }_2\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}{\displaystyle {\int }_{\Omega }{\left|\nabla u^{\frac{\sigma +m-1}{2}}\right|}^2}\text{d}x.\end{array}$ (3.18)

取 ${\tau }_2=\frac{2\sigma \left(2\sigma +N\left(m-1\right)\right)\left(m\left(\sigma -1\right)+1\right)}{2N{M}_1\left(m+2r-3\right){\left(\sigma +m-1\right)}^2}$ 并将(3.18)式代入(3.11)，可得

${\varphi }^{\prime }\left(t\right)\le \delta \beta \varphi \left(t\right)+M_3{k}^{\delta }\left(t\right){\left({\displaystyle {\int }_{\Omega }{u}^{\sigma }}\text{d}x\right)}^{\frac{2\sigma +N\left(m-1\right)-\left(m+2r-3\right)\left(N-2\right)}{2\sigma +2N\left(1-r\right)}}+M_4$ ， (3.19)

其中

$M_3=2M_1\left(\frac{2\sigma +2N\left(1-r\right)}{2\sigma +N\left(m-1\right)}{\left(\frac{2N{C}^2\left(m+2r-3\right)}{2\sigma +N\left(m-1\right)}\right)}^{\frac{N\left(m+2r-3\right)}{2\sigma +2N\left(1-r\right)}}+\frac{2\sigma +2N\left(1-r\right)}{2\sigma +N\left(m-1\right)}{\left(C^2{\tau }_2^{-1}\right)}^{\frac{N\left(m+2r-3\right)}{2\sigma +2N\left(1-r\right)}}\right)$ ，

$M_4=\frac{4M_1{k}^{\delta }\left(t\right)\left(r-1\right)}{\sigma +m+2r-3}{\left(\frac{\sigma +m+2r-3}{2\left(\sigma +m-1\right)}\right)}^{-\frac{\left(\sigma +m-1\right)}{2\left(r-1\right)}}\left|\Omega \right|+M_2$ 。 (3.20)

由(3.19)，可得

${\varphi }^{\prime }\left(t\right)\le \delta \beta \varphi \left(t\right)+M_5{\left(\varphi \left(t\right)\right)}^A+M_4$ ， (3.21)

其中

$M_5=M_3{\left(k\left(t\right)\right)}^{\delta -\delta A}$ ， $A=\frac{2\sigma +N\left(m-1\right)-\left(m+2r-3\right)\left(N-2\right)}{2\sigma +2N\left(1-r\right)}$ 。 (3.22)

对(3.21)式两边从0到t积分，可得

$t^{\ast }\ge {\displaystyle {\int }_{\varphi \left(0\right)}^{+\infty }\frac{\text{d}\eta }{\delta \beta \eta +M_5{\eta }^A+M_4}}$ 。