具有非线性非局部边界条件的非散度型退化抛物方程的定性分析Qualitative Analysis of Nondivergent Degraded Parabolic Equations with Nonlinear Nonlocal Boundary Condition

DOI: 10.12677/PM.2019.92021, PDF, HTML, XML, 下载: 414  浏览: 595  科研立项经费支持

Abstract: In this paper, we consider the qualitative analysis of a class of degenerate parabolic equation of non-divergence type with non-linear and non-local boundary conditions. Under the condition of generalized exponential terms, the global existence and blow-up properties of solutions of the equation under various conditions are discussed by using the upper and lower solutions method.

1. 引言

$\left\{\begin{array}{l}{u}_{t}=f\left(u\right)\left(\Delta u+a{\int }_{\Omega }{u}^{\gamma }\left(x,t\right)\text{d}x\right),x\in \Omega ,t>0,\\ u\left(x,t\right)={\int }_{\Omega }g\left(x,y\right){u}^{l}\left(y,t\right)\text{d}y,x\in \partial \Omega ,t>0,\\ u\left(x,0\right)={u}_{0}\left(x\right),x\in \Omega ,\end{array}$ (1.1)

(H1) ${u}_{0}\left(x\right)\in {C}^{2+\alpha }\left(\Omega \right)\cap C\left(\stackrel{¯}{\Omega }\right),\alpha \in \left(0,1\right),{u}_{0}\left(x\right)>0,x\in \Omega$ ，且

${u}_{0}\left(x\right)={\int }_{\Omega }g\left(x,y\right){u}_{0}{}^{l}\left(y\right)\text{d}y,x\in \partial \Omega$

(H2) 当 $x\in \partial \Omega ,y\in \stackrel{¯}{\Omega }$ 时， $g\left(x,y\right)$ 是连续非负函数，且 $g\left(x,y\right)\overline{)\equiv }0$

(H3) $f\left(s\right)\in C\left(0,\infty \right)\cap {C}^{1}\left(0,\infty \right),f\left(s\right)>0,{f}^{\prime }\left(s\right)\ge 0,s\in \left(0,\infty \right)$

${u}_{t}=f\left(u\right)\left(\Delta u+a{\int }_{\Omega }u\text{d}x\right),$ (1.2)

$\phi \left(x\right)$ 是如下线性椭圆问题的唯一正解

$-\Delta \phi =1,x\in \Omega ;\phi \left(x\right)=0,x\in \partial \Omega .$ (1.3)

$\left\{\begin{array}{l}{u}_{t}=\Delta u+g\left(x,u\right),x\in \Omega ,t>0,\\ u\left(x,t\right)={\int }_{\Omega }f\left(x,y\right)u\left(y,t\right)\text{d}y,x\in \partial \Omega ,t>0,\end{array}$ (1.4)

$\left\{\begin{array}{l}{u}_{t}=f\left(u\right)\left(\Delta u+au\left({x}_{0},t\right)\right),x\in \Omega ,t>0,\\ u\left(x,t\right)={\int }_{\Omega }g\left(x,y\right)u\left(y,t\right)\text{d}y,x\in \partial \Omega ,t>0,\\ u\left(x,0\right)={u}_{0}\left(x\right),x\in \Omega ,\end{array}$ (1.5)

1) 如果 $a$ 充分小，那么问题(1.1)的解整体存在；

2) 如果 $a$ 充分大，且存在某个正数 $\delta$ 使得 ${\int }_{\delta }^{+\infty }1/\left(sf\left(s\right)\right)\text{d}s=+\infty$ ，那么问题(1.1)的解整体存在。

(1.6)

2. 比较原理

(2.1)

(2.2)

，其中。由于内有界且连续，可知。定义，由以上变换我们易知。事实上，从(2.2)的第二式和第三式，利用转换式。假定，从而的最小值在内取得。不失一般性，我们假定最小值在处取得，从而对于任一带入(2.2)第一式，得到

，则

，则由已知条件，且在

3. 整体存在和有限时刻爆破

1) 设是线性椭圆问题

(3.1)

， (3.2)

(3.3)

(3.4)

2) 考虑常微分方程

(3.5)

，那么整体存在且

(3.6. A)

，令足够大且满足，则只需取

(3.6. B)

，令满足，且取，仍得

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

，则对于，有

(3.15)

(3.16)

. (3.17)

。证讫。

，则

(3.18)

.

,

(3.19)

。定义

(3.20)

(3.21)

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