Pure Mathematics
Vol. 09  No. 02 ( 2019 ), Article ID: 29225 , 10 pages
10.12677/PM.2019.92021

Qualitative Analysis of Nondivergent Degraded Parabolic Equations with Nonlinear Nonlocal Boundary Condition

Shuwang Xu, Zhuangzhuang Li, Jinzhong Qiu, Meng Li

School of Mathematics and Statistics, Shandong Normal University, Jinan Shandong

Received: Feb. 20th, 2019; accepted: Mar. 6th, 2019; published: Mar. 13th, 2019

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.

Keywords:Comparison Principle, Upper and Lower Solutions, Degenerate Parabolic Equation of Non-Divergence Type

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) ${u}_{0}\left(x\right)\in {C}^{2+\alpha }\left(\Omega \right)\cap C\left(\overline{\Omega }\right),\alpha \in \left(0,1\right),{u}_{0}\left(x\right)>0,x\in \Omega$ ，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 $x\in \partial \Omega ,y\in \overline{\Omega }$ 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)

Qualitative Analysis of Nondivergent Degraded Parabolic Equations with Nonlinear Nonlocal Boundary Condition[J]. 理论数学, 2019, 09(02): 164-173. https://doi.org/10.12677/PM.2019.92021

1. 1. Deng, W., Li, Y. and Xie, C. (2003) Existence and Nonexistence of Global Solutions of Some Nonlocal Degenerate Parabolic Equa-tions. Applied Mathematics Letters, 16, 803-808.
https://doi.org/10.1016/S0893-9659(03)80118-0

2. 2. Friedman, A. (1986) Monotonic Decay of Solutions of Parabolic Equations with Nonlocal Boundary Conditions. Quarterly of Applied Mathematics, 44, 401-407.
https://doi.org/10.1090/qam/860893

3. 3. Lin, Z. and Liu, Y. (2004) Uniform Blow up Profiles for Diffusion Equations with Nonlocal Source and Nonlocal Boundary. Acta Mathematica Scientia, 24, 443-450.
https://doi.org/10.1016/S0252-9602(17)30168-6

4. 4. Chen, Y. (2011) Blow-Up for a localized Singular Parabolic Equation with Weighted Nonlocal Nonlinear Boundary Conditions. Journal of Mathematical Analysis & Applications, 384, 421-430.
https://doi.org/10.1016/j.jmaa.2011.05.079

5. 5. 钟光胜. 几类非线性抛物方程(组)解的性质研究[D]: [博士学位论文]. 镇江: 江苏大学, 2016.

6. 6. Deng, K. (1992) Comparison Principle for Some Nonlocal Problems. Quarterly of Applied Mathematics, 50, 517-522.
https://doi.org/10.1090/qam/1178431