Pure Mathematics
Vol.05 No.02(2015), Article ID:14934,6
pages
10.12677/PM.2015.52009
Blowup of Solutions for a Class of Doubly Nonlinear Parabolic Equations
Jing Su, Longfei Qi, Qingying Hu
College of Science, Henan University of Technology, Zhengzhou Henan
Email: slxhqy@163.com
Received: Feb. 27th, 2015; accepted: Mar. 8th, 2015; published: Mar. 12th, 2015
Copyright © 2015 by authors and Hans Publishers Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
ABSTRACT
This paper is concerned with a class of doubly nonlinear parabolic systems. Under the homogeneous Dirichlet conditions and suitable conditions on the nonlinearity and certain initial datum, a sufficient condition for finite time blowup of its solution in a bounded domain is gave by using a modification of Levine’s concavity method.
Keywords:Blowup of Solution, Doubly Nonlinear Parabolic Equations, Levine’s Concavity Method
多重非线性抛物方程组解的爆破
苏璟,齐龙飞,呼青英
河南工业大学理学院,河南 郑州
Email: slxhqy@163.com
收稿日期:2015年2月27日;录用日期:2015年3月8日;发布日期:2015年3月12日
摘 要
本文研究了一类多重非线性抛物方程组解的爆破,利用修正的Levine凸性方法,对齐次Dirichlet边界和非线性项和初始条件的适当条件下,给出了解爆破时间的充分条件。
关键词 :爆破,多重非线性抛物方程组,Levine凸性方法
1. 引言
本文研究如下非线性抛物方程组解的爆破性
(1.1)
(1.2)
(1.3)
(1.4)
其中是上的有界区域且有光滑边界,是上的Laplace 算子,,为以后给定的函数。
型如(1.1)的单个多重非线性抛物方程
(1.5)
就是经典的所谓双非线性抛物方程,这类方程可以描述诸多化学反应、热传导过程和种群动力学过程(详细见文献[1] )。方程(1.5)的初值问题或初边值问题已经有许多文献研究其局部和整体可解性[2] -[6] ,文献[7] -[12] 则研究了其整体吸引子的存在性和正则性。最近几十年,该类非线性抛物方程的爆破问题吸引了许多人的注意,基于Levine [13] [14] 凸性方法这一开创性的证明爆破的结果,Iami和Mochizuki [15] 则给出了方程(1.5)带Neumann初边值问题解爆破的充分条件,该凸性方法还被Levine [16] [17] 用于如下渗流方程
(1.6)
Sacks [18] 研究了如下包含方程解的爆破问题
(1.7)
Zhang [19] 和Ding和Guo [20] -[22] 通过构造适当的辅助函数,利用一阶微分不等式考虑了下面带梯度项和Neumann (或Robin)初边值问题解的爆破条件
(1.8)
Korpusou和Sveshnikov [23] [24] 给出了如下方程初边值问题弱解爆破的充分条件
(1.9)
最后,还应提及Ouardi和 Hachimi [25] [26] 研究了如下多重非线性抛物方程组
得到了其整体吸引子的存在性和正则性以及Hausdorff维数估计。
本文用修正的Levine凸性方法证明问题(1.1)~(1.4)的解在有限时刻爆破,该方法比原始的Levine凸性方法更简洁,其基本技巧是Korpousov [24] 给出的一个微分不等式,本文把[24] 的方法用于多重非线性抛物方程组。据作者所知,关于多重非线性抛物方程组的爆破问题的研究还比较少。本文的安排如下:第二节将给出一些假设和基本引理,第三节给出主要结果和证明。
2. 假设和基本引理
本文用和表示通常的Soblev空间,其范数分别记为和,特别是当时,记,这些符号的含义和记法同文献[2] 。
本文始终假设。关于非线性项,的假设如下:
(A1),存在函数使得
,
且存在常数使得
,
其中,当时,当时。
注:满足条件(A1)的函数是存在的。事实上,一个典型的例子是取
且,即
,
,
这时,,其中,,,。该例的详细情况可见文献[27] 。
利用Galerkin方法,结合单调性理论和紧性方法[2] ,类似文献[24] 可得问题(1.1)~(1.4)解的局部存在性。
定理2.1:假设条件(A1)成立,,,则问题(1.1)~(1.4)存在弱解,即,存在使得
, ,.
且对任意成立:
,
,
以及。
下面给出本文的基本引理。
引理2.2 [13] [24] [28] :设是R上非负二次连续可导函数且满足不等式
其中为常数。若,,则必存在时刻,使当时有,其中
。
3. 主要结果及证明
首先引入泛函
(3.1)
(3.2)
(3.3)
(3.4)
现给出主要引理。
引理3.1:对任意,下面不等式成立
(3.5)
证明 注意到
(3.6)
而由Holder不等式得
(3.7)
(3.8)
(3.9)
(3.10)
考虑到(3.7)~(3.10),则由(3.6)得
再利用不等式
,
得
于是,引理得证。
下面,给出主要定理。
定理3.2:设定理2.1的条件成立, 且
(3.11)
则问题(1.1)~(1.4)的弱解必在某有限时刻爆破,即
证明:方程(1.1),(1.2)两边分别同乘和,然后关于x积分并相加,得
(3.12)
方程(1.1),(1.2)两边分别同乘和,然后关于x积分并相加,得
(3.13)
(3.13)关于t积分得
(3.14)
再利用条件(A1)得
(3.15)
(3.12)结合(3.15),并用到,得
即
(3.16)
注意到得
(3.17)
利用引理3.1,得
(3.18)
其中。
如果,由(3.1),(3.6)和(3.11)知
, ,
于是,由引理2.2得结论。如果,取,则(3.18)变为
于是,由标准的凸性引理得结论。
文章引用
苏 璟,齐龙飞,呼青英, (2015) 多重非线性抛物方程组解的爆破
Blowup of Solutions for a Class of Doubly Nonlinear Parabolic Equations. 理论数学,02,59-65. doi: 10.12677/PM.2015.52009
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