几类非线性吸收热方程组奇性解的渐近行为
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摘要
本文讨论具非线性吸收的几类热方程组奇性解的渐近行为.这里包括两个专题,其一是研究由奇异吸收耦合的几类热方程组的quenching行为,其二是讨论具非线性吸收的多重耦合热方程组奇性解的多重blow-up速率.
本文分以下四个章节:
第一章概述本文所研究问题的实际背景和国内外发展现状,并简要介绍本文的主要工作.
第二章考虑以u~(-p1)v~(-q1)和u~(-p2)v~(-q2)作为奇异边界吸收耦合项的一维热方程组的有限时刻quenching问题.这里除了起耦合作用的v~(-p2)和u~(-p2)而外,分量u和v的奇异边界源还分别有两个奇性因子u~(-p1)和v~(-p2).我们首先提出一个区分同时与非同时quenching的准则,然后决定模型在不同非线性机制占优情形下所有可能的quenching速率:三种同时quenching速率和两个非同时quenching速率.与已有结果对照,这些结果显示出两个非耦合奇性因子对解的quenching行为的影响.
第三章主要研究由加和形式奇性吸收u~(-m)+v~(-p)和u~(-q)+v~(-n)耦合的热方程组解的quenching行为.分两部分讨论.首先考虑内吸收耦合情形,然后边界吸收耦合情形.两者都是先区分同时与非同时quenching准则,然后建立四种不同的同时quenching速率.我们引入特征代数方程组统一给出这些速率的简洁刻画.特别发现这对应于两种不同的同时quenching速率的指标区域,真实速率由初值决定.
第四章讨论具内吸收的多重耦合热方程组解的有限时刻blow-up问题.第一部分考虑平衡吸收情形,得到不同非线性占优情形下的四种与吸收无关的同时blow-up速率.这里引入两个特征代数方程组,分别用以刻画blow-up准则和四种blow-up速率.第二部分考虑非平衡吸收情形(即一个分量的吸收较强而另一个较弱),除前述四种与吸收无关的blow-up速率,继而得到八种与吸收有关的blow-up速率.由于具有吸收的单个方程的奇性解的blow-up速率全都与吸收无关,故与吸收有关的blow-up速率将是具吸收的耦合组问题的特有现象,而本质区别于单个方程问题.
This thesis deals with asymptotic behavior of singular solutions for heat systems with nonlinear absorptions.There are two topics included:Firstly,quenching behavior of solutions for heat systems coupled via singular absorptions;Secondary,multiple blow-up rates for multi-coupled heat systems with absorptions.
The thesis composes of four chapters:
In Chapter 1 we summarize the background of the related issues and state the main results of the present thesis.
Chapter 2 deals with finite time quenching for one-dimensional heat systems coupled via singular boundary absorptions of the form u~(-p1)v~(-q1) and u~(-p2)v~(-q2).Besides the coupling factors v~(-q1) and u~(-p2),there are two additional singular boundary absorption factors, namely,u~(-p1) and v~(-q2) for the components u and v respectively.We at first propose a criterion to distinguish simultaneous and non-simultaneous quenching,and then determine all possible quenching rates in the model,three for simultaneous quenching and two for non-simultaneous quenching,under different nominations.This shows real contributions of the two additional singular factors to the quenching behavior of solutions.
Chapter 3 mainly discuss quenching behaviors of solutions to heat equations with coupling singular absorptions u~(-m) + v~(-p) and u~(-q) + v~(-n).We study them in two cases respectively, inner absorption couplings and boundary absorption couplings.For both cases, we at first identify simultaneous and non-simultaneous quenching,and then establish four kinds of simultaneous quenching rates,which are uniformly represented via the characteristic algebraic system introduced for the model.It is interesting to find that there are two simultaneous quenching rates which share a common subregion of parameters,determined by the initial data assumed.
Chapter 4 studies finite time blow-up problem to a multi-coupled heat system with absorptions.In the first part we consider the case of balanced absorptions,and obtain four absorption-independent simultaneous blow-up rates under different dominations of nonlinearities.Two characteristic algebraic systems associated to the problem are introduced to get very simple descriptions for the four simultaneous blow-up rates as well as the critical blow-up criterion,respectively.The second part deals with unbalanced absorption case(i.e.,stronger absorption for one component and weaker absorption for the other).We prove that if the positive coupling sources dominate the system with unbalanced absorptions,then there are also eight possible absorption-related blow-up rates for the model,besides the four absorption-independent ones shown above.This is substantially different from the scalar equations with absorptions,whose blow-up rates are all known as absorption-independent.
本文分以下四个章节:
第一章概述本文所研究问题的实际背景和国内外发展现状,并简要介绍本文的主要工作.
第二章考虑以u~(-p1)v~(-q1)和u~(-p2)v~(-q2)作为奇异边界吸收耦合项的一维热方程组的有限时刻quenching问题.这里除了起耦合作用的v~(-p2)和u~(-p2)而外,分量u和v的奇异边界源还分别有两个奇性因子u~(-p1)和v~(-p2).我们首先提出一个区分同时与非同时quenching的准则,然后决定模型在不同非线性机制占优情形下所有可能的quenching速率:三种同时quenching速率和两个非同时quenching速率.与已有结果对照,这些结果显示出两个非耦合奇性因子对解的quenching行为的影响.
第三章主要研究由加和形式奇性吸收u~(-m)+v~(-p)和u~(-q)+v~(-n)耦合的热方程组解的quenching行为.分两部分讨论.首先考虑内吸收耦合情形,然后边界吸收耦合情形.两者都是先区分同时与非同时quenching准则,然后建立四种不同的同时quenching速率.我们引入特征代数方程组统一给出这些速率的简洁刻画.特别发现这对应于两种不同的同时quenching速率的指标区域,真实速率由初值决定.
第四章讨论具内吸收的多重耦合热方程组解的有限时刻blow-up问题.第一部分考虑平衡吸收情形,得到不同非线性占优情形下的四种与吸收无关的同时blow-up速率.这里引入两个特征代数方程组,分别用以刻画blow-up准则和四种blow-up速率.第二部分考虑非平衡吸收情形(即一个分量的吸收较强而另一个较弱),除前述四种与吸收无关的blow-up速率,继而得到八种与吸收有关的blow-up速率.由于具有吸收的单个方程的奇性解的blow-up速率全都与吸收无关,故与吸收有关的blow-up速率将是具吸收的耦合组问题的特有现象,而本质区别于单个方程问题.
This thesis deals with asymptotic behavior of singular solutions for heat systems with nonlinear absorptions.There are two topics included:Firstly,quenching behavior of solutions for heat systems coupled via singular absorptions;Secondary,multiple blow-up rates for multi-coupled heat systems with absorptions.
The thesis composes of four chapters:
In Chapter 1 we summarize the background of the related issues and state the main results of the present thesis.
Chapter 2 deals with finite time quenching for one-dimensional heat systems coupled via singular boundary absorptions of the form u~(-p1)v~(-q1) and u~(-p2)v~(-q2).Besides the coupling factors v~(-q1) and u~(-p2),there are two additional singular boundary absorption factors, namely,u~(-p1) and v~(-q2) for the components u and v respectively.We at first propose a criterion to distinguish simultaneous and non-simultaneous quenching,and then determine all possible quenching rates in the model,three for simultaneous quenching and two for non-simultaneous quenching,under different nominations.This shows real contributions of the two additional singular factors to the quenching behavior of solutions.
Chapter 3 mainly discuss quenching behaviors of solutions to heat equations with coupling singular absorptions u~(-m) + v~(-p) and u~(-q) + v~(-n).We study them in two cases respectively, inner absorption couplings and boundary absorption couplings.For both cases, we at first identify simultaneous and non-simultaneous quenching,and then establish four kinds of simultaneous quenching rates,which are uniformly represented via the characteristic algebraic system introduced for the model.It is interesting to find that there are two simultaneous quenching rates which share a common subregion of parameters,determined by the initial data assumed.
Chapter 4 studies finite time blow-up problem to a multi-coupled heat system with absorptions.In the first part we consider the case of balanced absorptions,and obtain four absorption-independent simultaneous blow-up rates under different dominations of nonlinearities.Two characteristic algebraic systems associated to the problem are introduced to get very simple descriptions for the four simultaneous blow-up rates as well as the critical blow-up criterion,respectively.The second part deals with unbalanced absorption case(i.e.,stronger absorption for one component and weaker absorption for the other).We prove that if the positive coupling sources dominate the system with unbalanced absorptions,then there are also eight possible absorption-related blow-up rates for the model,besides the four absorption-independent ones shown above.This is substantially different from the scalar equations with absorptions,whose blow-up rates are all known as absorption-independent.
引文
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[20]WANG M X.Blow-up rates for semilinear parabolic systems with nonlinear boundary conditions[J].Appl.Math.Lett.,2003,16:543-549.
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