具有非线性源或边界流的双重退化抛物型方程
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摘要
本文研究带有非线性源或边界流的双重退化抛物型偏微分方程(组)中多重非线性之间的相互作用及其对解的整体存在与有限时刻blow-up行为的影响。这是一类具有重要意义的非线性抛物型方程(组),在渗流理论、相变理论、物理学、化学、生物群体动力学、图像处理等诸多领域有着广泛的应用。而模型的双重退化性以及源或流的非线性给问题的研究带来较大的复杂性和一定的本质性困难。
本文所考虑的双重退化模型一般不具有古典解,而只能从弱解及相应的比较原理出发。所关心的是来自双重退化及源和(或)流的多重非线性项之间复杂的相互作用,运用适合的比较原理讨论解的整体存在与不存在条件,并通过精细、完全的指标分类,建立模型问题的临界指标。本文还运用试验函数的方法建立了一类双重退化抛物型不等式问题解的Liouville型定理。
第一章叙述与本文相关的研究工作的背景和发展现状,并概述本文主要工作。
第二章考虑一类具有非线性内部源与非线性边界流的双重退化抛物型方程。充分讨论了四种非线性机制之间的相互作用,利用比较原理,推导出该问题的非负弱解整体存在的充分必要条件。
第三章研究一类具有非线性内吸收项(负源)与非线性边界流的双重退化抛物型方程。从弱解的定义和比较原理出发,通过细致而完全的指标分类,建立刻画整体弱解存在与不存在性的临界边界流指标,这里还特别包含关于吸收项系数对解的临界性质影响的精细分析。
第四章首先考虑一类双重退化抛物型不等式,通过适当选用试验函数的办法得到此抛物不等式弱解的Liouville型定理。其次,对一类具有非线性边界流的双重退化抛物型方程半无限问题,运用比较原理给出了解的整体存在临界指标和Fujita临界指标。
第五章讨论由非线性内部源与边界流多重耦合的双重退化抛物型模型。从问题的弱解和弱比较原理出发,详细讨论了模型中多重非线性项之间的相互作用,通过对八个非线性项参数的完全分类,建立了该问题的临界指标,即解整体存在的充分必要条件。
This thesis studies interactions among multi-nonlinearities and their influences to global existence and finite time blow-up of solutions in doubly degenerate parabolic equations (systems) with nonlinear source(absorption) and(or) boundary flux.This is a class of important nonlinear parabolic equations(systems) possessing wide applications in many fields such as filtration,phase transition,physics,chemistry,biochemistry and dynamics of biological groups,image processing.The double degeneracy as well as the nonlinearities from the source and(or) boundary flux make the studies more complicated and difficult.
In general the doubly degenerate models considered here do not admit classical solutions, we have to start from weak solutions and related comparison principles.We are interested in the complicated interactions among multi-nonlinearities from double degeneracy and source and(or) flux.We will consider the existence and non-existence of global solutions by suitable comparison principle,and establish the critical exponents via precise and complete classifications for nonlinear parameters.In addition,we obtain a Liouville-type theorem for a doubly degenerate parabolic inequality by means of the test function method.This thesis is organized as follows.
Chapter 1 is an introduction to recall the background and the current development of the related topics and to summarize the main results of the present thesis.
Chapter 2 studies a doubly degenerate parabolic equation subject to nonlinear inner source and boundary flux.We discuss interaction among the four nonlinear mechanisms in the model,and establish the sufficient and necessary condition for global non-negative weak solutions by weak comparison principle.
Chapter 3 considers a doubly degenerate parabolic equation subject to inner absorption and boundary flux.The weak comparison principle for weak solutions and a careful classification for the four nonlinear parameters give us the critical boundary exponent to describe the sufficient and necessary condition for global non-negative weak solutions of the model,where,in particular,a precise analysis is included to show the significant contribution of the absorption coefficient to the critical property of solutions.
Chapter 4 at first concerns a doubly degenerate parabolic inequality.A Liouville-type theorem is obtained by carefully choosing desired test functions for weak solutions of the inequality problem.Next,critical global existence exponent and critical Fujita exponent are proved for the half space problem to a doubly degenerate parabolic equation with nonlinear flux condition.
Chapter 5 focuses on a system of doubly degenerate parabolic equations coupled via inner source and boundary flux.Under the framework of weak solutions and weak comparison principle,a detail discussion is made for the interaction among the multi-nonlinearities. The critical exponent(i.e.the sufficient and necessary conditions for global weak solutions) is determined via a complete classification to all the eight nonlinear parameters included.
本文所考虑的双重退化模型一般不具有古典解,而只能从弱解及相应的比较原理出发。所关心的是来自双重退化及源和(或)流的多重非线性项之间复杂的相互作用,运用适合的比较原理讨论解的整体存在与不存在条件,并通过精细、完全的指标分类,建立模型问题的临界指标。本文还运用试验函数的方法建立了一类双重退化抛物型不等式问题解的Liouville型定理。
第一章叙述与本文相关的研究工作的背景和发展现状,并概述本文主要工作。
第二章考虑一类具有非线性内部源与非线性边界流的双重退化抛物型方程。充分讨论了四种非线性机制之间的相互作用,利用比较原理,推导出该问题的非负弱解整体存在的充分必要条件。
第三章研究一类具有非线性内吸收项(负源)与非线性边界流的双重退化抛物型方程。从弱解的定义和比较原理出发,通过细致而完全的指标分类,建立刻画整体弱解存在与不存在性的临界边界流指标,这里还特别包含关于吸收项系数对解的临界性质影响的精细分析。
第四章首先考虑一类双重退化抛物型不等式,通过适当选用试验函数的办法得到此抛物不等式弱解的Liouville型定理。其次,对一类具有非线性边界流的双重退化抛物型方程半无限问题,运用比较原理给出了解的整体存在临界指标和Fujita临界指标。
第五章讨论由非线性内部源与边界流多重耦合的双重退化抛物型模型。从问题的弱解和弱比较原理出发,详细讨论了模型中多重非线性项之间的相互作用,通过对八个非线性项参数的完全分类,建立了该问题的临界指标,即解整体存在的充分必要条件。
This thesis studies interactions among multi-nonlinearities and their influences to global existence and finite time blow-up of solutions in doubly degenerate parabolic equations (systems) with nonlinear source(absorption) and(or) boundary flux.This is a class of important nonlinear parabolic equations(systems) possessing wide applications in many fields such as filtration,phase transition,physics,chemistry,biochemistry and dynamics of biological groups,image processing.The double degeneracy as well as the nonlinearities from the source and(or) boundary flux make the studies more complicated and difficult.
In general the doubly degenerate models considered here do not admit classical solutions, we have to start from weak solutions and related comparison principles.We are interested in the complicated interactions among multi-nonlinearities from double degeneracy and source and(or) flux.We will consider the existence and non-existence of global solutions by suitable comparison principle,and establish the critical exponents via precise and complete classifications for nonlinear parameters.In addition,we obtain a Liouville-type theorem for a doubly degenerate parabolic inequality by means of the test function method.This thesis is organized as follows.
Chapter 1 is an introduction to recall the background and the current development of the related topics and to summarize the main results of the present thesis.
Chapter 2 studies a doubly degenerate parabolic equation subject to nonlinear inner source and boundary flux.We discuss interaction among the four nonlinear mechanisms in the model,and establish the sufficient and necessary condition for global non-negative weak solutions by weak comparison principle.
Chapter 3 considers a doubly degenerate parabolic equation subject to inner absorption and boundary flux.The weak comparison principle for weak solutions and a careful classification for the four nonlinear parameters give us the critical boundary exponent to describe the sufficient and necessary condition for global non-negative weak solutions of the model,where,in particular,a precise analysis is included to show the significant contribution of the absorption coefficient to the critical property of solutions.
Chapter 4 at first concerns a doubly degenerate parabolic inequality.A Liouville-type theorem is obtained by carefully choosing desired test functions for weak solutions of the inequality problem.Next,critical global existence exponent and critical Fujita exponent are proved for the half space problem to a doubly degenerate parabolic equation with nonlinear flux condition.
Chapter 5 focuses on a system of doubly degenerate parabolic equations coupled via inner source and boundary flux.Under the framework of weak solutions and weak comparison principle,a detail discussion is made for the interaction among the multi-nonlinearities. The critical exponent(i.e.the sufficient and necessary conditions for global weak solutions) is determined via a complete classification to all the eight nonlinear parameters included.
引文
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[7]Chen Y,DiBenedetto E.H(o|¨)lder estimates of solutions of singular parabolic equations with measurable coefficients.Arch.Rationgl Mech.Anal.,1992,118:257-271.
[8]Crandall M,Pierre M.Regularization effects for u_t-δ_φ(u)=0.Trans.Amer.Math.Soc.,1982,274(1):159-168.
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[10]L(?)pez-Gomez J,Marquez V,Wolanski N.Blow up results and localization of blow up points for the heat equations with a nonlinear boundary conditions.J.Differential Equations,1991,92:384-401.
[11]陈亚浙.二阶抛物型偏微分方程.北京大学出版社,2003.
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[18]Hayakawa K.On nonexistence of global solutions of some semilinear parabolic differential equations.Proc.Japan Acad.Ser.A,1973,49:503-505.
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[23]Escobedo M,Herrero M.Boundedness and blow up for a semilinear reaction-diffusion system.J.Differential Equations,1991,89:176-202.
[24]Fila M,Quittner P.The blow-up rate for a semilinear parabolic system.J.Math.Anal.Appl,1999,238:468-476.
[25]Escobedo M,Levine H.Critical blow up and global existence numbers for weekly coupled system of reaction-diffusion equations.Arch.Rational Mech.Anal.,1995,129:47-100.
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