反应对流扩散方程的高维整体解及其应用
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摘要
非线性抛物型方程理论是现代数学的重要组成部分.本论文主要研究高维空间非线性抛物型方程的整体解(entire solution),这里所谓的整体解是指一类对所有时间t∈R都有定义的解.从动力系统的角度来看,一般意义上抛物型方程初值问题的解仅仅是半轨道,利用半轨道(t≥0)不能判定解的全部信息,而整体解(t∈R)实际上就是方程的一个全流,利用整体解可以确切的把握任何时刻有关方程解的信息.这使得研究非线性抛物型方程的整体解变得必要且有重要意义.本文主要研究行波解的交错作用,具体就是利用方程的单调行波解来构造新型整体解.我们知道,关于整体解的已有结果都是在一维齐次空间(或固定方向)下对单稳和双稳型非线性方程建立的.考虑到来自物理、化学、生态等领域的许多课题都是高维空间问题,因此本文试图建立高维空间反应扩散方程整体解理论.特别,我们需要指出的是,当空间变量为一维情形(或固定方向)时,相应的波方程是一个二阶常微分方程,而当空间变量为高维时,如果考虑非线性抛物型方程的曲面行波解,则相应的波方程为椭圆型方程.在椭圆型方程理论框架下利用曲面行波解研究高维空间变量方程的整体解变得比较困难而且有意义.
首先,我们研究了无穷柱体上单稳型和点火型反应对流扩散方程的整体解.对具有单稳型非线性项的方程,通过考虑两列沿柱体相向传播的行波解,并通过利用比较原理和上下解方法,建立了整体解的存在性.对于点火型非线性方程,利用方程唯一存在的沿柱体方向相向传播的行波解对,证明了方程整体解的存在性.并且证明了以上所有得到的整体解当时间t→—∞时表现为两列沿柱体方向相向传播的行波解,而且随着时间的推移,两列行波解在有限时间内相互碰撞并最终消失.
其次,我们考虑了无穷柱体上双稳型反应对流扩散方程整体解的存在性.双稳型反应对流扩散方程一般具有三个平衡点,其中两个为线性化稳定的,一个为线性化不稳定的.其三个平衡点中的任意两个之间有行波解连接.通过考虑连接不同平衡点的不同行波解,建立了三种不同类型的整体解,并给出了它们的渐近行为.进一步,通过考虑一个定义在无穷柱体上的拟不变流形,我们证明了当双稳型非线性方程任意的非平凡整体解满足一定条件时,其整体解是唯一的,并证明了所得到的唯一整体解是Liapunov稳定的.
最后,我们考虑了具有周期性介质单稳型空间各向异性方程的整体解.利用方程连接常数平衡态和周期函数形式平衡态的脉动行波解,证明了方程存在表现为两列相向传播的脉动行波解的脉动整体解,并且给出了在生物种群模型和化学反应模型中的应用.
The theory of nonlinear parabolic differential equations is an important component of the modern mathematic researches.In this paper,we concerned with the entire solutions of a reaction-advection-diffusion equation in higher dimensions.Here,the entire solutions are defined in the whole space and for all time t∈R.In fact,entire solution is a full-flow of the equation.By using entire solution,we can know the exact moment of the relevant information about the equations.Thus,the study of the entire solutions is necessary and practical significance.Indeed,front propagation occurs in many applied problems,such as chemical kinetics,combustion,transport in porous media and biology.For specific front propagation--traveling wave solution,are special kinds of entire solutions.In addition to traveling wave solutions,the interaction between them is also an important topic in the study of reaction-diffusion equations,which is crucially related to the pattern formation problem,specially to the time evolutional process of localized patterns,where more important information on the evolutional process of patterns are given and there are important application in physical,Chemical,biological,physiological systems.From the dynamical points of view,the study of entire solution is essential for a full understanding of the transient dynamics and the structure of the global attractors.Also,entire solutions can be used to imply that the dynamics of two solutions can have distinct histories in the configuration.
By considering a combination of any two of those different traveling wave fronts and constructing appropriate subsolutions and supersolutions,we establish entire solutions of nonlinear parabolic differential equations in higher dimensions.The main techniques are to characterize the asymptotic behavior of the traveling wave solutions as t→-∞. Though there are many well-known results of entire solutions of reaction-diffusion equations in the one-dimensional space,the issue of the existence of entire solutions of reactionadvection -diffusion equations which admits nonplanar traveling wave solutions,is still open.Comparing with the case of one-dimensional space,which is related to a second order ordinary differential equations,the case of study entire solutions in higher dimensions actually related to elliptic equations,which is not only more meaningful and valuable in theory and practice,but also more challengeable in mathematics to study such equations. Especially,for the ignition temperature nonlinearity,we need to establish the theory of entire solutions.These are the mainly motivations of this thesis.
First,we consider the existence of entire solutions of a reaction-advection-diffusion equation with monostable and ignition temperature nonlinearities in infinite-cylinders.A comparison argument is employed to prove the existence of entire solutions which behave as two traveling wave fronts coming from both directions.In order to illustrate our main results,a passive-reaction-diffusion equation model arising from propagation of fronts is considered.
In next chapter,we deal with entire solutions and the interaction of traveling wave fronts of bistable reaction-advection-diffusion equation with infinite cylinders.Assume that the equation admits three equilibria:two stable equilibria 0 and 1,and an unstable equilibriumθ.It is well known that there are different wave fronts connecting any two of those three equilibria.By considering a combination of any two of those different traveling wave fronts and constructing appropriate subsolutions and supersolutions,we establish three different types of entire solutions.In the sequel,we study the uniqueness and Liapunov stability of entire solutions for bistable reaction-advection-diffusion equation in heterogeneous media.By using traveling curved fronts connecting a constant unstable stationary state and a stable stationary state,we proved that there exist entire solutions behaving as two traveling curved fronts coming from both directions,and we prove that such an entire solution is unique and is Liapunov stable.
At last,we establish the existence of pulsating entire solutions of reaction-advection-diffusion equations with monostable nonlinearities in periodic framework.By studying a pulsating traveling front connecting a constant unstable stationary state to a stable stationary state which is allowed to be a positive function,we proved that there exist pulsating entire solutions behaving as two pulsating fronts coming from both directions, and approaching each other.
首先,我们研究了无穷柱体上单稳型和点火型反应对流扩散方程的整体解.对具有单稳型非线性项的方程,通过考虑两列沿柱体相向传播的行波解,并通过利用比较原理和上下解方法,建立了整体解的存在性.对于点火型非线性方程,利用方程唯一存在的沿柱体方向相向传播的行波解对,证明了方程整体解的存在性.并且证明了以上所有得到的整体解当时间t→—∞时表现为两列沿柱体方向相向传播的行波解,而且随着时间的推移,两列行波解在有限时间内相互碰撞并最终消失.
其次,我们考虑了无穷柱体上双稳型反应对流扩散方程整体解的存在性.双稳型反应对流扩散方程一般具有三个平衡点,其中两个为线性化稳定的,一个为线性化不稳定的.其三个平衡点中的任意两个之间有行波解连接.通过考虑连接不同平衡点的不同行波解,建立了三种不同类型的整体解,并给出了它们的渐近行为.进一步,通过考虑一个定义在无穷柱体上的拟不变流形,我们证明了当双稳型非线性方程任意的非平凡整体解满足一定条件时,其整体解是唯一的,并证明了所得到的唯一整体解是Liapunov稳定的.
最后,我们考虑了具有周期性介质单稳型空间各向异性方程的整体解.利用方程连接常数平衡态和周期函数形式平衡态的脉动行波解,证明了方程存在表现为两列相向传播的脉动行波解的脉动整体解,并且给出了在生物种群模型和化学反应模型中的应用.
The theory of nonlinear parabolic differential equations is an important component of the modern mathematic researches.In this paper,we concerned with the entire solutions of a reaction-advection-diffusion equation in higher dimensions.Here,the entire solutions are defined in the whole space and for all time t∈R.In fact,entire solution is a full-flow of the equation.By using entire solution,we can know the exact moment of the relevant information about the equations.Thus,the study of the entire solutions is necessary and practical significance.Indeed,front propagation occurs in many applied problems,such as chemical kinetics,combustion,transport in porous media and biology.For specific front propagation--traveling wave solution,are special kinds of entire solutions.In addition to traveling wave solutions,the interaction between them is also an important topic in the study of reaction-diffusion equations,which is crucially related to the pattern formation problem,specially to the time evolutional process of localized patterns,where more important information on the evolutional process of patterns are given and there are important application in physical,Chemical,biological,physiological systems.From the dynamical points of view,the study of entire solution is essential for a full understanding of the transient dynamics and the structure of the global attractors.Also,entire solutions can be used to imply that the dynamics of two solutions can have distinct histories in the configuration.
By considering a combination of any two of those different traveling wave fronts and constructing appropriate subsolutions and supersolutions,we establish entire solutions of nonlinear parabolic differential equations in higher dimensions.The main techniques are to characterize the asymptotic behavior of the traveling wave solutions as t→-∞. Though there are many well-known results of entire solutions of reaction-diffusion equations in the one-dimensional space,the issue of the existence of entire solutions of reactionadvection -diffusion equations which admits nonplanar traveling wave solutions,is still open.Comparing with the case of one-dimensional space,which is related to a second order ordinary differential equations,the case of study entire solutions in higher dimensions actually related to elliptic equations,which is not only more meaningful and valuable in theory and practice,but also more challengeable in mathematics to study such equations. Especially,for the ignition temperature nonlinearity,we need to establish the theory of entire solutions.These are the mainly motivations of this thesis.
First,we consider the existence of entire solutions of a reaction-advection-diffusion equation with monostable and ignition temperature nonlinearities in infinite-cylinders.A comparison argument is employed to prove the existence of entire solutions which behave as two traveling wave fronts coming from both directions.In order to illustrate our main results,a passive-reaction-diffusion equation model arising from propagation of fronts is considered.
In next chapter,we deal with entire solutions and the interaction of traveling wave fronts of bistable reaction-advection-diffusion equation with infinite cylinders.Assume that the equation admits three equilibria:two stable equilibria 0 and 1,and an unstable equilibriumθ.It is well known that there are different wave fronts connecting any two of those three equilibria.By considering a combination of any two of those different traveling wave fronts and constructing appropriate subsolutions and supersolutions,we establish three different types of entire solutions.In the sequel,we study the uniqueness and Liapunov stability of entire solutions for bistable reaction-advection-diffusion equation in heterogeneous media.By using traveling curved fronts connecting a constant unstable stationary state and a stable stationary state,we proved that there exist entire solutions behaving as two traveling curved fronts coming from both directions,and we prove that such an entire solution is unique and is Liapunov stable.
At last,we establish the existence of pulsating entire solutions of reaction-advection-diffusion equations with monostable nonlinearities in periodic framework.By studying a pulsating traveling front connecting a constant unstable stationary state to a stable stationary state which is allowed to be a positive function,we proved that there exist pulsating entire solutions behaving as two pulsating fronts coming from both directions, and approaching each other.
引文
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