伪抛物型方程的定性理论
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摘要
本文是一篇读书报告,介绍了伪抛物型方程的定性理论,包括解的存在唯一性和解的长时间的渐近行为,全文共分三章.
我们在第一章先给出了一类非经典的扩散方程—伪抛物型方程的概念和物理背景,以及这类方程的特点.
在本文的第二章,我们讨论含三阶项△ut的齐次、非齐次半线性伪抛物型方程全空间上的Cauchy问题以及外区域上的第二初边值问题的定性理论.作为粘性项,三阶项△ut具有保持原方程某些特征的特点.我们从所了解的知识中得知研究伪抛物型方程定性理论时遇到了很多的困难.首先,在抛物型方程的研究中最常用的是构造自相似上、下解的方法,但是因为这里含有三阶项,所以这种方法在这里并不适用.其次,虽然可以写出伪抛物型方程的基本解,但是由于其Green函数形式极其复杂,故想简单的对基本解进行估计来得到爆破也行不通.本文第二章中介绍的是用于证明拟线性抛物型方程解的爆破的方法——能量(积分)爆破法,并给出一些适用的比较原理.
在本文最后一章中,我们将介绍如BBMB方程,含Pseudo-Laplace项的BBMB方程,非线性广义Boussinesq方程,非线性双曲型粘滞弹性方程,含P-Laplace扩散项伪抛物型方程等一些其他类型的伪抛物型方程的定性理论.我们介绍的重点是这些方程解的存在唯一性理论以及解的长时间渐近行为,包括解的整体存在性、解的爆破准则和爆破时间估计.而由于篇幅问题,我们并未介绍详细的证明,仅列出了一些必须的条件和其中某些定理的简要证明.
Diffusion equations, as an important class of partial differential equations, come from a variety of diffusion phenomena appeared widely in nature. With the intensive study by many researchers, diffusion equations have been thought of as an impor-tant branch of partial differential equations after developing in the past half center. We know that the research of nonclassical diffusion equations and classical diffu-sion equations add radiance and beauty to each other. In this article, we mainly introduce the long time behavior of solutions of a class of non-classical diffusion equations——pseudo-parabolic equations. We firstly introduce the critical exponents of the Cauchy problems in the whole space Rn and the second initial-boundary value problems in the exterior space of the homogeneous and inhomogeneous semilinear pseudo-parabolic equations. Then, we give some results about some other classes of pseudo-parabolic equations.
In the poineering paper of Sobolev [51] (1954), the equation for small oscialla-tions in a rotation liquid was obtained, it has the form It is easy to see that, the highest-order term of the above equation has mixed time and space derivatives. The equation with this character is said to be a Sobolev type equation. Gal'pern [47] studied the Cauchy problem for the equation of the form where M and L are linear elliptic operators; Showalter [44] investigated the ex-istence and uniqueness of strong solutions to the initial-boundary value problem when M and L are second order linear elliptic operators; Equations of the above form have been called pseudo-parabolic by Showalter and Ting [45], because well-posed initial-boundary value problems for parabolic equations are also well-posed for the corresponding pseudo-parabolic equation. Moreover, in certain cases, the solution of a parabolic initial-boundary value problem can be obtained as a limit of solutions to the corresponding problem for the pseudo-parabolic equation. Pseudo-parabolic equation is an example of a general class of equations of Sobolev type, sometimes referred to as Sobolev-Galpern type. From then on, people define that a pseudo-parabolic equation is an arbitrary higher-order partial differential equation with the first-order derivative with respect to time. In this monograph, we introduce the case with the third order term Aut.
Pseudo-parabolic equations, as a type of non-classical diffusion equations, de-scribe a variety of important physical processes, such as the seepage of homogeneous fluids through a fissured rock [15] (where the coefficient of third order term corre-sponds to a reduction in block dimension and an increase in the degree of fissuring), the unidirectional propagation of nonlinear, dispersive, long waves [53,54] (where u is typically the amplitude or velocity), the aggregation of populations [55] (where u is represents the population density), the heat conduction involving a thermo-dynamic temperatures-kΔu and a conductive temperatures [40]. Specially, the following pseudo-parabolic equation with sources can be used in the analysis of nonstationary processes in semiconductors in the pres-ence of sources[39,36], where k((?)Δu)/((?)t)-((?)u)/((?)t) corresponds to the free electron density rate,Δucorrespond to the linear dissipation of the free charge current and up describes a source of free electron current. From the physical point of view, the blow-up of solu-tions corresponds to electric breakdown in semiconductors or magnetic breakdown in magnetics, which occur in experiments.
In the past several decades, pseudo-parabolic equations have been studied by many scientists; some of them studied wave pseudo-parabolic equations, others—dissipative pseudo-parabolic equations. With thorough research, people found that there are close connections and essential differences between pseudo-parabolic equa-tions and parabolic equations.
With the results given by some others about the pseudo-parabolic equations, we can find the global existence and finite time blow-up results of the pseudo-parabolic equations with sources only in Levine [31], Kozhanov [2,1], several papers of Kor-pusov et al ([37] and the references therein) and Kaikina et al [12,10]. Kozhanov investigated the following initial-boundary value problem of nonlinear Boussinesq equation with sources Through constructing upper and lower solutions and using the comparison principle, he proved the following existence and nonexistence results
(i)when p≥m≥2, then there exist global solutions if the initial values are sufficient small;
(ii)when p≥m≥2, then no nontrivial global solutions exist if the initial values are sufficient large.
For the strong nonlinear pseudo-parabolic equations, especially for the case that the mixed terms are nonlinear elliptic operators such as only Korpusov et al have studied carefully, see the survey paper [31] and the refer-ences therein.
It was Kaikina et al [12] who considered the superlinear case of the Cauchy problem with p>1 and proved the existence and uniqueness of solutions using the integral representation and the contraction-mapping principle. Furthermore, it was shown that the Cauchy problem has a unique global solution ifσ>2/n for u0 being sufficiently small. Subsequently, Kaikina [10] considered the initial-boundary value problem for (1) on a half-line and proved the global existence of solutions whenσ>1 and the initial data is small enough. We find from the existing results, the work of Kaikina et al only restricted to the global existence.
In the second chapter, we introduce Cao's work([4]) about the Cauchy problem and Neumann boundary problem of semilinear pseudo-parabolic equations with in-terior sources up. In the first section, we show the chapter's introducation. In the second section, we introduce the Cauchy problem and Neumann boundary problem of homogeneous pseudo-parabolic equations with nonlinear interior sources. For the Cauchy problem, just like the corresponding Cauchy problem of the semilinear heat equation, there still exist tow critical exponents p0=1,pc=1+2/n, such that
(ⅰ)there exist global solutions for each initial datum in the case 0p0;
(ⅱ)any nontrivial solution blows up in a finite time in the case p0pc. Here p0 and pc are called to be the critical global existence exponent and the crit-ical Fujita exponent, respectively. For the Nuemann boundary problem, we can obtain that the Fujita exponent is still pc=1+2/n. In the last section, we de-tailedly introduce the Cauchy problem of semilinear pseudo-parabolic equations with inhomogeneous interior sources and the Neumann boundary problem of semilinear pseudo-parabolic equations with inhomogeneous boundary condition. We know that the Fujita exponents are pc=n/(n-2), this indicates that the appearance of inho-mogeneous term make the blow-up interval become larger.
From the results in Chapter 2, we can easily discover that the appearance of third order Aut does not influence the magnitude of critical exponents. Seemingly there is little significance of third order term, but if we refer to the explanation in Barenblatt et al [3]:third order termΔut is a viscous term, has viscous relaxation effect, then our results review that the effect of this viscous term is not strong enough to change the critical exponents, and will not influence the essence of the equation to change the properties of the solutions. In addition, the estimation of the upper bound of the blow-up time shows that the third order term will delay the blow-up time.
In the last chapter, we introduce the qualitative theory of some other classes of the pseudo-parabolic equations, such as, the BBMB equation, the BBMB equa-tion with the Pseudo-Laplacian, the nonlinear generalized Boussinesq equation, the nonlinear viscoelastic hyperbolic equation, the p-Laplacian pseudo-parabolic equa-tions, and so on. The existence and uniqueness of these equations'solutions and the asymptotic behavior of the solutions for a long time are main things we introduce in this chapter, including the local existence and uniqueness of the solution, the global existence of the solution, the blow-up of the sloution and the estimate time of the blow-up of the solution. The conclusions in this chapter are given in the form of theorems and lemmas without detailed proofs, Specific process of these conclusions can refer the corresponding references.
我们在第一章先给出了一类非经典的扩散方程—伪抛物型方程的概念和物理背景,以及这类方程的特点.
在本文的第二章,我们讨论含三阶项△ut的齐次、非齐次半线性伪抛物型方程全空间上的Cauchy问题以及外区域上的第二初边值问题的定性理论.作为粘性项,三阶项△ut具有保持原方程某些特征的特点.我们从所了解的知识中得知研究伪抛物型方程定性理论时遇到了很多的困难.首先,在抛物型方程的研究中最常用的是构造自相似上、下解的方法,但是因为这里含有三阶项,所以这种方法在这里并不适用.其次,虽然可以写出伪抛物型方程的基本解,但是由于其Green函数形式极其复杂,故想简单的对基本解进行估计来得到爆破也行不通.本文第二章中介绍的是用于证明拟线性抛物型方程解的爆破的方法——能量(积分)爆破法,并给出一些适用的比较原理.
在本文最后一章中,我们将介绍如BBMB方程,含Pseudo-Laplace项的BBMB方程,非线性广义Boussinesq方程,非线性双曲型粘滞弹性方程,含P-Laplace扩散项伪抛物型方程等一些其他类型的伪抛物型方程的定性理论.我们介绍的重点是这些方程解的存在唯一性理论以及解的长时间渐近行为,包括解的整体存在性、解的爆破准则和爆破时间估计.而由于篇幅问题,我们并未介绍详细的证明,仅列出了一些必须的条件和其中某些定理的简要证明.
Diffusion equations, as an important class of partial differential equations, come from a variety of diffusion phenomena appeared widely in nature. With the intensive study by many researchers, diffusion equations have been thought of as an impor-tant branch of partial differential equations after developing in the past half center. We know that the research of nonclassical diffusion equations and classical diffu-sion equations add radiance and beauty to each other. In this article, we mainly introduce the long time behavior of solutions of a class of non-classical diffusion equations——pseudo-parabolic equations. We firstly introduce the critical exponents of the Cauchy problems in the whole space Rn and the second initial-boundary value problems in the exterior space of the homogeneous and inhomogeneous semilinear pseudo-parabolic equations. Then, we give some results about some other classes of pseudo-parabolic equations.
In the poineering paper of Sobolev [51] (1954), the equation for small oscialla-tions in a rotation liquid was obtained, it has the form It is easy to see that, the highest-order term of the above equation has mixed time and space derivatives. The equation with this character is said to be a Sobolev type equation. Gal'pern [47] studied the Cauchy problem for the equation of the form where M and L are linear elliptic operators; Showalter [44] investigated the ex-istence and uniqueness of strong solutions to the initial-boundary value problem when M and L are second order linear elliptic operators; Equations of the above form have been called pseudo-parabolic by Showalter and Ting [45], because well-posed initial-boundary value problems for parabolic equations are also well-posed for the corresponding pseudo-parabolic equation. Moreover, in certain cases, the solution of a parabolic initial-boundary value problem can be obtained as a limit of solutions to the corresponding problem for the pseudo-parabolic equation. Pseudo-parabolic equation is an example of a general class of equations of Sobolev type, sometimes referred to as Sobolev-Galpern type. From then on, people define that a pseudo-parabolic equation is an arbitrary higher-order partial differential equation with the first-order derivative with respect to time. In this monograph, we introduce the case with the third order term Aut.
Pseudo-parabolic equations, as a type of non-classical diffusion equations, de-scribe a variety of important physical processes, such as the seepage of homogeneous fluids through a fissured rock [15] (where the coefficient of third order term corre-sponds to a reduction in block dimension and an increase in the degree of fissuring), the unidirectional propagation of nonlinear, dispersive, long waves [53,54] (where u is typically the amplitude or velocity), the aggregation of populations [55] (where u is represents the population density), the heat conduction involving a thermo-dynamic temperatures-kΔu and a conductive temperatures [40]. Specially, the following pseudo-parabolic equation with sources can be used in the analysis of nonstationary processes in semiconductors in the pres-ence of sources[39,36], where k((?)Δu)/((?)t)-((?)u)/((?)t) corresponds to the free electron density rate,Δucorrespond to the linear dissipation of the free charge current and up describes a source of free electron current. From the physical point of view, the blow-up of solu-tions corresponds to electric breakdown in semiconductors or magnetic breakdown in magnetics, which occur in experiments.
In the past several decades, pseudo-parabolic equations have been studied by many scientists; some of them studied wave pseudo-parabolic equations, others—dissipative pseudo-parabolic equations. With thorough research, people found that there are close connections and essential differences between pseudo-parabolic equa-tions and parabolic equations.
With the results given by some others about the pseudo-parabolic equations, we can find the global existence and finite time blow-up results of the pseudo-parabolic equations with sources only in Levine [31], Kozhanov [2,1], several papers of Kor-pusov et al ([37] and the references therein) and Kaikina et al [12,10]. Kozhanov investigated the following initial-boundary value problem of nonlinear Boussinesq equation with sources Through constructing upper and lower solutions and using the comparison principle, he proved the following existence and nonexistence results
(i)when p≥m≥2, then there exist global solutions if the initial values are sufficient small;
(ii)when p≥m≥2, then no nontrivial global solutions exist if the initial values are sufficient large.
For the strong nonlinear pseudo-parabolic equations, especially for the case that the mixed terms are nonlinear elliptic operators such as only Korpusov et al have studied carefully, see the survey paper [31] and the refer-ences therein.
It was Kaikina et al [12] who considered the superlinear case of the Cauchy problem with p>1 and proved the existence and uniqueness of solutions using the integral representation and the contraction-mapping principle. Furthermore, it was shown that the Cauchy problem has a unique global solution ifσ>2/n for u0 being sufficiently small. Subsequently, Kaikina [10] considered the initial-boundary value problem for (1) on a half-line and proved the global existence of solutions whenσ>1 and the initial data is small enough. We find from the existing results, the work of Kaikina et al only restricted to the global existence.
In the second chapter, we introduce Cao's work([4]) about the Cauchy problem and Neumann boundary problem of semilinear pseudo-parabolic equations with in-terior sources up. In the first section, we show the chapter's introducation. In the second section, we introduce the Cauchy problem and Neumann boundary problem of homogeneous pseudo-parabolic equations with nonlinear interior sources. For the Cauchy problem, just like the corresponding Cauchy problem of the semilinear heat equation, there still exist tow critical exponents p0=1,pc=1+2/n, such that
(ⅰ)there exist global solutions for each initial datum in the case 0
(ⅱ)any nontrivial solution blows up in a finite time in the case p0
From the results in Chapter 2, we can easily discover that the appearance of third order Aut does not influence the magnitude of critical exponents. Seemingly there is little significance of third order term, but if we refer to the explanation in Barenblatt et al [3]:third order termΔut is a viscous term, has viscous relaxation effect, then our results review that the effect of this viscous term is not strong enough to change the critical exponents, and will not influence the essence of the equation to change the properties of the solutions. In addition, the estimation of the upper bound of the blow-up time shows that the third order term will delay the blow-up time.
In the last chapter, we introduce the qualitative theory of some other classes of the pseudo-parabolic equations, such as, the BBMB equation, the BBMB equa-tion with the Pseudo-Laplacian, the nonlinear generalized Boussinesq equation, the nonlinear viscoelastic hyperbolic equation, the p-Laplacian pseudo-parabolic equa-tions, and so on. The existence and uniqueness of these equations'solutions and the asymptotic behavior of the solutions for a long time are main things we introduce in this chapter, including the local existence and uniqueness of the solution, the global existence of the solution, the blow-up of the sloution and the estimate time of the blow-up of the solution. The conclusions in this chapter are given in the form of theorems and lemmas without detailed proofs, Specific process of these conclusions can refer the corresponding references.
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