有界区域或紧支集反应项的Fujita型定理
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摘要
周知,关于Fujita指标的研究通常都是针对无界区域的.本文讨论有界区域或反应项具紧支集时非线性抛物方程(组)的Fujita现象,主要考虑了以下两类问题：一是有界区域的耦合组的Fujita型定理,二是源仅作用在有界域上的Cauchy问题的Fujita指标.部分情形还得到解的blow-up速率与blow-up集等.
本文分为以下四个章节：
第一章主要概述本文所研究问题的背景和国内外发展现状,并简要介绍了本文的主要工作.
第二章考虑局部化源α(X)up影响下的一维快扩散方程的Cauchy问题ut=(um)xx+a(x)up这里的源α(x)Up中·α(X)具有紧支集.我们知道,源为up的快扩散方程的Cauchy问题具有Fujita指标pc=m+2,而相应的有界区域不会出现Fujita现象.这里所讨论的问题介于以上两者之间,既不是简单的有界区域问题,也不是全空间的一般Cauchy问题,我们将证明此问题的Fujita指标为pc=m+1,从而推广了之前关于慢扩散方程的结论.我们特别指出,对应的高维问题不会有Fujita型结论.最后,我们一定初值条件下证明非整体解的blow-up速率为(T-t)-1/p-1,blow-up集为B(u)={0}.
第二章研究变指数源up(x)和ug(x)耦合的热方程组的Fujita型结论.先考虑解对任意初值整体存在的指标范围I,以及整体解与非整体解共存的指标范围II.在I和II之外,解是否整体存在还与区域大小有关：如果区域充分小使得包含在某个小球内,则解对小初值整体存在；如果区域足够大使得包含某个大球,则存在p(x),q(x)使得解对任意初值爆破.
第四章考虑有界区域、变系数反应项影响下的耦合组ut=△u+eqtvp,vt=△v+eβtuq的Dirichlet问题,其中αβ∈R,p,q>0,证明其临界Fujita曲线为(pq)c=1+max{α+βp,β+αq,0}/λ1,其中λ1>0是齐次Dilchlet特征值问题一△4=λ4的主特征值.作为一个直接推论,我们还得到一个有趣的Fujita型结论：耦合组Ut=△U+mU+Vp, Vt=△V+nV+Uq的Fujita临界曲线表现为Fujita临界系数max{m,n}=λ1,即不存在非平凡整体解的充要条件为：max{m,n}≥λ1据我们所知,以往关于耦合组的Fujita曲线的研究(特别是临界情形)都依赖于Jensen不等式和／或Kaplan方法,这样就需要讨论p,q与1的大小关系.与此不同,本章所采用的方法是引入热半群来辅助研究Fujita曲线,使得我们可以将p,q的所有情况一起讨论,避免了指数分类,从而使Fujita型定理的证明过程得到了简化.作为(简化证明)的例证,我们除了应用木章的方法讨论几个相关的问题外,还对第三章的部分重要结论重新给出新的简洁证明.
It is known that in general the study on the critical Fujita admit unbounded domains only. This thesis discusses Fujita phenomena of nonlinear parabolic equations for bounded domains or compactly supported reactions. There are two topics included:The Fujita exponent of nonlinear parabolic systems in bounded domain; The Cauchy problem with sources compactly supported. The topics of blow-up set and blow-up rate etc are involved as well.
The thesis composes of four chapters:
In Chapter1we summarize the background of the related issues and state the main results of the present thesis.
Chapter2deals with the Cauchy problem ut=(um)xx+a(x)up with localized reaction a(x)up, where the reaction a(x)up is compactly supported. As we know, the Fujita exponent of Cauchy problem with source up is pc=m+2, and there is no Fujita type conclusion with bounded domains. The problem studied here is between them, neither the problem with bounded domains, nor general Cauchy problem on the whole space, and its critical Fujita exponent will be shown as pc=m+1. This extends the related result for the slow diffusion situation. It is pointed out that there is no Fujita type conclusion for the high dimension case of this problem. In addition, we prove that the problem admits blow-up rate (T-t)-1/p-1and the blow-up set{0} under suitable initial data.
Chapter3studies the Fujita type conclusion for a parabolic system coupled via variable sources up(x) and vg(x).At first consider a region I for global solutions with any initial data, and a coexistence region II of global and non-global solutions. Outside of I and Ⅱ, the existence or not of global solutions is related to the size of the domain:the solutions are global with small initial data if the domain is small to be contained in a small ball, and there is p(x),q(x) such that the solutions are non-global with large initial data if the domain is large to contain a big ball.
Chapter4treats the coupled parabolic system with time-weighted sources in a bounded domain:ut=△u+eαtvp, ut=△u+eβtuq in Ω×(0,T) with α,β∈R and p,q>0, subject to null Dirichlet boundary value condition. The critical Fujita curve is determined as (pq)c=1+max{α+βp,β+αq,0} where λ1is the first eigenvalue of the Lapla-cian with null Dirichlet boundary condition, and there is no any additional restriction on α,β,p,q. Next, as an extension, an interesting Fujita phenomenon is observed for another coupled system Ut=△U+mU+Vp,△t=△V+nV+Uq in Ω×(0, T) with pq>1that the critical Fujita curve is represented via the Fujita critical coefficient max{m,n}=λ1namely, any nontrivial solutions blow up in finite time if and only if max{m,n}≥λ1As for the techniques used in this Chapter, it is mentioned that the current studies of critical Fujita curves for coupled systems (especially in critical cases) seem to be heavenly relying upon Jensen's inequality and/or the Kaplan method, for which one has to deal with complicated discussions on the exponents p, q being greater or less than one. Differ-ently, in the framework of this paper, the heat semigroup is introduced to study critical Fujita curves for coupled system problems, where various superlinear and sublinear cases will be treated uniformly by estimates involved. This greatly simplifies the arguments for establishing the Fujita type theorems. Finally, as applications of the framework of the paper, a new and simpler proof is proposed to some previous results of the authors in Chapter3.
本文分为以下四个章节：
第一章主要概述本文所研究问题的背景和国内外发展现状,并简要介绍了本文的主要工作.
第二章考虑局部化源α(X)up影响下的一维快扩散方程的Cauchy问题ut=(um)xx+a(x)up这里的源α(x)Up中·α(X)具有紧支集.我们知道,源为up的快扩散方程的Cauchy问题具有Fujita指标pc=m+2,而相应的有界区域不会出现Fujita现象.这里所讨论的问题介于以上两者之间,既不是简单的有界区域问题,也不是全空间的一般Cauchy问题,我们将证明此问题的Fujita指标为pc=m+1,从而推广了之前关于慢扩散方程的结论.我们特别指出,对应的高维问题不会有Fujita型结论.最后,我们一定初值条件下证明非整体解的blow-up速率为(T-t)-1/p-1,blow-up集为B(u)={0}.
第二章研究变指数源up(x)和ug(x)耦合的热方程组的Fujita型结论.先考虑解对任意初值整体存在的指标范围I,以及整体解与非整体解共存的指标范围II.在I和II之外,解是否整体存在还与区域大小有关：如果区域充分小使得包含在某个小球内,则解对小初值整体存在；如果区域足够大使得包含某个大球,则存在p(x),q(x)使得解对任意初值爆破.
第四章考虑有界区域、变系数反应项影响下的耦合组ut=△u+eqtvp,vt=△v+eβtuq的Dirichlet问题,其中αβ∈R,p,q>0,证明其临界Fujita曲线为(pq)c=1+max{α+βp,β+αq,0}/λ1,其中λ1>0是齐次Dilchlet特征值问题一△4=λ4的主特征值.作为一个直接推论,我们还得到一个有趣的Fujita型结论：耦合组Ut=△U+mU+Vp, Vt=△V+nV+Uq的Fujita临界曲线表现为Fujita临界系数max{m,n}=λ1,即不存在非平凡整体解的充要条件为：max{m,n}≥λ1据我们所知,以往关于耦合组的Fujita曲线的研究(特别是临界情形)都依赖于Jensen不等式和／或Kaplan方法,这样就需要讨论p,q与1的大小关系.与此不同,本章所采用的方法是引入热半群来辅助研究Fujita曲线,使得我们可以将p,q的所有情况一起讨论,避免了指数分类,从而使Fujita型定理的证明过程得到了简化.作为(简化证明)的例证,我们除了应用木章的方法讨论几个相关的问题外,还对第三章的部分重要结论重新给出新的简洁证明.
It is known that in general the study on the critical Fujita admit unbounded domains only. This thesis discusses Fujita phenomena of nonlinear parabolic equations for bounded domains or compactly supported reactions. There are two topics included:The Fujita exponent of nonlinear parabolic systems in bounded domain; The Cauchy problem with sources compactly supported. The topics of blow-up set and blow-up rate etc are involved as well.
The thesis composes of four chapters:
In Chapter1we summarize the background of the related issues and state the main results of the present thesis.
Chapter2deals with the Cauchy problem ut=(um)xx+a(x)up with localized reaction a(x)up, where the reaction a(x)up is compactly supported. As we know, the Fujita exponent of Cauchy problem with source up is pc=m+2, and there is no Fujita type conclusion with bounded domains. The problem studied here is between them, neither the problem with bounded domains, nor general Cauchy problem on the whole space, and its critical Fujita exponent will be shown as pc=m+1. This extends the related result for the slow diffusion situation. It is pointed out that there is no Fujita type conclusion for the high dimension case of this problem. In addition, we prove that the problem admits blow-up rate (T-t)-1/p-1and the blow-up set{0} under suitable initial data.
Chapter3studies the Fujita type conclusion for a parabolic system coupled via variable sources up(x) and vg(x).At first consider a region I for global solutions with any initial data, and a coexistence region II of global and non-global solutions. Outside of I and Ⅱ, the existence or not of global solutions is related to the size of the domain:the solutions are global with small initial data if the domain is small to be contained in a small ball, and there is p(x),q(x) such that the solutions are non-global with large initial data if the domain is large to contain a big ball.
Chapter4treats the coupled parabolic system with time-weighted sources in a bounded domain:ut=△u+eαtvp, ut=△u+eβtuq in Ω×(0,T) with α,β∈R and p,q>0, subject to null Dirichlet boundary value condition. The critical Fujita curve is determined as (pq)c=1+max{α+βp,β+αq,0} where λ1is the first eigenvalue of the Lapla-cian with null Dirichlet boundary condition, and there is no any additional restriction on α,β,p,q. Next, as an extension, an interesting Fujita phenomenon is observed for another coupled system Ut=△U+mU+Vp,△t=△V+nV+Uq in Ω×(0, T) with pq>1that the critical Fujita curve is represented via the Fujita critical coefficient max{m,n}=λ1namely, any nontrivial solutions blow up in finite time if and only if max{m,n}≥λ1As for the techniques used in this Chapter, it is mentioned that the current studies of critical Fujita curves for coupled systems (especially in critical cases) seem to be heavenly relying upon Jensen's inequality and/or the Kaplan method, for which one has to deal with complicated discussions on the exponents p, q being greater or less than one. Differ-ently, in the framework of this paper, the heat semigroup is introduced to study critical Fujita curves for coupled system problems, where various superlinear and sublinear cases will be treated uniformly by estimates involved. This greatly simplifies the arguments for establishing the Fujita type theorems. Finally, as applications of the framework of the paper, a new and simpler proof is proposed to some previous results of the authors in Chapter3.
引文
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[2]ARONSON D J, WEINBERGER H F. Multidimensional nonlinear diffusion arising in population genetics [J]. Adv. in Math.,1978,30:33-76.
[3]HAYAKAWA K. On nonexistence of global solutions of some semilinear parabolic equations [J]. Proc. Japan Acad.,1973,49:503-525.
[4]KOBAYASHI K, SIARO T, TANAKA H. On the blowing up problem for semilinearr heat equations [J]. J. Math. Soc. Japan,1977,29:407-424.
[5]WEISSLER F B. Existence and nonexistence of global solutions for a semilinear heat equation [J]. Israel J. Math.,1981,38:29-40.
[6]GALAKTIONOV V, KURDYUMOV S, P M A, SAMARSKII A A. Unbounded solutions of the cauchy problem for the parabolic equation ut=▽(uσ▽u)+uβ [J]. Dokl. Akad. Nauk SSR.252:1362-1364(in Russian):English translation Sov. Phys. Dokl.,1980,25:458-459.
[7]GALAKTIONOV V A. Conditions for global nonexistence and localization for a class of nonlinear parabolic equations [J]. U.S.S.R. Comput. Math. Math. Phys., 1983,23:35-44.
[8]MOCHIZUKI K, MUKAI K. Existence and nonexistence of global solutions to fast diffusions with source [J]. Methods and Applications of Analysisl,1995,2:92-102.
[9]LIU X, WANG M. The critical exponent of doubly singular parabolic equations [J]. J. Math. Anal. Appl,2001,257:170-188.
[10]AGUIRRE J, ESCOBEDO M. On the blow-up of solutions of a convective reaction diffusion equation [J]. Proc. Roy. Soc. Edinburgh Sect. A,1993,123:433-460.
[11]BANDLE C, LEVINE H. Fujita type phenomena for reaction-diffusion equations with convection like terms [J]. Trans. Amer. Math. Soc,1994,7:1169-1193.
[12]GALAKTIONOV V, LEVINE H. On critical fujita exponents for heat equations with nonlinear flux conditions on the boundary [J]. Israel J.Math.,1996,94:125-146.
[13]DENG K, FILA M, LEVINE HA. On critical exponent for a system of heat equations coupled in the boundary conditions [J]. Acta Math. Univ. Comenianae,1994,63:169-192.
[14]DENG K. Nonexistence of global solutions of a nonlinear hyperbolic system [J]. Trans. Amer. Math. Soc,1997,349:1685-1696.
[15]PINSKY P. Existence and nonexistence of global solutions for ut=△u+a(x)up in Rd [J]. J. Differential Equations,1997,133:152-177.
[16]Qi Y. The critical exponents of parabolic equations and blow-up in Rn [J]. Proc. Roy. Soc. Edinburgh Sect.A,1998,128:123-136.
[17]FERREIRA R, DE PABLO A, VAZQUEZ J L. Classification of blow-up with nonlin-ear diffusion and localized reaction [J]. J. Differential Equations,2006,231:195-211.
[18]GALAKTIONOV V, LEVINE H. A general approach to critical Fujita exponents and systems [J]. Nonlinear Anal. TMA,1998,34:1005-1027.
[19]SOUPLET P. Uniform blow-up profiles and boundary behavior for diffusion equa-tions with nonlocal nonlinear source [J]. J.Differential Equations,1999,153:374-406.
[20]DENG K, LEVINE H. The role of critical exponents in blow-up theorems:the sequel [J]. J. Math. Anal. Appl,2000,243:85-126.
[21]LEVINE H. The role of critical exponents in blow-up theorems [J]. SI AM Rev. 1990,32:262-288.
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