几类非经典扩散方程的渐近行为
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摘要
本文研究几类发展型非线性偏微分方程的渐近行为,涉及Fujita (?)临界指标与第二临界指标、整体解的渐近profile与非整体解的life span等问题.论文所考虑的四类非线性偏微分方程(组)特别包括伪抛物祸合组与非局部扩散方程这两类非经典扩散方程.我们着重考虑伪抛物耦合组中高阶粘性项,以及非局部扩散方程的源的局部化对解的渐近行为的特殊影响.
本文分为以下四个章节：
第一章介绍本文研究问题的实际背景及国内外发展情况,并概述主要内容和结果.
第二章考虑一类非线性伪抛物耦合组的Cauchy (?)司题.借助压缩不动点定理得到温和解的局部存在性,进而得到古典解的局部存在性.借助于伪抛物方程基本解的正性性质建立该耦合组问题的比较原理.在这些准备工作的基础上,深入研究该问题的Fujita临界指标、第二临界指标,以及整体解的渐近profile.结果表明,高阶粘性项的存在形式上并没有改变对应经典抛物问题的两个临界指标以及渐近profile.然而，这一高阶项的出现给问题的研究带来了本质性困难,例如解的自相似性与正则性的缺失,基本解形式的复杂性等.
第三章讨论源的非齐次性对解的渐近行为的影响.第一节研究具有局部化源的非局部扩散方程.结果表明,尽管扩散为非局部的,源的局部化仍然保持了对解的渐近行为的巨大影响.在空间维数大于等于二时,不再存在Fujita现象.即使在一维情形，与非局部扩散方程的局部源情形相比,局部化因子的存在不仅使Fuiita (?)旨标变小(亦即使解对任意初值爆破的指标范围缩小),而且使第二临界指标也变小(从而提高了在整体解和非整体解共存区域爆破对于初值的门槛要求).第二节研究一类具有径向加权非线性源的p-Laplace方程.在已有Fujita (?)临界指标结果的基础上,进一步研究其第二临界指标，并对齐次源情形得到关于非整体解life span的一致性估计.
第四章研究一类含有非齐次项的快扩散非线性耦合组的Fujita (?)临界指标与第二临界指标.结果表明,非齐次项存在,就会使问题的Fujita (?)临界指标发生改变,具体说,使解对任意初值爆破的指标范围扩大.但另一方面,小的非齐次项不改变第二临界指标.
This thesis deals with asymptotic behavior for some evolution nonlinear PDEs, in-volving critical and second critical Fujita exponents, asymptotic profiles for global solu-tions, and life spans for nonglobal solutions. Two of the four nonlinear PDEs considered, i.e., the coupled nonlinear pseudo-parabolic equations and the nonlinear nonlocal diffu-sion equations, belong to nonclassical parabolic equations. We will study the contributions from the high-order viscosity terms in pseudo-parabolic equation and the localization of the source in the nonlocal diffusion problem, respectivcly, to the asymptotic behavior of solutions.
The thesis is composed of four chapters:
Chapter1introduces the background of the related issues, and briefly summarizes the main results of the present thesis.
In Chapter2, we consider the Cauchy problem to a coupled nonlinear pseudo-parabolic system. The local existence of the mild solution is obtained via the fixed-point theorem technique, which implies the existence of the classical solutions under smooth initial data. Furthermore, the comparison principle comes from the positivity of the fun-damental solution associated. Based on these preliminaries, we study the critical Fujita exponent, the second critical exponent, and the asymptotic profile of global solutions. It is observed that the high-order viscosity terms do not change the forms of the two critical exponents and the asymptotic profile. However, the presence of the high order terms bring some substantial difficulties, such as the lost of self-similarity and regularity of solutions, and the complicated form of the fundamental solution involved.
In Chapter3, we study the influence of the inhomogeneity of source to the asymptotic behavior of solutions. The first section considers a nonlocal diffusion equation with local-ized source. It is shown that the localization of source significantly affects the asymptotic behavior of solutions, even under nonlocal diffusion. More precisely, there is no Fujita phenomenon if the space dimension N≥2, and the Fujita phenomenon happens only for N=1. In the case of N=1, compared with the situation with local source under nonlo-cal diffusion, both the critical Fujita exponent and the second critical exponent decrease because of the localization factor, which implies that the parameter region for blowing up under any initial data shrinks, and the threshold of initial data for blowing up (in the coexistence parameter region) is enlarged. Next, we consider the Cauchy problem o an evolution p-Laplacian equation with weighted source in the second section, the critical Fujita exponent of which was known. We establish the second critical exponent, as well as the uniform estimate on the life span of non-global solutions for the case of homogeneous source.
Chapter4concerns the Fujita phenomenon for the Cauchy problem of an inhomoge-neous fast diffusion system. Both the critical exponent and the second exponent are ob-tained. We observe that the inhomogeneous terms in the system substantially contribute to the critical exponent, in that the blow-up exponent region is obviously enlarged, with keeping the second critical exponent unchanged.
本文分为以下四个章节：
第一章介绍本文研究问题的实际背景及国内外发展情况,并概述主要内容和结果.
第二章考虑一类非线性伪抛物耦合组的Cauchy (?)司题.借助压缩不动点定理得到温和解的局部存在性,进而得到古典解的局部存在性.借助于伪抛物方程基本解的正性性质建立该耦合组问题的比较原理.在这些准备工作的基础上,深入研究该问题的Fujita临界指标、第二临界指标,以及整体解的渐近profile.结果表明,高阶粘性项的存在形式上并没有改变对应经典抛物问题的两个临界指标以及渐近profile.然而，这一高阶项的出现给问题的研究带来了本质性困难,例如解的自相似性与正则性的缺失,基本解形式的复杂性等.
第三章讨论源的非齐次性对解的渐近行为的影响.第一节研究具有局部化源的非局部扩散方程.结果表明,尽管扩散为非局部的,源的局部化仍然保持了对解的渐近行为的巨大影响.在空间维数大于等于二时,不再存在Fujita现象.即使在一维情形，与非局部扩散方程的局部源情形相比,局部化因子的存在不仅使Fuiita (?)旨标变小(亦即使解对任意初值爆破的指标范围缩小),而且使第二临界指标也变小(从而提高了在整体解和非整体解共存区域爆破对于初值的门槛要求).第二节研究一类具有径向加权非线性源的p-Laplace方程.在已有Fujita (?)临界指标结果的基础上,进一步研究其第二临界指标，并对齐次源情形得到关于非整体解life span的一致性估计.
第四章研究一类含有非齐次项的快扩散非线性耦合组的Fujita (?)临界指标与第二临界指标.结果表明,非齐次项存在,就会使问题的Fujita (?)临界指标发生改变,具体说,使解对任意初值爆破的指标范围扩大.但另一方面,小的非齐次项不改变第二临界指标.
This thesis deals with asymptotic behavior for some evolution nonlinear PDEs, in-volving critical and second critical Fujita exponents, asymptotic profiles for global solu-tions, and life spans for nonglobal solutions. Two of the four nonlinear PDEs considered, i.e., the coupled nonlinear pseudo-parabolic equations and the nonlinear nonlocal diffu-sion equations, belong to nonclassical parabolic equations. We will study the contributions from the high-order viscosity terms in pseudo-parabolic equation and the localization of the source in the nonlocal diffusion problem, respectivcly, to the asymptotic behavior of solutions.
The thesis is composed of four chapters:
Chapter1introduces the background of the related issues, and briefly summarizes the main results of the present thesis.
In Chapter2, we consider the Cauchy problem to a coupled nonlinear pseudo-parabolic system. The local existence of the mild solution is obtained via the fixed-point theorem technique, which implies the existence of the classical solutions under smooth initial data. Furthermore, the comparison principle comes from the positivity of the fun-damental solution associated. Based on these preliminaries, we study the critical Fujita exponent, the second critical exponent, and the asymptotic profile of global solutions. It is observed that the high-order viscosity terms do not change the forms of the two critical exponents and the asymptotic profile. However, the presence of the high order terms bring some substantial difficulties, such as the lost of self-similarity and regularity of solutions, and the complicated form of the fundamental solution involved.
In Chapter3, we study the influence of the inhomogeneity of source to the asymptotic behavior of solutions. The first section considers a nonlocal diffusion equation with local-ized source. It is shown that the localization of source significantly affects the asymptotic behavior of solutions, even under nonlocal diffusion. More precisely, there is no Fujita phenomenon if the space dimension N≥2, and the Fujita phenomenon happens only for N=1. In the case of N=1, compared with the situation with local source under nonlo-cal diffusion, both the critical Fujita exponent and the second critical exponent decrease because of the localization factor, which implies that the parameter region for blowing up under any initial data shrinks, and the threshold of initial data for blowing up (in the coexistence parameter region) is enlarged. Next, we consider the Cauchy problem o an evolution p-Laplacian equation with weighted source in the second section, the critical Fujita exponent of which was known. We establish the second critical exponent, as well as the uniform estimate on the life span of non-global solutions for the case of homogeneous source.
Chapter4concerns the Fujita phenomenon for the Cauchy problem of an inhomoge-neous fast diffusion system. Both the critical exponent and the second exponent are ob-tained. We observe that the inhomogeneous terms in the system substantially contribute to the critical exponent, in that the blow-up exponent region is obviously enlarged, with keeping the second critical exponent unchanged.
引文
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[20]ZENG X Z. The critical exponents for the quasi-linear parabolic equations with inhomogeneous terms. J. Math. Anal. Appl,2007,2:1408-1424.
[21]ZHANG Q S. Blow up and global existence of solutions to an inhomogeneous parabolic system. J. Differential Equations,1998,147:155-183.
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[28]KARCH G. Asymptotic behaviour of solutions to some pesudoparabolic equations. Math. Methods Appl. Sci,1997,20:271-289.
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[35]QUITTNER P, SOUPLET P. Superlinear parabolic problems:blow-up, globlal ex-istence and steady states. Birkhauser Verlag, Basel,2007.
[36]SOUPLET P, WINKLER M. The influence of space dimension on the large-time behavior in a reaction-diffusion system modeling diallelic selection. J. Math. Biol, 2011,62:391-421.
[37]CARRILLO C, FIFE P. Spatial effects in discrete generation population models. J. Math. Biol,2005,50:161-188.
[38]KINDERMANN S, OSHER S, JONES P. Deblurring and denoising of images by nonlocal functionals. Multiscale Model. Simul,2005,4:1091-1115.
[39]PINSKY R. Existence and nonexistence of global solution for ut= Δu+a(x)up in Rn. J. Differential Equations,1997,133:152-177.
[40]SUZUKI R. Existence and nonexistence of global solutions of quasilinear parabolic equations. J. Math. Soc. Japan,2002,54:747-792.
[41]FERREIRA R, PABLO A, VAZQUEZ J. Classification of blow-up with nonlinear diffusion and localized reaction. J. Differential Equations,2006,231:195-211.
[42]BRANDLE C, CHASSEIGNE E, FERREIRA R. Unbounded solutions of the non-local heat equation. Commun. Pure Appl. Anal,2011,10:1663-1686.
[43]CHEN W X. LI C M, OU B. Classification of solutions for an integral equation. Comm. Pure Appl. Math,2006,59:330-343.
[44]WANG C P, ZHENG S N. Critical Fujita exponents of degenerate and singular parabolic equations. Proc. Roy. Soc. Edinburgh Sect. A,2006,136:415-430.
[45]BERTOZZI A, GARNETT J, LAURENT T. Characterization of radially symmetric finite time blowup in multidimensional aggregation equations. SIAM J. Math. Anal, 2011,44:651-681.
[46]TERRA J. WOLANSKI N. Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data. Discrete and Continuous Dynamical Systems, 2011,31:581-605.
[47]GARCIA-MELIAN J, ROSSI J D. On the principal eigenvalue of some nonlocal diffusion problems. J. Differential Equations,2009,246:21-38.
[48]GARCIA-MELIAN J. ROSSI J D. A nonlocal p-laplacian evolution equation with neumann boundary conditions. J. Math. Pures Appl,2008,90:201-227.
[49]TERRA J. WOLANSKI N. Asymptotic behavior for a nonlocal diffusion equation with absorption and nonintegrable initial data. the supercritical case. Proc. Am. Math. Soc,2011,139:1421-1432.
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[54]LIU X F, WANG M X. The critical exponent of doubly singular parabolic equations. J. Math. Anal. Appl,2001,257:170-188.
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[61]LEI P D, ZHENG S N. Global and nonglobal weak solutions to a degenerate parabolic system. J. Math. Anal. Appl,2006,324:177-198.
[62]KAPLAN S. On the growth of solutions quasi-linear parabolic equations. Comm. Pure Appl. Math,1963,16:305-330.
[2]WEISSLER F B. Existence and non-existence of global solutions for semilinear equatiuon. Israel J. Math,1981,6:29-40.
[3]LEE T Y, NI W M. Global existence, large time behavior and life span on solution of a semilinear parabolic Cauchy problem. Trans. Amer. Math. Soc,1992,333:365-378.
[4]GUI C F, WANG X F. Life span of solutions of the Cauchy problem for a semilinear heat equation. J. Differential Equations,1995,115:166-172.
[5]BENACHOUR S, KARCH G, LAURENQOT P. Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations. J. Math. Pures Appl,2004,83:1275-1308.
[6]NARAZAKI T, NISHIHARA K. Asymptotic behavior of solutions for the damped wave equation with slowly decaying data. J. Math. Anal. Appl,2008,338:803-819.
[7]ESCOBEDO M, HERRERO A. Boundedness and blow-up for a semilinear reaction-diffusion system. J. Differential Equations,1991,89:176-202.
[8]MOCHIZUKI K. Life span and large time behavior of solutions of a weakly coupled system reaction-diffusion equations. Adv. Math. Appl. Sci,1998.48:175-198.
[9]LEVINE H A. The role of critical exponents in blowup theorems. SIAM Rev,1990, 32:262-288.
[10]DENG K, LEVINE H A. The role of critical exponents in blow-up theorems:The sequel. J. Math. Anal. Appl,2000,243:85-126.
[11]KAIKINA E I, NAUMKIN P I, A S I. The Cauchy problem for a sobolev type equation with power like nonlinearity. Izv. Math,2005,69:59-111.
[12]CAO Y, YIN J X, WANG C P. Cauchy problems of semilinear pseudo-parabolic equations. J. Differential Equations,2009,246:4568-4590.
[13]CHASSEIGNE E, CHAVES M, ROSSI J. Asymptotic behavior for nonlocal diffusion equations. J. Math. Pures Appl.,2006,86:271-291.
[14]CARR P, GEMAN H, MADAN D B, YOR M. Stochastic volatility for Levy processes. Math. Finance,2003,13:345-382.
[15]SILVESTRE L. Holder estimates for solutions of integro-differential equations like the fractional Laplace. Indiana Univ. Math. J.,2006,55:1155-1174.
[16]GARCIA-MELIAN J, QUIROS F. Fujita exponents for evolution problems with nonlocal diffusion. J. Evol. Equ,2010.10:147-161.
[17]QI Y W, LEVINE H A. The critical exponent of degenerate parabolic systems. Z. Angew. Math. Phys,1993,44:249-265.
[18]ZHANG Q S. A new critical phenomenon for semilinear parabolic problems. J. Math. Anal. Appl,1998,219:125-139.
[19]ZHANG Q S. Blow-up results for nonlinear parabolic equations on manifolds. Duke Math. J,1999,97:515-539.
[20]ZENG X Z. The critical exponents for the quasi-linear parabolic equations with inhomogeneous terms. J. Math. Anal. Appl,2007,2:1408-1424.
[21]ZHANG Q S. Blow up and global existence of solutions to an inhomogeneous parabolic system. J. Differential Equations,1998,147:155-183.
[22]QI Y W, WANG M X. Critical exponents of quasilinear parabolic equations. J. Math. Anal. Appl,2002,267:264-280.
[23]CHEN P J, GURTIN M E. On a theory of heat conduction involving two tempera-tures. Z. Angew. Math. Phys,1998.19:614-627.
[24]AMICK C J, BONA J L. Decay of solutions of some nonlinear wave equations. J. Differential Equations,1989,81:1-49.
[25]BENJAMIN T B, BONA J L. Model equations for long waves in nonlinear dispersive systems. Phil. Trans. Roy. Soc. London Ser. A,1972,272:47-78.
[26]BARENBLAT G, ZHELTOV I, I K. Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech,1960,24:1286-1303.
[27]RAO V R G, TING T W. Solutions of pseudo-heat equations in the whole space. Arch. Ration. Mech. Anal,1972/73,49:57-78.
[28]KARCH G. Asymptotic behaviour of solutions to some pesudoparabolic equations. Math. Methods Appl. Sci,1997,20:271-289.
[29]KORPUSOV M O, SVESHNIKOV A G. Blow-up of solutions of strongly nonlinear equations of pseudoparabolic type. Journal of Mathematical Sciences,2008,148:1-142.
[30]YANG C X, CAO Y, ZHENG S N. Life span and second critical exponent for semi-linear pseudo-parabolic equation. J. Differential Equations.2012,253:3286-3303.
[31]MITIDIERI E, POHOZAEV S I. A priori estimates and blow-up of solutions of nonlineari partial differential equations and inequalities. Proc. Steklov Inst. Math, 2001,234:1-362.
[32]KAMIN S, PELETIER L A. Large time behaviour of solutions of the heat equation with absorption. Ann. Scuola Norm. Sup. Pisa Cl. Sci,1985,12:393-408.
[33]GUO J S, GUO Y Y. On a fast diffusion equation with source. Tohoku Math. J, 2001,53:571-579.
[34]MUKAI K, MOCHIZUKI K, Qing H. Large time behavior and life span for a quasi-linear parabolic equation with slowly decaying initial values. Nonlinear Anal.2000, 53:33-45.
[35]QUITTNER P, SOUPLET P. Superlinear parabolic problems:blow-up, globlal ex-istence and steady states. Birkhauser Verlag, Basel,2007.
[36]SOUPLET P, WINKLER M. The influence of space dimension on the large-time behavior in a reaction-diffusion system modeling diallelic selection. J. Math. Biol, 2011,62:391-421.
[37]CARRILLO C, FIFE P. Spatial effects in discrete generation population models. J. Math. Biol,2005,50:161-188.
[38]KINDERMANN S, OSHER S, JONES P. Deblurring and denoising of images by nonlocal functionals. Multiscale Model. Simul,2005,4:1091-1115.
[39]PINSKY R. Existence and nonexistence of global solution for ut= Δu+a(x)up in Rn. J. Differential Equations,1997,133:152-177.
[40]SUZUKI R. Existence and nonexistence of global solutions of quasilinear parabolic equations. J. Math. Soc. Japan,2002,54:747-792.
[41]FERREIRA R, PABLO A, VAZQUEZ J. Classification of blow-up with nonlinear diffusion and localized reaction. J. Differential Equations,2006,231:195-211.
[42]BRANDLE C, CHASSEIGNE E, FERREIRA R. Unbounded solutions of the non-local heat equation. Commun. Pure Appl. Anal,2011,10:1663-1686.
[43]CHEN W X. LI C M, OU B. Classification of solutions for an integral equation. Comm. Pure Appl. Math,2006,59:330-343.
[44]WANG C P, ZHENG S N. Critical Fujita exponents of degenerate and singular parabolic equations. Proc. Roy. Soc. Edinburgh Sect. A,2006,136:415-430.
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