局部化—局部源的相互作用与奇性解的渐近行为
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摘要
本文研究几类具有局部与局部化源的非线性抛物方程(组)奇性解的渐近行为,具体涉及五个同时具有局部源和局部化源的模型.我们的兴趣在于这两类源相互作用对解的奇性的产生与传播的影响,重点讨论blow-up集.首先考虑两类分别由局部化源和局部源耦合的热方程组解的全局与单点blow-up问题,通过比较发现这两类耦合源对解的blow-up集等性质的影响有本质性区别;接着我们讨论局部化源与局部源相互作用对非线性扩散问题解的blow-up速率,blow-up profile,blow-up集的影响;最后,我们通过研究局部化源项对抛物型方程(组)解Fujita指标的影响发现:与通常的局部源情形不同,局部化源的存在可以使相关模型具有无限Fujita指标(亦即,排除解对大初值blow-up、小初值整体存在的指标情形).
本文的主要结果概述如下:
(Ⅰ)关于全局与单点blow-up
第二章考虑具有局部源和耦合局部化源的热方程组u_t =△u + u~m + v~p(0, t),v_t =△v + v~n + u~q(0, t), (x, t)∈Ω×(0, T)的齐次Diriehlet问题.解的性质依赖于局部源、耦合局部化源,以及扩散和零边值之间的相互作用.我们得到关于非整体解全局与单点blow-up的完全的指标分类.讨论还涉及不同占优机制所导致的解的同时与非同时blow-up,以及解的多重blow-up速率.顺便指出,此前的已有文献中未曾发现有人讨论过方程组情形解的全局与单点blow-up问题.
第三章研究具有局部化源和耦合局部源的热方程组u_t =△u + u~m(0.t) + v~p,v_t =△v + v~n(0,t) + u~q, (x, t)∈Ω×(0, T)的齐次Dirichlet问题,进行了与第二章模型的平行讨论,亦即奇性解的全局与单点blow-up,同时与非同时blow-up等.特别地,将本章与上一章的结果进行比较,可以看出这两类不同的耦合关系所造成的关于解的奇性产生与传播的某些本质不同.例如,第二章模型指标分类中解的一个分量全局blow-up而另一个分量单点blow-up的现象对本章所讨论的模型却没有出现.
在第四章,我们研究具有齐次Dirichlet边界条件的局部非线性扩散问题u_t =△u~m+λ_1u~p+λ_2u~q(0, t) ,其中p,q≥0,max{p,q}>m>1,且λ_1,λ_2>0.我们通过研究局部化源、局部源、非线性扩散以及齐次Dirichlet边界条件之间的相互作用给出解在不同占优机制下的blow-up速率和一致blow-up profiles.关于解的blow-up集,我们发现非线性扩散对解的全局与单点blow-up无影响.
(Ⅱ)关于Fujita指标
第五章考虑非线性扩散模型u_t =△u~m +λ_1u~(p_1)(x,t) +λ_2u~(p_2)(x~*(t),t)的Cauchy问题,其中m≥1,p_i,λ_i≥0(i=1,2)并且x~*(t)H(?)lder连续.我们发现这样一个新现象:当λ_2>0时该模型的临界Fujita指标p_c=+∞.也就是说,只要p=max{p_1,p_2}>1,则解对非平凡非负初值必发生blow-up.我们进一步证明,这一结果对于其他形式的局部化源情形(哪怕是衰减的)以及具有局部化源的耦合组均成立.
第六章研究具有局部化源与局部源耦合的反应扩散方程组u_t =△u+v~p(x~*(t),t), v_t =△v+u~q.为了研究局部源与局部化源间的相互作用,我们分别考虑Cauchy问题以及具有齐次Dirichlet边界的初边值问题.对于初边值问题我们证明了解在整个区域内处处blow-up,并具有一致的blow-up profiles.对于Cauchy问题,我们给出一个有趣的结论:它所对应的Fujita指标为无穷大,亦即,只要pq>1,则解对于任意非平凡非负初值都blow-up.
This thesis deals with asymptotic behavior of singular solutions for multi-nonlinear parabolic equation (system) with local and localized sources. Five models with local and localized sources of the sum forms are involved. We will study interactions between the two kinds of sources and their influences to the occurrence and propagation of singularities of solutions, and pay more attentions to the topic of blow-up sets. At first, we consider global versus single point blow-up of solutions to two models with coupling localized and local sources respectively. Comparing the two models, we find substantial different influences of the two kinds of couplings to the blow-up sets of solutions. Then, we study in what way the interactions between localized and local sources affect the blow-up rate, blow-up profile, and blow-up set in a nonlinear diffusion problem. Finally, we will show via models how the localized sources substantially influence the critical Fujita exponents. It is interesting to find they may admit an infinite Fujita exponent because of the localized sources. This excludes the situation where the solutions are non-global for large initial data and global with small initial data.
The main results obtained in this thesis can be summarized as follows:
(Ⅰ) Total and single point blow-up
Chapter 2 considers heat equations with local and coupling localized sources u_t =△u + u~m + v~p(0, t), v_t =△v + v~n + u~q(0, t), (x, t)∈Ω×(0, T) subject to null Dirichlet boundary conditions. The behavior of solutions depends on the interactions among the local and localized sources as well as the diffusions with the null boundary conditions in the model. We obtain a complete classification of parameters to distinguish total and single point blow-up for the non-global solutions. In addition, simultaneous versus nonsimultaneous blow-up of solutions under different dominations are determined also with four possible simultaneous blow-up rates. To our knowledge, this is the first study on total versus single point blow-up for the case of coupled systems.
Chapter 3 treats homogeneous Dirichlet problem to heat system with localized and coupling local sources u_t =△u + u~m(0,t) + v~p, v_t =△v + v~n(0,t)+ u~q, (x, t)∈Ω×(0, T) with a parallel discussion as that in Chapter 2, i.e., total versus single point blow-up, simultaneous versus non-simultaneous blow-up etc. In particular, comparing with the results of Chapter 2, we find some substantial differences on occurrence and propagation of singularities of solutions due to the two kinds of couplings. For example, the situation of global and single blow-up for the two components respectively, included in the classification to the model in Chapter 2, does not appear in the classification to the present model.
Chapter 4 studies a localized nonlinear diffusion equation u_t =△u~m+λ_1u~p+λ_2u~q(0, t) subject to null Dirichlet boundary condition with p, q≥0, max{p, q} > m > 1, andλ_1,λ_2 > 0. By investigating the interactions among the localized and local sources, the nonlinear diffusion with the zero boundary value condition, we establish blow-up rates and uniform blow-up profiles of solutions under different dominations. In addition, as for the blow-up sets of solutions, we find that nonlinear diffusion has no contributions to the total and single point blow-up of solutions.
(Ⅱ) Fujita exponents
Chapter 5 deals with Cauchy problem to nonlinear diffusion model u_t =△u~m +λ_1u~(p_1)(x,t) +λ_2u~(p_2)(x~*(t),t) with m≥1, p_i,λ_i≥0 (i = 1,2) and x~*(t) H(?)lder continuous. A new phenomenon is observed that the critical Fujita exponent p_c= +∞wheneverλ_2 > 0. More precisely, the solution blows up under any nontrivial and nonnegative initial data whenever p = max{p_1,p_2} > 1. This result is then extended to a coupled system with localized sources as well as the cases with other nonlinearities (even the decay ones).
Chapter 6 focuses on the asymptotic behavior of solutions for reaction-diffusion equations coupled via localized and local sources: u_t =△u+v~p(x~*(t),t). v_t =△v+u~q Both the initial-boundary problem with null Dirichlet boundary condition and the Cauchy problem are considered to study the interaction between the localized and the local sources. For the initial-boundary problem we prove that the solutions blow up everywhere in the domain with uniform blow-up profiles. In addition, it is interesting to show that the Cauchy problem admits an infinity Fujita exponent, namely, the solutions blow up under any nontrivial and nonnegative initial data whenever pq > 1.
本文的主要结果概述如下:
(Ⅰ)关于全局与单点blow-up
第二章考虑具有局部源和耦合局部化源的热方程组u_t =△u + u~m + v~p(0, t),v_t =△v + v~n + u~q(0, t), (x, t)∈Ω×(0, T)的齐次Diriehlet问题.解的性质依赖于局部源、耦合局部化源,以及扩散和零边值之间的相互作用.我们得到关于非整体解全局与单点blow-up的完全的指标分类.讨论还涉及不同占优机制所导致的解的同时与非同时blow-up,以及解的多重blow-up速率.顺便指出,此前的已有文献中未曾发现有人讨论过方程组情形解的全局与单点blow-up问题.
第三章研究具有局部化源和耦合局部源的热方程组u_t =△u + u~m(0.t) + v~p,v_t =△v + v~n(0,t) + u~q, (x, t)∈Ω×(0, T)的齐次Dirichlet问题,进行了与第二章模型的平行讨论,亦即奇性解的全局与单点blow-up,同时与非同时blow-up等.特别地,将本章与上一章的结果进行比较,可以看出这两类不同的耦合关系所造成的关于解的奇性产生与传播的某些本质不同.例如,第二章模型指标分类中解的一个分量全局blow-up而另一个分量单点blow-up的现象对本章所讨论的模型却没有出现.
在第四章,我们研究具有齐次Dirichlet边界条件的局部非线性扩散问题u_t =△u~m+λ_1u~p+λ_2u~q(0, t) ,其中p,q≥0,max{p,q}>m>1,且λ_1,λ_2>0.我们通过研究局部化源、局部源、非线性扩散以及齐次Dirichlet边界条件之间的相互作用给出解在不同占优机制下的blow-up速率和一致blow-up profiles.关于解的blow-up集,我们发现非线性扩散对解的全局与单点blow-up无影响.
(Ⅱ)关于Fujita指标
第五章考虑非线性扩散模型u_t =△u~m +λ_1u~(p_1)(x,t) +λ_2u~(p_2)(x~*(t),t)的Cauchy问题,其中m≥1,p_i,λ_i≥0(i=1,2)并且x~*(t)H(?)lder连续.我们发现这样一个新现象:当λ_2>0时该模型的临界Fujita指标p_c=+∞.也就是说,只要p=max{p_1,p_2}>1,则解对非平凡非负初值必发生blow-up.我们进一步证明,这一结果对于其他形式的局部化源情形(哪怕是衰减的)以及具有局部化源的耦合组均成立.
第六章研究具有局部化源与局部源耦合的反应扩散方程组u_t =△u+v~p(x~*(t),t), v_t =△v+u~q.为了研究局部源与局部化源间的相互作用,我们分别考虑Cauchy问题以及具有齐次Dirichlet边界的初边值问题.对于初边值问题我们证明了解在整个区域内处处blow-up,并具有一致的blow-up profiles.对于Cauchy问题,我们给出一个有趣的结论:它所对应的Fujita指标为无穷大,亦即,只要pq>1,则解对于任意非平凡非负初值都blow-up.
This thesis deals with asymptotic behavior of singular solutions for multi-nonlinear parabolic equation (system) with local and localized sources. Five models with local and localized sources of the sum forms are involved. We will study interactions between the two kinds of sources and their influences to the occurrence and propagation of singularities of solutions, and pay more attentions to the topic of blow-up sets. At first, we consider global versus single point blow-up of solutions to two models with coupling localized and local sources respectively. Comparing the two models, we find substantial different influences of the two kinds of couplings to the blow-up sets of solutions. Then, we study in what way the interactions between localized and local sources affect the blow-up rate, blow-up profile, and blow-up set in a nonlinear diffusion problem. Finally, we will show via models how the localized sources substantially influence the critical Fujita exponents. It is interesting to find they may admit an infinite Fujita exponent because of the localized sources. This excludes the situation where the solutions are non-global for large initial data and global with small initial data.
The main results obtained in this thesis can be summarized as follows:
(Ⅰ) Total and single point blow-up
Chapter 2 considers heat equations with local and coupling localized sources u_t =△u + u~m + v~p(0, t), v_t =△v + v~n + u~q(0, t), (x, t)∈Ω×(0, T) subject to null Dirichlet boundary conditions. The behavior of solutions depends on the interactions among the local and localized sources as well as the diffusions with the null boundary conditions in the model. We obtain a complete classification of parameters to distinguish total and single point blow-up for the non-global solutions. In addition, simultaneous versus nonsimultaneous blow-up of solutions under different dominations are determined also with four possible simultaneous blow-up rates. To our knowledge, this is the first study on total versus single point blow-up for the case of coupled systems.
Chapter 3 treats homogeneous Dirichlet problem to heat system with localized and coupling local sources u_t =△u + u~m(0,t) + v~p, v_t =△v + v~n(0,t)+ u~q, (x, t)∈Ω×(0, T) with a parallel discussion as that in Chapter 2, i.e., total versus single point blow-up, simultaneous versus non-simultaneous blow-up etc. In particular, comparing with the results of Chapter 2, we find some substantial differences on occurrence and propagation of singularities of solutions due to the two kinds of couplings. For example, the situation of global and single blow-up for the two components respectively, included in the classification to the model in Chapter 2, does not appear in the classification to the present model.
Chapter 4 studies a localized nonlinear diffusion equation u_t =△u~m+λ_1u~p+λ_2u~q(0, t) subject to null Dirichlet boundary condition with p, q≥0, max{p, q} > m > 1, andλ_1,λ_2 > 0. By investigating the interactions among the localized and local sources, the nonlinear diffusion with the zero boundary value condition, we establish blow-up rates and uniform blow-up profiles of solutions under different dominations. In addition, as for the blow-up sets of solutions, we find that nonlinear diffusion has no contributions to the total and single point blow-up of solutions.
(Ⅱ) Fujita exponents
Chapter 5 deals with Cauchy problem to nonlinear diffusion model u_t =△u~m +λ_1u~(p_1)(x,t) +λ_2u~(p_2)(x~*(t),t) with m≥1, p_i,λ_i≥0 (i = 1,2) and x~*(t) H(?)lder continuous. A new phenomenon is observed that the critical Fujita exponent p_c= +∞wheneverλ_2 > 0. More precisely, the solution blows up under any nontrivial and nonnegative initial data whenever p = max{p_1,p_2} > 1. This result is then extended to a coupled system with localized sources as well as the cases with other nonlinearities (even the decay ones).
Chapter 6 focuses on the asymptotic behavior of solutions for reaction-diffusion equations coupled via localized and local sources: u_t =△u+v~p(x~*(t),t). v_t =△v+u~q Both the initial-boundary problem with null Dirichlet boundary condition and the Cauchy problem are considered to study the interaction between the localized and the local sources. For the initial-boundary problem we prove that the solutions blow up everywhere in the domain with uniform blow-up profiles. In addition, it is interesting to show that the Cauchy problem admits an infinity Fujita exponent, namely, the solutions blow up under any nontrivial and nonnegative initial data whenever pq > 1.
引文
[1]Chadam J M,Peirce A,Yin H M.The blowup property of solutions to some diffusion equations with localized nonlinear reactions.J.Math.Anal.Appl.,1992,169:313-328.
[2]Chen Q,Xiang Z Y,Mu C L.Blow-up and asymptotic behavior of solutions to a semilinear integrodifferential system.Comm.Pure Appl.Anal.,2006,5:435-446.
[3]Deng K,Kwong M K,Levine H A.The influence of nonlocal nonlinearities on the long time behavior of solutions of burgers equation.Quart.Appl.Math.,1992,50:1237-1245.
[4]Du L L.Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources.J.Comput.Appl.Math.,2007,202:237-247.
[5]Jiang L J,Li H L.Uniform blow-up profiles and boundary layer for a parabolic system with nonlocal sources.Math.Comput.Modelling,2007,45:814-824.
[6]Li F C,Huang S X,Xie C H.Global existence and blow-up of solutions to a nonlocal reaction-diffusion system.Discrete Contin.Dyn.Syst.,2003,9:1519-1532.
[7]Li H L,Wang M X.Properties of blow-up solutions to a parabolic system with nonliner localized terms.Discrete Contin.Dyn.Syst.,2005,13:683-700.
[8]Souplet P.Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source.J.Differential Equations,1999,153:374-406.
[9]Souplet P.Blow-up in non-local reaction-diffusion equations.SIAM J.Math.Appl.,1998,29:1301-1334.
[10]Souplet P.Uniform blow-up profile and boundary behavior for a nonlocal reactiondiffusion equation with critical damping.Math.Methods Appl.Sci.,2004,27:1819-1829.
[11]Zhao L Z,Zheng S N.Critical exponents and asymptotic estimates of solutions to parabolic systems with localized nonlinear sources.J.Math.Anal.Appl.,2002,292:621-635.
[12]Zheng S N,Wang J H.Total versus single point blow-up in heat equations with coupled localized sources.Asymptot.Anal.,2007,51:133-156.
[13]Chen Y P.Blow-up for a system of heat equations with nonlocal sources and absorptions.Comput.Math.Appl,2004,48(3-4):361-372.
[14]Chen Y P.Global blow-up for a heat system with localized sources and absorptions.Appl.Math.J.Chinese Univ.B,2007,22(2):213-225.
[15]Li H L,Wang M X.Uniform blow-up profiles and boundary layer for a parabolic system with localized nonlinear reaction terms.Sci.China Ser.A,2005,48:185-197.
[16]Liu Q L,Li YX,Xie C H.The blow-up property of solutions to degenerate parabolic equation with localized nonlinear reactions.Acta Math.Sinica(Chin.Ser.),2003,46(6):1135-1142.
[17]Zheng S N,Su H.A quasilinear reaction-diffusion system coupled via nonlocal sources.Appl.Math.Comput.,2006,180:295-308.
[18]Deng K.Comparison principle for some nonlocal problems.Quart.Appl.Math.,1992,50:517-522.
[19]Friedman A.Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions.Quart.Appl.Math.,1986,44:401-407.
[20]Lin Z G,Liu Y R.Uniform blow-up profiles for diffusion equations with nonlocal source and nonlocal boundary.Acta Math.Sci.,2004,24B:443-450.
[21]Pao C V.Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions.J.Comput.Appl.Math.,1998,88(1):225-238.
[22]Pao C V.Blowing-up of solution for a nonlocal reaction-diffusion problem in combustion theory.J.Math.Anal.Appl,1992,166(2):591-600.
[23]Pao C V.Dynamics of reaction-diffusion equations with nonlocal boundary conditions.J.Math.Anal.Appl.,1995,195:702-718.
[24]Pao C V.Reaction diffusion equations with nonlocal boundary and nonlocal initial conditions.Quart.Appl.Math.,1995,LⅢ(1):173-186.
[25]Liu Q L,Li Y X,Gao H J.Uniform blow-up rate for diffusion equations with nonlocal nonlinear source.Nonlinear Anal.,2007,67:1947-1957.
[26]Okada A,Fukuda I.Total versus single point blow-up of a semilinear parabolic heat equation with localized reaction.J.Math.Anal.Appl.,2003,281:485-500.
[27]Yin H M.On a class of parabolic equations with nonlocal boundary conditions.J.Math.Anal.Appl.,2004,294(2).712-728.
[28]Fukuda I,Suzuki R.Blow-up behavior for a nonlinear heat equation with a localized source in a ball.J.Differential Equations,2005,218:273-291.
[29]Ferreira R,de Pablo A,Vazquez J L.Classification of blow-up with nonlinear diffusion and localized reaction.J.Differential Equations,2006,231:195-211.
[30]Liu Q L,Li Y X,Gao H G.Uniform blow-up rate for diffusion equations with localized nonliear source.J.Math.Anal.Appl,2007,320:771-778.
[31]Gladkov A L,Kim K I.Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition.J.Math.Anal.Appl.,2008,338(1):264-273.
[32]Wang M X,Wang Y M.Properties of positive solutions for nonlocal reaction-diffusion problems.Math.Methods Appl.Sci.,1996,19:1141-1156.
[33]Chen Y J,Gao H J.Asymptotic blow-up behavior for a nonlocal degenerate parabolic equation.J.Math.Anal.Appl.,2007,330(2):852-863.
[34]Zheng S N,Wang L D.Blow-up rate and profile for a degenerate parabolic system coupled via nonlocal sources.Comput.Math.Appl,2006,52(10-11).
[35]Lu H H,Wang M X.Global solutions and blow-up problems for a nonlinear degenerate parabolic system coupled via nonlocal sources.J.Math.Anal.Appl.,2007,333(2).
[36]Wang L W,Chen Q Y.The asymptotic behavior of blow-up solution of localized nonlinear equation.J.Math.Anal.Appl,1996,200:315-321.
[37]Weissler E B.Single point blow-up for a semilinear initial value problem.J.Differential Equations,1984,55:204-224.
[38]Chen Y G.On blow-up solutions of semilinear parabolic equations;analytical and numerical studies,Thesis.Tokyo University,1987.
[39]Fujita H,Chen Y G.On the set of blow-up points and asymptotic behaviors of blow-up solutions to a semilinear parabolic equation.Analyse Mathematique et Applications,1988,Gauthier-Villars,Montrouge:181-201.
[40]Friedman A,Mcleod B.Blow-up of positive solutions of semilenear heat equations.Indiana Univ.Math.J.,1985,34:425-447.
[41]Giga Y,Kohn R V.Asymptotic self-similar blow-up of semilinear heat equations.Comm.Pure Appl.Math.,1985,38:297-319.
[42]Samarskii A A,Galaktionov V A,Kurdyumov S P,Mikhailov A P.Blow-up in problems for quarsilinear parabolic equations.Berlin:Walter de Gruyter,1995.
[43]Galaktionov V A.On asymptotic self-similar behaviour for a quasilinear heat equation:single point blow-up.Math.Methods Appl.Sci.,1995,26:675-693.
[44]Gui C F.Symmetry of the blow-up set of a porous medium type equation.Comm.Pure Appl.Math.,1995,XLⅧ:471-500.
[45]Du L L,Xiang Z Y.A further blow-up analysis for a localized porous medium equation.Appl.Math.Comput.,2006,179:200-208.
[46]Friedman A,Giga Y.A single point blow-up for solutions of semilenear parabolic systems.J.Fac.Sci.Univ.Tokyo,Sect.I.,1987,34:65-79.
[47]Pedersen M,Lin Z G.Coupled diffusion systems with localized nonlinear reactions.Comput.Math.Appl.,2001,42:807-816.
[48]Du L L.Blow-up for a degenerate reaction-diffusion system with nonlinear localized sources.J.Math.Anal.Appl.,2006,324:304-320.
[49]Wang M X,Wei Y F.Blow-up properties for a degenerate parabolic system with nonliner localized sources.J.Math.Anal.Appl,2008,343:621-635.
[50]Fujita H.On the blowing up of solutions of the Cauchy problem for u_t= △u + u~(1+α).J.Fac.Sci.Univ.Tokyo Sect.I.,1966,16:105-113.
[51]Aronson D,Weinberger H F.Multidimensional nonlinear diffusion arising in population genetics.Adv.Math.,1978,30:33-76.
[52]Hayakawa K.On the nonexistence of global solutions of some semilinear parabolic equations.Proc.Japan Acad.,1973,49:503-525.
[53]Kobayashi K,Sirao T,Tanaka H.On the blowing up problem for semilinear heat equations.J.Math.Soc.Japan,1977,29:407-424.
[54]Weissler E B.Existence and non-existence of global solutions for semilinear equation.Israel J.Math.,1981,6:29-40.
[55]Deng K,Levine H A.The role of critical exponents in blow-up theorems:The sequel.J.Math.Anal.Appl.,2000,243:85-126.
[56]Levine H A.The role of critical exponents in blow-up theorems.SIAM Rev.,1990,32:262-288.
[57]Galaktionov V A,Levine H A.A general approach to critical Fujita exponents in nonlinear parabolic problems.Nonlinear Anal.,1998,34:1005-1027.
[58]Escobedo M,Herrero M A.Boundedness and blow up for a semilinear reactiondiffusion system.J.Differential Equations,1991,89:176-202.
[59]Aris R.The mathematical theory of diffusion and reaction in permeable catalysts Ⅰ and Ⅱ.Clarendon press,London,1975.
[60]Bebernes J W,Eberly D.Mathematical problems from combustion theory.Applied Mathematics Series,Springer-Verlag,New York,1989.
[61]Bimpong-Bota K,Ortoleva P,Rossi J D.Far-from-equilibrium phenomena at loca sites of reaction.J.Chem.Phys.,1974,60:3124-3133.
[62]Ortoleva P,Ross J.Local structures in chemical reactions with heterogeneous catalysis.J.Chem.Phys.,1972,56:4397-4400.
[63]Aronson D,Crandall M G,Peletier L A.Stabilization of solutions of a degenerate nonlinear diffusion problem.Nonlinear Anal.,1982,6:1001-1022.
[64] Brunell J G, Lacey A A, Wake G C. Steady states of reaction-diffusion equations. partⅠ: Questions of existence and continuity of solution branches. J. Aust. Math. Soc. Ser., 1983, B 24:374-391.
[65] Pao C V. On nonlinear reaction-diffusion systems. J. Math. Anal. Appl., 1982, 87:165-198.
[66] Zheng S N. Global existence and global non-existence of solutions to a reaction-diffusion system. Nonlinear Anal., 2000, 39:327-340.
[67] Zheng S N, Liu B C, Li F J. Simultaneous and non-simultaneous blow-up for a cross-coupled parabolic system. J. Math. Anal. Appl., 2007, 326:414-431.
[68] Zheng S N, Wang W. Critical exponents for a nonlinear diffusion system. Nonlinear Anal., 2007, 67:1190-1210.
[69] Rossi J D, Souplet P. Coexistence of simultaneous and non-simultaneous blow-up in a semilinear parabolic system. Differential Integral Equations, 2005, 18:405-418.
[70] Li F J, Liu B C, Zheng S N. Simultaneous and non-simultaneous blow-up for heat equations with coupled nonlinear boundary fluxes. Z. Angew. Math. Phys., 2007, 58:717-735.
[71] Souplet P, Tayachi S. Optimal condition for non-simultaneous blow-up in a reaction-diffusion system. J. Math. Soc. Japan, 2004, 56:571-584.
[72] Brandle C, Quiros F, Rossi J D. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Commun. Pure Appl. Anal., 2005, 4:523-536.
[73] Friedman A. Partial Differential Equations of Parabolic Type, Prentice-Hall, Engle-wood Cliffs, New Jersey,. Reprinted by Krieger, 1964.
[74] Bebernes J, Bressan A, Lacey A. Total blow-up versus single point blow-up. J. Differential Equations, 1988, 73:30-44.
[75] Chen Y, Xie C H. Blow-up for a porous medium equation with a localized source. Appl. Math. Comput., 2004, 159:79-93.
[76] Li F C, Xie C H. Global existence and blow-up for a nonlinear porous medium equation. Appl. Math. Lett., 2003, 2:185-192.
[77]Giga Y,Kohn R V.Characterizing blowup using similarity variables.Indiana Univ.Math.J.,1987,36:1-40.
[78]Hu B,Yin H M.The profile near blow-up time for solutions of the heat equation with a nonlinear boundary condition.Trans.Amer.Math.Soc,1994,346:117-135.
[79]Ferreira R,de Pablo A,Rossi J D.Blow-up with logarithmic nonlinearities.J.Differential Equations,2007,240:196-215.
[80]Fife P.Mathematical aspects of reacting and diffusing systems.Lecture Notes in Biomath.28,Springer-Verlag,Berlin,New York,1975.
[81]Gavalas G R.Nonlinear differential equations of chemically reacting systems.Springer Tracts in Natural Philosophy,Springer-Verlag,New York,1968.
[82]Straughan B.Instability,nonexistence and weighted energy methods in fluid dynamics and related theories.Res.Notes in Math.,1982,74:Pitman,Boston.
[83]Newell A C,Rand D A,Russell D.Turbulent transport and the radom occurrence of coherent events.Phys.D,32:281-303.
[84]Kong L H,Wang J H,Zheng S N.Asymptotic analysis to a parabolic equation with a weighted localized source.Appl.Math.Comput.,2008,197(2):819-827.
[2]Chen Q,Xiang Z Y,Mu C L.Blow-up and asymptotic behavior of solutions to a semilinear integrodifferential system.Comm.Pure Appl.Anal.,2006,5:435-446.
[3]Deng K,Kwong M K,Levine H A.The influence of nonlocal nonlinearities on the long time behavior of solutions of burgers equation.Quart.Appl.Math.,1992,50:1237-1245.
[4]Du L L.Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources.J.Comput.Appl.Math.,2007,202:237-247.
[5]Jiang L J,Li H L.Uniform blow-up profiles and boundary layer for a parabolic system with nonlocal sources.Math.Comput.Modelling,2007,45:814-824.
[6]Li F C,Huang S X,Xie C H.Global existence and blow-up of solutions to a nonlocal reaction-diffusion system.Discrete Contin.Dyn.Syst.,2003,9:1519-1532.
[7]Li H L,Wang M X.Properties of blow-up solutions to a parabolic system with nonliner localized terms.Discrete Contin.Dyn.Syst.,2005,13:683-700.
[8]Souplet P.Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source.J.Differential Equations,1999,153:374-406.
[9]Souplet P.Blow-up in non-local reaction-diffusion equations.SIAM J.Math.Appl.,1998,29:1301-1334.
[10]Souplet P.Uniform blow-up profile and boundary behavior for a nonlocal reactiondiffusion equation with critical damping.Math.Methods Appl.Sci.,2004,27:1819-1829.
[11]Zhao L Z,Zheng S N.Critical exponents and asymptotic estimates of solutions to parabolic systems with localized nonlinear sources.J.Math.Anal.Appl.,2002,292:621-635.
[12]Zheng S N,Wang J H.Total versus single point blow-up in heat equations with coupled localized sources.Asymptot.Anal.,2007,51:133-156.
[13]Chen Y P.Blow-up for a system of heat equations with nonlocal sources and absorptions.Comput.Math.Appl,2004,48(3-4):361-372.
[14]Chen Y P.Global blow-up for a heat system with localized sources and absorptions.Appl.Math.J.Chinese Univ.B,2007,22(2):213-225.
[15]Li H L,Wang M X.Uniform blow-up profiles and boundary layer for a parabolic system with localized nonlinear reaction terms.Sci.China Ser.A,2005,48:185-197.
[16]Liu Q L,Li YX,Xie C H.The blow-up property of solutions to degenerate parabolic equation with localized nonlinear reactions.Acta Math.Sinica(Chin.Ser.),2003,46(6):1135-1142.
[17]Zheng S N,Su H.A quasilinear reaction-diffusion system coupled via nonlocal sources.Appl.Math.Comput.,2006,180:295-308.
[18]Deng K.Comparison principle for some nonlocal problems.Quart.Appl.Math.,1992,50:517-522.
[19]Friedman A.Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions.Quart.Appl.Math.,1986,44:401-407.
[20]Lin Z G,Liu Y R.Uniform blow-up profiles for diffusion equations with nonlocal source and nonlocal boundary.Acta Math.Sci.,2004,24B:443-450.
[21]Pao C V.Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions.J.Comput.Appl.Math.,1998,88(1):225-238.
[22]Pao C V.Blowing-up of solution for a nonlocal reaction-diffusion problem in combustion theory.J.Math.Anal.Appl,1992,166(2):591-600.
[23]Pao C V.Dynamics of reaction-diffusion equations with nonlocal boundary conditions.J.Math.Anal.Appl.,1995,195:702-718.
[24]Pao C V.Reaction diffusion equations with nonlocal boundary and nonlocal initial conditions.Quart.Appl.Math.,1995,LⅢ(1):173-186.
[25]Liu Q L,Li Y X,Gao H J.Uniform blow-up rate for diffusion equations with nonlocal nonlinear source.Nonlinear Anal.,2007,67:1947-1957.
[26]Okada A,Fukuda I.Total versus single point blow-up of a semilinear parabolic heat equation with localized reaction.J.Math.Anal.Appl.,2003,281:485-500.
[27]Yin H M.On a class of parabolic equations with nonlocal boundary conditions.J.Math.Anal.Appl.,2004,294(2).712-728.
[28]Fukuda I,Suzuki R.Blow-up behavior for a nonlinear heat equation with a localized source in a ball.J.Differential Equations,2005,218:273-291.
[29]Ferreira R,de Pablo A,Vazquez J L.Classification of blow-up with nonlinear diffusion and localized reaction.J.Differential Equations,2006,231:195-211.
[30]Liu Q L,Li Y X,Gao H G.Uniform blow-up rate for diffusion equations with localized nonliear source.J.Math.Anal.Appl,2007,320:771-778.
[31]Gladkov A L,Kim K I.Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition.J.Math.Anal.Appl.,2008,338(1):264-273.
[32]Wang M X,Wang Y M.Properties of positive solutions for nonlocal reaction-diffusion problems.Math.Methods Appl.Sci.,1996,19:1141-1156.
[33]Chen Y J,Gao H J.Asymptotic blow-up behavior for a nonlocal degenerate parabolic equation.J.Math.Anal.Appl.,2007,330(2):852-863.
[34]Zheng S N,Wang L D.Blow-up rate and profile for a degenerate parabolic system coupled via nonlocal sources.Comput.Math.Appl,2006,52(10-11).
[35]Lu H H,Wang M X.Global solutions and blow-up problems for a nonlinear degenerate parabolic system coupled via nonlocal sources.J.Math.Anal.Appl.,2007,333(2).
[36]Wang L W,Chen Q Y.The asymptotic behavior of blow-up solution of localized nonlinear equation.J.Math.Anal.Appl,1996,200:315-321.
[37]Weissler E B.Single point blow-up for a semilinear initial value problem.J.Differential Equations,1984,55:204-224.
[38]Chen Y G.On blow-up solutions of semilinear parabolic equations;analytical and numerical studies,Thesis.Tokyo University,1987.
[39]Fujita H,Chen Y G.On the set of blow-up points and asymptotic behaviors of blow-up solutions to a semilinear parabolic equation.Analyse Mathematique et Applications,1988,Gauthier-Villars,Montrouge:181-201.
[40]Friedman A,Mcleod B.Blow-up of positive solutions of semilenear heat equations.Indiana Univ.Math.J.,1985,34:425-447.
[41]Giga Y,Kohn R V.Asymptotic self-similar blow-up of semilinear heat equations.Comm.Pure Appl.Math.,1985,38:297-319.
[42]Samarskii A A,Galaktionov V A,Kurdyumov S P,Mikhailov A P.Blow-up in problems for quarsilinear parabolic equations.Berlin:Walter de Gruyter,1995.
[43]Galaktionov V A.On asymptotic self-similar behaviour for a quasilinear heat equation:single point blow-up.Math.Methods Appl.Sci.,1995,26:675-693.
[44]Gui C F.Symmetry of the blow-up set of a porous medium type equation.Comm.Pure Appl.Math.,1995,XLⅧ:471-500.
[45]Du L L,Xiang Z Y.A further blow-up analysis for a localized porous medium equation.Appl.Math.Comput.,2006,179:200-208.
[46]Friedman A,Giga Y.A single point blow-up for solutions of semilenear parabolic systems.J.Fac.Sci.Univ.Tokyo,Sect.I.,1987,34:65-79.
[47]Pedersen M,Lin Z G.Coupled diffusion systems with localized nonlinear reactions.Comput.Math.Appl.,2001,42:807-816.
[48]Du L L.Blow-up for a degenerate reaction-diffusion system with nonlinear localized sources.J.Math.Anal.Appl.,2006,324:304-320.
[49]Wang M X,Wei Y F.Blow-up properties for a degenerate parabolic system with nonliner localized sources.J.Math.Anal.Appl,2008,343:621-635.
[50]Fujita H.On the blowing up of solutions of the Cauchy problem for u_t= △u + u~(1+α).J.Fac.Sci.Univ.Tokyo Sect.I.,1966,16:105-113.
[51]Aronson D,Weinberger H F.Multidimensional nonlinear diffusion arising in population genetics.Adv.Math.,1978,30:33-76.
[52]Hayakawa K.On the nonexistence of global solutions of some semilinear parabolic equations.Proc.Japan Acad.,1973,49:503-525.
[53]Kobayashi K,Sirao T,Tanaka H.On the blowing up problem for semilinear heat equations.J.Math.Soc.Japan,1977,29:407-424.
[54]Weissler E B.Existence and non-existence of global solutions for semilinear equation.Israel J.Math.,1981,6:29-40.
[55]Deng K,Levine H A.The role of critical exponents in blow-up theorems:The sequel.J.Math.Anal.Appl.,2000,243:85-126.
[56]Levine H A.The role of critical exponents in blow-up theorems.SIAM Rev.,1990,32:262-288.
[57]Galaktionov V A,Levine H A.A general approach to critical Fujita exponents in nonlinear parabolic problems.Nonlinear Anal.,1998,34:1005-1027.
[58]Escobedo M,Herrero M A.Boundedness and blow up for a semilinear reactiondiffusion system.J.Differential Equations,1991,89:176-202.
[59]Aris R.The mathematical theory of diffusion and reaction in permeable catalysts Ⅰ and Ⅱ.Clarendon press,London,1975.
[60]Bebernes J W,Eberly D.Mathematical problems from combustion theory.Applied Mathematics Series,Springer-Verlag,New York,1989.
[61]Bimpong-Bota K,Ortoleva P,Rossi J D.Far-from-equilibrium phenomena at loca sites of reaction.J.Chem.Phys.,1974,60:3124-3133.
[62]Ortoleva P,Ross J.Local structures in chemical reactions with heterogeneous catalysis.J.Chem.Phys.,1972,56:4397-4400.
[63]Aronson D,Crandall M G,Peletier L A.Stabilization of solutions of a degenerate nonlinear diffusion problem.Nonlinear Anal.,1982,6:1001-1022.
[64] Brunell J G, Lacey A A, Wake G C. Steady states of reaction-diffusion equations. partⅠ: Questions of existence and continuity of solution branches. J. Aust. Math. Soc. Ser., 1983, B 24:374-391.
[65] Pao C V. On nonlinear reaction-diffusion systems. J. Math. Anal. Appl., 1982, 87:165-198.
[66] Zheng S N. Global existence and global non-existence of solutions to a reaction-diffusion system. Nonlinear Anal., 2000, 39:327-340.
[67] Zheng S N, Liu B C, Li F J. Simultaneous and non-simultaneous blow-up for a cross-coupled parabolic system. J. Math. Anal. Appl., 2007, 326:414-431.
[68] Zheng S N, Wang W. Critical exponents for a nonlinear diffusion system. Nonlinear Anal., 2007, 67:1190-1210.
[69] Rossi J D, Souplet P. Coexistence of simultaneous and non-simultaneous blow-up in a semilinear parabolic system. Differential Integral Equations, 2005, 18:405-418.
[70] Li F J, Liu B C, Zheng S N. Simultaneous and non-simultaneous blow-up for heat equations with coupled nonlinear boundary fluxes. Z. Angew. Math. Phys., 2007, 58:717-735.
[71] Souplet P, Tayachi S. Optimal condition for non-simultaneous blow-up in a reaction-diffusion system. J. Math. Soc. Japan, 2004, 56:571-584.
[72] Brandle C, Quiros F, Rossi J D. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Commun. Pure Appl. Anal., 2005, 4:523-536.
[73] Friedman A. Partial Differential Equations of Parabolic Type, Prentice-Hall, Engle-wood Cliffs, New Jersey,. Reprinted by Krieger, 1964.
[74] Bebernes J, Bressan A, Lacey A. Total blow-up versus single point blow-up. J. Differential Equations, 1988, 73:30-44.
[75] Chen Y, Xie C H. Blow-up for a porous medium equation with a localized source. Appl. Math. Comput., 2004, 159:79-93.
[76] Li F C, Xie C H. Global existence and blow-up for a nonlinear porous medium equation. Appl. Math. Lett., 2003, 2:185-192.
[77]Giga Y,Kohn R V.Characterizing blowup using similarity variables.Indiana Univ.Math.J.,1987,36:1-40.
[78]Hu B,Yin H M.The profile near blow-up time for solutions of the heat equation with a nonlinear boundary condition.Trans.Amer.Math.Soc,1994,346:117-135.
[79]Ferreira R,de Pablo A,Rossi J D.Blow-up with logarithmic nonlinearities.J.Differential Equations,2007,240:196-215.
[80]Fife P.Mathematical aspects of reacting and diffusing systems.Lecture Notes in Biomath.28,Springer-Verlag,Berlin,New York,1975.
[81]Gavalas G R.Nonlinear differential equations of chemically reacting systems.Springer Tracts in Natural Philosophy,Springer-Verlag,New York,1968.
[82]Straughan B.Instability,nonexistence and weighted energy methods in fluid dynamics and related theories.Res.Notes in Math.,1982,74:Pitman,Boston.
[83]Newell A C,Rand D A,Russell D.Turbulent transport and the radom occurrence of coherent events.Phys.D,32:281-303.
[84]Kong L H,Wang J H,Zheng S N.Asymptotic analysis to a parabolic equation with a weighted localized source.Appl.Math.Comput.,2008,197(2):819-827.