【作者】 王金环；

【导师】 郑斯宁；

【作者基本信息】 大连理工大学 ， 基础数学， 2009， 博士

【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 + vⁿ + 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 + vⁿ（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+λ₁u^p+λ₂u^q（0, t） ,其中p,q≥0,max{p,q}>m>1,且λ₁,λ₂>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 +λ₁u^p₁（x,t） +λ₂u^p₂（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λ₂>0时该模型的临界Fujita指标p_c=+∞.也就是说,只要p=max{p₁,p₂}>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.更多还原

【Abstract】 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-upChapter 2 considers heat equations with local and coupling localized sources u_t =△u + u^m + v^p（0, t）, v_t =△v + vⁿ + 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 + vⁿ（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+λ₁u^p+λ₂u^q（0, t） subject to null Dirichlet boundary condition with p, q≥0, max{p, q} > m > 1, andλ₁,λ₂ > 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 exponentsChapter 5 deals with Cauchy problem to nonlinear diffusion model u_t =△u^m +λ₁u^p₁（x,t） +λ₂u^p₂（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= +∞wheneverλ₂ > 0. More precisely, the solution blows up under any nontrivial and nonnegative initial data whenever p = max{p₁,p₂} > 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.更多还原