【作者】 乔岚；

【导师】 郑斯宁；

【作者基本信息】 大连理工大学 ， 计算数学， 2008， 博士

【摘要】 本文研究几类具有内部吸收和耦合边界流的多重非线性抛物方程组奇性解的渐近行为，包括由不同非线性机制占优而导致的多重blow-up速率，非同时与同时blow-up等．首先讨论了一个具有混合型内部吸收与耦合边界流的多重非线性模型．通过引入特征代数方程组得到其blow-up临界指标的清晰刻画．特别引入包含两个新参数的另一类特征代数方程组用以刻画多重blow-up速率．非常有趣的是，这里得到了两个与吸收有关的blow-up速率，而现有文献中的所有blow-up速率结果都是与吸收无关的．为进一步分析这个现象到底来自混合型非线性还是耦合机制，又对比考虑了单一非线性耦合情形，同样得到与吸收有关的blow-up速率，从而确认耦合机制在这里所起的关键作用．此外，本文还分别讨论了（对称的）负-负源和（非对称的）正-负源的边界流耦合抛物方程组的非同时与同时blow-up问题，由此确定源的符号对引发非同时blow-up的影响．本文得到的主要结果概述如下：（Ⅰ）关于多重blow-up速率在第二章中，对于具有混合型内部吸收的方程u_t=△u-a₁u^m，v_t=△v-a₂e^nv，（x，t）∈Ω×（0，T），附加边界条件（?）=e^pv，（?）=u^q，（x，t）∈（?）Ω×（0，T）的问题，先根据比较原理建立blow-up临界指标，再利用Green表示公式和多重形式下的Scaling方法得到了上述问题在N=1（N表示维数）时的blow-up速率．需要提及的是，在现有文献中，吸收项会影响临界指标、blow-up时间以及blow-up解所需初值等等，但不影响blow-up速率，而这里得到的多重blow-up速率中有两种是与吸收有关的．第三章研究了具有幂型内部吸收的方程u_t=u_xx-a₁u^m，v_t=v_xx-a₂vⁿ，（x，t）∈（0，1）×（0，T）经由边界条件u_x（1，t）=v^p（1，t），v_x（1，t）=v^q（1，t），u_x（0，t）=0，v_x（0，t）=0，t∈（0，T）耦合的初边值问题，得到基于非线性指标分类完全的多重blow-up速率，其中也有两种速率是与吸收有关的．这说明与吸收有关的blow-up速率是由耦合机制造成的．若p=q，m=n，初值u₀（x）=v₀（x），则方程组化为单个方程，而此时恰属于blow-up速率与吸收项无关的指标区域，进一步说明存在与吸收项有关的blow-up速率是耦合方程组区别于单个方程问题所特有的现象．（Ⅱ）关于非同时与同时blow-up第四章考虑了方程u_t=u_xx-λ₁u^α₁，v_t=v_xx-λ₂^β₁，（x，t）∈（0，1）×（0，T），附加边界条件u_x（1，t）=v^α₂v^p，v_x（1，t）=u^qv^β₂，u_x（0，t）=v_x（0，t）=0，t∈（0，T）的解的非同时blow-up．首先借助对辅助问题的研究并引入截断函数得到一个基本引理，继而，结合Green表示公式和Scaling方法最终得到了存在初值发生非同时blow-up的充分必要条件，以及所有blow-up必为非同时blow-up的充分条件．第五章研究了u_t=u_xx+u^α₁，v_t=v_xx-v^β₁，（x，t）∈（0，1）×（0，T）经由边界条件u_x（1，t）=u^α₂v^p，v_x（1，t）=u^qv^β₂，u_x（0，t）=（0，t）=0，t∈（0，T）耦合的方程组的解的非同时blow-up，得到了与第四章相对应的结果．两个分量u和v的非对称性导致了讨论过程和所得结果都较前一模型更为复杂．与第四章结果的对比可见源的符号对引发非同时blow-up的作用．更多还原

【Abstract】 This thesis deals with asymptotic behavior of singular solutions for multi-nonlinear parabolic systems with inner absorptions and coupled boundary fluxes, such as multiple blow-up rates under different dominations of nonlinearities, non-simultaneous versus simultaneous blow-up, and so on. Firstly, a multi-nonlinear model with inner absorptions and coupled boundary fluxes of mixed type nonlinearities is discussed. The critical exponent is obtained, and clearly described via a so called characteristic algebraic system. In particular, another characteristic algebraic system with two new parameters is introduced to simply show the multiple blow-up rates. It is interesting to observe that two of the multiple blow-up rates obtained here do depend on the absorption exponents, unlike all the known results that the blow-up rates in the current literatures, to our knowledge, are all absorption-independent. Furthermore, in order to explore the reason for the new phenomenon （i.e., it is due to the mixed type nonlinearities or the coupling mechanism）, the same problem is considered for the case of single type nonlinearities, and the blowup rates of the same propeties are obtained also. This is to confirm that the coupling mechanism plays the key role for the absorption-relevant blow-up rates. In addition, the thesis discusses non-simultaneous blow-up of solutions for coupled parabolic systems with negative-negative sources and positive-negative sources, respectively. Thereby, the influence of the sign of sources to non-simultaneous blow-up is determined.The main results obtained in this thesis can be summarized as follows:（Ⅰ） Multiple blow-up rateIn Chapter 2, for the system with mixed type nonlinearities u_t =Δu - a₁u^m, v_t =Δv-a₂e^nv in （0,1）×（0,T）, （?） =e^pv, （?） = u^q on d （?）Ω×（0, T）, the critical blow-up exponent is established by using the comparision principle. Furthermore, multiple simultaneous blow-up rates of solutions with N = 1 （N is the space dimension） are established by Green’s identity and the Scaling method . It should be mentioned that in previous literatures, the absorptions affect the blow-up criteria, the blow-up time, as well as the initial data required for the blow-up of solutions, all without changing the blow-up rates, while here some absorption-erlevant simultaneous blow-up rates are obtained. In Chapter 3, the system u_t = u_xx - a₁u^m, v_t = v_xx - a₂vⁿ in （0,1）×（0, T） with coupled boundary fluxes u_x（1,t） = v^p, v_x（1,t） = u^q, u_x（0,t） = v_x（0,t） = 0 is considered. The multiple simultaneous blow-up rates obtained with a complete classification for all the nonlinear parameters of the model, where two absorption-relevant ones are observed also. This is to say that the absorption-relevant blow-up rates should be caused by the coupling mechanism. If p = q, m = n with u₀（x） = v₀（x）, the system reduces to a scalar problem, which belongs to the class of absorption-independent blow-up rate. This shows a substantial difference between the coupled systems and the scalar equations with inner absorptions.（Ⅱ） Non-simultaneous versus simultaneous blow-upChapter 4 deals with the initial-boundary problem for u_t = u_xx -λ₁u^α₁, v_t = v_xx -λ₂v^β₁ in （0,1）×（0, T） coupled via u_x（1, t） = u^α₂v^p, v_x（1, t） = u^qv^β₂, u_x（0, t） = v_x（0, t） = 0, t∈（0, T）. Firstly, by introducing an auxilary problem and a cut-off function, a basic lemma is proved. Then, combining with Green’s identity and the Scaling method, the necessary-sufficient conditions for non-simultaneous blow-up of solutions under suitable initial data as well as the sufficient conditions under which any blow-up of solutions would be non-simultaneous are established.Chapter 5 considers non-simultaneous blow-up of solutions for the system with positive-negative sources u_t = u_xx + u^α₁ v_t = v_xx - v^β₁ in （0,1）×（0,T）, coupled via boundary conditions u_x（1,t） = u^α₂v^p, v_x（1,t） = u^qv^β₂, u_x（0,t） = vx_0,t = 0, t∈（0,T）. The non-symmetry of components u and v leads to a more complicated discussion. Comparing with the corresponding conclusions for the two models in Chapters 4 and 5, the contributions of the signs of sources to the non-simultaneous blow-up of solutions are shown here clearly.更多还原