【作者】 王巍；

【导师】 郑斯宁；

【作者基本信息】 大连理工大学 ， 基础数学， 2008， 博士

【摘要】 本文主要研究几类多重非线性抛物方程（组）奇性解的渐近行为．所讨论的问题包括确定非线性扩散方程组的blow-up临界指标、考查梯度项对非线性抛物方程解的blow-up性质的影响，以及研究带有奇异非线性反应项的抛物问题解的quenching现象等．首先讨论一个具有内部吸收项及耦合边界流的非线性扩散方程组．通过对模型中非线性机制之间相互作用的精确分析，我们确定了其blow-up临界指标．其次考虑含梯度项的非线性抛物模型，探究其中对流项是否并且以何种程度影响解的blow-up行为．最后我们研究有限时刻quenching问题．对于具有耦合吸收项并附加正Dirichlet边界条件的非线性抛物方程组，我们考虑了解的同时与非同时quenching；而对于具有加权非线性吸收的热方程，我们描述了解的quenching时间和quenching集的渐近行为．第一章概述本文所研究问题的实际背景和国内外的发展现状，并简要介绍本文的主要工作．第二章考虑内吸收非线性扩散方程（u^m）_t=△u-a₁u^α₁，（vⁿ）_t=△v-a₂v^β₁经由边界流（?）耦合的初边值问题．通过引入特征代数方程组以及对所有八个非线性指标的完全分类，我们得到对该问题blow-up临界指标的简明而清晰的刻画．由于所考虑模型的一般性，这一工作包含了关于blow-up临界指标的许多已有结果．与单个方程结果的比较可见耦合机制对blow-up临界指标的本质性影响．第三章致力于含梯度项的非线性抛物方程解的blow-up分析，其目的在于研究梯度项对解的渐近行为的影响．对于具有内部吸收项及正性梯度项的半线性抛物方程u_t=△u+|▽u|^r-ae^pu,附加边界条件（?）的初边值问题，我们证明当且仅当r≥2时，梯度项对blow-up的形成起本质作用．进一步，当r>2时，对流项还将显著影响blow-up速率，并且使得blow-up速率具有关于模型参数的不连续性（discontinuous）．然而，梯度项对空间blow-up profile并无本质性影响．对非线性扩散方程w_t=(e^（m-1）w)_xx-λe^（p-1）w附加Neumann边界条件w_x（0，t）=0，w_x（1，t）=e^{（q-m）w（1,t）}的初边值问题，利用scaling方法，我们建立了解的blow-up速率估计．通过适当变换，该方程等价于一含对流项的多孔介质类型方程．虽然此模型中的对流项不改变解的blow-up速率，但它给问题的讨论带来一定的困难．第四章研究有限时刻quenching问题，包括非线性抛物方程组u_t=△u-v^-p，v_t=△u-u^-q，（x，t）∈Ω×（0，T）具有正Dirichlet边界条件的初边值问题，以及带有加权非线性吸收的热方程u_t=u_xx-Mf（x）u^-p,（x，t）∈（-1，1）×（0，T）附加边界条件u（-1，t）=u（1，t）=1和初值φ（x）的模型．对于前一问题，我们建立了Ω=B_R径向解的非同时quenching准则：当p，q≥1时quenching必为同时，当p<1≤q或q<1≤p时必为非同时；若p，q<1且R>（?），则quenching依赖于初值既可能同时也可能非同时．需要提及的是，在对共存性结果的证明中，我们工作的新意在于恰当地定义一个初值集合以使quenching解具有某些一致下界估计．对于后一问题，我们利用局部能量估计方法，得到当M→+∞时解的quenching时间与quenching集的渐近性态．我们有，当M-→+∞时，quenching时间T～（m/（p+1））·1/M，其中m=（?）；并且当M充分大时，quenching点将集中于f／φ^p+1的最大值点．更多还原

【Abstract】 This thesis deals with asymptotic behavior of singular solutions for multi-nonlinear parabolic equations （systems）. The topics include the critical exponent for a nonlinear diffusion system, the influences of the gradient perturbations on the blow-up properties of solutions for nonlinear parabolic equations, and the quenching behavior of solutions for parabolic problems with singular absorptions. Firstly, we consider a nonlinear diffusion system with inner absorptions and coupled nonlinear boundary fluxes. A precise analysis on interactions among the multi-nonlinearities in the system is given to determine the critical exponent. Secondly, we concern nonlinear parabolic models with convection so as to explore whether and in what extent the gradient terms influence blow-up behavior of solutions. Finally, in the studying of quenching phenomena, we determine simultaneous versus non-simultaneous quenching for a nonlinear parabolic system with coupled absorptions subject to positive Dirichlet boundary conditions, and characterize the asymptotic behavior of quenching time and set of solutions for heat equations with weighted nonlinear absorptions.Chapter 1 is to summarize the background of the related issues and to briefly introduce the main results of the present thesis.Chapter 2 deals with the initial-boundary problem for （u^m）_t =△u -α₁u^α₁, （vⁿ）_t =△v -α₂v^β₁ coupled via boundary flux （?）. We introduce a so called characteristic algebraic system together with a complete classification for all the eight nonlinear parameters to obtain a simple and clear description to the critical exponent of the problem. Due to the generality of the model considered, this covers many known results on critical blow-up exponents. Comparing with those for scalar cases, the substantial effects of the coupling mechanism on critical exponents can be observed.Chapter 3 is devoted to the blow-up analysis for nonlinear parabolic equations with convection. The aim is to investigate the influences of gradient perturbations on the asymptotic behavior of solutions. For the semilinear parabolic equation u_t =△u + |▽u|^r - ae^pu subject to nonlinear boundary flux （?） = e^qu, we obtain that the gradient term makes a substantial contribution to the formation of blow-up if and only if r≥2. In addition, the gradient term would significantly affect the blow-up rate as well whenever r > 2. Moreover, the blow-up rate may be discontinuous with respect to parameters included in the problem due to convection. However, the gradient perturbations have no essential effects on the spatial blow-up profile. For the nonlinear diffusion equation w_t = (e^（m-1）w)_xx -λe^（p-1）w with Neumann boundary conditions w_x（0,t） = 0, w_x（1,t） = e^{（q-m）w（1,t）} , using the scaling method, we establish the blow-up rate estimates for blow-up solutions. Under a transformation, this equation is equivalent to a porous medium type one with convection. We find that the gradient term just leads to a more complicated discussion without changing the blow-up rate of solutions.Chapter 4 studies two quenching problems, namely, coupled nonlinear parabolic system u_t =△u-v^-p, v_t =△v-u^-qinΩ×（0, T） with positive Dirichlet boundary conditions, and scalar heat equations with weighted nonlinear absorptions u_t = u_xx - Mf（x）u^-p subject to boundary conditions u（-1,t） = u（1,t） = 1 and initial dataφ（x）. For the former problem, we characterize the non-simultaneous quenching criteria for radial quenching solutions withΩ= B_r: The quenching is simultaneous if p,q≥1, and non-simultaneous if p < 1≤q or q < 1≤p; If p,q < 1 with R > （?）, then both simultaneous and non-simultaneous quenching may happen, depending on the initial data. It should be mentioned that to get the coexistence result, we have to skillfully construct a set of initial data admitting required uniform lower estimates on quenching solutions. For the latter model, the asymptotic behavior of quenching time and set of solutions as M→+∞is established by local energy estimates. It is obtained that the quenching time T - （m/（p+1））. M^-1 with （?） as M→+∞. It is shown also how the quenching set concentrates near the maximum points of f/φ^p+1 for large M.更多还原