Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications
Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications
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DOI：
10.1201/9780203998069
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发表时间：
2004-05
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影响因子：
--
通讯作者：
V. Galaktionov
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文献类型：
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作者：
V. Galaktionov
导论：Sturm定理与非线性奇异抛物方程Sturm定理与线性抛物方程的交点比较。第一个Sturm定理：符号变化数的不增加。第二个Sturm定理：多重零的演化形成和坍缩。非线性抛物方程解的交点比较的第一个方面。几何有序流：横向性和凹性技术。演化b方程保持Sturmian性质：横向性、凹性和符号不变性。线性不变子空间上的解简介：过滤方程和凹凸性通过与行波的交点比较的横截性和凹凸性的证明过滤方程的最终凹性具有低阶项的过滤方程的凹性具有p-拉普拉斯算子的奇异方程的凹性保持b -凹性和b -凸性的概念。多孔介质方程的各种b -凹凸性和热方程的符号不变量b -凹凸性和带源b -凹凸性的多孔介质方程指数非线性方程的符号不变量b -凹凸性和横向性径向n维几何中奇异抛物扩散方程的一般b -凹凸性通过线性不变子空间的解拟线性热方程的基本方程和凹性估计通过行波解的局部凹性分析具有吸收的p- laplace方程的凹性b -相对于行波的凹性b -过滤方程的凹性b -相对于不完全泛函子集的凹性最终的b -凹性最终的b -凸性：拟线性热方程的完全爆破和消光的判据爆破问题奇异爆破行波的存在与不存在。爆破条件的讨论。病态方程的完全和不完全爆破证明拟线性热方程奇异行波完全和不完全爆破问题不完全爆破的第一性质显式固有爆破行波和爆破传播的第一估计不变子空间上的显式爆破解爆破界面的低速估计爆破界面的动力学方程爆破界面不是C2函数固有爆破解的大时间行为带源的p-拉普拉斯方程的爆破界面一般非线性方程的爆破界面IRN在若干空间维度上的完全爆破与不完全爆破简介：IRN中的爆破问题与爆破解的临界指数构造单调半群的扩展非平凡固有解的整体延拓极限情况下的爆破集p = 2_m上的完全爆破到临界Sobolev指数的完全爆破次临界情况下聚焦解的完全爆破临界Sobolev情况下的完全爆破非聚焦解的完全爆破超临界情况下的完全爆破带p-拉普拉斯算子方程的完全爆破和不完全爆破IRN中的消光问题及完备准则非线性奇异抛物方程几何理论。最大解介绍：奇异行波的几何理论集B的主要步骤和概念及相关的几何概念：压力、斜率、界面算子、tw -图关于固有极大解的构造存在：IR和IRN中的不完全奇点IR和IRN中的完全奇点。进一步的几何概念：b -凹性，符号不变量，b类的b数正则性梯度估计，瞬时平滑，Lipschitz界面，连续最优模，IRN径向几何中的横向和平滑，IRN径向几何中的凹性，接口算子和方程，唯一性应用于IR中各种具有消光和爆破奇点的非线性模型和IRN广义自由边界问题的几何理论。非极大解单相自由边界Stefan和Florin问题热方程自由边界问题的分类二次多孔介质方程自由边界问题的分类一般单相自由边界问题含吸收的多孔介质方程高阶自由边界问题含奇异吸收的双多孔介质方程高阶自由边界问题广义两相自由边界问题文献变号解的规律性介绍：变号解与奇异传播现象应用：具有奇异吸收的符号多孔介质方程奇异曲线的传播具有完全奇异和不完全奇异的广义模型张方程Cauchy问题中的完全奇异一维间断极限半群中张方程Dirichlet问题中的瞬时形状化简和IRN中奇异方程形状化简算子具有爆破奇点和空间奇点的IRN中的半群方程：不连续半群和奇异初始层常用缩写索引参考文献列表每章还包括对文献部分的评论和评论。
Introduction: Sturm Theorems and Nonlinear Singular Parabolic Equations Sturm Theorems for Linear Parabolic Equations and Intersection Comparison. B-equations First Sturm Theorem: Nonincrease of the number of sign changes Second Sturm Theorem: Evolution formation and collapse of multiple zeros First aspects of intersection comparison of solutions of nonlinear parabolic equations Geometrically ordered flows: Transversality and concavity techniques Evolution B-equations preserving Sturmian properties Transversality, Concavity and Sign-Invariants. Solutions on Linear Invariant Subspaces Introduction: Filtration equation and concavity properties Proofs of transversality and concavity estimates by intersection comparison with travelling waves Eventual concavity for the filtration equation Concavity for filtration equations with lower-order terms Singular equations with the p-Laplacian operator preserving concavity Concepts of B-concavity and B-convexity. First example of sign-invariants Various B-concavity properties for the porous medium equation and sign-invariants B-concavity and sign-invariants for the heat equation B-concavity and transversality for the porous medium equation with source B-convexity for equations with exponential nonlinearities Singular parabolic diffusion equations in the radialN-dimensional geometry On general B-concavity via solutions on linear invariant subspaces B-Concavity and Transversality on Nonlinear Subsets for Quasilinear Heat Equations Introduction: Basic equations and concavity estimates Local concavity analysis via travelling wave solutions Concavity for the p-Laplacian equation with absorption B-concavity relative to travelling waves B-concavity for the filtration equation B-concavity relative to incomplete functional subsets Eventual B-concavity Eventual B-convexity: a Criterion of Complete Blow-up and Extinction for Quasilinear Heat Equation Introduction: The blow-up problem Existence and nonexistence of singular blow-up travelling waves Discussion of the blow-up conditions. Pathological equations Proof of complete and incomplete blow-up The extinction problem Complete and incomplete extinction via singular travelling waves Blow-up Interfaces for Quasilinear Heat Equations Introduction: First properties of incomplete blow-up Explicit proper blow-up travelling waves and first estimates of blow-up propagation Explicit blow-up solutions on an invariant subspace Lower speed estimate of blow-up interfaces Dynamical equation of blow-up interfaces Blow-up interfaces are not C2 functions Large time behaviour of proper blow-up solutions Blow-up interfaces for the p-Laplacian equation with source Blow-up interfaces for equations with general nonlinearities Examples of blow-up surfaces in IRN Complete and Incomplete Blow-up in Several Space Dimensions Introduction: The blow-up problem in IRN and critical exponents Construction of the proper blow-up solution: extension of monotone semigroups Global continuation of nontrivial proper solutions On blow-up set in the limit case p = 2_m Complete blow-up up to critical Sobolev exponent Complete blow-up of focused solutions in the subcritical case Complete blow-up in the critical Sobolev case Complete blow-up of unfocused solutions Complete blow-up in the supercritical case Complete and incomplete blow-up for the equation with the p-Laplacian operator Extinction problems in IRN and the criteria of complete and incomplete singularities Geometric Theory of Nonlinear Singular Parabolic Equations. Maximal Solutions Introduction:Main steps and concepts of the geometric theory Set B of singular travelling waves and related geometric notions: pressure, slopes, interface operators, TW-diagram On construction of proper maximal solutions Existence: incomplete singularities in IR and IRN Complete singularities in IR and IRN. Infinite propagation and pathological equations Further geometric notions: B-concavity, sign-invariants, B-number Regularity in B-classes by transversality: gradient estimates, instantaneous smoothing, Lipschitz interfaces, optimal moduli of continuity Transversality and smoothing in the radial geometry in IRN B-concavity in the radial geometry in IRN Interface operators and equations, uniqueness Applications to various nonlinear models with extinction and blow-up singularities in IR and IRN Geometric Theory of Generalized Free-Boundary Problems. Non-Maximal Solutions Introduction: One-phase free-boundary Stefan and Florin problems Classification of free-boundary problems for the heat equation Classification of free-boundary problems for the quadratic porous medium equation On general one-phase free-boundary problems Higher-order free-boundary problems for the porous medium equation with absorption Higher-order free-boundary problems for the dual porous medium equation with singular absorption On generalized two-phase free-boundary problems Remarks and comments on the literature Regularity of Solutions of Changing Sign Introduction: Solutions of changing sign and the phenomenon of singular propagation Application: the sign porous medium equation with singular absorption On propagation of singularity curves Discontinuous Limit Semigroups for the Singular Zhang Equation Introduction: New nonlinear models with discontinuous semigroups Existence and nonexistence results for the hydrodynamic version A generalized model with complete and incomplete singularities Complete singularity in the Cauchy problem for the Zhang equation Instantaneous shape simplification in the Dirichlet problem for the Zhang equation in one dimension Discontinuous limit semigroups and operator of shape simplification for singular equations in IRN Further Examples of Discontinuous and Continuous Limit Semigroups Equations in IRN with blow-up and spatial singularities: discontinuous semigroups and singular initial layers When do singular interfaces not move? References List of Frequently Used Abbreviations Index Each chapter also includes a Remarks and Comments on the Literature section.