Geometric Sturmian Theory of Nonlinear Parabolic Equations with Applications

Geometric Sturmian Theory of Nonlinear Parabolic Equations with Applications

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Unlike the classical Sturm theorems on the zeros of solutions of second-order ODEs, Sturm's evolution zero set analysis for parabolic PDEs did not attract much attention in the 19th century, and, in fact, it was lost or forgotten for almost a century. Briefly revived by Polya in the 1930's and rediscovered in part several times since, it was not until the 1980's that the Sturmian argument for PDEs began to penetrate into the theory of parabolic equations and was found to have several fundamental applications. Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications focuses on geometric aspects of the intersection comparison for nonlinear models creating finite-time singularities. After introducing the original Sturm zero set results for linear parabolic equations and the basic concepts of geometric analysis, the author presents the main concepts and regularity results of the geometric intersection theory (G-theory). Here he considers the general singular equation and presents the geometric notions related to the regularity and interface propagation of solutions. In the general setting, the author describes the main aspects of the ODE-PDE duality, proves existence and nonexistence theorems, establishes uniqueness and optimal Bernstein-type estimates, and derives interface equations, including higher-order equations. The final two chapters explore some special aspects of discontinuous and continuous limit semigroups generated by singular parabolic equations. Much of the information presented here has never before been published in book form. Readable and self-contained, this book forms a unique and outstanding reference on second-order parabolic PDEs used as models for a wide range of physical more

Product details

  • Hardback | 384 pages
  • 160 x 238.8 x 27.9mm | 680.4g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • New.
  • 29 black & white illustrations
  • 1584884622
  • 9781584884620

Table of contents

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 more