1st Edition

Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications

By Victor A. Galaktionov Copyright 2004
    384 Pages 29 B/W Illustrations
    by Chapman & Hall

    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 Pólya 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 problems.

    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.

    Biography

    Victor A. Galaktionov