Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications

Summary

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.

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 section.