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# Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications

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## Book Description

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.