Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrodinger Equations: 1st Edition (Hardback) book cover

Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrodinger Equations

1st Edition

By Victor A. Galaktionov, Enzo L. Mitidieri, Stanislav I. Pohozaev

Chapman and Hall/CRC

569 pages | 178 B/W Illus.

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Description

Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrödinger Equations shows how four types of higher-order nonlinear evolution partial differential equations (PDEs) have many commonalities through their special quasilinear degenerate representations. The authors present a unified approach to deal with these quasilinear PDEs.

The book first studies the particular self-similar singularity solutions (patterns) of the equations. This approach allows four different classes of nonlinear PDEs to be treated simultaneously to establish their striking common features. The book describes many properties of the equations and examines traditional questions of existence/nonexistence, uniqueness/nonuniqueness, global asymptotics, regularizations, shock-wave theory, and various blow-up singularities.

Preparing readers for more advanced mathematical PDE analysis, the book demonstrates that quasilinear degenerate higher-order PDEs, even exotic and awkward ones, are not as daunting as they first appear. It also illustrates the deep features shared by several types of nonlinear PDEs and encourages readers to develop further this unifying PDE approach from other viewpoints.

Reviews

"This volume gives a collection of results on self-similar singular solutions for nonlinear partial differential equations (PDEs), with special emphasis on ‘exotic’ equations of higher order …"

Zentralblatt MATH 1320

Table of Contents

Introduction: Self-Similar Singularity Patterns for Various Higher-Order Nonlinear Partial Differential Equations

Complicated Self-Similar Blow-up, Compacton, and Standing Wave Patterns for Four Nonlinear PDEs: A Unified Variational Approach to Elliptic Equations

Introduction: higher-order evolution models, self-similar blowup, compactons, and standing wave solutions

Problem "blow-up": parabolic and hyperbolic PDEs

Problem "existence": variational approach to countable families of solutions by the Lusternik–Schnirel’man category and Pohozaev’s fibering theory

Problem "oscillations": local oscillatory structure of solutions close to interfaces

Problem "numerics": a first classification of basic types of localized blow-up or compacton patterns for m = 2

Problem "numerics": patterns for m ≥ 3

Toward smoother PDEs: fast diffusion

New families of patterns: Cartesian fibering

Problem "Sturm index": a homotopy classification of patterns via ε-regularization

Problem "fast diffusion": extinction and blow-up phenomenon in the Dirichlet setting

Problem "fast diffusion": L–S and other patterns

Non-L–S patterns: "linearized" algebraic approach

Problem "Sturm index": R-compression

Quasilinear extensions: a gradient diffusivity

Classification of Global Sign-Changing Solutions of Semilinear Heat Equations in the Subcritical Fujita Range: Second- and Higher-Order Diffusion

Semilinear heat PDEs, blow-up, and global solutions

Countable set of p-branches of global self-similar solutions: general strategy

Pitchfork p-bifurcations of profiles

Global p-bifurcation branches: fibering

Countable family of global linearized patterns

Some structural properties of the set of global solutions via critical points: blow-up, transversality, and connecting orbits

On evolution completeness of global patterns

Higher-order PDEs: non-variational similarity and centre subspace patterns

Global similarity profiles and bifurcation branches

Numerics: extension of even p-branches of profiles

Odd non-symmetric profiles and their p-branches

Second countable family: global linearized patterns

Global and Blow-up Solutions for Kuramoto–Sivashinsky, Navier–Stokes, and Burnett Equations

Introduction: Kuramoto–Sivashinsky, Navier–Stokes, and Burnett equations

Interpolation: global existence for the KSE

Method of eigenfunctions: blow-up

Global existence by weighted Gronwall’s inequalities

Global existence and L∞-bounds by scaling techniques

L∞-bounds for the Navier–Stokes equations in IRN and wellposed Burnett equations

Regional, Single-Point, and Global Blow-up for a Fourth-Order Porous Medium-Type Equation with Source

Semilinear and quasilinear blow-up reaction–diffusion models

Fundamental solution and spectral properties: n = 0

Local properties of solutions near interfaces

Blow-up similarity solutions

Regional blow-up profiles for p = n + 1

Single-point blow-up for p > n + 1

Global blow-up profiles for p ∈ (1, n + 1)

Semilinear Fourth-Order Hyperbolic Equation: Two Types of Blow-up Patterns

Introduction: semilinear wave equations and blow-up patterns

Fundamental solution of the linear PDE and local existence

Rescaled equation and related Hermitian spectral theory

Construction of linearized blow-up patterns

Self-similar blow-up: nonlinear eigenfunctions

Quasilinear Fourth-Order Hyperbolic Boussinesq Equation: Shock, Rarefaction, and Fundamental Solutions

Introduction: quasilinear Boussinesq (wave) model and shocks

Shock formation blow-up similarity solutions

Fundamental solution as a nonlinear eigenfunction

Blow-up and Global Solutions for Korteweg–de Vries-Type Equations

Introduction: KdV equation and blow-up

Method of investigation: blow-up via nonlinear capacity

Proofs of blow-up results

The Cauchy problem for the KdV equation

Higher-Order Nonlinear Dispersion PDEs: Shock, Rarefaction, and Blow-Up Waves

Introduction: nonlinear dispersion PDEs and main problems

First blow-up results by two methods

Shock and rarefaction waves for S∓(x), H(±)(x), etc.

Unbounded shocks and other singularities

TWs and generic formation of moving shocks

The Cauchy problem for NDEs: smooth deformations, compactons, and extensions to higher orders

Conservation laws: smooth δ-deformations

On δ-entropy solutions (a test) of the NDE

On extensions to other related NDEs

On related higher-order in time NDEs

On shocks for spatially higher-order NDEs

Changing sign compactons for higher-order NDEs

NDE–3: gradient blow-up and nonuniqueness

Gradient blow-up similarity solutions

Nonunique extensions beyond blow-up

NDE–3: parabolic approximation

Fifth-order NDEs and main problems

Problem "blow-up": shock S− solutions

Riemann problems S±: rarefactions and shocks

Nonuniqueness after shock formation

Shocks for NDEs with the Cauchy–Kovalevskaya theorem

Problem "oscillatory compactons" for fifth- and seventh-order NDEs

Higher-Order Schrödinger Equations: From "Blow-up" Zero Structures to Quasilinear Operators

Introduction: duality of "global" and "blow-up" scalings, Hermitian spectral theory, and refined scattering

The fundamental solution and the convolution

Discrete real spectrum and eigenfunctions of B

Spectrum and polynomial eigenfunctions of B∗

Application I: evolution completeness of _ in L2_∗(IRN), sharp estimates in IRN+1+ , extensions

Applications II and III: local structure of nodal sets and unique continuation by blow-up scaling

Application IV: a boundary point regularity via a blow-up micro-analysis

Application V: toward countable families of nonlinear eigenfunctions of the QLSE

Extras: eigenfunction expansions and little Hilbert spaces

References

List of Frequently Used Abbreviations

About the Series

Chapman & Hall/CRC Monographs and Research Notes in Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT007000
MATHEMATICS / Differential Equations
MAT012000
MATHEMATICS / Geometry / General