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

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

Biography

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

"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