1st Edition
Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrodinger Equations
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.
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
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
