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 epsilon-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 delta-deformations On delta-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 Schrodinger 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