Galaktionov / Mitidieri / Pohozaev Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrodinger Equations

1. Auflage 2014
ISBN: 978-1-4822-5173-9
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 569 Seiten

Reihe: Monographs and Research Notes in Mathematics

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.

Zielgruppe

Researchers and PhD students in applied mathematics; mathematical physicists; electrical engineers.

Autoren/Hrsg.

Galaktionov, Victor A.

Mitidieri, Enzo L.

Pohozaev, Stanislav I.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

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 e-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
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