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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