Pokhozhaev's Legacy

The Art of Nonexistence, Blow-Up, and Fibering in Nonlinear PDEs and Inequalities
1. Auflage 2027
ISBN: 978-1-041-47039-7
Verlag: Taylor & Francis Ltd

Buch, Englisch, 496 Seiten, Format (B × H): 178 mm x 254 mm

Reihe: Chapman & Hall/CRC Monographs and Research Notes in Mathematics

Pokhozhaev’s Legacy explores nonexistence, blow-up and pattern formation for nonlinear partial differential equations and inequalities. Centred on Stanislav Pokhozhaev’s mathematical ideas, it brings together methods for studying problems across different equation types and differential orders.

The book develops Pokhozhaev nonlinear capacity and test-function techniques, and the fibering method, alongside invariant manifold, Sturmian and spectral analysis. It connects nonexistence results with global sign-changing solutions and the classification of nonlinear patterns. Written for researchers and advanced graduate students, it assumes a grounding in functional analysis and partial differential equations.

Features

- Treatment of nonexistence and blow-up for second-order and higher-order equations, including polyharmonic, quasilinear and doubly nonlinear problems.

- Applications of Pokhozhaev’s fibering method and Lusternik–Schnirel’man theory to the existence and classification of nonlinear patterns.

- Material from Pokhozhaev’s unpublished manuscripts, accompanied by historical recollections and bibliographic notes.

- An illustrated dictionary of more than seventy model problems, examining patterns in fourth-order and sixth-order equations of Cahn–Hilliard type and related models.

- Nine chapters with introductory discussions, a bibliography of 337 references, an analytical index and an author index, supporting use as a research reference and a guide to open problems.

Zielgruppe

Postgraduate

Autoren/Hrsg.

Galaktionov, Victor A.

Mitidieri, Enzo

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

1. Quasilinear Parabolic PDEs and Inequalities 2. Nonexistence of Weak Solutions for Degenerate and Singular Hyperbolic Problems 3. Nonexistence of Weak Solutions for Degenerate Elliptic and Parabolic Problems on R^N 4. Global and Blow-up Sign-changing Solutions of the Semilinear Heat Equation: Second-order Diffusion 5. Global Sign-changing Solutions of the Higher-Order Semilinear Heat Equation: Polyharmonic Diffusion 6. Global Solutions, Blow-up, and Well-posed Burnett Equations: Kuramoto–Sivashinsky Type Models 7. The Porous Medium Equation: Branching of Similarity Solutions 8. Algebraically Solvable Piece-wise Linear PDEs and First Applications of Pokhozhaev’s Fibering Method 9. A Dictionary: Patterns in Cahn–Hilliard PDEs

Über Autor(innen)

Victor A. Galaktionov is a mathematician at the Frome Mathematical Institute, United Kingdom. His research concerns nonlinear partial differential equations, with particular emphasis on blow-up, self-similar solutions, higher-order evolution equations and the asymptotic behaviour of sign-changing solutions. His books include Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications and, with Sergey R. Svirshchevskii, Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics. He coauthored the 2015 monograph Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrödinger Equations with Enzo Mitidieri and Stanislav Pokhozhaev.

Enzo Mitidieri is a professor of mathematical analysis at the University of Trieste, Italy. His research focuses on nonlinear partial differential equations and inequalities, including Liouville-type theorems, a priori estimates, nonexistence, blow-up and qualitative properties of solutions. His work also addresses Hardy–Rellich inequalities, nonlinear potential theory and higher-order problems. His long collaboration with Stanislav Pokhozhaev led to the 2001 monograph A Priori Estimates and the Absence of Solutions of Nonlinear Partial Differential Equations and Inequalities. With Victor A. Galaktionov and Pokhozhaev, he coauthored Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrödinger Equations, published in 2015.