This book shows how four types of higher-order nonlinear evolution PDEs have many commonalities through their special quasilinear degenerate representations. The authors present a unified approach to deal with these quasilinear PDEs, describe many properties of the equations, and examine traditional questions of existence/nonexistence, uniqueness/nonuniqueness, global asymptotics, regularizations, shock-wave theory, and various blow-up singularities. The book illustrates how complex PDEs are used in a variety of applications and describes new nonlinear phenomena for the equations.

Victor A. Galaktionov, Enzo L. Mitidieri, Stanislav I. Pohozaev

Introduction. Complicated Self-Similar Blow-Up, Compacton, and Standing Wave Patterns for Four Nonlinear PDEs: A Unified Variational Approach to Elliptic Equations. Classification of Global Sign-Changing Solutions of Semilinear Heat Equations in the Subcritical Fujita Range: Second- and Higher-Order Diffusion. Global and Blow-Up Solutions for Kuramoto-Sivashinsky, Navier-Stokes, and Burnett Equations. Regional, Single-Point, and Global Blow-Up for a Fourth-Order Porous Medium-Type Equation with Source. Semilinear Fourth-Order Hyperbolic Equation: Two Types of Blow-Up Patterns. Quasilinear Fourth-Order Hyperbolic Boussinesq Equation: Shock, Rarefaction, and Fundamental Solutions. Blow-Up and Global Solutions for Korteweg-de Vries-Type Equations. Higher-Order Nonlinear Dispersion PDEs: Shock, Rarefaction, and Blow-Up Waves. Higher-Order Schrödinger Equations: From "Blow-Up" Zero Structures to Quasilinear Operators. References.