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.
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
Inhaltsangabe
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.
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.
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