Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrodinger Equations Chapman & Hall/CRC Monographs and Research Notes in Mathematics Series
Auteurs : Galaktionov Victor A., Mitidieri Enzo L., Pohozaev Stanislav I.
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.
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.
Date de parution : 10-2014
15.6x23.4 cm
Thèmes de Blow-up for Higher-Order Parabolic, Hyperbolic... :
Mots-clés :
IRN; Cauchy Problem; higher-order nonlinear evolution partial differential equations; PDE; quasilinear PDEs; Smooth; self-similar singularity solutions of PDEs; NDE; shock-wave theory; Rarefaction Waves; blow-up singularities; Linear PDE; parabolic; hyperbolic; dispersion; and Schrödinger equations; Generalized Hermite Polynomials; nonlinear capacity and generalized eigenfunction methods; Nonlinear Dispersion Equations; homotopy and branching approaches; Eigenfunction Expansion; nonvariational elliptic problems; Holds; compacton theory; Odd; Initial Data U0; D Dt; T− 1; Nonlinear Evolution PDEs; Saddle Node Bifurcation; Bvp4c Solver; Ode Problem; Shock Similarity; Countable Set; Smooth Solutions; Riemann Problem; Parabolic PDEs; Nonlinear PDEs