Beyond Partial Differential Equations: On Linear and Quasi-Linear Abstract Hyperbolic Evolution Equations

Horst Reinhard Beyer (auth.)3540711287, 9783540711292, 9783540711285

The present volume is self-contained and introduces to the treatment of linear and nonlinear (quasi-linear) abstract evolution equations by methods from the theory of strongly continuous semigroups. The theoretical part is accessible to graduate students with basic knowledge in functional analysis. Only some examples require more specialized knowledge from the spectral theory of linear, self-adjoint operators in Hilbert spaces. Particular stress is on equations of the hyperbolic type since considerably less often treated in the literature. Also, evolution equations from fundamental physics need to be compatible with the theory of special relativity and therefore are of hyperbolic type. Throughout, detailed applications are given to hyperbolic partial differential equations occurring in problems of current theoretical physics, in particular to Hermitian hyperbolic systems. This volume is thus also of interest to readers from theoretical physics.

Table of contents :
Front Matter….Pages i-xiv
Conventions….Pages 1-3
Mathematical Introduction….Pages 5-12
Prerequisites….Pages 13-39
Strongly Continuous Semigroups….Pages 41-69
Examples of Generators of Strongly Continuous Semigroups….Pages 71-103
Intertwining Relations, Operator Homomorphisms….Pages 105-121
Examples of Constrained Systems….Pages 123-135
Kernels, Chains, and Evolution Operators….Pages 137-163
The Linear Evolution Equation….Pages 165-176
Examples of Linear Evolution Equations….Pages 177-214
The Quasi-Linear Evolution Equation….Pages 215-234
Examples of Quasi-Linear Evolution Equations….Pages 235-263
Back Matter….Pages 265-287