This book mainly serves as an elementary, self-contained
introduction to several important aspects of the theory of global
solutions to initial value problems for nonlinear evolution
equations. The book employs the classical method of continuation of
local solutions with the help of a priori estimates obtained for
small data. The existence and uniqueness of small, smooth solutions
that are defined for all values of the time parameter are
investigated. Moreover, the asymptotic behaviour of the solutions
is described as time tends to infinity. The methods for nonlinear
wave equations are discussed in detail. Other examples include the
equations of elasticity, heat equations, the equations of
thermoelasticity, Schroedinger equations, Klein-Gordon equations,
Maxwell equations and plate equations. To emphasize the importance
of studying the conditions under which small data problems offer
global solutions, some blow-up results are briefly described.
Moreover, the prospects for corresponding initial boundary value
problems and for open questions are provided. In this second
edition, initial-boundary value problems in waveguides are
additionally considered.
General
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