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, Schrödinger 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.