The familiar wave equation is the most fundamental
hyperbolic partial differential equation. Other hyperbolic equations,
both linear and nonlinear, exhibit many wave-like phenomena. The
primary theme of this book is the mathematical investigation of such
wave phenomena.
The exposition begins with derivations of some wave equations,
including waves in an elastic body, such as those observed in
connection with earthquakes. Certain existence results are proved
early on, allowing the later analysis to concentrate on properties of
solutions. The existence of solutions is established using methods
from functional analysis. Many of the properties are developed using
methods of asymptotic solutions. The last chapter contains an
analysis of the decay of the local energy of solutions. This analysis
shows, in particular, that in a connected exterior domain,
disturbances gradually drift into the distance and the effect of a
disturbance in a bounded domain becomes small after sufficient time
passes.
The book is geared toward a wide audience interested in PDEs.
Prerequisite to the text are some real analysis and elementary
functional analysis. It would be suitable for use as a text in PDEs or
mathematical physics at the advanced undergraduate and graduate
level.
Readership
Advanced undergraduate and graduate students and
researchers interested in partial differential equations and
mathematical physics.
