This concise text is based on a series of lectures held only a few
years ago and originally intended as an introduction to known
results on linear hyperbolic and parabolic equations. Yet the topic
of differential equations on graphs, ramified spaces, and more
general network-like objects has recently gained significant
momentum and, well beyond the confines of mathematics, there is a
lively interdisciplinary discourse on all aspects of so-called
complex networks. Such network-like structures can be found in
virtually all branches of science, engineering and the humanities,
and future research thus calls for solid theoretical foundations.
This book is specifically devoted to the study of evolution
equations – i.e., of time-dependent differential equations such
as the heat equation, the wave equation, or the Schrödinger
equation (quantum graphs) – bearing in mind that the majority of
the literature in the last ten years on the subject of differential
equations of graphs has been devoted to elliptic equations and
related spectral problems. Moreover, for tackling the most general
settings - e.g. encoded in the transmission conditions in the
network nodes - one classical and elegant tool is that of operator
semigroups. This book is simultaneously a very concise introduction
to this theory and a handbook on its applications to differential
equations on networks. With a more interdisciplinary readership in
mind, full proofs of mathematical statements have been frequently
omitted in favor of keeping the text as concise, fluid and
self-contained as possible. In addition, a brief chapter devoted to
the field of neurodynamics of the brain cortex provides a concrete
link to ongoing applied research.
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