This course explores
the semigroup approach to partial differential equations. The
abstract semigroup theory allows one to treat, by a unified
approach, standard parabolic and hyperbolic equations---such as
those considered in [L.C. Evans, Partial differential equations],
for example---along with equation that are not of the standard type.
Also, the semigroup approach to nonlinear equations is best suited
for qualitative analysis of such equations, as in, for example,
stability theory and invariant manifolds.
We will discuss basic elements of the theory, including strongly
continuous semigroups and more specific analytic semigroups,
interpolation spaces, and generation of semigroups by elliptic
differential operators. More advanced topics will include optimal
regularity results for solutions of linear equations and spectral
mapping theorems (relations between the spectra of the semigroup and
its generator). Building on the linear theory, we will
establish basic existence-uniqueness-regularity properties of
solutions of nonlinear equations.
Abstract results will be applied to specific evolution equations of
parabolic and hyperbolic types.
The course will then continue with a selection of topics from
the qualitative theory of evolution equations.
More detailed description of the course and its prerequisites can be
found in this pdf file.