We consider two different classes of models which arise from the study of microwave heating of ceramics in a single-mode resonant cavity. The stability and dynamics of hot-spot solutions to the two classes of scalar, nonlocal, singularly perturbed reaction-diffusion equations are analyzed. For the first model, where the coefficients in the differential operator are spatially homogeneous, an explicit characterization of metastable(exponetially slow motion) hot-spot behaviour is given in the limit of small thermal diffusivity. For the second model, where the differential operator has a spatially inhomogeneous term resulting from the variation in the electric field along the ceramic sample, a hot-spot solution is shown to propagate on an algebraically long time-scale towards the point of maximum field strength.

The goal of the talk is to discuss some recent results
(obtained in
joint work with Bernhard Lamel) concerning the structure of the
local and
global groups of CR automorphisms of real-analytic CR manifolds,
whose
levi-form is allowed to degenerate. We will mainly focus on the
class of
real-analytic hypersurfaces containing no holomorphic curves and
show that in
such a setting the local automorphism groups can be analytically
parametrized
by a finite jet at any given point.