Pisa Course Outline
1 March 06

Title: Dispersive behavior of hyperbolic partial
differential equations.

Outline.  One of the central themes of the theory
of partial differential equations is to 
create a dictionary which reads properties
of the equations from properties of the
characteristic variety.  For hyperbolic
differential equations, dispersive behavior
 in the sense that solutions spread out and
 decay, is read from the curvature of the 
 characteristic variety.  In this short
 course several results of this sort will be
 presented.  The main estimates concern
 constant coefficient linear systems with
 proofs using Fourier analysis.  No special
 expertise in hyperbolic equations required.
 
 1.  Rates of decay for some model problems.
 Group velocity and stationary phase.
 
 2.  The characterisation of nondispersive
 systems as those reducable to scalar trasnport
 equations.
 
 3.  Homogeneous systems  all of whose solutions 
 (compactly supported in x) tend to zero as t 
 tends to infinity are those whose variety
 contains no hyperplanes.
 
 4.  The fact (P. Brenner) that symmetric systems
 whose solution operator is an L^p multiplier
 fall under heading 2.
 
 5.  Strichartz estimates.  Application to the
 nopnlinear Klein-Gordon equation.