View source for Finite speed of propagation

'''Finite speed of propagation''' refers to the phenomenon for certain equations that information (such as support or singularities of solutions) only propagate at a bounded speed.  In general, relativisitic equations such as [[wave equations|nonlinear wave and Klein-Gordon equations]] enjoy finite speed of
propagation, whereas highly [[dispersive]] equations such as [[Schrodinger equations]] or [[KdV-type equations]] do not (see [[infinite speed of propagation]]).

One can determine finite or infinite speed of propagation in a number of ways.  One is via the [[dispersion relation]].  Another is by inspection of a [[fundamental solution]].  A third is
by [[energy estimate]]s, for instance by exploiting the positivity properties of the [[stress-energy tensor]].

== Relativistic equations ==

In relativistic equations, information only propagates at the speed of light ''c'' (typically normalized to 1) or slower; for massless equations such as the [[free wave equation]], information in fact propagates at ''exactly'' 'c'.  Singularities also tend to [[propagation of singularities|propagate]] at exactly ''c''.

Finite speed of propagation allows one to localize space whenever time is localized. Because of this, there is usually no distinction between periodic and non-periodic wave equations. Another application is to convert local existence results for large data to that of small data (though in sub-critical situations this is often better achieved by scaling or similar arguments). Also, the behaviour of blowup at a point is only determined by the solution in the backwards light cone from that point; thus to avoid blowup one needs to show that the solution cannot concentrate into a backwards light cone. One can also use finite speed of propagation to truncate constant-in-space solutions (which evolve by some simple ODE) to obtain localized solutions. This is often useful to demonstrate blowup for various focussing equations.

[[Category:concept]]