# 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 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 estimates, 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 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.