Infinite speed of propagation

Infinite speed of propagation is of course the negation of finite speed of propagation. It is a feature of highly dispersive equations such as Schrodinger equations and KdV-type equations, though not of relativistic equations such as wave equations. Indeed infinite speed of propagation occurs whenever the dispersion relation grows faster than linearly at infinity.

Equations which have infinite speed of propagation tend to enjoy local smoothing properties, due to the high frequencies escaping to infinity very quickly. However if the underlying manifold contains trapped geodesics then the local smoothing effect may not occur.

Infinite speed of propagation can also cause even linear equations to be illposed in the $C^{\infty }$ category, if one does not temper the growth at infinity. However if one works in categories such as tempered distributions, or Sobolev spaces $H_{x}^{s}(\mathbb {R} ^{d})$, then linear dispersive and wave equation are well-posed even in the presence of infinite speed of propagation.