Liouville type phenomenon for the Klein-Gordon equation

Liouville’s theorem is the basic “collapse” theorem, where an entire bounded function must be constant. A similar phenomenon is the Phragmen-Lindelof principle which says that if a holomorphic function does not grow sufficiently near a boundary point, it must be bounded there. Here we extend the realm of such statements to the Klein-Gordon equation in $1+1$ dimensions, a prototypical hyperbolic partial differential equation. This reports on joint work with A. Bakan.