On Propagation Speed of Evolution Equations

Abstract

Consider the evolution equation [formula] where a p ( t) are complex value functions and a p ( t) ∈ L 1 loc ( R). We prove that if u ∈ C( R; L 2( R n )) is a solution of (∗) (in the weak sense) and it has compact support in the space R n at t = t 0 for some t 0 ∈ R, then in order that u( x, t) has compact support at another time t = t 1, it is necessary that ∫ t 1 t 0 a p(t) dt = 0, for all p ∈ N n with |p| ≥ 2. (∗∗) With a few more assumptions on the coefficients a p , we show that (∗∗) is also a sufficient condition for the solution u( x, t) to have compact support at t = t 1. Then, based on the above result, the necessary and sufficient conditions are given for evolution equation (∗) to have finite propagation speed or infinite propagation speed.