Abstract :
Consider the evolution equation [formula] where ap(t) are complex value functions and ap(t) L1loc(R). We prove that if u C(R; L2(Rn)) is a solution of (*) (in the weak sense) and it has compact support in the space Rn at t = t0 for some t0 R, then in order that u(x, t) has compact support at another time t = t1, it is necessary that ∫t1t0ap(t) DT = 0, for all p Nn with p ≥ 2. (**)With a few more assumptions on the coefficients ap, we show that (**) is also a sufficient condition for the solution u(x, t) to have compact support at t = t1. 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.
