ON PROPAGATION SPEED OF EVOLUTION-EQUATIONS

Consider the evolution equation [GRAPHICS] where a(p)(t) are complex value functions and a(p)(t) is-an-element-of L(loc)1(R). We prove that if u is-an-element-of C(R; L2(R(n))) is a solution of (*) (in the weak sense) and it has compact support in the space R(n) at t = t0 for some t0 is-an-element-of R, then in order that u(x, t) has compact support at another time t = t1, it is necessary that integral-t1/t0 a(p)(t)dt = 0, for all p is-an-element-of N(n) with \p\ greater-than-or-equal-to 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 = 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. (C) 1994 Academic Press, Inc.