For p > 2, the equation u(t )= u(p )u(xx), x is an element of R, t is an element of R, is shown to admit positive and spatially increasing smooth solutions on all of R x R which are precisely of the form of an accelerating wave for t < 0, and of a wave slowing down for t > 0. These solutions satisfy u(& sdot;, t) -> 0 in L-loc(infinity )(R) as t -> +infinity and as t -> -infinity, and exhibit a yet apparently undiscovered phenomenon of transient rapid spatial growth, in the sense that lim(x -> +infinity )x(-1 )u(x, t) exists for all t < 0, that lim(x -> +infinity )x(-2/p )u(x, t) exists for all t > 0, but that u(x, 0) = Ke(alpha x) for all x is an element of R with some K > 0 and alpha > 0.
