Properties of the support of solutions of a class of nonlinear evolution equations

In this work we consider equations of the form(sic)(t)u + P((sic)(x))u + G(u, (sic)(x)u, ... , (sic)(l)(x)u) = 0,where P is any polynomial without constant term, and G is any polynomial without constant or linear terms. We prove that if u is a sufficiently smooth solution of the equation, such that supp u(0), supp u(T)subset of (-infinity, B] for some B > 0, then there exists R-0 > 0 such that supp u(t) c (-oo, R0] for every t is an element of [0, T]. Then, as an example of the application of this result, we employ it to show a unique continuation principle for the Kawahara equation,(sic)(t)u + (sic)(5)(x)u + (sic)(x)(3)u + u(sic)(x)u = 0,and for the generalized KdV hierarchy(sic)(t)u + (-1)(k+1)(sic)(2k+1) (x )u + G(u, (sic)(x)u, ... , (sic)(x)(2k)u) = 0.