  We study the well-posedness of the initial value problem on periodic
intervals for linear and quasilinear evolution equations for which the
leading-order terms have three spatial derivatives. In such equations, there is
a competition between the dispersive effects which stem from the leading-order
term, and anti-diffusion which stems from the lower-order terms with two
spatial derivatives. We show that the dispersive effects can dominate the
backwards diffusion: we find a condition which guarantees well-posedness of the
initial value problem for linear, variable coefficient equations of this kind,
even when such anti-diffusion is present. In fact, we show that even in the
presence of localized backwards diffusion, the dispersion will in some cases
lead to an overall effect of parabolic smoothing. By contrast, we also show
that when our condition is violated, the backwards diffusion can dominate the
dispersive effects, leading to an ill-posed initial value problem. We use these
results on linear evolution equations as a guide when proving well-posedness of
the initial value problem for some quasilinear equations which also exhibit
this competition between dispersion and anti-diffusion: a Rosenau-Hyman
compacton equation, the Harry Dym equation, and equations which arise in the
numerical analysis of finite difference schemes for dispersive equations. For
these quasilinear equations, the well-posedness theorem requires that the
initial data be uniformly bounded away from zero.
