  The question is studied whether weak solutions of linear partial
integrodifferential equations approach a constant spatial profile after
rescaling, as time goes to infinity. The possible limits and corresponding
scaling functions are identified and are shown to actually occur. The limiting
equations are fractional diffusion equations which are known to have
self-similar fundamental solutions. For an important special case, is is shown
that the asymptotic profile is Gaussian and convergence holds in $L^2$, that
is, solutions behave like fundamental solutions of the heat equation to leading
order. Systems of integrodifferential equations occurring in viscoelasticity
are also discussed, and their solutions are shown to behave like fundamental
solutions of a related Stokes system. The main assumption is that the integral
kernel in the equation is regularly varying in the sense of Karamata.
