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The question is studied whether weak solutions of linear partialintegrodifferential equations approach a constant spatial profile afterrescaling, as time goes to infinity. The possible limits and correspondingscaling functions are identified and are shown to actually occur. The limitingequations are fractional diffusion equations which are known to haveself-similar fundamental solutions. For an important special case, is is shownthat the asymptotic profile is Gaussian and convergence holds in $L^2$, thatis, solutions behave like fundamental solutions of the heat equation to leadingorder. Systems of integrodifferential equations occurring in viscoelasticityare also discussed, and their solutions are shown to behave like fundamentalsolutions of a related Stokes system. The main assumption is that the integralkernel in the equation is regularly varying in the sense of Karamata.

Asymptotic Self-Similarity for Solutions of Partial Integro-Differential Equations