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Parabolic integro-differential model Cauchy problem is considered in the scale of L p  -spaces of functions whose regularity is defined by a scalable Levy measure. Existence and uniqueness of a solution is proved by deriving apriori estimates. Some rough probability density function estimates of the associated Levy process are used as well.

On the Cauchy Problem for Integro-Differential Equations in the Scale of Spaces of Generalized Smoothness