Regularity properties of degenerate convolution-elliptic equations

Abstract

The coercive properties of degenerate abstract convolution-elliptic equations are investigated. Here we find sufficient conditions that guarantee the separability of these problems in Lp spaces. It is established that the corresponding convolution-elliptic operator is positive and is also a generator of an analytic semigroup. Finally, these results are applied to obtain the maximal regularity properties of the Cauchy problem for a degenerate abstract parabolic equation in mixed Lp norms, boundary value problems for degenerate integro-differential equations, and infinite systems of degenerate elliptic integro-differential equations.