Fundamental Solutions for Abstract Fractional Evolution Equations with Generalized Convolution Operators

Abstract

We investigate a class of abstract fractional evolution equations governed by convolution-type derivatives associated with Sonine kernels. These generalized derivatives encompass several known fractional operators, including the Caputo--Dzhrbashyan and distributed-order derivatives. We analyze the Cauchy problem \[ \partial_t(k \ast (u - u_0))(t) = -A^\alpha u(t), \] where \( k \) is a Sonine kernel, \( A \) is a closed linear operator generating a bounded analytic semigroup, and \( \alpha \in (0,1) \). Using functional analytic techniques and subordination theory, we establish well-posedness in the space of infinitely smooth vectors and derive explicit representations for the solution via Laplace transforms and fractional semigroup theory. Several examples involving the Laplacian on different function spaces are discussed to illustrate the theory.