Fractional Cauchy problem with memory effects

Abstract

We give an original representation integral formula for the resolvent families associated to the fractional Cauchy equation with memory effects () − () + ∫ 0 (−) () = (, ()), ∈ [0, ], > 0, where (0) = 0 ∈ and is a sectorial operator on a Banach space. Moreover, we get spatial bounds for the resolvent families in order to study global or blow up mild solutions when the nonlinearity satisfies a locally Lipschitz condition. The case of critical nonlinearities is also treated.