Regularity of mild solutions for fractional abstract Cauchy problem with order \({\alpha \in (1, 2)}\) (Q903238)

The author considers the fractional linear differential equation \[ D_t^{\alpha}u(t)+Au(t)=f(t) \] with \(t\in (0,T]\), \(u(0)\), \(u'(0)\) given and where the solution takes values in a Banach space \(X\). Here, \(D_t^{\alpha}\) denotes the fractional \(t\)-derivative of order \(\alpha\) and \(\alpha \in (1,2)\). Thus, the equation lies between a pure diffusion equation and a pure wave equation. \(A\) is linear and admits an analytic resolvent operator. The function \(f\) satisfies at least \(f\in L^1((0,T);X)\). Mild solutions are given by the resolvent operating on the initial values and \(f\). The paper gives conditions under which the mild solution satisfies additional regularity, at least on an interval \([\epsilon, T]\). The conditions involve the \(\alpha\) and \(p\) where \(f\in L^p ((0,T];X)\) and in some cases conditions on the given initial values.