Regular solutions for time dependent abstract integrodifferential equations with singular kernel (Q1099368)

The problems of existence, uniqueness and regularity of solutions to the linear integrodifferential equation \[ u'(t)=A(t)u(t)+ \int^{t}_{0} B(t,s)u(s)ds+f(t),\quad 0<t\leq T \] depending on the regularity of the initial value \(u(0)=u_ 0\) are considered. For each \(t\in [0,T]\) the operator \(A(t): F\subset E\to E\) generates an analytic semigroup \(e^{sA(t)}\) in E and \(B(t,s)\) \((0\leq s\leq t\leq T)\) is a linear operator from F to E. A(t) and f(t) are Hölder continuous with exponent \(\alpha\in (0,1)\) and, for each \(y\in F\) \(B(t,s)y\) is \(\alpha\)-Hölder continuous, with respect to t and \(L^ p\), with respect to s, for some \(p>1\). As an example of applications a linear parabolic Volterra equation is given.