A semilinear parabolic Volterra integrodifferential equation (Q1101641)

The authors study the existence of solutions of the equation \[ u'(t)+A(t)u(t)=\int^{t}_{t_ 0}a(t,s)g(s,u(s))ds+f(t,u(t)),\quad t\geq t_ 0,\quad u(t_ 0)=u_ 0, \] in a Banach space X. Here -A(t) is the generator of an analytic semigroup and the nonlinear operator g(t,\(\cdot)\) is Lipschitz continuous on the domain of A(0) in the graph norm. The function f satisfies a Hölder condition in both variables. By using a different method of proof the authors extend some results of the first author [SIAM J. Math. Anal. 13, 81-105 (1982; Zbl 0477.45008)] and \textit{G. F. Webb} [Lect. Notes Math. 737, 295-303 (1979; Zbl 0428.45008)]. In the case where X is a Hilbert space uniqueness is established. Finally the authors give some results on the asymptotic behavior of the solution in the case where A(t)\(\equiv A\) and \(a(t,s)=a(t-s)\).