Weak solutions for a class of parabolic Volterra integrodifferential equations (Q1823421)

Let A: D(A)\(\subseteq X\to X\) be the generator of an analytic semigroup T(t) of negative type in a Banach space X, K(t,s) a smooth scalar function, q(t,u) and f(t,u) two X-valued functions, lipschitz continuous on bounded sets of \({\mathbb{R}}_+\times D(A^{1/2})\). The authors prove the local existence and uniqueness of a maximally defined weak solution of the integrodifferential problem \[ u'(t)=Au(t)+\int^{t}_{0}K(t,s)g(s,u(s))ds+f(t,u(t)),\quad t\geq 0;\quad u(0)=u_ 0, \] where \(g(t,u)=A^{1/2}q(t,u)\). A weak solution is defined as a \(D(A^{1/2})\)-valued continuous function, solution of the integral equation \[ u(t)=T(t-t_ 0)u_ 0+\int^{t}_{t_ 0}A^{1/2}T(t-s)[\int^{s}_{t_ 0}K(s,r\quad)q(r,u(r))dr]ds+\int^{t}_{t_ 0}T(t-s)f(s,u(s))ds \] for \(t\geq t_ 0\). Applications are given to initial boundary value problems for semilinear partial integrodifferential equations of parabolic type whose asymptotic properties are also investigated.