An abstract semilinear hyperbolic Volterra integrodifferential equation

AbstractWe consider semilinear integrodifferential equations of the form u′(t) + A(t) u(t) = ∝0t g(t, s, u(s)) ds + f(t), u(0) = u0. For each t ⩾ 0, the operator A(t) is assumed to be the negative generator of a strongly continuous semigroup in a Banach space X. The domain D(A(t)) of A(t) is allowed to vary with t. Thus our models are Volterra integrodifferential equations of “hyperbolic type.” These types of equations arise naturally in the study of viscoelasticity. Our main results are the proofs of existence, uniqueness, continuation and continuous dependence of the solutions