Weak solutions for a class of parabolic Volterra integrodifferential equations

Abstract

We define weak solutions for a class of Volterra integrodifferential equations of the form u ′( t ) + Au ( t ) = ∝ 0 t a ( t , s ) g ( s , u ( s )) ds + f ( t , u ( t )), t ⩾ 0, u (0) = 0. The operator A is the negative infinitesimal generator of an analytic semigroup in a Banach space X . The operator g ( t , u ) is related to A by a special form g(t, u) = A 1 2 q(t, u) , where q ( t , u ) is an appropriate “lower order” operator. We show the existence and uniqueness of weak solutions and their continuability to infinity under suitable conditions. Using our results we study the asymptotic behavior, as time goes to infinity, of strong solutions of a second order initial-boundary value problem.