An integrodifferential wave equation

Abstract

This paper is devoted to the study of the integrodifferential equation $$ u'(t) =Au (t) +\int^t_0 a(t-s) A_1 u(s) ds +f (t) , \quad t\ge 0, $$ where $A$ is a Hille-Yosida operator in a Banach space $X$, $A_1 \in {\mathcal L} (D (A);$ $ X)$ and $a$ has bounded variation. Existence, uniqueness and estimates of strict and weak solutions are proved by extrapolation methods and the Miller scheme. Applications are given to the Cauchy-Dirichlet problem for the integrodifferential wave equation $$ w_{tt} (t,x) =w_{xx} (t,x)+\int^t_0 a (t-s) w_{xx} (s,x) ds +f(t,x), \quad t\ge 0, \quad x\in [0, \ell]. $$