## An integrodifferential wave equation.(English)Zbl 1143.45004

Summary: This paper is devoted to the study of the integrodifferential equation $u'(t)=Au(t)+\int_{0}^{t} a(t-s)A_{1} u(s)\,ds+f(t), \quad t \geq 0,$ 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
$w_{tt} (t,x) = w_{xx} (t,x) + \int_{0}^{t} a(t-s) w_{xx} (s,x)\,ds + f(t,x), \quad t \geq 0, \quad x \in [0,\ell].$

### MSC:

 45N05 Abstract integral equations, integral equations in abstract spaces 45J05 Integro-ordinary differential equations 45K05 Integro-partial differential equations