An integrodifferential wave equation. (English) Zbl 1143.45004

Sinestrari, Eugenio

Adv. Differ. Equ. 11, No. 7, 751-779 (2006).

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