## Advances in Differential Equations

- Adv. Differential Equations
- Volume 11, Number 7 (2006), 751-779.

### 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 $$ w_{tt} (t,x) =w_{xx} (t,x)+\int^t_0 a (t-s) w_{xx} (s,x) ds +f(t,x), \quad t\ge 0, \quad x\in [0, \ell]. $$

#### Article information

**Source**

Adv. Differential Equations Volume 11, Number 7 (2006), 751-779.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867675

**Mathematical Reviews number (MathSciNet)**

MR2236581

**Zentralblatt MATH identifier**

1143.45004

**Subjects**

Primary: 45J05: Integro-ordinary differential equations [See also 34K05, 34K30, 47G20]

Secondary: 34G10: Linear equations [See also 47D06, 47D09] 35L20: Initial-boundary value problems for second-order hyperbolic equations 35R10: Partial functional-differential equations 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] 45N05: Abstract integral equations, integral equations in abstract spaces 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]

#### Citation

Sinestrari, Eugenio. An integrodifferential wave equation. Adv. Differential Equations 11 (2006), no. 7, 751--779. https://projecteuclid.org/euclid.ade/1355867675