Advances in Differential Equations

An integrodifferential wave equation

Eugenio Sinestrari

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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

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

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]


Sinestrari, Eugenio. An integrodifferential wave equation. Adv. Differential Equations 11 (2006), no. 7, 751--779.

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