Singular perturbations of integrodifferential equations in Banach space

Liu, James H.

doi:10.1090/S0002-9939-1994-1287101-0

Singular perturbations of integrodifferential equations in Banach space
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Proc. Amer. Math. Soc. 122 (1994), 791-799 Request permission

Abstract:

Let $\varepsilon > 0$ and consider \[ \begin {array}{*{20}{c}} {{\varepsilon ^2}u''(t;\varepsilon ) + u’ (t;\varepsilon ) = Au(t;\varepsilon ) + \int _0^t {K(t - s)Au(s;\varepsilon )\;ds + f(t;\varepsilon ),\quad t \geq 0,} } \\ {u(0;\varepsilon ) = {u_0}(\varepsilon ),\quad u’ (0;\varepsilon ) = {u_1}(\varepsilon ),} \\ \end {array} \] and \[ w’ (t) = Aw(t) + \int _0^t {K(t - s)Aw(s) ds + f(t)\quad t \geq 0,w(0) = {w_0},} \] in a Banach space X when $\varepsilon \to 0$. Here A is the generator of a strongly continuous cosine family and a strongly continuous semigroup, and $K(t)$ is a bounded linear operator for $t \geq 0$. With some convergence conditions on initial data and $f(t;\varepsilon )$ and smoothness conditions on $K( \bullet )$, we prove that if $\varepsilon \to 0$, then $u(t;\varepsilon ) \to w(t)$ in X uniformly for $t \in [0,T]$ for any fixed $T > 0$. We will apply this to an equation in viscoelasticity.

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