Electron.n. J. Diff. Eqns. Vol. 1993(1993), No. 02, pp. 1-10.

A singular perturbation problem in integrodifferential equations

James H. Liu

Abstract:
Consider the singular perturbation problem for
$$\varepsilon ^2 u'' (t;\varepsilon ) + u'(t;\varepsilon ) = Au(t;\varepsilon )+\int_0^t K(t-s)Au(s;\varepsilon)\,ds+ f(t;\varepsilon )\,,$$
where $t\geq 0$, $u(0;\varepsilon ) = u_0 (\varepsilon )$, $u'(0;\varepsilon ) = u_1(\varepsilon )$, and
$$w'(t) = Aw(t)+\int_0^t K(t-s)Aw(s)\,ds+f(t)\,,\quad t\geq 0\,,\quad w(0) = w_0\,, $$
in a Banach space X when $\varepsilon \rightarrow 0$. 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 $K(\cdot)$, we prove that when $\varepsilon \rightarrow 0$, one has $u(t;\varepsilon)\rightarrow w(t)$ and $u'(t;\varepsilon)\rightarrow w'(t)$ in X uniformly on [0,T] for any fixed $T$ positive. An application to viscoelasticity is given.

Submitted June 14, 1993. Published September 16, 1993.
Math Subject Classification: 45D, 45J, 45N.
Key Words: Singular perturbation, Convergence in solutions and derivatives.

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James H. Liu
Department of Mathematics, James Madison University
Harrisonburg, VA 22807, USA
e-mail: liu@math.jmu.edu
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