Electr. J. Diff. Eqns. Vol. 1993(1993), No. 02, pp. 1--10.
Title: A singular perturbation problem in integrodifferential equations
Author: James H. Liu (James Madison Univ., Harrisonburg, VA, USA)
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 $T > 0$. 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.