## A singular perturbation problem in integrodifferential equations

### Summary

Summary: 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 $$w'(t) = Aw(t)+\int_0^t K(t-s)Aw(s)\,ds+f(t)\,,\quad t\geq 0\,,\quad w(0) = w_0\,,$$ 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 $T$. An application to viscoelasticity is given.

45D, 45J, 45N

### Keywords/Phrases

singular perturbation, convergence in solutions and derivatives