The problem
\[
u
t
t
=
a
(
0
)
σ
(
u
x
)
x
+
∫
0
t
a
˙
(
t
−
τ
)
σ
(
u
x
)
x
d
τ
+
f
,
0
>
x
>
1
,
t
>
0
,
u
(
0
,
t
)
≡
u
(
1
,
t
)
≡
0
,
u
(
x
,
0
)
=
u
o
(
x
)
,
u
t
(
x
,
0
)
=
u
1
(
x
)
{u_{tt}} = a\left ( 0 \right )\sigma {\left ( {{u_x}} \right )_x} + \int _0^t {\dot a} \left ( {t - \tau } \right )\sigma {\left ( {{u_x}} \right )_x}d\tau + f, \qquad 0 > x > 1, \qquad t > 0, \\ u\left ( {0, t} \right ) \equiv u\left ( {1, t} \right ) \equiv 0, \\ u\left ( {x, 0} \right ) = {u_o}\left ( x \right ), \qquad {u_t}\left ( {x, 0} \right ) = {u_1}\left ( x \right )
\]
is considered. The essential hypotheses are that
\[
a
(
t
)
=
a
∞
+
A
(
t
)
,
a
∞
>
0
,
A
∈
L
1
(
0
,
∞
)
,
(
−
1
)
k
a
(
k
)
(
t
)
≥
0
,
k
=
0
,
1
,
2
,
σ
(
0
)
=
0
,
σ
′
(
ξ
)
≥
ϵ
>
0
a\left ( t \right ) = {a_\infty } + A\left ( t \right ), {a_\infty } > 0, A \in {L^1}\left ( {0, \infty } \right ), \\ {\left ( { - 1} \right )^k}{a^{\left ( k \right )}}\left ( t \right ) \ge 0, k = 0, 1, 2, \sigma \left ( 0 \right ) = 0, \sigma ’\left ( \xi \right ) \ge \epsilon > 0
\]
. It is shown that the problem has a unique classical solution for all
t
t
if the data are sufficiently small and, if
f
f
is suitably restricted, this solution tends to zero as
t
t
tends to infinity. It is shown that the problem provides a special model for elastic materials which exhibit a memory effect.