The problem
\[
u
t
(
x
,
t
)
=
∫
0
t
a
(
t
−
τ
)
∂
∂
x
σ
(
u
x
(
x
,
τ
)
)
d
τ
+
f
(
x
,
t
)
,
0
>
x
>
1
,
t
>
0
,
u
(
0
,
t
)
≡
u
(
1
,
t
)
≡
0
u
(
x
,
0
)
=
u
0
(
x
)
{u_t}\left ( {x, t} \right ) = \int _0^t {} a\left ( {t - \tau } \right )\frac {\partial }{{\partial x}}\sigma \left ( {{u_x}\left ( {x,\tau } \right )} \right )d\tau + f\left ( {x, t} \right ), \qquad 0 > x > 1, \qquad t > 0, \\ u\left ( {0,t} \right ) \equiv u\left ( {1,t} \right ) \equiv 0 \qquad u\left ( {x, 0} \right ) = {u_0}\left ( x \right )
\]
is considered. Asymptotic stability theorems for the solution are established under appropriate conditions on
a
a
,
σ
\sigma
and
f
f
. The conditions on
a
a
are of frequency domain type and are related to ones used previously in the study of Volterra integral equations,
\[
u
˙
=
−
∫
0
t
a
(
t
−
τ
)
g
(
u
(
τ
)
)
d
τ
+
f
(
t
)
\dot u = - \int _0^t a \left ( {t - \tau } \right )g\left ( {u\left ( \tau \right )} \right )d\tau + f\left ( t \right )
\]
on a Hilbert space. An existence theorem for the problem is established under smallness assumptions on
f
f
and
u
0
{u_0}
This theorem is related to one by Nishida for the damped non-linear wave equation,
\[
u
t
t
+
α
u
t
−
∂
∂
x
σ
(
u
x
)
=
0
{u_{tt}} + \alpha {u_t} - \frac {\partial }{{\partial x}}\sigma \left ( {{u_x}} \right ) = 0
\]
. It is shown that the problem is related to a theory of heat flow in materials with memory.