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 − ∂

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