A model for one-dimensional, nonlinear viscoelasticity

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