An integro-differential equation with application in heat flow

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