An integro-differential equation with application in heat flow

The problem \[ u t ( x , t ) = ∫ 0 t a ( t − τ ) ∂ ∂ x σ ( u x ( x , τ ) ) d τ + f ( x , t ) , 0 > x > 1 , t > 0 , u ( 0 , t ) ≡ u ( 1 , t ) ≡ 0 u ( x , 0 ) = u 0 ( x ) {u_t} ( {x, t} ) = ∫ _0^t {} a ( {t - τ } ) {∂ }{{∂ x}}σ ( {{u_x} ( {x,τ } )} )dτ + f ( {x, t} ), 0 > x > 1, t > 0, \\ u ( {0,t} ) ≡ u ( {1,t} ) ≡ 0 u ( {x, 0} ) = {u_0} ( x ) \] is considered. Asymptotic stability theorems for the solution are established under appropriate conditions on a a ,