Global Solutions to the Equation of Viscoelasticity with Fading Memory

The initial-history value problem for the one-dimensional equation of viscoelasticity with fading memory is studied in a situation that allows the kernel function to have integrable singularities in the first order derivative. It is proved that if the data are smooth and small, then a unique solution exists globally in time and converges to the equilibrium as time goes to infinity, provided that the kernel is strongly positive definite. This is an improvement on the previous result by W. J. Hrusa and J. A. Nohel (J. Differential Equations59, 1985, 388-412). Our proof is based on an energy method which makes use of properties of strongly positive definite kernels.