The Cauchy problem in one-dimensional nonlinear viscoelasticity

The initial value problem for a nonlinear hyperbolic Volterra equation which models the motion of an unbounded viscoelastic bar is studied. Under physically motivated assumptions, the existence of a unique, globally defined, classical solution is established provided the initial data are sufficiently smooth and small. Boundedness and asymptotic behavior are also discussed. This analysis is based on energy estimates in conjunction with properties of strongly positive definite kernels.