A Model Equation for Viscoelasticity with a Strongly Singular Kernel

In much of the mathematical work on nonlinear viscoelasticity it is assumed that the kernel (or memory function) is smooth on [0,∞ ). There are, however, theoretical and experimental indications that certain viscoelastic materials may be described by equations involving kernels that are singular at zero. In this paper we establish local (in time) existence of smooth solutions to a nonlinear integrodifferential equation with a singular kernel. This equation provides a model for the motion of a certain class of viscoelastic materials. Our analysis is based on energy estimates and properties of positive definite kernels.