Nonlinear Viscoelastic Materials with Fading Memory.

The equations governing the motion of viscoelastic materials with fading memory incorporate a nonlinear elastic-type response with a natural dissipative mechanical. Our purpose is to discuss the subtle effects of this mechanism in viscoelastic materials of Boltzmann type. Recent results on the global existence and decay of classical solutions for smooth and small data in one space dimension are reviewed for smooth and singular memory kernels for smooth kernels a number of such results can be generalized to several space dimensions. A recent result on the development of singularities in finite time for large data is discussed several open problems are formulated. A program for studying weak solutions for such systems, including the development of numerical algorithms, is outlined. Keywords hyperbolic equations Volterra equations.