Class of Quasilinear Partial Integrodifferential Equations with Singular Kernels

摘要

In a recent paper, Dafermos and Nohel considered a model equation for nonlinear viscoelasticity. They proved that smooth solutions exist locally in time and also globally in time for small data. For large data, globally defined smooth solutions will not exist in general, and formation of shocks is expected. In the analysis of Dafermos and Nohel, and in other papers showing related results, it is essential that the viscoelastic memory function is absolutely continuous. There are, however, some indications, on both a theoretical and an experimental basis, that certain viscoelastic materials may be adequately described by models with singular memory functions. The mathematical existence properties for such models should in fact be better than for regular memory functions, since a singular memory function precludes the formation of shocks. However, the methods used in previous existence proofs cannot be generalized to singular kernels. In this paper, its author provide an existence theory for such models. They approximate the equation by equations with regular kernels, for which existence is known. They then use energy estimates to show that these approximate solutions converge to a limit.