Lp-Maximal Regularity for a Class of Degenerate Integro-differential Equations with Infinite Delay in Banach Spaces

Using the theory of operator-valued Fourier multipliers, we establish characterizations for well-posedness of a large class of degenerate integro-differential equations of second order in time in Banach spaces. We are concerned with the spaces Lp(R,X), 1 <= p<infinity where X is a given Banach space. When X is a UMD space and 1<p<infinity, we obtain concrete conditions for well-posedness based on the concept of R-boundedness (or Rademacher boundedness) for operator families. We rely on a transfer to weighted vector-valued Lp-spaces in order to prove the results. The results are applied to some concrete equations. The equation that we consider appear in several models in the applied sciences, particularly physics, rheology, material science and more generally in phenomena where memory effects play an important role.