Besov maximal regularity for a class of degenerate integro‐differential equations with infinite delay in Banach spaces

Abstract

The theory of operator‐valued Fourier multipliers is used to obtain characterizations for well‐posedness of a large class of degenerate integro‐differential equations of second order in time in Banach spaces. Specifically, we treat the case of vector‐valued Besov spaces on the real line. It is important to note that in particular, the results are applicable to the more familiar scale of vector‐valued Hölder spaces. The equations under consideration are important in several applied problems in physics and material science, in particular for phenomena where memory effects are important. Several models in the area of viscoelasticity, including heat conduction and wave propagation correspond to the general class of integro‐differential equations considered here. The importance of the results is that they can be used to treat nonlinear equations.