ABSTRACT
The study of maximal regularity for linear parabolic mixed problems is systematically based on real interpolation theory started with the fundamental papers of Grisvard and Da Prato in the sixties and seventies. It is well known that maximal regularity results establish linear and topological isomorphisms between certain Banach spaces of functions and are very useful to treat nonlinear problems. As inverse problems, even if concerning linear equations, are essentially nonlinear in nature, it is not surprising that maximal regularity can be very effective to study them. This chapter illustrates some applications of maximal regularity to inverse problems for integrodifferential equations of parabolic type. It considers linear problems together with some abstract generalizations of them. The chapter examines some cases where the memory kernel may depend on some of the space variables. It considers some quite general nonlinear problems and also considers the well-known Lotka-Volterra model with diffusion, arising in population dynamics.
