Degenerate Parabolic Equations : Some Alternative Approaches

The degenerate parabolic operators treated in Chapter III, Example 3.10, have had in last decades a relevant place in the mathematical research for their applications, both in physical, chemical, economical sciences and in probability theory. Our preceding results were obtained on reducing the resolvent (λ + BA)⁻¹ to the form (λB ⁻¹ + A)⁻¹ B ⁻¹ and this forced us to assumptions on the multiplication operator B that do not hold in some important concrete cases. This is due to the fact that the natural domain of BA is too small. Hence in this chapter we shall describe more direct approaches to degenerate parabolic operators that allow us to handle those situations and to establish in some cases the analyticity of the semigroups generated by them in spaces L ² with weight, in spaces L^p , in spaces W ^{1, p} , and above all, in spaces of continuous functions.