The Geometric Setting and an Alternative

For degenerate or singular equations, the energy and logarithmic estimates are not homogeneous, as we have seen in the previous chapter. They involve integral norms corresponding to different powers, namely the powers 2 and p. To go about this difficulty, the equation has to be analyzed in a geometry dictated by its own degenerate structure. This amounts to rescale the standard parabolic cylinders by a factor that depends on the oscillation of the solution. This procedure of intrinsic scaling, which can be seen as an accommodation of the degeneracy, allows for the restoration of the homogeneity in the energy estimates, when written over the rescaled cylinders. We can say heuristically that the equation behaves in its own geometry like the heat equation. Let us make this idea precise.