Nonlinear diffusion from a delocalized source: affine self-similarity,time reversal, & nonradial focusing geometries

A family of explicit solutions is described, to the porous medium equation in its full range of nonlinearities (plus some analogous fourth-order diffusions), in which the pressure is given by a quadratic function of space at each instant in time. These include spreading solutions whose source is concentrated on any conic region of dimension lower than the ambient space, and solutions which focus at conic regions. The singular limiting distributions are affine projections of Barenblatt type solutions (with arbitrary signature) onto lower dimensional subspaces. All affine images of Barenblatt solutions form an invariant space on which the dynamics can be integrated explicitly. A time-reversal symmetry is revealed for the pressure equation which transforms spreading solutions to focusing solutions, and vice-versa. This yields new information about the long and short time asymptotics of finite-mass solutions, about the instability of focusing, and about singularity geometry.