Radial Equivalence of Nonhomogeneous Nonlinear Diffusion Equations

We establish one-to-one transformations and self-maps between nonlinear diffusion equations in nonhomogeneous media, where the density function is given by a power. We use these transformations to deduce new interesting self-similar, radially symmetric solutions of the equations. In particular, Barenblatt, dipole and focusing Aronson-Graveleau type solutions are deduced, and some equations with singular potentials are studied. The new solutions are example of interesting or unexpected mathematical features of these equations, providing also natural candidates for the asymptotic behavior.