Non-linear versus linear diffusion from classical solutions to metasolutions

Abstract

In this paper we analyze the existence, the uniqueness and the stability of the positive steady-states of a sublinear logistic porous media equation involving a weight function vanishing on a region of the support domain. Quite surprisingly, the model possesses a unique positive steady state which attracts to all positive solutions within the range of values of the parameters for which the dynamics of the associated problem with linear diffusion is fully described by metasolutions. Metasolutions are extensions by infinity of large solutions in the support of the weight function. One of the main results of this paper shows how the classical solutions of the porous media equation approximate metasolutions as the nonlinear diffusion rate $m$ approaches one. The main technical difficulties that we must overcome in our analysis are the lost of a priori bounds in the vanishing region of the weight function and the fact that most of the underlying potentials are unbounded on the boundary of the support domain.