Combining fast, linear and slow diffusion

Abstract

Although the pioneering studies of G. I. Barenblatt [< i> On some unsteady motions of a liquid or a gas in a porous medium< /i> , Prikl. Mat. Mekh. < b> 16< /b> (1952), 67–68] and A. G. Aronson and L. A. Peletier [< i> Large time behaviour of solutions of some porous medium equation in bounded domains< /i> , J. Differential Equations < b> 39< /b> (1981), 378–412] did result into a huge industry around the porous media equation, none further study analyzed the effect of combining fast, slow, and linear diffusion simultaneously, in a spatially heterogeneous porous medium. Actually, it might be this is the first work where such a problem has been addressed. Our main findings show how the heterogeneous model possesses two different regimes in the presence of a priori bounds. The minimal steady-state of the model exhibits a genuine {\it fast diffusion behavior}, whereas the remaining states are rather reminiscent of the purely {\it slow diffusion model}. The mathematical treatment of these heterogeneous problems should deserve a huge interest from the point of view of its applications in fluid dynamics and population evolution.