Combining fast, linear and slow diffusion

Julian López-Gómez, Antonio Suárez



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.


Heterogeneous nonlinear diffusion; fast; slow and linear diffusion

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