  We study the long-time asymptotics of prototypical non-linear diffusion
equations. Specifically, we consider the case of a non-degenerate diffusivity
function that is a (non-negative) polynomial of the dependent variable of the
problem. We motivate these types of equations using Einstein's random walk
paradigm, leading to a partial differential equation in non-divergence form. On
the other hand, using conservation principles leads to a partial differential
equation in divergence form. A transformation is derived to handle both cases.
Then, a maximum principle (on both an unbounded and a bounded domain) is
proved, in order to obtain bounds above and below for the time-evolution of the
solutions to the non-linear diffusion problem. Specifically, these bounds are
based on the fundamental solution of the linear problem (the so-called
Aronson's Green function). Having thus sandwiched the long-time asymptotics of
solutions to the non-linear problems between two fundamental solutions of the
linear problem, we prove that, unlike the case of degenerate diffusion, a
non-degenerate diffusion equation's solution converges onto the linear
diffusion solution at long times. Select numerical examples support the
mathematical theorems and illustrate the convergence process. Our results have
implications on how to interpret asymptotic scalings of potentially anomalous
diffusion processes (such as in the flow of particulate materials) that have
been discussed in the applied physics literature.
