The asymptotic behavior of gas in an $n$-dimensional porous medium

Abstract:

Consider the flow of gas in an n-dimensional porous medium with initial density ${u_0}(x) \geqslant 0$. The density $u(x, t)$ then satisfies the nonlinear degenerate parabolic equation ${u_t} = \Delta {u^m}$ where $m > 1$ is a physical constant. Assuming that $I \equiv \int { {u_0}(x)} dx < \infty$ it is proved that $u(x, t)$ behaves asymptotically, as $t \to \infty$, like the special (explicitly given) solution $V(|x|, t)$ which is invariant by similarity transformations and which takes the initial values $\delta (x)I (\delta (x) =$ the Dirac measure) in the distribution sense.