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 ) ⩾ 0 {u_0}(x)\, \geqslant \,0 . The density u ( x , t ) u(x,\,t) then satisfies the nonlinear degenerate parabolic equation u t = Δ u m {u_t}\, = \,\Delta {u^m} where m > 1 m\, > \,1 is a physical constant. Assuming that I ≡ ∫ u 0 ( x ) d x > ∞ I\, \equiv \,\int {\,{u_0}(x)} dx\, > \,\infty it is proved that u ( x , t ) u(x,\,t) behaves asymptotically, as t → ∞ t\, \to \,\infty , like the special (explicitly given) solution V ( | x | , t ) V(|x|,\,t) which is invariant by similarity transformations and which takes the initial values δ ( x ) I ( δ ( x ) = \delta (x)I\,(\delta (x)\, = \, the Dirac measure) in the distribution sense.