The asymptotic behavior of gas in an -dimensional porous medium

Consider the flow of gas in an «-dimensional porous medium with initial density u0(x) > 0. The density u(x, l) then satisfies the nonlinear degenerate parabolic equation u, = Awm where m > 1 is a physical constant. Assuming that I = S "o(x)<tx < oo it is proved that u(x, t) behaves asymptotically, as Z -» oo, like the special (explicitly given) solution V(\x\, t) which is invariant by similarity transformations and which takes the initial values 8(x)I ($(x) = the Dirac measure) in the distribution sense. 1. Statement of the main result. Consider the Cauchy problem for u(x, t): u,=Aum (x E R",t >0), (1.1) u(x, 0) = u0(x) (x E R"). (1.2) The function u represents the density of a gas in a porous medium and m is a physical constant, m > 1. We assume that u0(x) is continuous, u0(x) > 0, u0(x) ^ 0, u0(x) < M, u0 e Ll(R") n L2(7T) (M constant), (1.3) and set I = f u0(x)dx. (1.4) J R" By a weak solution of (1.1), (1.2) we mean a function u satisfying, for any T > 0, fT f \(u(x,t))2+\Vxum(x,t)\2]dxdt<<* J0 JRn<J and fj iRh it ~ v-"m 'v J)dx dt+fRMx)f{x)dx=° for any continuously differentiable function / with compact support in 7? " X [0, T). It is well known [14], [11] that given u0 satisfying (1.3), (1.4) there exists a unique generalized solution « of (1.1), (1.2) and f u(x, t)dx = 7 for all z > 0; (1.5) JRn Received by the editors August 27, 1979. AMS (MOS) subject classifications (1970). Primary 35K55, 76S05; Secondary 35K15. 'This work is partially supported by National Science Foundation Grant MCS-7817204 and AFOSR Grant 78-3602. © 1980 American Mathematical Society 0002-9947/80/0000-0 565/$04.2 S