Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium

The one-dimensional porous media equation u, = (um)xx, m > 1, is considered for x 6E R, r>0 with initial conditions u(x,Q) = u0(x) integrable, nonnegative and with compact support. We study the behaviour of the solutions as l -» oo proving that the expressions for the density, pressure, local velocity and interfaces converge to those of a model solution. In particular the first term in the asymptotic development of the free-boundary is obtained. 0. Introduction. Suppose we have a certain distribution of gas whose density at time t = 0 is given by a function u0(x) of one spatial direction (x E R). If the gas flows through a homogeneous porous medium the density u = m(x, t) at time t > 0 is governed by the equation (0.1) u,= iu">)xx for x E R and t > 0; m is a. physical constant, m > 1, and we have scaled out other physical constants (see [1] for a physical derivation), u satisfies the initial condition (0.2) k(x,0) = «„(*) where u0 satisfies the following assumptions: (0.3) w0GF'(R), «„>0,ii„z0, and u0 is compactly supported, i.e. if ß0 = {x E R: u0ix) > 0} we have (0.4) a, = essinf ß0 >-oo, a2 — esssupß0 < oo. Sticking to the above application we define the pressure by v = mum~x/im — 1) on Q = R X (0, oo ) and the local velocity by V = -vx on the domain of dependence (0.5) ß = ß[w] = {(x,t) E Q: u{x,t)>0}. The total mass at time i>0 is Af(i) = / w(x, t) dx and the center of mass is xcit) — Mit)'1 j w(x, t)x dx. Set M0 = / u0ix) dx and x0 — M0~'/ u0ix)x dx: M0 > 0 and ax < x0< a2. l0 = a2 — ax measures the dispersion of the initial data. Much is already known for problem (0.1)-(0.4); see [19] for a survey of results up to 1980, where the «-dimensional case is considered, n > 1. In particular (0.1)-(0.4) Received by the editors July 29, 1981 and, in revised form, February 1, 1982. This paper has been presented at the Symposium on Free-Boundary Problems Theory and Applications held in June, 1981, in Montecatini (Italy). 1980 Mathematics Subject Classification. Primary 35K65, 76S05, 35B40.