The porous medium equation in one dimension

We consider a second order nonlinear degenerate parabolic partial differential equation known as the porous medium equation, restricting our attention to the case of one space variable and to the Cauchy problem where the initial data are nonnegative and have compact support consisting of a bounded interval. Solutions are known to have compact support for each fixed time. In this paper we study the lateral boundary, called the interface, of the support P [u] of the solution in RI x (0, T). It is shown that the interface consists of two monotone Lipschitz curves which satisfy a specified differential equation. We then prove results concerning the behavior of the interface curves as t approaches zero and as t approaches infinity, and prove that the interface curves are strictly monotone except possibly near t = 0. We conclude by proving some facts about the behavior of the solution in P[u]. Introduction. Consider the parabolic Cauchy problem (0.1) u, =a2c(u)/8x2 inR1 X (0, T), (0.2) u(x, 0) = uo(x) inR', where uo(x) is a bounded continuous nonnegative function and 0(u0) E C 0"(R 1). The function p belongs to a class (Do of smooth positive functions that are like a functibn ?(u) = Ur for some real constant m > 1. In particular, we assume p(0) = 4'(0) = 0 and that ?(u), ?'(u) and ?"(u) are positive when u > 0. Also, f'(()/t dX and sup0 1), equation (0.1) governs the flow of a gas in a porous medium where u(x, t) is proportional to the denisty of the gas at (x, t). Several authors have studied (0.1), (0.2) and other boundary value problems of (0.1). Oleinik, Kalashnikov and Yui-Lin [14] established the existence and uniqueness of a class of continuous weak solutions u(x, t) of (0.1), (0.2) and derived some elementary properties. They proved that u is a classical solution of (0.1) in the open set P [u] wherein u > 0, and that u(xo, t) > 0 for Received by the editors January 27, 1976. AMS (MOS) subject classifications (1970). Primary 35K15, 35K55; Secondary 35B99.