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BULLETIN (New Series) of the American Mathematical Society MANUSCRIPT, PROOF AND COPYING INFORMATION

The prototypical degenerate diffusion equation is the porous medium equation (PME) ∂tu = ∆(u ), where u(x, t) ≥ 0, m > 1 is a constant, and ∆ is the Laplace operator in R for d ≥ 1. Since ∆(u) = ∇ · (mum−1∇(u)), the quantitymum−1 plays the role of the diffusion coefficient. The PME is parabolic for u > 0, but degenerates at u = 0 since the diffusion coefficient vanishes there. The earliest occurrence of the PME in the scientific literature seems to be in 1903– 04 in the work of J. Boussinesq on ground water flow [5]. In an approximate theory of almost horizontal flow, he obtained the PME with m = 2 for the height of the water mound. Subsequently, in the 1930s the petroleum engineers L. S. Leibenzon [9] and M. Muskat [10] independently derived the PME for the evolution of the density of an ideal gas flowing isentropically in a homogeneous porous medium. This flow is characterized by the ideal gas law, the conservation of mass, and an empirical relationship between the pressure and the velocity known as Darcy’s law. Solving these equations for the density and scaling out the constants leads to the PME with m = 1 + γ, where γ ≥ 1 is the exponent in the ideal gas law. In both the ground water and ideal gas flows m ≥ 2, but various values of m > 1 occur in other applications. Much more information can be found in Vazquez’s monograph [14]. One of the most striking manifestations of the degeneracy of the PME is the finite speed of propagation of disturbances from rest. Let u(x, t;u0) denote the solution to the Cauchy (initial value) problem