Interfaces in Solutions of Diffusion-absorption Equations in Arbitrary Space Dimension

Abstract

We study the Cauchy-Dirichlet problem for the degenerate parabolic equation

with the parameters a ∊, m > 1, p > 0, satisfying the condition m + p ≥ 2. The problem domain ɛ is the exterior of the cylinder bounded by a simple-connected surface S, supp u ₀ is an annular domain. We show that the velocity of the outer interface Γ = ∂ {supp u(x, t)} is given by the formulawhere II(x, t) is a solution of the degenerate elliptic equation

,depending on t as a parameter. It is proved that the solution and its interface Γ preserve their initial regularity with respect to the space variables, and that they are real analytic functions of time t. We also show that the regularity of the velocity v is better than it was at the initial instant. For the space dimensions n = 1, 2, 3, these results were established in [8]. We propose a modification of the method of [8] that makes it applicable to equations with an arbitrary number of independent variables.

The author was supported by the Spanish Research Project BFM2000-1324 and by the European RTN Programme HPRN-CT-2002-00274.