Interfaces in Solutions of Diffusion-absorption Equations in Arbitrary Space Dimension
Sergey Shmarev · Progress in nonlinear differential equations and their applications · 2005
We study the Cauchy-Dirichlet problem for the degenerate parabolic equation Open image in new window with the parameters a ∊ Open image in new window , 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 Open image in new window . We show that the velocity of the outer interface Γ = ∂ {supp u(x, t)} is given by the formula Open image in new window where II(x, t) is a solution of the degenerate elliptic equation Open image in new window ,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.