SECOND-ORDER INTERFACE EQUATIONS FOR NONLINEAR DIFFUSION WITH VERY STRONG ABSORPTION
Victor A. Galaktionov, Sergey Shmarev, JUAN L. VAZQUEZ · Communications in Contemporary Mathematics · 1999
We derive the interface equations for weak (or maximal) nonnegative solutions u(x,t) of the porous medium equation with strong absorption in one dimension [Formula: see text] We consider here the very singular range where the exponents m > 1 and p < 1 satisfy 0 < m + p < 2. Unlike the range m + p≥2, where we have shown that the movement of the interface x =η(t) is described by a first-order equation, we prove that in this case there is actually a system of two equations: (i) a universal law N1(u(·,t))=a0, where a0=a0(m,p) is a fixed constant and N1(u)=(u(m - p)/2)x calculated at the interface (again a first-order operator), and (ii) a specific movement law, D+η(t) = N2(u(·,t)), where N2 is of the second order and D+η(t) is the right-hand derivative. We establish the instantaneous smoothing effect and prove optimal gradient bounds on the solutions as well as the second-order estimate on the interface. The analysis is based on intersection comparison with the set of the travelling wave solutions. The results apply to the linear diffusion m=1 with p∈(-1,1) and for fast diffusion, m∈(0,1), when p∈(-m,m). They can be also applied to equations of diffusion-convection type.