Second-order interface equations for nonlinear diffusion with very strong absorption

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 u(t) = (u(m))xx - u(P), x is an element of R, t > 0. 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 greater than or equal to 2, where we have shown that the movement of the interface x = eta(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 N-1(u(., t)) = a(0), where a(0) = a(0)(m,p) is a fixed constant and N-1(u) = (u((m-p)/2))(x) calculated at the interface (again a first-order operator), and (ii) a specific movement law, D(+)eta(t) = N-2(u(., t)), where N-2 is of the second order and D(+)eta(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 is an element of (-1, 1) and for fast diffusion, m is an element of (0, 1), when p is an element of (-m, m). They can be also applied to equations of diffusion-convection type.