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

We derive the interface equations for weak (or maximal) nonnegative solutio ns u(x, t) of the porous medium equation with strong absorption in one dime nsion 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 descr ibed by a first-order equation, we prove that in this case there is actuall y 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) calculat ed at the interface (again a first-order operator), and (ii) a specific mov ement 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 smo othing effect and prove optimal gradient bounds on the solutions as well as the second-order estimate on the interface. The analysis is based on inter section comparison with the set of the travelling wave solutions. The resul ts 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.