Regularity of Interfaces in Diffusion Processes under the Influence of Strong Absorption

Abstract

.We present a method of analysis which allows us to establish the interface equation and to prove Lipschitz continuity of interfaces and solutions which appear in a large class of nonlinear parabolic equations and conservation laws posed in one space dimension. Its main feature is intersection comparison with travelling waves. The method is explained on the following study case: We consider the Cauchy problem for the diffusion-absorption model: $u_t = \left( u^m \right)_{xx} - u^p, \quad u\ge 0,$ in the range of parameters \(m>1,\ 0<p<1,\ m+p\ge2$, i.e., we have slow diffusion combined with strong absorption. Contrary to the case $p\ge 1$, or the purely diffusive equation $u_t=(u^m)_{xx},\ m>1$, where the support of the solution expands with time and the motion is governed by Darcy's law, in the strong absorption range there might appear shrinking interfaces and the interface evolution obeys a different mechanism. Previous methods have failed to provide an adequate analysis of the interface motion and regularity in such a situation.