Tracking the Interface of the Diffusion-Absorption Equation: Theoretical Analysis

Abstract

This work is devoted to the theoretical study of the Cauchy problem for the degenerate parabolic equation of the diffusion-absorption type $u_t=\varDelta u^m-au^q$ with the exponents $m>1$, $q>0$, $m+q\ge 2$ and constant $a>0$. We propose an algorithm for tracking the interface in the case of arbitrary $m>1$ and $q>0$. Based on the idea of Shmarev (Nonlinear Anal 53:791–828, 2003; Progr Nonlinear Diff Eqn Appl Birkhäuser, Basel 61:257–273, 2005), we transform the moving support of the solution into a time-independent domain by means of introduction of a local system of Lagrangian coordinates. In the new coordinate system the problem converts into a system of nonlinear differential equations, which describes the motion of a continuous medium. This system is solved by means of the modified Newton method, which allows one to reduce the nonlinear problem to a sequence of linear degenerate problems. We formulate the problem in the framework of Sobolev spaces and prove the convergence of the sequence of approximate solutions to the solution of the original problem.