LIMIT BEHAVIOR OF FOCUSING SOLUTIONS TO NONLINEAR DIFFUSIONS

The paper is concerned with the behaviour of focusing solutions to non linear diffusion problems. These solutions describe the movement of a flow filling a hole and have consequences for the qualitative theory o f degenerate nonlinear parabolic equations. The general equation under study is the so-called doubly nonlinear diffusion equation u(t)(x, t) = Delta(p)(u(m))(x, t), with parameters m > 0 and p > 1 such that m(p - 1) > 1 so that the finite propagation property holds and free bound aries occur. Well-known particular cases are the Porous Medium Equatio n and the evolutionary p-Laplacian Equation. We study the behaviour of the families of selfsimilar focusing solutions as the parameters m an d p tend to their limiting values and identify the limit problems thes e limits solve. In the case m(p - 1) --> 1 we find as appropriate asym ptotic problems a family of Hamilton-Jacobi equations, When we let m - -> infinity we obtain in the limit the Hele-Shaw problem. When p --> i nfinity we obtain linear travelling waves with arbitrary speed, soluti ons of a certain infinity-Laplacian evolution problem.