Limit behaviour of focusing solutions to nonlinear diffusions

Abstract

The paper is concerned with the behaviour of focusing solutions to nonlinear diffusion problems. These solutions describe the movement of a flow filling a hole and have consequences for the qualitative theory of degenerate nonlinear parabolic equations. The general equation under study is theso-called doubly nonlinear diffusion equation a²with parameters m > 0 and p > 1 such that m(p - 1) > 1 so that the finite propagation property holds and free boundaries occur. Well-known particular cases are the Porous Medium Equation and the evolutionary p-

Laplacian Equation. We study the behaviour of the families of selfsimilar focusing solutions as the parameters m and p tend to their limiting values and identify the limit problems these limits solve. In the case m(p - 1) -+ 1 we find as appropriate asymptotic problems a family of Hamilton-Jacobi equations. When we let m + ffi we obtain in the limit the Hele-Shaw problem. When p + cc we

obtain linear travelling waves with arbitrary speed, solutions of a certain ∞-Laplacian evolution problem.