Dynamical Structure of Some Nonlinear Degenerate Diffusion Equations

Abstract

We consider degenerate reaction diffusion equations of the form ut = Δum + f(x, u), where f(x, u) ~ a(x)up with 1 ≤ p 0 at least in some part of the spatial domain, so that \({u \equiv 0}\) is an unstable stationary solution. We prove that the unstable manifold of the solution \({u \equiv 0 }\) has infinite Hausdorff dimension, even if the spatial domain is bounded. This is in marked contrast with the case of non-degenerate semilinear equations. The above result follows by first showing the existence of a solution that tends to 0 as \({t\to -\infty}\) while its support shrinks to an arbitrarily chosen point x* in the region where a(x) > 0, then superimposing such solutions, to form a family of solutions of arbitrarily large number of free parameters. The construction of such solutions will be done by modifying self-similar solutions for the case where a is a constant.