Dynamical Structure of Some Nonlinear Degenerate Diffusion Equations

We consider degenerate reaction diffusion equations of the form u(t) = Delta u(m) + f(x, u), where f (x, u) similar to a(x) u(p) with 1 <= p < m. We assume that a(x) > 0 at least in some part of the spatial domain, so that u = 0 is an unstable stationary solution. We prove that the unstable manifold of the solution u = 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 -> -infinity 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.