Dynamical Structure of Some Nonlinear Degenerate Diffusion Equations |
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Authors: | Hiroshi Matano M Assunta Pozio |
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Institution: | 1. Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153-8914, Japan 2. Dipartimento di Matematica ??G. Castelnuovo??, Sapienza Universit?? di Roma, P.le A. Moro, 2, 00185, Rome, Italy
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Abstract: | We consider degenerate reaction diffusion equations of the form u t ?=???u m ?+?f(x, u), where f(x, u) ~ 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 ${u \equiv 0}$ 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. |
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