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| THE NAGUMO EQUATION ON SELF-SIMILAR FRACTAL SETS |
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Citation： |
HU Jiaxin.THE NAGUMO EQUATION ON SELF-SIMILAR FRACTAL SETS[J].Chinese Annals of Mathematics B,2002,23(4):519~530 |
| Page view： 1163
Net amount： 785 |
Authors： |
HU Jiaxin; |
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| Abstract： |
The Nagumo equation
$$
u_{t}=\Delta u + bu(u-a)(1-u),\quad t>0
$$
is investigated with initial data and zero Neumann boundary conditions on post-critically finite (p.c.f.) self-similar fractals that have regular harmonic structures and satisfy the separation condition. Such a nonlinear diffusion equation has no travelling wave solutions because of the “pathological” property of the fractal. However, it is shown that a global Holder continuous solution in spatial variables exists on the fractal considered. The Sobolev-type inequality plays a crucial role, which holds on such a class of p.c.f self-similar fractals. The heat kernel has an eigenfunction expansion and is well-defined due to a Weyl's formula. The large time asymptotic behavior of the solution is discussed, and the solution tends exponentially to the equilibrium state of the Nagumo equation as time tends to infinity if b is small. |
Keywords： |
Fractal set, Spectral dimension, Sobolev-type inequality, Strong (Weak) solution |
Classification： |
35L28 |
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