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《数学年刊B辑(英文版)》 2002-04
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HU JIAXINDepartment of Mathematics, Tsinghua University, Beijing 100084, China.  
The Nagumo equationut = △u + bu(u - a)(1 - u), t 0is 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.
【Key Words】: Fractal set Spectral dimension Sobolev-type inequality Strong (Weak) solution
【CateGory Index】: O175.29
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