THE NAGUMO EQUATION ON SELF-SIMILAR FRACTAL SETS

作     者：HU JIAXINDepartment of Mathematics, Tsinghua University, Beijing 100084, China. HU JIAXINDepartment of Mathematics, Tsinghua University, Beijing 100084, China.

作者机构：Department of Mathematics Tsinghua University Beijing 100084 China 

出 版 物：《Chinese Annals of Mathematics,Series B》 (数学年刊（B辑英文版）)

年 卷 期：2002年第23卷第4期

页      面：519-530页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

主　　题：Nagumo方程 自相似分形集 非线性扩散方程 行波解 Holer连续解 Sobolev不等式 特征函数 Weyl公式 谱 非对称行为 

摘      要：The Nagumo equation ut = △u+ bu(u-a)(1-u), t0is 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.