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Zeitschrift für Analysis und ihre Anwendungen

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Volume 20, Issue 2, 2001, pp. 331–345
DOI: 10.4171/ZAA/1019

Published online: 2001-06-30

Nonlinear Diffusion Equations on Bounded Fractal Domains

Jiaxin Hu[1]

(1) Tsinghua University, Beijing, China

We investigate nonlinear diffusion equations $\frac {\partial u}{\partial t} = \Delta ¢u +f(u)$ with initial data and zero boundary conditions on bounded fractal domains. We show that these equations possess global solutions for suitable $f$ and small initial data by employing the iteration scheme and the maximum principle that we establish on the bounded fractals under consideration. The Sobolev-type inequality is the starting point of this work, which holds true on a large class of bounded fractal domains and gives rise to an eigenfunction expansion of the heat kernel.

Keywords: Diffusion equations, fractals, Laplacian, Sobolev-type inequality, heat kernel, iteration scheme, maximum principle

Hu Jiaxin: Nonlinear Diffusion Equations on Bounded Fractal Domains. Z. Anal. Anwend. 20 (2001), 331-345. doi: 10.4171/ZAA/1019