Nonlinear Diffusion Equations on Bounded Fractal Domains

Abstract

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.