HEAT EQUATIONS DEFINED BY SELF-SIMILAR MEASURES WITH OVERLAPS

We study the heat equation on a bounded open set U subset of R-d supporting a Borel measure. We obtain asymptotic bounds for the solution and prove the weak parabolic maximum principle. We mainly consider self-similar measures defined by iterated function systems with overlaps. The structures of these measures are in general complicated and intractable. However, for a class of such measures that we call essentially of finite type, important information about the structure of the measures can be obtained. We make use of this information to set up a framework to study the associated heat equations in one dimension. We show that the heat equation can be discretized and the finite element method can be applied to yield a system of linear differential equations. We show that the numerical solutions converge to the actual solution and obtain the rate of convergence. We also study the propagation speed problem.

Automated summary

We study the heat equation on a bounded open set U subset of R-d supporting a Borel measure. We obtain asymptotic bounds for the solution and prove the weak parabolic maximum principle. We mainly consider self-similar measures defined by iterated function systems with overlaps. Important information about the structure of these measures can be obtained for a class of measures that we call essentially of finite type. We make use of this information to set up a framework to study the associated heat equations in one dimension. We discretize the heat equation and apply the finite element method to yield a system of linear differential equations. We show the convergence of numerical solutions to the actual solution and provide the rate of convergence. We also investigate the propagation speed problem.