Heat equation on the arithmetic von Koch snowflake
The author studies the Dirichlet problem for the heat equation \[ u_t'=\Delta u,\;x\in D,t>0,\quad u(0,x)=0,\;x\in D,\quad u(t,x)=1,\;x\in\partial D\tag{1} \] in a bounded domain \(D\subset\mathbb{R}^n\) with a boundary \(\partial D=K_s,\) where \(K_s\) is the \(s\)-adic arithmetic von Koch snowflake. Using the probabilistic representation of the solution to (1) he derives an approximate functional equation for the total amount \(E_{K_s}(t)= \int_{K_s}^{}u(t,x)dx\) of heat in \(K_s\) at time \(t\). This functional equation takes after some transformations the form of an inhomogeneous renewal equation which is further analyzed in the paper. Some asymptotic formulas for \(E_{K_s}(t)\) are derived.
probabilistic representation of the solution
scientific article; zbMATH DE number 1547920