A Comparison Estimate for the Heat Equation with an Application to the Heat Content of the S ‐Adic Von Koch Snowflake

Let D be an open set in Euclidean space R m with boundary ∂ D , and let φ:∂ D →[0, ∞) be a bounded, measurable function. Let u : D ∪∂ D ×[0, ∞)→[0, ∞) be the unique weak solution of the heat equation [formula] with initial condition [formula] and with inhomogeneous Dirichlet boundary condition [formula] Then u ( x ; t ) represents the temperature at a point x ∈ D at time t if D has initial temperature 0, while the temperature at a point x ∈∂ D is kept fixed at φ( x ) for all t >0. We define the total heat content (or energy) in D at time t by [formula] In this paper we wish to examine the effect of imposing additional cooling on some subset C on both u and E D . 1991 Mathematics Subject Classification 35K05, 60J65, 28A80.