The Heat Content Asymptotics of a Time Dependent Process

Let M be a compact manifold with smooth boundary. We study the heat content asymptotics on M defined by a time dependent heat source and time dependent boundary conditions. We adopt the following notational conventions: let M be a compact Riemannian manifold with smooth boundary ∂M , let ∆ = δd be the scalar Laplacian on M , let x be a point of the interior of M , let y be a point of the boundary of M , and let t ≥ 0 be the time parameter. Decompose ∂M = CN ∪ CD as the union of two disjoint closed sets where we permit CN or CD to be empty. If u is a smooth function on M , let u;m be the inward unit normal derivative of u on CN . Let B be the boundary operator defined by an auxiliary function ψ; on the Neumann boundary component we pump heat into the manifold at a rate determined by ψ and on the Dirichlet component we keep the boundary at temperature ψ. Then u satisfies the boundary conditions (i.e. Bu = ψ) means that for t > 0 we have: (1) u(y; t) = ψ;m(y; t) if y ∈ CN and u(y; t) = ψ(y, t) if y ∈ CD. Let p(x; t) be a heat source on M and let φ(x) be the initial temperature. The temperature distribution u = up,φ,ψ(x; t) defined by these data is the solution to the equations: (2) (∂t + ∆)u(x; t) = p(x; t), u(x; 0) = φ(x), and Bu = ψ. Let ρ(x; t) be the specific heat. Let dx and dy be the Riemannian measures on M and on ∂M determined by the metric. The heat energy content β is defined by: (3) β(p, φ, ψ, ρ)(t) := ∫ M up,φ,ψ(x; t)ρ(x; t)dx. We assume that (p, φ, ψ, ρ) are smooth. We refer to Grubb [9] for details concerning the calculus of pseudo-differential operators for manifolds with boundary. Standard elliptic methods show that β has an asymptotic expansion as t ↓ 0 of the form: (4) β(p, φ, ψ, ρ)(t) ∼ ∑ n≥0 βn(p, φ, ψ, ρ)t . Version S05v6.tex printed 2 October 2001