Spatial decay in transient heat conduction for general elongated regions

Abstract

Zanaboni’s procedure for establishing Saint-Venant’s principle is extended to anisotropic homogeneous transient heat conduction on regions that are successively embedded in each other to become indefinitely elongated. No further geometrical restrictions are imposed. The boundary of each region is maintained at zero temperature apart from the common surface of intersection which is heated to the same temperature assumed to be of bounded time variation. Heat sources are absent. Subject to these conditions, the thermal energy, supposed bounded in each region, becomes vanishingly small in those parts of the regions sufficiently remote from the heated common surface. As with the original treatment, the proof involves certain monotone bounded sequences, and does not depend upon differential inequalities or the maximum principle. A definition of an elongated region is presented.