Spatial decay estimates in transient heat conduction

The spatial decay of solutions to initial-boundary value problems for the heat equation in a three-dimensional cylinder, subject to non-zero boundary conditions only on the ends, is investigated. It is shown that the spatial decay of end effects in the transient problem is faster than that for the steady-state case. Qualitative methods involving second-order partial differential inequalities for quadratic functionals are first employed. The explicit spatial decay estimates are then obtained by using comparison principle arguments involving solutions of the one-dimensional heat equation . The results give rise to versions of Saint-Venant’s principle in transient heat conduction.