Lifespan of effective boundary conditions for the heat equation

Thermal barrier coatings are used to protect from overheating isotropically conducting bodies. Compared to the physical nature of the bodies, the coating layers are thin and anisotropic, and their thermal conductivity is small. To resolve the issue caused by multiscale in the scenario, the asymptotic behavior of the Dirichlet problem for the heat equation is studied and effective boundary conditions (boundary conditions satisfied by the limiting function on the boundary of the isotropically body) are derived [Li et al., Proc. Am. Math. Soc. 137, 1711-1721 (2009)] as the thickness of the coating layer shrinks. It is shown that the convergence holds in any fixed finite time interval. This paper is devoted to finding the maximal time interval (called lifespan) in which the uniform convergence still holds when the effective boundary condition is of the Dirichlet or Robin type. In particular, we show that the lifespan is, indeed, infinite, provided that the source term itself or the source term minus a time-independent function is square integrable over the half space. Convergence rates are also obtained. Published under an exclusive license by AIP Publishing.