Homogenization of the Heat Equation with a Vanishing Volumetric Heat Capacity

Abstract

This paper is a study of the homogenization of the heat conduction equation, with a homogeneous Dirichlet boundary condition, having a periodically oscillating thermal conductivity and a vanishing volumetric heat capacity. In particular, the volumetric heat capacity equals ε ^q and the thermal conductivity oscillates with period ε in space and ε ^r in time, where 0 < q < r are real numbers. By using certain evolution settings of multiscale and very weak multiscale convergence we investigate, as ε tends to zero, how the relation between the volumetric heat capacity and the microscopic structure affects the homogenized problem and its associated local problem. It turns out that this relation gives rise to certain special effects in the homogenization result.