  This paper explores the concept of near-cloaking in the context of
time-dependent heat propagation. We show that after the lapse of a certain
threshold time instance, the boundary measurements for the homogeneous heat
equation are close to the cloaked heat problem in a certain Sobolev space norm
irrespective of the density-conductivity pair in the cloaked region. A
regularised transformation media theory is employed to arrive at our results.
Our proof relies on the study of the long time behaviour of solutions to the
parabolic problems with high contrast in density and conductivity coefficients.
It further relies on the study of boundary measurement estimates in the
presence of small defects in the context of steady conduction problem. We then
present some numerical examples to illustrate our theoretical results.
