Decay rates of Saint-Venant type for functionally graded heat-conducting materials

This paper investigates decay rates for the spatial behaviour of solutions for functionally graded heat-conducting materials. From a mathematical point of view, we obtain a new inequality of Poincaré type. This new result allows us to give new decay rates for functionally graded materials when the inhomogeneity depends on the radial variable and the axial variable of the cylinder. The case when the cross-section is increasing is also considered. Besides, we propose to obtain estimates for the case of mixtures. Some pictures illustrate our estimates.