| This dissertation is devoted to the study of several types of reaction diffusion equations on domains with thin layers. The common feature is that the thermal tensor or diffusion rate differs significantly in size and/or nature on the large component of the entire domain and on the thin layer. The physical situations include the protection of spacecrafts and turbine engine blades (of metallic nature) by thermal barrier coatings (of ceramic nature), and the effect of a road or a buffer zone in a nature reserve. Such types of PDEs are associated with at least two issues. It is hard to see the effect of the thin layers, and it is numerically challenging to solve them due to the small scales involved. Our resolution here is to study the asymptotic behavior of the solutions to the PDEs and seek the effective boundary conditions (EBCs) on the boundary of the large components, as the thickness of the thin layers shrink. EBCs enable us to see the effect of the thin layers transparently. Furthermore, with the help of EBCs, we can simply solve the PDEs on the large components of the domains, which involve no small scales any more.;We first study the asymptotic behavior and EBCs of the linear heat equation on a body coated by functionally graded material (FGM). The motivation is that, practically, in thermal barrier coatings, there is often coating failure due to the large stress between the ceramic topcoat and the metallic surface, while FGM is meant to replace the sharp interface with a gradient interface that makes a smooth transition from one material to the other. We model the FGM coating by assuming that the material is graded in the normal direction of the boundary of the metallic body and generalize some previous results. In particular, it is shown that there are more options in order to perfectly protect the body due to the introduction of FGM coating.;Next, we investigate the coating problem for the logistic diffusion equation in the context of ecology. Here we interpret the body as a nature reserve, and the coating layer a buffer zone where small diffusion rate occurs. It turns out that the asymptotic behavior of the density function ;Finally, we consider the logistic diffusion equation on the entire plane, including a horizontal strip of width ;+;{-1}right)... |
