Diiusion and Bifurcation Problems in Singularly Perturbed Domains

Diiusion problems under singular perturbations of the domain or the boundary conditions are analyzed. The rst problem that we consider is the diiusion of a material from a domain that is nearly impermeable, having only several small patches on the boundary where the material can slowly leak out. The second problem that is studied is the diiusion of a material that originates from some localized regions in a two or three-dimensional domain. Steady-state solutions and the long-time behavior of solutions are analyzed in detail. Finally, the analysis is extended to determine the change in bifurcation values associated with nonlinear diiusion equations under singular perturbations of the domain. The results are then applied to a model in resource management.