Other Boundary Conditions, Nonlinear Diffusion, Asymptotics

In this Chapter, we consider various extensions of the theories in the last chapter in order to include more realistic and general problems. In section 3.2, we study the situation where the boundary condition is coupled, mixed and nonlinear. In prey-predator interaction for example, the predator may be under control at the boundary of the medium, while the prey cannot move across the boundary. Diffusion of predators at the boundary may be adjusted nonlinearly according to populations present, and there might also be some physical limitations to the process. In section 3.3, we analyze the problem when the diffusion rate is density dependent, and thus the Laplacian operator will be modified to become nonlinear and u-dependent. Moreover, the nonlinear nonhomogeneous terms become highly spatially dependent. In section 3.4, we consider the case when diffusion rate of some component is small. More thorough results concerning large-time behavior can be obtained by asympptotic methods. Estimates can be obtained by using an appropriate “reduced”problem.