ON SOME REACTION-DIFFUSION SYSTEMS WITH NONLINEAR DIFFUSION ARISING IN BIOLOGY

摘要

We consider a class of reaction-diffusion systems with resource-consumer interaction. Such systems have been thoroughly studied in a number of mathematical articles such as those of Alikakos, Masuda, Haraux and Youkana, Hoshino and Kanel. Their main results concern the well-posedness of the parabolic problems, L~∞ bounds on the solutions which do not depend on time and a study of their large time behavior; it turns out that the solution pairs converge to constants as t tends to infinity. However their transient behavior may be very complex and phenomena such as tip splitting may occur at intermediate times. Recently mathematical biologists have introduced nonlinear diffusion in several models and they have observed new instability phenomena. In particular replacing linear by nonlinear diffusion may cause the destabi-lization of planar interfaces. Our purpose is to revisit the mathematical results obtained for the systems with linear diffusion and show how they can be extended to the case of nonlinear diffusion. We cannot use anymore arguments based on Green potentials and linear semi-group theory and our methods of proof are based on energy estimates and on the construction of sub- and supersolutions.