Communications in Mathematical Sciences
Volume 9 (2011)
A strongly degenerate parabolic aggregation equation
Pages: 711 – 742
This paper is concerned with a strongly degenerate convection-diffusion equation in one space dimension whose convective flux involves a nonlinear function of the total mass to one side of the given position. This equation can be understood as a model of aggregation of the individuals of a population with the solution representing their local density. The aggregation mechanism is balanced by a degenerate diffusion term describing the effect of dispersal. In the strongly degenerate case, solutions of the nonlocal problem are usually discontinuous and need to be defined as weak solutions. A finite difference scheme for the nonlocal problem is formulated and its convergence to the unique weak solution is proved. This scheme emerges from taking divided differences of a monotone scheme for the local PDE for the primitive. Some numerical examples illustrate the behaviour of solutions of the nonlocal problem, in particular the aggregation phenomenon.
Aggregation, strongly degenerate convection-diffusion equation, nonlocal flux, well-posedness, finite difference scheme
2010 Mathematics Subject Classification
35K65, 65N06, 92Cxx