# Preprint 2010-010

# A strongly degenerate parabolic aggregation equation

## Fernando Betancourt, Raimund Bürger and Kenneth H. Karlsen

**Abstract:**
This paper is concerned with a strongly degenerate
convection-diffusion equation in one space dimension
whose convective flux involves a non-linear 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 accounting for dispersal.
In the strongly degenerate case,
solutions of the non-local problem are usually discontinuous
and need to be defined as weak solutions
satisfying an entropy condition.
A finite difference scheme for the non-local problem is formulated
and its convergence to the unique entropy solution is proved.
The scheme emerges from taking divided differences
of a monotone scheme for the local PDE for the primitive.
Numerical examples illustrate the behaviour
of entropy solutions of the non-local problem,
in particular the aggregation phenomenon.