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# Communications in Mathematical Sciences

## Volume 9 (2011)

### Number 3

### A strongly degenerate parabolic aggregation equation

Pages: 711 – 742

DOI: https://dx.doi.org/10.4310/CMS.2011.v9.n3.a4

#### Authors

#### Abstract

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.

#### Keywords

Aggregation, strongly degenerate convection-diffusion equation, nonlocal flux, well-posedness, finite difference scheme

#### 2010 Mathematics Subject Classification

35K65, 65N06, 92Cxx

Published 11 March 2011