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A strongly degenerate parabolic aggregation equation. (English) Zbl 1272.35125

Summary: 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.

MSC:

35K65 Degenerate parabolic equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35R09 Integro-partial differential equations
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