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Existence, uniqueness and approximation of a doubly-degenerate nonlinear parabolic system

by J.W. Barrett, K. Deckelnick.

Series: 2006-25, Preprints

35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
65M15 Error bounds
65M60 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods

We consider a nonlinear parabolic system which models the
spatiotemporal evolution of a bacterium on a thin film of
nutrient. The nutrient concentration satisfies a reaction
diffusion equation while the equation for the bacterial cell density is of porous medium type. The diffusion coefficient
in that equation also depends in a degenerate way on the nutrient concentration making the system possibly doubly
degenerate. We prove existence and uniqueness of a weak
solution and obtain error bounds for a fully practical
finite element method for the above system.

doubly-degenerate parabolic system, bacterial pattern formation, existence, uniqueness, finite elements, error analysis