Convergence of a time discrete scheme for a chemotaxis-consumption model
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In the present work we propose and study a time discrete scheme for the following chemotaxis-consumption model (for any s≥ 1), ∂_t u - Δ u = - ∇· (u ∇ v), ∂_t v - Δ v = - u^s v in (0,T)×Ω, endowed with isolated boundary conditions and initial conditions, where (u,v) model cell density and chemical signal concentration. The proposed scheme is defined via a reformulation of the model, using the auxiliary variable z = √(v + α^2) combined with a Backward Euler scheme for the (u,z) problem and a upper truncation of u in the nonlinear chemotaxis and consumption terms. Then, two different ways of retrieving an approximation for the function v are provided. We prove the existence of solution to the time discrete scheme and establish uniform in time a priori estimates, yielding to the convergence of the scheme towards a weak solution (u,v) of the chemotaxis-consumption model.
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