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A free boundary problem for aggregation by short range sensing and differentiated diffusion

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  • On the $d$-dimensional torus we consider the drift-diffusion equation, where the diffusion coefficient may take one of two possible values depending on whether the locally sensed density is below or above a given threshold. This can be interpreted as an aggregation model for particles like insect populations or freely diffusing proteins which slow down their dynamics within dense aggregates. This leads to a free boundary model where the free boundary separates densely packed aggregates from areas with a loose particle concentration.
        The paper has a rigorous part and a formal part. In the rigorous part we prove existence of solutions to the distributional formulation of the model. In the second, formal, part we derive the strong formulation of the model including the free boundary conditions and characterize stationary solutions giving necessary conditions for the emergence of stationary plateaus. We conclude that stationary aggregation plateaus in this situation are either spherical, complements of sphericals or stripes, which has implications for biological applications.
        Finally, numerical simulations in one and two dimensions are used to give evidence for the long time convergence to stationary states which feature aggregations.
    Mathematics Subject Classification: Primary: 35Q92; Secondary: 35R35, 35B36.

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