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Cell polarisation model: The 1D case
Authors:Thomas Lepoutre  Nicolas Meunier  Nicolas Muller
Institution:1. Institut Camille Jordan, Université Claude Bernard Lyon 1, France;2. MAP5, CNRS UMR 8145, Université Paris Descartes, 45 rue des Saints Pères, 75006 Paris, France
Abstract:We study the dynamics of a one-dimensional non-linear and non-local drift-diffusion equation set in the half-line, with the coupling involving the trace value on the boundary. The initial mass M of the density determines the behaviour of the equation: attraction to self-similar profile, to a steady state of finite time, blow-up for supercritical mass. Using the logarithmic Sobolev and the HWI inequalities we obtain a rate of convergence for the sub-critical and critical mass cases. Moreover, we prove a comparison principle on the equation obtained after space integration. This concentration-comparison principle allows proving blow-up of solutions for large initial data without any monotonicity assumption on the initial data.
Keywords:Blow-up  Asymptotic convergence  Logarithmic Sobolev inequality  HWI inequality
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