Cell polarisation model: The 1D case |
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Authors: | Thomas Lepoutre Nicolas Meunier Nicolas Muller |
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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 |
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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. |
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Keywords: | Blow-up Asymptotic convergence Logarithmic Sobolev inequality HWI inequality |
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