Wellposedness and regularity for a degenerate parabolic equation arising in a modelof chemotaxis with nonlinear sensitivity

Alexandre Montaru

doi:10.3934/dcdsb.2014.19.231

Article Contents

2014, Volume 19, Issue 1: 231-256. Doi: 10.3934/dcdsb.2014.19.231

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Wellposedness and regularity for a degenerate parabolic equation arising in a model of chemotaxis with nonlinear sensitivity

Alexandre Montaru^1,

1.
Département de Mathématiques - Institut Galilée, Université Paris 13, 99, avenue Jean-Baptiste Clément, 93430 Villetaneuse

Received: November 30, 2012

Revised: July 31, 2013

Published: December 2013

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Abstract

We study a one-dimensional parabolic PDE with degenerate diffusion and non-Lipschitz nonlinearity involving the derivative. This evolution equation arises when searching radially symmetric solutions of a chemotaxis model of Patlak-Keller-Segel type. We prove its local in time wellposedness in some appropriate space, a blow-up alternative, regularity results and give an idea of the shape of solutions. A transformed and an approximate problem naturally appear in the way of the proof and are also crucial in [22] in order to study the global behaviour of solutions of the equation for a critical parameter, more precisely to show the existence of a critical mass.

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Mathematics Subject Classification: 35A01, 35A02, 35A09, 35B44, 35K40, 35K51, 35K65, 35Q92.

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