In this work, we investigate the dynamics of a nonlocal model describing spontaneous cell polarization. It consists of a drift-diffusion equation set in the half-space, with the coupling involving the trace value on the boundary. We characterize the following behaviors in the one-dimensional case: solutions are global if the mass is below the critical mass and they blow up in finite time above the critical mass. The higher-dimensional case is also discussed. The results are reminiscent of the classical Keller–Segel system, but critical spaces are different ($L^N$ instead of $L^{N/2}$ due to the coupling on the boundary). In addition, in the one-dimensional case we prove quantitative convergence results using relative entropy techniques. This work is complemented with a more realistic model that takes into account dynamical exchange of molecular content at the boundary. In the one-dimensional case we prove that blow-up is prevented. Furthermore, density converges toward a nontrivial stationary configuration.

MSC codes

  1. 35B60
  2. 35B44
  3. 35Q92
  4. 92C17
  5. 92B05


  1. cell polarization
  2. global existence
  3. blow-up
  4. asymptotic convergence
  5. entropy method
  6. Keller–Segel system

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Information & Authors


Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 594 - 622
ISSN (online): 1095-712X


Submitted: 23 May 2011
Accepted: 25 January 2012
Published online: 12 April 2012

MSC codes

  1. 35B60
  2. 35B44
  3. 35Q92
  4. 92C17
  5. 92B05


  1. cell polarization
  2. global existence
  3. blow-up
  4. asymptotic convergence
  5. entropy method
  6. Keller–Segel system



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