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

Haskovec, Jan and Oelz, Dietmar (2015) A free boundary problem for aggregation by short range sensing and differentiated diffusion. Discrete and Continuous Dynamical Systems - Series B, 20 5: 1461-1480. doi:10.3934/dcdsb.2015.20.1461

Author Haskovec, Jan
Oelz, Dietmar
Title A free boundary problem for aggregation by short range sensing and differentiated diffusion
Journal name Discrete and Continuous Dynamical Systems - Series B   Check publisher's open access policy
ISSN 1553-524X
Publication date 2015-07-01
Sub-type Article (original research)
DOI 10.3934/dcdsb.2015.20.1461
Open Access Status Not yet assessed
Volume 20
Issue 5
Start page 1461
End page 1480
Total pages 20
Place of publication Springfield, MO, United States
Publisher American Institute of Mathematical Sciences
Language eng
Formatted abstract
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.
Keyword Parabolic equation with discontinuous coefficients
Piecewise constant volatility
Differentiated diffusion
Parabolic free boundary problem
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status Non-UQ

Document type: Journal Article
Sub-type: Article (original research)
Collection: School of Mathematics and Physics
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Created: Sat, 14 Jan 2017, 01:56:47 EST by Kay Mackie on behalf of School of Mathematics & Physics