Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models

Ansgar Jüngel; Oliver Leingang

doi:10.3934/dcdsb.2019029

Article Contents

2019, Volume 24, Issue 9: 4755-4782. Doi: 10.3934/dcdsb.2019029

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Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models

Ansgar Jüngel and
Oliver Leingang

Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria

Received: August 31, 2017

Revised: October 31, 2017

Early access: February 2019

Published: July 2019

The authors acknowledge partial support from the Austrian Science Fund (FWF), grants P27352, P30000, F65, and W1245

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Abstract

The existence of weak solutions and upper bounds for the blow-up time for time-discrete parabolic-elliptic Keller-Segel models for chemotaxis in the two-dimensional whole space are proved. For various time discretizations, including the implicit Euler, BDF, and Runge-Kutta methods, the same bounds for the blow-up time as in the continuous case are derived by discrete versions of the virial argument. The theoretical results are illustrated by numerical simulations using an upwind finite-element method combined with second-order time discretizations.

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Mathematics Subject Classification: Primary: 35B44; Secondary: 35Q92, 65L06, 65M20, 92C17.

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Figure 1. Cell density computed from the BDF-2 scheme at times $ t = 0 $ (top left), $ t = 0.005 $ (top right), $ t = 0.007 $ (bottom left), $ t = 0.02 $ (bottom right)

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Figure 2. Cell density computed from the BDF-2 scheme at times $ t = 0 $ (top left), $ t = 0.005 $ (top right), $ t = 0.02 $ (bottom left), $ t = 0.1001 $ (bottom right)

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Figure 3. $ L^p $ error $ e_p $ for $ p = 1,2,4,\infty $ at time $ T = 0.01 $ for various time step sizes $ \tau_k = \tau $ (left: BDF-2 discretization; right: midpoint discretization)

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Figure 4. Cell density at time step $ k = 0 $ (left) and $ k = k_{\rm max} = 44 $. The mesh size is $ h = 0.02 $

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Figure 5. The residuum $ R_k $ for time steps 1 to 81 (left) and time steps 30 to 81 (right) versus time steps k

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Figure 6. $ L^\infty $ norm $ \|n_k\|_{L^\infty(\Omega)} $ (left) and second moment $ I_k $ (right) versus time. The vertical line marks the upper bound $ k_{\rm max} $ defined in (18)

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Figure 7. Cell density computed from the BDF-2 scheme with a coarse mesh at times $ t = 0 $ (top left), $ t = 0.006 $ (top right), $ t = 0.021 $ (bottom left) and the $ L^1 $ norm of $ n_k $ (bottom right)

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