September  2019, 24(9): 4755-4782. doi: 10.3934/dcdsb.2019029

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

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

Received  September 2017 Revised  November 2017 Published  February 2019

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

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.

Citation: Ansgar Jüngel, Oliver Leingang. Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4755-4782. doi: 10.3934/dcdsb.2019029
References:
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M. Akhmouch and M. Amine, A time semi-exponentially fitted scheme for chemotaxis-growth models, Calcolo, 54 (2017), 609-641. doi: 10.1007/s10092-016-0201-4. Google Scholar

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B. AndreianovM. Bendahmane and M. Saad, Finite volume methods for degenerate model, J. Comput. Appl. Math., 235 (2011), 4015-4031. doi: 10.1016/j.cam.2011.02.023. Google Scholar

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A. BlanchetJ. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $ {\mathbb R}^2$, Commun. Pure Appl. Math., 61 (2008), 1449-1481. doi: 10.1002/cpa.20225. Google Scholar

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A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electr. J. Diff. Eqs., 2006 (2006), article 44, 32 pages. Google Scholar

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L. Bonaventura and A. Della Rocca, Unconditionally strong stability preserving extensions of the TR-BDF2 method, J. Sci. Comput., 70 (2017), 859-895. doi: 10.1007/s10915-016-0267-9. Google Scholar

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C. BuddR. Carretero-González and R. Russell, Precise computations of chemotactic collapse using moving mesh methods, J. Comput. Phys., 202 (2005), 463-487. doi: 10.1016/j.jcp.2004.07.010. Google Scholar

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V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $ {\mathbb R}^2$, Commun. Math. Sci., 6 (2008), 417-447. doi: 10.4310/CMS.2008.v6.n2.a8. Google Scholar

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V. CalvezL. Corrias and M. A. Ebde, Blow-up, concentration phenomena and global existence for the Keller-Segel model in high dimensions, Commun. Partial Diff. Eqs., 37 (2012), 561-584. doi: 10.1080/03605302.2012.655824. Google Scholar

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J. A. CarrilloH. Ranetbauer and M.-T. Wolfram, Numerical simulation of continuity equations by evolving diffeomorphisms, J. Comput. Phys., 327 (2016), 186-202. doi: 10.1016/j.jcp.2016.09.040. Google Scholar

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F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model, Numer. Math., 104 (2006), 457-488. doi: 10.1007/s00211-006-0024-3. Google Scholar

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J. Haskovec and C. Schmeiser, Stochastic particle approximation for measure valued solutions of the 2D Keller-Segel system, J. Stat. Phys., 135 (2009), 133-151. doi: 10.1007/s10955-009-9717-1. Google Scholar

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A. Jüngel and J.-P. Milišić, Entropy dissipative one-leg multistep time approximations of nonlinear diffusive equations, Numer. Meth. Partial Diff. Eqs., 31 (2015), 1119-1149. doi: 10.1002/num.21938. Google Scholar

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A. Jüngel and S. Schuchnigg, Entropy-dissipating semi-discrete Runge-Kutta schemes for nonlinear diffusion equations, Commun. Math. Sci., 15 (2017), 27-53. doi: 10.4310/CMS.2017.v15.n1.a2. Google Scholar

[27]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[28]

D. Ketcheson, Step sizes for strong stability preservation with downwind-biased operators, SIAM J. Numer. Anal., 49 (2011), 1649-1660. doi: 10.1137/100818674. Google Scholar

[29]

H. Kozono and Y. Sugiyama, Local existence and finite time blow-up in the 2-D Keller-Segel system, J. Evol. Eqs., 8 (2008), 353-378. doi: 10.1007/s00028-008-0375-6. Google Scholar

[30]

N. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Amer. Math. Soc., Providence, Rhode Island, USA, 2008. doi: 10.1090/gsm/096. Google Scholar

[31]

E. Lieb and M. Loss, Analysis, 2nd edition, Amer. Math. Soc., Providence, USA, 2001. doi: 10.1090/gsm/014. Google Scholar

[32]

J.-G. LiuL. Wang and Z. Zhou, Positivity-preserving and asymptotic preserving method for the 2D Keller-Segel equations, Math. Comput., 87 (2018), 1165-1189. doi: 10.1090/mcom/3250. Google Scholar

[33]

T. Nagai and T. Ogawa, Global existence of solutions to a parabolic-elliptic system of drift-diffusion type in $ {\mathbb R}^2$, Funkcialaj Ekvacioj, 59 (2016), 67-112. doi: 10.1619/fesi.59.67. Google Scholar

[34]

E. Nakaguchi and A. Yagi, Fully discrete approximation by Galerkin Runge-Kutta methods for quasilinear parabolic systems, Hokkaido Math. J., 31 (2002), 385-429. doi: 10.14492/hokmj/1350911871. Google Scholar

[35]

C. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407. Google Scholar

[36]

B. Perthame, Transport Equations in Biology, Birkhäuser, Basel, 2007. Google Scholar

[37]

N. Saito, Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis, IMA J. Numer. Anal., 27 (2007), 332-365. doi: 10.1093/imanum/drl018. Google Scholar

[38]

N. Saito, Conservative numerical schemes for the Keller-Segel system and numerical results, RIMS Kôkyûroku Bessatsu, B15 (2009), 125–146. Google Scholar

[39]

N. Saito, Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis, Commun. Pure Appl. Anal., 11 (2012), 339-364. doi: 10.3934/cpaa.2012.11.339. Google Scholar

[40]

N. Saito and T. Suzuki, Notes on finite difference schemes to a parabolic-elliptic system modelling chemotaxis, Appl. Math. Comput., 171 (2005), 72-90. doi: 10.1016/j.amc.2005.01.037. Google Scholar

[41]

T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Meth. Appl. Anal., 8 (2001), 349-367. doi: 10.4310/MAA.2001.v8.n2.a9. Google Scholar

[42]

M. Smiley, A monotone conservative Eulerian-Lagrangian scheme for reaction-diffusion-convection equations modeling chemotaxis, Numer. Meth. Partial Diff. Eqs., 23 (2007), 553-586. doi: 10.1002/num.20185. Google Scholar

[43]

M. N. Spijker, Contractivity in the numerical solution of initial value problems, Numer. Math., 42 (1983), 271-290. doi: 10.1007/BF01389573. Google Scholar

[44]

R. StrehlA. SokolovD. Kuzmin and S. Turek, A flux-corrected finite element method for chemotaxis problems, Comput. Meth. Appl. Math., 10 (2010), 219-232. doi: 10.2478/cmam-2010-0013. Google Scholar

[45]

R. StrehlA. SokolovD. KuzminD. Horstmann and S. Turek, A positivity-preserving finite element method for chemotaxis problems in 3D, J. Comput. Appl. Math., 239 (2013), 290-303. doi: 10.1016/j.cam.2012.09.041. Google Scholar

[46]

T. Suzuki, Free Energy and Self-Interacting Particles, Birkhäuser, Boston, 2005. doi: 10.1007/0-8176-4436-9. Google Scholar

[47]

J. Valenciano and M. Chaplain, Computing highly accurate solutions of a tumour angiogenesis model, Math. Models Meth. Appl. Sci., 13 (2003), 747-766. doi: 10.1142/S0218202503002702. Google Scholar

[48]

R. ZhangJ. ZhuA. Loula and X. Yu, Operator splitting combined with positivity-preserving discontinuous Galerkin method for the chemotaxis model, J. Comput. Appl. Math., 302 (2016), 312-326. doi: 10.1016/j.cam.2016.02.018. Google Scholar

[49]

G. Zhou and N. Saito, Finite volume methods for a Keller-Segel system: discrete energy, error estimates and numerical blow-up analysis, Numer. Math., 135 (2017), 265-311. doi: 10.1007/s00211-016-0793-2. Google Scholar

show all references

References:
[1]

M. Akhmouch and M. Amine, A time semi-exponentially fitted scheme for chemotaxis-growth models, Calcolo, 54 (2017), 609-641. doi: 10.1007/s10092-016-0201-4. Google Scholar

[2]

B. AndreianovM. Bendahmane and M. Saad, Finite volume methods for degenerate model, J. Comput. Appl. Math., 235 (2011), 4015-4031. doi: 10.1016/j.cam.2011.02.023. Google Scholar

[3]

M. Bessemoulin-Chatard and A. Jüngel, A finite volume scheme for a Keller-Segel model with additional cross-diffusion, IMA J. Numer. Anal., 34 (2014), 96-122. doi: 10.1093/imanum/drs061. Google Scholar

[4]

A. BlanchetV. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model, SIAM J. Numer. Anal., 46 (2008), 691-721. doi: 10.1137/070683337. Google Scholar

[5]

A. BlanchetJ. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $ {\mathbb R}^2$, Commun. Pure Appl. Math., 61 (2008), 1449-1481. doi: 10.1002/cpa.20225. Google Scholar

[6]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electr. J. Diff. Eqs., 2006 (2006), article 44, 32 pages. Google Scholar

[7]

C. Bolley and M. Crouzeix, Conservation de la positivité lors de la discrétisation des problèmes d'évolution paraboliques, RAIRO Anal. Numér., 12 (1978), 237-245. doi: 10.1051/m2an/1978120302371. Google Scholar

[8]

L. Bonaventura and A. Della Rocca, Unconditionally strong stability preserving extensions of the TR-BDF2 method, J. Sci. Comput., 70 (2017), 859-895. doi: 10.1007/s10915-016-0267-9. Google Scholar

[9]

C. BuddR. Carretero-González and R. Russell, Precise computations of chemotactic collapse using moving mesh methods, J. Comput. Phys., 202 (2005), 463-487. doi: 10.1016/j.jcp.2004.07.010. Google Scholar

[10]

V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $ {\mathbb R}^2$, Commun. Math. Sci., 6 (2008), 417-447. doi: 10.4310/CMS.2008.v6.n2.a8. Google Scholar

[11]

V. CalvezL. Corrias and M. A. Ebde, Blow-up, concentration phenomena and global existence for the Keller-Segel model in high dimensions, Commun. Partial Diff. Eqs., 37 (2012), 561-584. doi: 10.1080/03605302.2012.655824. Google Scholar

[12]

J. A. CarrilloH. Ranetbauer and M.-T. Wolfram, Numerical simulation of continuity equations by evolving diffeomorphisms, J. Comput. Phys., 327 (2016), 186-202. doi: 10.1016/j.jcp.2016.09.040. Google Scholar

[13]

M. ChapwanyaJ. Lubuma and R. Mickens, Positivity-preserving nonstandard finite difference schemes for cross-diffusion equations in biosciences, Comput. Math. Appl., 68 (2014), 1071-1082. doi: 10.1016/j.camwa.2014.04.021. Google Scholar

[14]

A. Chertock and A. Kurganov, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models, Numer. Math., 111 (2008), 169-205. doi: 10.1007/s00211-008-0188-0. Google Scholar

[15]

A. ChertockY. EpshteynH. Hu and A. Kurganov, High-order positivity-preserving hybrid finite-volume finite-difference methods for chemotaxis systems, Adv. Comput. Math., 44 (2018), 327-350. doi: 10.1007/s10444-017-9545-9. Google Scholar

[16]

L. CorriasB. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x. Google Scholar

[17]

S. DejakD. EgliP. Lushnikov and I. Sigal, On blowup dynamics in the Keller-Segel model of chemotaxis, Algebra i Analiz, 25 (2013), 47-84. doi: 10.1090/S1061-0022-2014-01306-4. Google Scholar

[18]

Y. Epshtyn, Discontinuous Galerkin methods for the chemotaxis and haptotaxis models, J. Comput. Appl. Math., 224 (2009), 168-181. doi: 10.1016/j.cam.2008.04.030. Google Scholar

[19]

M. A. Farina, M. Marras and G. Viglialoro, On explicit lower bounds and blow-up times in a model of chemotaxis, Dyn. Sys., Diff. Eqs. and Appl., AIMS Proceedings, (2015), 409–417. doi: 10.3934/proc.2015.0409. Google Scholar

[20]

F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model, Numer. Math., 104 (2006), 457-488. doi: 10.1007/s00211-006-0024-3. Google Scholar

[21]

R. Garg and S. Spector, On regularity of solutions to Poisson's equation, Comptes Rendus Math., 353 (2015), 819-823. doi: 10.1016/j.crma.2015.07.001. Google Scholar

[22]

S. GottliebC.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Review, 43 (2001), 89-112. doi: 10.1137/S003614450036757X. Google Scholar

[23]

J. Haskovec and C. Schmeiser, Stochastic particle approximation for measure valued solutions of the 2D Keller-Segel system, J. Stat. Phys., 135 (2009), 133-151. doi: 10.1007/s10955-009-9717-1. Google Scholar

[24]

A. Jüngel and J.-P. Milišić, Entropy dissipative one-leg multistep time approximations of nonlinear diffusive equations, Numer. Meth. Partial Diff. Eqs., 31 (2015), 1119-1149. doi: 10.1002/num.21938. Google Scholar

[25]

A. Jüngel and S. Schuchnigg, A discrete Bakry-Emery method and its application to the porous-medium equation, Discrete Cont. Dyn. Sys., 37 (2017), 5541-5560. doi: 10.3934/dcds.2017241. Google Scholar

[26]

A. Jüngel and S. Schuchnigg, Entropy-dissipating semi-discrete Runge-Kutta schemes for nonlinear diffusion equations, Commun. Math. Sci., 15 (2017), 27-53. doi: 10.4310/CMS.2017.v15.n1.a2. Google Scholar

[27]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[28]

D. Ketcheson, Step sizes for strong stability preservation with downwind-biased operators, SIAM J. Numer. Anal., 49 (2011), 1649-1660. doi: 10.1137/100818674. Google Scholar

[29]

H. Kozono and Y. Sugiyama, Local existence and finite time blow-up in the 2-D Keller-Segel system, J. Evol. Eqs., 8 (2008), 353-378. doi: 10.1007/s00028-008-0375-6. Google Scholar

[30]

N. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Amer. Math. Soc., Providence, Rhode Island, USA, 2008. doi: 10.1090/gsm/096. Google Scholar

[31]

E. Lieb and M. Loss, Analysis, 2nd edition, Amer. Math. Soc., Providence, USA, 2001. doi: 10.1090/gsm/014. Google Scholar

[32]

J.-G. LiuL. Wang and Z. Zhou, Positivity-preserving and asymptotic preserving method for the 2D Keller-Segel equations, Math. Comput., 87 (2018), 1165-1189. doi: 10.1090/mcom/3250. Google Scholar

[33]

T. Nagai and T. Ogawa, Global existence of solutions to a parabolic-elliptic system of drift-diffusion type in $ {\mathbb R}^2$, Funkcialaj Ekvacioj, 59 (2016), 67-112. doi: 10.1619/fesi.59.67. Google Scholar

[34]

E. Nakaguchi and A. Yagi, Fully discrete approximation by Galerkin Runge-Kutta methods for quasilinear parabolic systems, Hokkaido Math. J., 31 (2002), 385-429. doi: 10.14492/hokmj/1350911871. Google Scholar

[35]

C. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407. Google Scholar

[36]

B. Perthame, Transport Equations in Biology, Birkhäuser, Basel, 2007. Google Scholar

[37]

N. Saito, Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis, IMA J. Numer. Anal., 27 (2007), 332-365. doi: 10.1093/imanum/drl018. Google Scholar

[38]

N. Saito, Conservative numerical schemes for the Keller-Segel system and numerical results, RIMS Kôkyûroku Bessatsu, B15 (2009), 125–146. Google Scholar

[39]

N. Saito, Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis, Commun. Pure Appl. Anal., 11 (2012), 339-364. doi: 10.3934/cpaa.2012.11.339. Google Scholar

[40]

N. Saito and T. Suzuki, Notes on finite difference schemes to a parabolic-elliptic system modelling chemotaxis, Appl. Math. Comput., 171 (2005), 72-90. doi: 10.1016/j.amc.2005.01.037. Google Scholar

[41]

T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Meth. Appl. Anal., 8 (2001), 349-367. doi: 10.4310/MAA.2001.v8.n2.a9. Google Scholar

[42]

M. Smiley, A monotone conservative Eulerian-Lagrangian scheme for reaction-diffusion-convection equations modeling chemotaxis, Numer. Meth. Partial Diff. Eqs., 23 (2007), 553-586. doi: 10.1002/num.20185. Google Scholar

[43]

M. N. Spijker, Contractivity in the numerical solution of initial value problems, Numer. Math., 42 (1983), 271-290. doi: 10.1007/BF01389573. Google Scholar

[44]

R. StrehlA. SokolovD. Kuzmin and S. Turek, A flux-corrected finite element method for chemotaxis problems, Comput. Meth. Appl. Math., 10 (2010), 219-232. doi: 10.2478/cmam-2010-0013. Google Scholar

[45]

R. StrehlA. SokolovD. KuzminD. Horstmann and S. Turek, A positivity-preserving finite element method for chemotaxis problems in 3D, J. Comput. Appl. Math., 239 (2013), 290-303. doi: 10.1016/j.cam.2012.09.041. Google Scholar

[46]

T. Suzuki, Free Energy and Self-Interacting Particles, Birkhäuser, Boston, 2005. doi: 10.1007/0-8176-4436-9. Google Scholar

[47]

J. Valenciano and M. Chaplain, Computing highly accurate solutions of a tumour angiogenesis model, Math. Models Meth. Appl. Sci., 13 (2003), 747-766. doi: 10.1142/S0218202503002702. Google Scholar

[48]

R. ZhangJ. ZhuA. Loula and X. Yu, Operator splitting combined with positivity-preserving discontinuous Galerkin method for the chemotaxis model, J. Comput. Appl. Math., 302 (2016), 312-326. doi: 10.1016/j.cam.2016.02.018. Google Scholar

[49]

G. Zhou and N. Saito, Finite volume methods for a Keller-Segel system: discrete energy, error estimates and numerical blow-up analysis, Numer. Math., 135 (2017), 265-311. doi: 10.1007/s00211-016-0793-2. Google Scholar

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)
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)
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)
Figure 4.  Cell density at time step $ k = 0 $ (left) and $ k = k_{\rm max} = 44 $. The mesh size is $ h = 0.02 $
Figure 5.  The residuum $ R_k $ for time steps 1 to 81 (left) and time steps 30 to 81 (right) versus time steps k
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)
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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