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

Previous Article
Dynamics of a predator-prey model with state-dependent carrying capacity
DCDS-B Home
This Issue
Next Article
Global dynamics of a virus infection model with repulsive effect

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.

Keywords: Time discretization, chemotaxis, finite-time blow-up of solutions, existence of weak solutions, higher-order schemes, BDF scheme, Runge-Kutta scheme.

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

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:

[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. Andreianov, M. 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. Blanchet, V. 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. Blanchet, J. 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. Budd, R. 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. Calvez, L. 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. Carrillo, H. 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. Chapwanya, J. 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. Chertock, Y. Epshteyn, H. 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. Corrias, B. 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. Dejak, D. Egli, P. 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. Gottlieb, C.-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, 2^nd edition, Amer. Math. Soc., Providence, USA, 2001. doi: 10.1090/gsm/014. Google Scholar
[32]	J.-G. Liu, L. 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. Strehl, A. Sokolov, D. 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. Strehl, A. Sokolov, D. Kuzmin, D. 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. Zhang, J. Zhu, A. 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. Andreianov, M. 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. Blanchet, V. 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. Blanchet, J. 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. Budd, R. 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. Calvez, L. 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. Carrillo, H. 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. Chapwanya, J. 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. Chertock, Y. Epshteyn, H. 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. Corrias, B. 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. Dejak, D. Egli, P. 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. Gottlieb, C.-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, 2^nd edition, Amer. Math. Soc., Providence, USA, 2001. doi: 10.1090/gsm/014. Google Scholar
[32]	J.-G. Liu, L. 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. Strehl, A. Sokolov, D. 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. Strehl, A. Sokolov, D. Kuzmin, D. 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. Zhang, J. Zhu, A. 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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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

Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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)

Figure Options

Download full-size image

Download as PowerPoint slide

[1]	Da Xu. Numerical solutions of viscoelastic bending wave equations with two term time kernels by Runge-Kutta convolution quadrature. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2389-2416. doi: 10.3934/dcdsb.2017122
[2]	Sihong Shao, Huazhong Tang. Higher-order accurate Runge-Kutta discontinuous Galerkin methods for a nonlinear Dirac model. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 623-640. doi: 10.3934/dcdsb.2006.6.623
[3]	Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure & Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243
[4]	Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843
[5]	Bin Guo, Wenjie Gao. Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 715-730. doi: 10.3934/dcds.2016.36.715
[6]	Akmel Dé Godefroy. Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 117-137. doi: 10.3934/dcds.2015.35.117
[7]	Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831
[8]	Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903
[9]	Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 233-255. doi: 10.3934/dcdss.2020013
[10]	Grzegorz Karch, Kanako Suzuki, Jacek Zienkiewicz. Finite-time blowup of solutions to some activator-inhibitor systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4997-5010. doi: 10.3934/dcds.2016016
[11]	Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54
[12]	Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535
[13]	Arno Berger. On finite-time hyperbolicity. Communications on Pure & Applied Analysis, 2011, 10 (3) : 963-981. doi: 10.3934/cpaa.2011.10.963
[14]	Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569
[15]	Yūki Naito, Takasi Senba. Blow-up behavior of solutions to a parabolic-elliptic system on higher dimensional domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3691-3713. doi: 10.3934/dcds.2012.32.3691
[16]	C. Y. Chan. Recent advances in quenching and blow-up of solutions. Conference Publications, 2001, 2001 (Special) : 88-95. doi: 10.3934/proc.2001.2001.88
[17]	Marek Fila, Hiroshi Matano. Connecting equilibria by blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 155-164. doi: 10.3934/dcds.2000.6.155
[18]	Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267
[19]	José M. Arrieta, Raúl Ferreira, Arturo de Pablo, Julio D. Rossi. Stability of the blow-up time and the blow-up set under perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 43-61. doi: 10.3934/dcds.2010.26.43
[20]	Arno Berger, Doan Thai Son, Stefan Siegmund. Nonautonomous finite-time dynamics. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 463-492. doi: 10.3934/dcdsb.2008.9.463

2018 Impact Factor: 1.008

Tools

Metrics

Tools

Article outline

Figures and Tables

2018 Impact Factor: 1.008

Tools

Figures and Tables

2018 Impact Factor: 1.008

American Institute of Mathematical Sciences