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July  2019, 39(7): 3789-3838. doi: 10.3934/dcds.2019154

Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions

Leo G. Rebholz 1, , Dehua Wang 2, , Zhian Wang 3, , Camille Zerfas 1, and  Kun Zhao 4,,

1. 

Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, USA
2. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
3. 

Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China
4. 

Department of Mathematics, Tulane University, New Orleans, LA 70118, USA

* Corresponding author

Received  March 2018 Revised  December 2018 Published  April 2019

We study the qualitative behavior of a system of parabolic conservation laws, derived from a Keller-Segel type chemotaxis model with singular sensitivity, on the unit square or cube subject to various types of boundary conditions. It is shown that for given initial data in $H^3(\Omega)$, under the assumption that only the entropic energy associated with the initial data is small, there exist global-in-time classical solutions to the initial-boundary value problems of the model subject to the Neumann-Stress-free and Dirichlet-Stress-free type boundary conditions; these solutions converge to equilibrium states, determined from initial and/or boundary data, exponentially rapidly as time goes to infinity. In addition, it is shown that the solutions of the fully dissipative model converge to those of the corresponding partially dissipative model as the chemical diffusion rate tends to zero under the Neumann-Stress-free type boundary conditions. Numerical analysis is performed for a discretization of the model with the Dirichlet-Stress-free type boundary conditions, and a monotonic exponential decay to the equilibrium solution (analogous to the continuous case) is proven. Numerical simulations are supplemented to illustrate the exponential decay, test the assumptions of the exponential decay theorem, and to predict boundary layer formation under the Dirichlet-Stress-free type boundary conditions.

Keywords: Parabolic conservation laws, Keller-Segel chemotaxis model, initial-boundary value problem, global well-posedness, long-time behavior, diffusion limit, boundary layer, numerical analysis, numerical simulation.
Mathematics Subject Classification: 35A01, 35B40, 35K61, 35L65, 35Q92, 92C17.
Citation: Leo G. Rebholz, Dehua Wang, Zhian Wang, Camille Zerfas, Kun Zhao. Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3789-3838. doi: 10.3934/dcds.2019154
References:
[1]

J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. 

[2]

M. Akbas, S. Kaya and L. Rebholz, On the stability at all times of linearly extrapolated BDF2 timestepping for multiphysics incompressible flow problems, Numer. Meth. Part. Diff. Equ., 33 (2017), 995-1017.  doi: 10.1002/num.22061.

[3]

N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[4]

P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359. 

[5]

J. P. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Func. Anal., 15 (1974), 341-363.  doi: 10.1016/0022-1236(74)90027-5.

[6]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, volume 15 of Texts in Applied Mathematics, Springer Science+Business Media, LLC, 2008. doi: 10.1007/978-0-387-75934-0.

[7]

F. W. Dahlquist, P. Lovely and D. E. Jr Koshland, Quantitative analysis of bacterial migration in chemotaxis, Nature, New Biol., 236 (1972), 120-123.  doi: 10.1038/newbio236120a0.

[8]

C. Deng and T. Li, Well-posedness of a 3D parabolic-hyperbolic Keller-Segel system in the Sobolev space framework, J. Differential Equations, 257 (2014), 1311-1332.  doi: 10.1016/j.jde.2014.05.014.

[9]

J. Fan and K. Zhao, Blow up criteria for a hyperbolic-parabolic system arising from chemotaxis, J. Math. Anal. Appl., 394 (2012), 687-695.  doi: 10.1016/j.jmaa.2012.05.036.

[10]

M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330-1355.  doi: 10.1137/S0036141001385046.

[11]

K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.  doi: 10.1016/j.jmaa.2014.11.045.

[12]

K. Fujie and T. Senba, Global existence and boundedness of radial solutions to a two dimensional fully parabolic chemotaxis system with general sensitivity, Nonlinearity, 29 (2016), 2417-2450.  doi: 10.1088/0951-7715/29/8/2417.

[13]

J. Guo, J. Xiao, H. Zhao and C. Zhu, Global solutions to a hyperbolic-parabolic coupled system with large initial data, Acta Math. Sci. Ser. B Engl. Ed, 29 (2009), 629-641.  doi: 10.1016/S0252-9602(09)60059-X.

[14]

C. Hao, Global well-posedness for a multidimensional chemotaxis model in critical Besov spaces, Z. Angew. Math. Phys., 63 (2012), 825-834.  doi: 10.1007/s00033-012-0193-0.

[15]

T. Heister, M. A. Olshanskii and L. Rebholz, Unconditional long-time stability of a velocity-vorticity method for the 2D Navier-Stokes equations, Numer. Math., 135 (2017), 143-167.  doi: 10.1007/s00211-016-0794-1.

[16]

T. Hillen and K. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[17]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresberichteder DMV, 105 (2003), 103-165. 

[18]

Q. Hou, Z. Wang and K. Zhao, Boundary layer problem on a hyperbolic system arising from chemotaxis, J. Differential Equations, 261 (2016), 5035-5070.  doi: 10.1016/j.jde.2016.07.018.

[19]

Q. Hou, C. Liu, Y. Wang and Z. Wang, Stability of boundary layers for a viscous hyperbolic system arising from chemotaxis: one dimensional case, SIAM J. Math. Anal., 50 (2018), 3058-3091.  doi: 10.1137/17M112748X.

[20]

H. Jin, J. Li and Z. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differential Equations, 255 (2013), 193-219.  doi: 10.1016/j.jde.2013.04.002.

[21]

Y. V. Kalinin, L. Jiang, Y. Tu and M. Wu, Logarithmic sensing in Escherichia coli bacterial chemotaxis, Biophysical J., 96 (2009), 2439-2448.  doi: 10.1016/j.bpj.2008.10.027.

[22]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.

[23]

J. Lankeit, A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 39 (2016), 394-404.  doi: 10.1002/mma.3489.

[24]

J. Lankeit and M. Winkler, A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: Global solvability for large nonradial data, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 49, 33 pp. doi: 10.1007/s00030-017-0472-8.

[25]

H. A. Levine and B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730.  doi: 10.1137/S0036139995291106.

[26]

H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton, A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. Ⅰ. The role of protease inhibitors in preventing angiogenesis, Math. Biosci., 168 (2000), 71-115.  doi: 10.1016/S0025-5564(00)00034-1.

[27]

D. Li, T. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis, Math. Models Methods Appl. Sci., 21 (2011), 1631-1650.  doi: 10.1142/S0218202511005519.

[28]

D. Li, R. Pan and K. Zhao, Quantitative decay of a one-dimensional hybrid chemotaxis model with large data, Nonlinearity, 28 (2015), 2181-2210.  doi: 10.1088/0951-7715/28/7/2181.

[29]

H. Li and K. Zhao, Initial boundary value problems for a system of hyperbolic balance laws arising from chemotaxis, J. Differential Equations, 258 (2015), 302-338.  doi: 10.1016/j.jde.2014.09.014.

[30]

T. Li, R. Pan and K. Zhao, Global dynamics of a hyperbolic-parabolic model arising from chemotaxis, SIAM J. Appl. Math., 72 (2012), 417-443.  doi: 10.1137/110829453.

[31]

T. Li and Z. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009/10), 1522-1541.  doi: 10.1137/09075161X.

[32]

T. Li and Z. Wang, Nonlinear stability of large amplitude viscous shock waves of a hyperbolic-parabolic system arising in chemotaxis, Math. Models Methods Appl. Sci., 20 (2010), 1967-1998.  doi: 10.1142/S0218202510004830.

[33]

T. Li and Z. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 1310-1333.  doi: 10.1016/j.jde.2010.09.020.

[34]

T. Li and Z. Wang, Steadily propagating waves of a chemotaxis model, Math. Biosci., 240 (2012), 161-168.  doi: 10.1016/j.mbs.2012.07.003.

[35]

V. Martinez, Z. Wang and K. Zhao, Asymptotic and viscous stability of large-amplitude solutions of a hyperbolic system arising from biology, Indiana Univ. Math. J., 67 (2018), 1383-1424.  doi: 10.1512/iumj.2018.67.7394.

[36]

T. Nishida, Nonlinear Hyperbolic Equations and Related Topics in Fluid Dynamics, Publ. Math. D'osay 78-02, Département de Mathématique, Université de Paris-Sud, Orsay, France, 1978.

[37]

H. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.  doi: 10.1137/S0036139995288976.

[38]

H. Peng, Z. Wang, K. Zhao and C. Zhu, Boundary layers and stabilization of the singular Keller-Segel system, Kinet. Relat. Models, 11 (2018), 1085-1123.  doi: 10.3934/krm.2018042.

[39]

H. Peng, H. Wen and C. Zhu, Global well-posedness and zero diffusion limit of classical solutions to 3D conservation laws arising in chemotaxis, Z. Angew. Math. Phys., 65 (2014), 1167-1188.  doi: 10.1007/s00033-013-0378-1.

[40]

S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Comm. Math. Phys., 104 (1986), 49-75.  doi: 10.1007/BF01210792.

[41]

C. Stinner and M. Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity, Nonlinear Analysis: Real World Applications, 12 (2011), 3727-3740.  doi: 10.1016/j.nonrwa.2011.07.006.

[42]

Y. Tao, L. Wang and Z. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension, Discrete Contin. Dyn. Syst.-Series B., 18 (2013), 821-845.  doi: 10.3934/dcdsb.2013.18.821.

[43]

Z. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods Appl. Sci., 31 (2008), 45-70.  doi: 10.1002/mma.898.

[44]

Z. Wang, Z. Xiang and P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 2225-2258.  doi: 10.1016/j.jde.2015.09.063.

[45]

Z. Wang and K. Zhao, Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model, Comm. Pure Appl. Anal., 12 (2013), 3027-3046.  doi: 10.3934/cpaa.2013.12.3027.

[46]

M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 34 (2011), 176-190.  doi: 10.1002/mma.1346.

[47]

M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: Global large data solutions and their relaxation properties, Math. Models Methods Appl. Sci., 26 (2016), 987-1024.  doi: 10.1142/S0218202516500238.

[48]

M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: Eventual smoothness and equilibration of small-mass solutions, preprint.

[49]

M. Winkler and T. Yokota, Stabilization in the logarithmic Keller-Segel system, Nonlinear Analysis, 170 (2018), 123-141.  doi: 10.1016/j.na.2018.01.002.

[50]

Y. Xiao and Z.-P. Xin, On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition, Comm. Pure Appl. Math., 60 (2007), 1027-1055.  doi: 10.1002/cpa.20187.

[51]

M. Zhang and C. Zhu, Global existence of solutions to a hyperbolic-parabolic system, Proc. Amer. Math. Soc., 135 (2006), 1017-1027.  doi: 10.1090/S0002-9939-06-08773-9.

[52]

X. Zhao and S. Zheng, Global boundedness of solutions in a parabolic-parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 443 (2016), 445-452.  doi: 10.1016/j.jmaa.2016.05.036.

show all references

References:
[1]

J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. 

[2]

M. Akbas, S. Kaya and L. Rebholz, On the stability at all times of linearly extrapolated BDF2 timestepping for multiphysics incompressible flow problems, Numer. Meth. Part. Diff. Equ., 33 (2017), 995-1017.  doi: 10.1002/num.22061.

[3]

N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[4]

P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359. 

[5]

J. P. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Func. Anal., 15 (1974), 341-363.  doi: 10.1016/0022-1236(74)90027-5.

[6]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, volume 15 of Texts in Applied Mathematics, Springer Science+Business Media, LLC, 2008. doi: 10.1007/978-0-387-75934-0.

[7]

F. W. Dahlquist, P. Lovely and D. E. Jr Koshland, Quantitative analysis of bacterial migration in chemotaxis, Nature, New Biol., 236 (1972), 120-123.  doi: 10.1038/newbio236120a0.

[8]

C. Deng and T. Li, Well-posedness of a 3D parabolic-hyperbolic Keller-Segel system in the Sobolev space framework, J. Differential Equations, 257 (2014), 1311-1332.  doi: 10.1016/j.jde.2014.05.014.

[9]

J. Fan and K. Zhao, Blow up criteria for a hyperbolic-parabolic system arising from chemotaxis, J. Math. Anal. Appl., 394 (2012), 687-695.  doi: 10.1016/j.jmaa.2012.05.036.

[10]

M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330-1355.  doi: 10.1137/S0036141001385046.

[11]

K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.  doi: 10.1016/j.jmaa.2014.11.045.

[12]

K. Fujie and T. Senba, Global existence and boundedness of radial solutions to a two dimensional fully parabolic chemotaxis system with general sensitivity, Nonlinearity, 29 (2016), 2417-2450.  doi: 10.1088/0951-7715/29/8/2417.

[13]

J. Guo, J. Xiao, H. Zhao and C. Zhu, Global solutions to a hyperbolic-parabolic coupled system with large initial data, Acta Math. Sci. Ser. B Engl. Ed, 29 (2009), 629-641.  doi: 10.1016/S0252-9602(09)60059-X.

[14]

C. Hao, Global well-posedness for a multidimensional chemotaxis model in critical Besov spaces, Z. Angew. Math. Phys., 63 (2012), 825-834.  doi: 10.1007/s00033-012-0193-0.

[15]

T. Heister, M. A. Olshanskii and L. Rebholz, Unconditional long-time stability of a velocity-vorticity method for the 2D Navier-Stokes equations, Numer. Math., 135 (2017), 143-167.  doi: 10.1007/s00211-016-0794-1.

[16]

T. Hillen and K. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[17]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresberichteder DMV, 105 (2003), 103-165. 

[18]

Q. Hou, Z. Wang and K. Zhao, Boundary layer problem on a hyperbolic system arising from chemotaxis, J. Differential Equations, 261 (2016), 5035-5070.  doi: 10.1016/j.jde.2016.07.018.

[19]

Q. Hou, C. Liu, Y. Wang and Z. Wang, Stability of boundary layers for a viscous hyperbolic system arising from chemotaxis: one dimensional case, SIAM J. Math. Anal., 50 (2018), 3058-3091.  doi: 10.1137/17M112748X.

[20]

H. Jin, J. Li and Z. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differential Equations, 255 (2013), 193-219.  doi: 10.1016/j.jde.2013.04.002.

[21]

Y. V. Kalinin, L. Jiang, Y. Tu and M. Wu, Logarithmic sensing in Escherichia coli bacterial chemotaxis, Biophysical J., 96 (2009), 2439-2448.  doi: 10.1016/j.bpj.2008.10.027.

[22]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.

[23]

J. Lankeit, A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 39 (2016), 394-404.  doi: 10.1002/mma.3489.

[24]

J. Lankeit and M. Winkler, A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: Global solvability for large nonradial data, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 49, 33 pp. doi: 10.1007/s00030-017-0472-8.

[25]

H. A. Levine and B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730.  doi: 10.1137/S0036139995291106.

[26]

H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton, A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. Ⅰ. The role of protease inhibitors in preventing angiogenesis, Math. Biosci., 168 (2000), 71-115.  doi: 10.1016/S0025-5564(00)00034-1.

[27]

D. Li, T. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis, Math. Models Methods Appl. Sci., 21 (2011), 1631-1650.  doi: 10.1142/S0218202511005519.

[28]

D. Li, R. Pan and K. Zhao, Quantitative decay of a one-dimensional hybrid chemotaxis model with large data, Nonlinearity, 28 (2015), 2181-2210.  doi: 10.1088/0951-7715/28/7/2181.

[29]

H. Li and K. Zhao, Initial boundary value problems for a system of hyperbolic balance laws arising from chemotaxis, J. Differential Equations, 258 (2015), 302-338.  doi: 10.1016/j.jde.2014.09.014.

[30]

T. Li, R. Pan and K. Zhao, Global dynamics of a hyperbolic-parabolic model arising from chemotaxis, SIAM J. Appl. Math., 72 (2012), 417-443.  doi: 10.1137/110829453.

[31]

T. Li and Z. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009/10), 1522-1541.  doi: 10.1137/09075161X.

[32]

T. Li and Z. Wang, Nonlinear stability of large amplitude viscous shock waves of a hyperbolic-parabolic system arising in chemotaxis, Math. Models Methods Appl. Sci., 20 (2010), 1967-1998.  doi: 10.1142/S0218202510004830.

[33]

T. Li and Z. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 1310-1333.  doi: 10.1016/j.jde.2010.09.020.

[34]

T. Li and Z. Wang, Steadily propagating waves of a chemotaxis model, Math. Biosci., 240 (2012), 161-168.  doi: 10.1016/j.mbs.2012.07.003.

[35]

V. Martinez, Z. Wang and K. Zhao, Asymptotic and viscous stability of large-amplitude solutions of a hyperbolic system arising from biology, Indiana Univ. Math. J., 67 (2018), 1383-1424.  doi: 10.1512/iumj.2018.67.7394.

[36]

T. Nishida, Nonlinear Hyperbolic Equations and Related Topics in Fluid Dynamics, Publ. Math. D'osay 78-02, Département de Mathématique, Université de Paris-Sud, Orsay, France, 1978.

[37]

H. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.  doi: 10.1137/S0036139995288976.

[38]

H. Peng, Z. Wang, K. Zhao and C. Zhu, Boundary layers and stabilization of the singular Keller-Segel system, Kinet. Relat. Models, 11 (2018), 1085-1123.  doi: 10.3934/krm.2018042.

[39]

H. Peng, H. Wen and C. Zhu, Global well-posedness and zero diffusion limit of classical solutions to 3D conservation laws arising in chemotaxis, Z. Angew. Math. Phys., 65 (2014), 1167-1188.  doi: 10.1007/s00033-013-0378-1.

[40]

S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Comm. Math. Phys., 104 (1986), 49-75.  doi: 10.1007/BF01210792.

[41]

C. Stinner and M. Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity, Nonlinear Analysis: Real World Applications, 12 (2011), 3727-3740.  doi: 10.1016/j.nonrwa.2011.07.006.

[42]

Y. Tao, L. Wang and Z. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension, Discrete Contin. Dyn. Syst.-Series B., 18 (2013), 821-845.  doi: 10.3934/dcdsb.2013.18.821.

[43]

Z. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods Appl. Sci., 31 (2008), 45-70.  doi: 10.1002/mma.898.

[44]

Z. Wang, Z. Xiang and P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 2225-2258.  doi: 10.1016/j.jde.2015.09.063.

[45]

Z. Wang and K. Zhao, Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model, Comm. Pure Appl. Anal., 12 (2013), 3027-3046.  doi: 10.3934/cpaa.2013.12.3027.

[46]

M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 34 (2011), 176-190.  doi: 10.1002/mma.1346.

[47]

M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: Global large data solutions and their relaxation properties, Math. Models Methods Appl. Sci., 26 (2016), 987-1024.  doi: 10.1142/S0218202516500238.

[48]

M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: Eventual smoothness and equilibration of small-mass solutions, preprint.

[49]

M. Winkler and T. Yokota, Stabilization in the logarithmic Keller-Segel system, Nonlinear Analysis, 170 (2018), 123-141.  doi: 10.1016/j.na.2018.01.002.

[50]

Y. Xiao and Z.-P. Xin, On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition, Comm. Pure Appl. Math., 60 (2007), 1027-1055.  doi: 10.1002/cpa.20187.

[51]

M. Zhang and C. Zhu, Global existence of solutions to a hyperbolic-parabolic system, Proc. Amer. Math. Soc., 135 (2006), 1017-1027.  doi: 10.1090/S0002-9939-06-08773-9.

[52]

X. Zhao and S. Zheng, Global boundedness of solutions in a parabolic-parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 443 (2016), 445-452.  doi: 10.1016/j.jmaa.2016.05.036.

Figure 1.  Shown above is decay in time of solution norms to Algorithm 5.1, with varying $ \varepsilon $
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Figure 2.  Shown above are plots of solutions for $ \varepsilon = 0.001 $ and varying times, for Numerical Test 3. Contours of $ p $ (top) and $ | {\bf q}| $ (middle row) are shown at various times, and the bottom row shows plots of $ p $ and $ {\bf q}_2 $ with fixed $ x = 0.5 $ and varying $ t $
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Figure 3.  Shown above is a plot of $ | {\bf q}| $ with fixed $ x = 0.5 $ and $ t = 0.2 $ (right), for varying $ \varepsilon $ in numerical experiment 3
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Figure 4.  Shown above are solution plots for $ \varepsilon = 0.01 $ at varying times, for numerical experiment 4. Across the top row are contour plots of solutions $ p $, in the middle row are contours for $ | {\bf q}| $, and in the bottom row we show plots of $ p $ (left) and $ {\bf q}_2 $ (right) along $ x = 0 $
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Figure 5.  Shown above are plots of $ (\| p(t)\|^2 + \| {\bf q}(t) \|^2) $ versus time $ t $ (left), and $ {\bf q}_2(0,y,0.1) $, for varying $ \varepsilon $, for numerical experiment 4
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Table 1.  Values of energy norms at the end time $ T = 1 $ for various $ P^* $ values
Backward Euler
$ P^* $ $ \left( \| p_h(T) \|^2 + \| {\bf q}_h(T) \|^2 \right)^{1/2} $
50 433.1920
51 913.7230
52 5.3532e+03
53 1.4410e+04
54 2.7783e+04
55 4.5390e+04
56 7.4976e+04
57 3.6508e+11
58 1.2408e+13
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Backward Euler
$ P^* $ $ \left( \| p_h(T) \|^2 + \| {\bf q}_h(T) \|^2 \right)^{1/2} $
50 433.1920
51 913.7230
52 5.3532e+03
53 1.4410e+04
54 2.7783e+04
55 4.5390e+04
56 7.4976e+04
57 3.6508e+11
58 1.2408e+13
Table 2.  Values of the critical $ P^* $ for blowup, for varying time step sizes
$ \Delta t $ $ P^*_{critical} $
0.01 57
0.005 57
0.0025 64
0.00125 90
0.000625 91
0.0003125 90
0.00015625 90
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$ \Delta t $ $ P^*_{critical} $
0.01 57
0.005 57
0.0025 64
0.00125 90
0.000625 91
0.0003125 90
0.00015625 90
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