September  2019, 39(9): 5365-5402. doi: 10.3934/dcds.2019220

On the logarithmic Keller-Segel-Fisher/KPP system

1. 

Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA

2. 

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

* Corresponding author: Kun Zhao

Received  October 2018 Revised  February 2019 Published  May 2019

Fund Project: Y. Zeng was partially supported by the Simons Foundation grant 244905. K. Zhao was partially supported by the Simons Foundation grant 413028.

We consider the Cauchy problem of a Keller-Segel type chemotaxis model with logarithmic sensitivity and logistic growth. We study the global well-posedness, long-time behavior, vanishing coefficient limit and decay rate of solutions in $ \mathbb{R} $. By utilizing energy methods, we show that for any given classical initial datum which is a perturbation around a constant equilibrium state with finite energy (not small), there exists a unique global-in-time solution to the Cauchy problem, and the solution converges to the constant equilibrium state, as time goes to infinity. Under the same initial condition, it is shown that the solution with positive chemical diffusion coefficient converges to the solution with zero chemical diffusion coefficient, as the coefficient goes to zero. Furthermore, for a slightly smaller class of initial data, we identify the algebraic decay rates of the solution to the constant equilibrium state by employing time-weighted energy estimates.

Citation: Yanni Zeng, Kun Zhao. On the logarithmic Keller-Segel-Fisher/KPP system. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5365-5402. doi: 10.3934/dcds.2019220
References:
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M. AidaK. OsakiT. TsujikawaA. Yagi and M. Mimura, Chemotaxis and growth system with singular sensitivity function, Nonlinear Anal. Real World Appl., 6 (2005), 323-336.  doi: 10.1016/j.nonrwa.2004.08.011.

[2]

N. BellomoA. BellouquidY. 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.

[3]

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

[4]

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.

[5]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 353-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.

[6]

M. A. FontelosA. 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.

[7]

A. Friedman, Partial Differential Equations of Parabolic Type, Reprint Ed., Robert E. Krieger Publishing, Malabar, FL, 1983.

[8]

J. GuoJ. XiaoH. 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.

[9]

X. He and S. Zheng, Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source, J. Math. Anal. Appl., 436 (2016), 970-982.  doi: 10.1016/j.jmaa.2015.12.058.

[10]

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.

[11]

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

[12]

Q. HouZ. 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.

[13]

H. JinJ. 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.

[14]

Y. V. KalininL. JiangY. 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.

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S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Doctoral thesis, Kyoto University, 1983.

[16]

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.

[17]

A. KolmogorovI. Petrovskii and N. Piskunov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Bull. Moscow Univ., Math. Mech., 1 (1937), 1-25. 

[18]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.

[19]

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.

[20]

H. A. LevineB. 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.

[21]

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

[22]

D. LiR. 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.

[23]

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.

[24]

T. LiR. 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.

[25]

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.

[26]

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.

[27]

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.

[28]

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.

[29]

V. MartinezZ. 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.

[30]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.

[31]

K. Osaki and A. Yagi, Global existence for a chemotaxis-growth system in $\mathbb{R}^2$, Adv. Math. Sci. Appl., 12 (2002), 587-606. 

[32]

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.

[33]

H. PengZ. WangK. 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.

[34]

Y. TaoL. 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.

[35]

J. I. Tello, Mathematical analysis and stability of a chemotaxis model with logistic term, Math. Methods Appl. Sci., 27 (2004), 1865-1880.  doi: 10.1002/mma.528.

[36]

G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.  doi: 10.1016/j.jmaa.2016.02.069.

[37]

G. Viglialoro and T. E. Woolley, Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth, Disc. Cont. Dyn. Syst. Ser. B, 23 (2018), 3023-3045.  doi: 10.3934/dcdsb.2017199.

[38]

G. Viglialoro, Boundedness properties of very weak solutions to a fully parabolic chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 34 (2017), 520-535.  doi: 10.1016/j.nonrwa.2016.10.001.

[39]

L. WangY. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Disc. Cont. Dyn. Syst. Ser. A, 34 (2014), 789-802.  doi: 10.3934/dcds.2014.34.789.

[40]

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.

[41]

M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.  doi: 10.1016/j.jmaa.2008.07.071.

[42]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.

[43]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.

[44]

M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), Art. 69, 40 pp. doi: 10.1007/s00033-018-0935-8.

[45]

M. Zhang and C. Zhu, Global existence of solutions to a hyperbolic-parabolic system, Proc. Amer. Math. Soc., 135 (2007), 1017-1027.  doi: 10.1090/S0002-9939-06-08773-9.

[46]

J. Zheng, Boundedness and global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with nonlinear logistic source, J. Math. Anal. Appl., 450 (2017), 1047-1061.  doi: 10.1016/j.jmaa.2017.01.043.

[47]

Y. Zeng, Global existence theory for general hyperbolic-parabolic balance laws with application, J. Hyperbolic Differ. Equ., 14 (2017), 359-391.  doi: 10.1142/S0219891617500126.

[48]

Y. Zeng, $L^p$ decay for general hyperbolic-parabolic systems of balance laws, Discrete Contin. Dyn. Syst. Ser. A, 38 (2018), 363-396.  doi: 10.3934/dcds.2018018.

[49]

Y. Zeng, Asymptotic behavior of solutions to general hyperbolic-parabolic systems of balance laws in multi-space dimensions, Pure Appl. Math.Quart., 14 (2018), 161-192.  doi: 10.4310/PAMQ.2018.v14.n1.a6.

[50]

Y. Zeng, $L^p$ time asymptotic decay for general hyperbolic-parabolic balance laws with applications., Preprint.

[51]

P. ZhengC. Mu and X. Hu, Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source, Disc. Cont. Dyn. Syst. Ser. A, 35 (2015), 2299-2323.  doi: 10.3934/dcds.2015.35.2299.

show all references

References:
[1]

M. AidaK. OsakiT. TsujikawaA. Yagi and M. Mimura, Chemotaxis and growth system with singular sensitivity function, Nonlinear Anal. Real World Appl., 6 (2005), 323-336.  doi: 10.1016/j.nonrwa.2004.08.011.

[2]

N. BellomoA. BellouquidY. 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.

[3]

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

[4]

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.

[5]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 353-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.

[6]

M. A. FontelosA. 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.

[7]

A. Friedman, Partial Differential Equations of Parabolic Type, Reprint Ed., Robert E. Krieger Publishing, Malabar, FL, 1983.

[8]

J. GuoJ. XiaoH. 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.

[9]

X. He and S. Zheng, Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source, J. Math. Anal. Appl., 436 (2016), 970-982.  doi: 10.1016/j.jmaa.2015.12.058.

[10]

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.

[11]

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

[12]

Q. HouZ. 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.

[13]

H. JinJ. 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.

[14]

Y. V. KalininL. JiangY. 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.

[15]

S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Doctoral thesis, Kyoto University, 1983.

[16]

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.

[17]

A. KolmogorovI. Petrovskii and N. Piskunov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Bull. Moscow Univ., Math. Mech., 1 (1937), 1-25. 

[18]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.

[19]

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.

[20]

H. A. LevineB. 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.

[21]

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

[22]

D. LiR. 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.

[23]

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.

[24]

T. LiR. 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.

[25]

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.

[26]

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.

[27]

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.

[28]

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.

[29]

V. MartinezZ. 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.

[30]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.

[31]

K. Osaki and A. Yagi, Global existence for a chemotaxis-growth system in $\mathbb{R}^2$, Adv. Math. Sci. Appl., 12 (2002), 587-606. 

[32]

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.

[33]

H. PengZ. WangK. 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.

[34]

Y. TaoL. 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.

[35]

J. I. Tello, Mathematical analysis and stability of a chemotaxis model with logistic term, Math. Methods Appl. Sci., 27 (2004), 1865-1880.  doi: 10.1002/mma.528.

[36]

G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.  doi: 10.1016/j.jmaa.2016.02.069.

[37]

G. Viglialoro and T. E. Woolley, Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth, Disc. Cont. Dyn. Syst. Ser. B, 23 (2018), 3023-3045.  doi: 10.3934/dcdsb.2017199.

[38]

G. Viglialoro, Boundedness properties of very weak solutions to a fully parabolic chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 34 (2017), 520-535.  doi: 10.1016/j.nonrwa.2016.10.001.

[39]

L. WangY. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Disc. Cont. Dyn. Syst. Ser. A, 34 (2014), 789-802.  doi: 10.3934/dcds.2014.34.789.

[40]

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.

[41]

M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.  doi: 10.1016/j.jmaa.2008.07.071.

[42]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.

[43]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.

[44]

M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), Art. 69, 40 pp. doi: 10.1007/s00033-018-0935-8.

[45]

M. Zhang and C. Zhu, Global existence of solutions to a hyperbolic-parabolic system, Proc. Amer. Math. Soc., 135 (2007), 1017-1027.  doi: 10.1090/S0002-9939-06-08773-9.

[46]

J. Zheng, Boundedness and global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with nonlinear logistic source, J. Math. Anal. Appl., 450 (2017), 1047-1061.  doi: 10.1016/j.jmaa.2017.01.043.

[47]

Y. Zeng, Global existence theory for general hyperbolic-parabolic balance laws with application, J. Hyperbolic Differ. Equ., 14 (2017), 359-391.  doi: 10.1142/S0219891617500126.

[48]

Y. Zeng, $L^p$ decay for general hyperbolic-parabolic systems of balance laws, Discrete Contin. Dyn. Syst. Ser. A, 38 (2018), 363-396.  doi: 10.3934/dcds.2018018.

[49]

Y. Zeng, Asymptotic behavior of solutions to general hyperbolic-parabolic systems of balance laws in multi-space dimensions, Pure Appl. Math.Quart., 14 (2018), 161-192.  doi: 10.4310/PAMQ.2018.v14.n1.a6.

[50]

Y. Zeng, $L^p$ time asymptotic decay for general hyperbolic-parabolic balance laws with applications., Preprint.

[51]

P. ZhengC. Mu and X. Hu, Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source, Disc. Cont. Dyn. Syst. Ser. A, 35 (2015), 2299-2323.  doi: 10.3934/dcds.2015.35.2299.

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