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doi: 10.3934/dcdsb.2021128
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Riemann problem for a non-strictly hyperbolic system in chemotaxis

Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA

* Corresponding author: Tong Li

Received  December 2020 Revised  March 2021 Early access April 2021

The Riemann problem is solved for a system arising in chemotaxis. The system is of mixed-type and transitions from a hyperbolic to an elliptic region. It is genuinely nonlinear in the $ u $-$ v $ plane except on the $ v $-axis, where it is linearly degenerate. We have solved the Riemann problem in the physically relevant region up to the boundary of the hyperbolic-elliptic region, which is non-strictly hyperbolic. We also solved the problem on the linearly degenerate region. While solving the Riemann problem, we found classical shock and rarefaction waves in the hyperbolic region and contact discontinuities in the linearly degenerate region.

Citation: Tong Li, Nitesh Mathur. Riemann problem for a non-strictly hyperbolic system in chemotaxis. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021128
References:
[1]

J. A. CarrilloJ. Li and Z.-A. Wang, Boundary spike-layer solutions of the singular Keller-Segel system: Existence and stability, Proc. Lond. Math. Soc. (3), 122 (2021), 42-68.  doi: 10.1112/plms.12319.  Google Scholar

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R. De la cruz, Riemann Problem for a $2\times 2$ hyperbolic system with linear damping, Acta Appl. Math., 170 (2020), 631-647.  doi: 10.1007/s10440-020-00350-w.  Google Scholar

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

[4]

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.  Google Scholar

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P. Goatin and P. G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 881-902.  doi: 10.1016/j.anihpc.2004.02.002.  Google Scholar

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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.  Google Scholar

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H. Holden, On the Riemann problem for a prototype of a mixed type conservation law, Comm. Pure Appl. Math., 40 (1987), 229-264.  doi: 10.1002/cpa.3160400206.  Google Scholar

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D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.   Google Scholar

[9]

Q. HouC.-J. LiuY.-G. 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.  Google Scholar

[10]

Q. Hou and Z. Wang, Convergence of boundary layers for the Keller-Segel system with singular sensitivity in the half-plane, J. Math. Pures Appl. (9), 130 (2019), 251-287.  doi: 10.1016/j.matpur.2019.01.008.  Google Scholar

[11]

Q. HouZ.-A. 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.  Google Scholar

[12]

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H.-Y. JinJ. Li and Z.-A. 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.  Google Scholar

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B. L. Keyfitz and H. C. Kranzer, The Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy, J. Differential Equations, 47 (1983), 35-65.  doi: 10.1016/0022-0396(83)90027-X.  Google Scholar

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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.  Google Scholar

[19]

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.  Google Scholar

[20]

T. LiH. Liu and L. Wang, Oscillatory traveling wave solutions to an attractive chemotaxis system, J. Differential Equations, 261 (2016), 7080-7098.  doi: 10.1016/j.jde.2016.09.012.  Google Scholar

[21]

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.  Google Scholar

[22]

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

[23]

T. Li and Z.-A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2010), 1522-1541.  doi: 10.1137/09075161X.  Google Scholar

[24]

T. Li and Z.-A. 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.  Google Scholar

[25]

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

[26]

D. Marchesin and P. J. Paes-Leme, A {R}iemann problem in gas dynamics with bifurcation, in Comput. Math. Appl. Part A, 12, Hyperbolic partial differential equations, III, (1986), 433-455.  Google Scholar

[27]

V. R. 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.  Google Scholar

[28]

H. G. 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.  Google Scholar

[29]

H. Peng, Z.-A. 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.  Google Scholar

[30]

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

[31]

M. Rascle, The Riemann problem for a nonlinear non-strictly hyperbolic system arising in biology, Hyperbolic partial differential equations. II, 11 (1985), 223-238.  doi: 10.1016/0898-1221(85)90148-8.  Google Scholar

[32]

L. G. RebholzD. WangZ. WangC. Zerfas and K. Zhao, Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions, Discrete Contin. Dyn. Syst., 39 (2019), 3789-3838.  doi: 10.3934/dcds.2019154.  Google Scholar

[33]

D. G. Schaeffer and M. Shearer, Riemann problems for non-strictly hyperbolic $2\times 2$ systems of conservation laws, Trans. Amer. Math. Soc., 304 (1987), 267-306.  doi: 10.2307/2000714.  Google Scholar

[34]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[35]

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

[36]

D. Wang, Z. Wang and K. Zhao, Cauchy problem of a system of parabolic conservation laws arising from the singular Keller-Segel model in multi-dimensions, Indiana Univ. Math. J., 70 (2021), 1-47. doi: 10.1512/iumj.2021.70.8075.  Google Scholar

[37]

Z. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods Appl. Sci., 31 (1008), 45-70.  doi: 10.1002/mma.898.  Google Scholar

[38]

Z.-A. WangZ. 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.  Google Scholar

[39]

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

[40]

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.  Google Scholar

[41]

Z. Zhang, The Existence and Decay of Solutions of a Class of Non-Strictly Hyperbolic Systems of Conservation Laws, Ph.D thesis, University of Houston in Houston, 1997.  Google Scholar

[42]

Y. Zheng, Globally smooth solutions to Cauchy problem of a quasilinear hyperbolic system arising in biology, Acta Math. Sci. Ser. B (Engl. Ed.), 21 (2001), 460-468.  doi: 10.1016/S0252-9602(17)30435-6.  Google Scholar

show all references

References:
[1]

J. A. CarrilloJ. Li and Z.-A. Wang, Boundary spike-layer solutions of the singular Keller-Segel system: Existence and stability, Proc. Lond. Math. Soc. (3), 122 (2021), 42-68.  doi: 10.1112/plms.12319.  Google Scholar

[2]

R. De la cruz, Riemann Problem for a $2\times 2$ hyperbolic system with linear damping, Acta Appl. Math., 170 (2020), 631-647.  doi: 10.1007/s10440-020-00350-w.  Google Scholar

[3]

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

[4]

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.  Google Scholar

[5]

P. Goatin and P. G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 881-902.  doi: 10.1016/j.anihpc.2004.02.002.  Google Scholar

[6]

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.  Google Scholar

[7]

H. Holden, On the Riemann problem for a prototype of a mixed type conservation law, Comm. Pure Appl. Math., 40 (1987), 229-264.  doi: 10.1002/cpa.3160400206.  Google Scholar

[8]

D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.   Google Scholar

[9]

Q. HouC.-J. LiuY.-G. 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.  Google Scholar

[10]

Q. Hou and Z. Wang, Convergence of boundary layers for the Keller-Segel system with singular sensitivity in the half-plane, J. Math. Pures Appl. (9), 130 (2019), 251-287.  doi: 10.1016/j.matpur.2019.01.008.  Google Scholar

[11]

Q. HouZ.-A. 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.  Google Scholar

[12]

E. L. IsaacsonD. Marchesin and B. J. Plohr, Transitional waves for conservation laws, SIAM J. Math. Anal., 21 (1990), 837-866.  doi: 10.1137/0521047.  Google Scholar

[13]

H.-Y. JinJ. Li and Z.-A. 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.  Google Scholar

[14]

E. F. Keller and L. A. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[15]

B. L. Keyfitz and H. C. Kranzer, The Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy, J. Differential Equations, 47 (1983), 35-65.  doi: 10.1016/0022-0396(83)90027-X.  Google Scholar

[16]

P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math., 10 (1957), 537-566.  doi: 10.1002/cpa.3160100406.  Google Scholar

[17]

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.  Google Scholar

[18]

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.  Google Scholar

[19]

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.  Google Scholar

[20]

T. LiH. Liu and L. Wang, Oscillatory traveling wave solutions to an attractive chemotaxis system, J. Differential Equations, 261 (2016), 7080-7098.  doi: 10.1016/j.jde.2016.09.012.  Google Scholar

[21]

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.  Google Scholar

[22]

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

[23]

T. Li and Z.-A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2010), 1522-1541.  doi: 10.1137/09075161X.  Google Scholar

[24]

T. Li and Z.-A. 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.  Google Scholar

[25]

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

[26]

D. Marchesin and P. J. Paes-Leme, A {R}iemann problem in gas dynamics with bifurcation, in Comput. Math. Appl. Part A, 12, Hyperbolic partial differential equations, III, (1986), 433-455.  Google Scholar

[27]

V. R. 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.  Google Scholar

[28]

H. G. 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.  Google Scholar

[29]

H. Peng, Z.-A. 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.  Google Scholar

[30]

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

[31]

M. Rascle, The Riemann problem for a nonlinear non-strictly hyperbolic system arising in biology, Hyperbolic partial differential equations. II, 11 (1985), 223-238.  doi: 10.1016/0898-1221(85)90148-8.  Google Scholar

[32]

L. G. RebholzD. WangZ. WangC. Zerfas and K. Zhao, Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions, Discrete Contin. Dyn. Syst., 39 (2019), 3789-3838.  doi: 10.3934/dcds.2019154.  Google Scholar

[33]

D. G. Schaeffer and M. Shearer, Riemann problems for non-strictly hyperbolic $2\times 2$ systems of conservation laws, Trans. Amer. Math. Soc., 304 (1987), 267-306.  doi: 10.2307/2000714.  Google Scholar

[34]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[35]

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

[36]

D. Wang, Z. Wang and K. Zhao, Cauchy problem of a system of parabolic conservation laws arising from the singular Keller-Segel model in multi-dimensions, Indiana Univ. Math. J., 70 (2021), 1-47. doi: 10.1512/iumj.2021.70.8075.  Google Scholar

[37]

Z. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods Appl. Sci., 31 (1008), 45-70.  doi: 10.1002/mma.898.  Google Scholar

[38]

Z.-A. WangZ. 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.  Google Scholar

[39]

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

[40]

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.  Google Scholar

[41]

Z. Zhang, The Existence and Decay of Solutions of a Class of Non-Strictly Hyperbolic Systems of Conservation Laws, Ph.D thesis, University of Houston in Houston, 1997.  Google Scholar

[42]

Y. Zheng, Globally smooth solutions to Cauchy problem of a quasilinear hyperbolic system arising in biology, Acta Math. Sci. Ser. B (Engl. Ed.), 21 (2001), 460-468.  doi: 10.1016/S0252-9602(17)30435-6.  Google Scholar

Figure 1.  Shock and Rarefaction Waves at $ U_{-} = (u_{-}, v_{-}) = (1, 3) $
Figure 2.  Shock and Rarefaction Waves at $ U_{-} = (u_{-}, v_{-}) = (1, 2) $
Figure 3.  Regions of $ U_{-} $ and $ U_{+} $
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