American Institute of Mathematical Sciences

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August  2016, 21(6): 2039-2056. doi: 10.3934/dcdsb.2016035

Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion

Pan Zheng 1,

1. 

Department of Applied Mathematics Chongqing University of Posts, and Telecommunications, Chongqing 400065

Received  August 2015 Revised  November 2015 Published  June 2016

This paper deals with a parabolic-elliptic-ODE chemotaxis-haptotaxis system with nonlinear diffusion \begin{eqnarray*}\label{1a} \left\{ \begin{split}{} &u_t=\nabla\cdot(\varphi(u)\nabla u)-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w), \\ &0=\Delta v-v+u, \\ &w_{t}=-vw, \end{split} \right. \end{eqnarray*} under Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{n}$ $(n\geq1)$, where $\chi$, $\xi$ and $\mu$ are positive parameters and $\varphi(u)$ is a nonlinear diffusion. Under the non-degenerate diffusion and some suitable assumptions on positive parameters $\chi,\xi,\mu$, it is shown that the corresponding initial boundary value problem possesses a unique global classical solution that is uniformly bounded in $\Omega\times(0,\infty)$. Moreover, under the degenerate diffusion, it is proved that the corresponding problem admits at least one nonnegative global bounded-in-time weak solution. Finally, for the suitably small initial data $w_{0}$, we give the decay estimate of $w$.
Keywords: Global boundedness, decay, chemotaxis-haptotaxis system, nonlinear diffusion..
Mathematics Subject Classification: Primary: 35K55; Secondary: 35B45, 35B33, 35K57, 92C1.
Citation: Pan Zheng. Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 2039-2056. doi: 10.3934/dcdsb.2016035
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N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113.

[2]

S. Aznavoorian, M. L. Stracke, H. Krutzsch, E. Schiffmann and L. A. Liotta, Signal transduction for chemotaxis and haptotaxis by matrix molecules in tumor cells, J. Cell Biol., 110 (1990), 1427-1438. doi: 10.1083/jcb.110.4.1427.

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D. Besser, P. Verde, Y. Nagamine and F. Blasi, Signal transduction and u-PA/u-PAR system, Fibrinolysis, 10 (1996), 215-237. doi: 10.1016/S0268-9499(96)80018-X.

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X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys., 67 (2016), p67, arXiv:1501.05383. doi: 10.1007/s00033-015-0601-3.

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X. Cao and S. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, Math. Methods Appl. Sci., 37 (2014), 2326-2330. doi: 10.1002/mma.2992.

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M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 15 (2005), 1685-1734. doi: 10.1142/S0218202505000947.

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M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med., 1 (2006), 399-439. doi: 10.3934/nhm.2006.1.399.

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T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076. doi: 10.1088/0951-7715/21/5/009.

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A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.

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K. Fujie, M. Winkler and T. Yokota, Blow-up prevention by logistic sources in a parabolic-elliptic Keller-Segel system with singular sensitivity, Nonlinear Anal., 109 (2014), 56-71. doi: 10.1016/j.na.2014.06.017.

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T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

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T. Hillen, K. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Methods Appl. Sci., 23 (2013), 165-198. doi: 10.1142/S0218202512500480.

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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.

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D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II. Jahresber. Deutsch. Math. -Verein., 106 (2004), 51-69.

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D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.

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W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6.

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E. F. Keller and L. A. 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.

[18]

J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527. doi: 10.3934/dcdsb.2015.20.1499.

[19]

Y. Li and J. Lankeit, Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), arXiv:1508.05846v1. doi: 10.1088/0951-7715/29/5/1564.

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L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733-737.

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

[22]

B. Perthame, Transport Equations in Biology, Birkhäser-BaselVerlag, Switzerland, 2007.

[23]

Y. Tao, Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source, J. Math. Anal. Appl., 354 (2009), 60-69. doi: 10.1016/j.jmaa.2008.12.039.

[24]

Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, arXiv:1407.7382v1, 2014.

[25]

Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion, Nonlinearity, 21 (2008), 2221-2238. doi: 10.1088/0951-7715/21/10/002.

[26]

Y. Tao and M. Wang, A combined chemotaxis-haptotaxis system: The role of logistic source, SIAM J. Math. Anal., 41 (2009), 1533-1558. doi: 10.1137/090751542.

[27]

Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 1 (2013), 1-36. doi: 10.1142/S0218202512500443.

[28]

Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704. doi: 10.1137/100802943.

[29]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.

[30]

Y. Tao and M. Winkler, Dominance of chemotaxis in a chemotaxis-haptotaxis model, Nonlinearity, 27 (2014), 1225-1239. doi: 10.1088/0951-7715/27/6/1225.

[31]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003.

[32]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Stud. Math. Appl., vol.2, North-Holland, Amsterdam, 1977.

[33]

L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872. doi: 10.1016/j.jde.2013.12.007.

[34]

Y. F. Wang, Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Differential Equations, 260 (2016), 1975-1989. doi: 10.1016/j.jde.2015.09.051.

[35]

Z. A. Wang, M. Winkler and D. Wrzosek, Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume-filling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 3502-3525. doi: 10.1137/110853972.

[36]

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.

[37]

M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855. doi: 10.1007/s00332-014-9205-x.

[38]

M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064. doi: 10.1016/j.na.2009.07.045.

[39]

J. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140. doi: 10.1016/j.jde.2015.02.003.

[40]

P. Zheng, C. Mu, X. Hu and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl., 424 (2015), 509-522. doi: 10.1016/j.jmaa.2014.11.031.

[41]

P. Zheng, C. Mu and X. Song, On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 1737-1757. doi: 10.3934/dcds.2016.36.1737.

show all references

References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113.

[2]

S. Aznavoorian, M. L. Stracke, H. Krutzsch, E. Schiffmann and L. A. Liotta, Signal transduction for chemotaxis and haptotaxis by matrix molecules in tumor cells, J. Cell Biol., 110 (1990), 1427-1438. doi: 10.1083/jcb.110.4.1427.

[3]

D. Besser, P. Verde, Y. Nagamine and F. Blasi, Signal transduction and u-PA/u-PAR system, Fibrinolysis, 10 (1996), 215-237. doi: 10.1016/S0268-9499(96)80018-X.

[4]

X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys., 67 (2016), p67, arXiv:1501.05383. doi: 10.1007/s00033-015-0601-3.

[5]

X. Cao and S. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, Math. Methods Appl. Sci., 37 (2014), 2326-2330. doi: 10.1002/mma.2992.

[6]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 15 (2005), 1685-1734. doi: 10.1142/S0218202505000947.

[7]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med., 1 (2006), 399-439. doi: 10.3934/nhm.2006.1.399.

[8]

T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076. doi: 10.1088/0951-7715/21/5/009.

[9]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.

[10]

K. Fujie, M. Winkler and T. Yokota, Blow-up prevention by logistic sources in a parabolic-elliptic Keller-Segel system with singular sensitivity, Nonlinear Anal., 109 (2014), 56-71. doi: 10.1016/j.na.2014.06.017.

[11]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

[12]

T. Hillen, K. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Methods Appl. Sci., 23 (2013), 165-198. doi: 10.1142/S0218202512500480.

[13]

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

[14]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II. Jahresber. Deutsch. Math. -Verein., 106 (2004), 51-69.

[15]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.

[16]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6.

[17]

E. F. Keller and L. A. 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.

[18]

J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527. doi: 10.3934/dcdsb.2015.20.1499.

[19]

Y. Li and J. Lankeit, Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), arXiv:1508.05846v1. doi: 10.1088/0951-7715/29/5/1564.

[20]

L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733-737.

[21]

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

[22]

B. Perthame, Transport Equations in Biology, Birkhäser-BaselVerlag, Switzerland, 2007.

[23]

Y. Tao, Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source, J. Math. Anal. Appl., 354 (2009), 60-69. doi: 10.1016/j.jmaa.2008.12.039.

[24]

Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, arXiv:1407.7382v1, 2014.

[25]

Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion, Nonlinearity, 21 (2008), 2221-2238. doi: 10.1088/0951-7715/21/10/002.

[26]

Y. Tao and M. Wang, A combined chemotaxis-haptotaxis system: The role of logistic source, SIAM J. Math. Anal., 41 (2009), 1533-1558. doi: 10.1137/090751542.

[27]

Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 1 (2013), 1-36. doi: 10.1142/S0218202512500443.

[28]

Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704. doi: 10.1137/100802943.

[29]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.

[30]

Y. Tao and M. Winkler, Dominance of chemotaxis in a chemotaxis-haptotaxis model, Nonlinearity, 27 (2014), 1225-1239. doi: 10.1088/0951-7715/27/6/1225.

[31]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003.

[32]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Stud. Math. Appl., vol.2, North-Holland, Amsterdam, 1977.

[33]

L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872. doi: 10.1016/j.jde.2013.12.007.

[34]

Y. F. Wang, Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Differential Equations, 260 (2016), 1975-1989. doi: 10.1016/j.jde.2015.09.051.

[35]

Z. A. Wang, M. Winkler and D. Wrzosek, Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume-filling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 3502-3525. doi: 10.1137/110853972.

[36]

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.

[37]

M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855. doi: 10.1007/s00332-014-9205-x.

[38]

M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064. doi: 10.1016/j.na.2009.07.045.

[39]

J. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140. doi: 10.1016/j.jde.2015.02.003.

[40]

P. Zheng, C. Mu, X. Hu and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl., 424 (2015), 509-522. doi: 10.1016/j.jmaa.2014.11.031.

[41]

P. Zheng, C. Mu and X. Song, On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 1737-1757. doi: 10.3934/dcds.2016.36.1737.

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