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

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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

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

Received August 2015 Revised November 2015 Published June 2016

Abstract
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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 & Continuous Dynamical Systems - B, 2016, 21 (6) : 2039-2056. doi: 10.3934/dcdsb.2016035

References:

[1]	N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations,, Comm. Partial Differential Equations, 4 (1979), 827. 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. 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. 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). 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. 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. 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. 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. doi: 10.1088/0951-7715/21/5/009.
[9]	A. Friedman, Partial Differential Equations,, Holt, (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. 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. 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. doi: 10.1142/S0218202512500480.
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[14]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II., Jahresber. Deutsch. Math. -Verein., 106 (2004), 51.
[15]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52. 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. 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. 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. 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). 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.
[21]	C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311. doi: 10.1007/BF02476407.
[22]	B. Perthame, Transport Equations in Biology,, Birkhäser-BaselVerlag, (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. doi: 10.1016/j.jmaa.2008.12.039.
[24]	Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system,, , (2014).
[25]	Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion,, Nonlinearity, 21 (2008), 2221. 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. 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. 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. 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. 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. 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. doi: 10.1080/03605300701319003.
[32]	R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Stud. Math. Appl., (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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). 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. 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. 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. 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. doi: 10.1088/0951-7715/21/5/009.
[9]	A. Friedman, Partial Differential Equations,, Holt, (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. 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. 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. 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.
[14]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II., Jahresber. Deutsch. Math. -Verein., 106 (2004), 51.
[15]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52. 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. 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. 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. 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). 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.
[21]	C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311. doi: 10.1007/BF02476407.
[22]	B. Perthame, Transport Equations in Biology,, Birkhäser-BaselVerlag, (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. doi: 10.1016/j.jmaa.2008.12.039.
[24]	Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system,, , (2014).
[25]	Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion,, Nonlinearity, 21 (2008), 2221. 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. 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. 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. 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. 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. 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. doi: 10.1080/03605300701319003.
[32]	R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Stud. Math. Appl., (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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.3934/dcds.2016.36.1737.

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