March  2016, 36(3): 1737-1757. doi: 10.3934/dcds.2016.36.1737

On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion

1. 

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

2. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331

3. 

College of Mathematic and Information, China West Normal University, Nanchong 637002, China

Received  December 2014 Revised  May 2015 Published  August 2015

This paper deals with a parabolic-parabolic-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), \\ &v_{t}=\Delta v-v+u, \\ &w_{t}=-vw, \end{split} \right. \end{eqnarray*} under Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{2}$, where $\chi$, $\xi$ and $\mu$ are positive parameters and $\varphi(u)$ is a nonlinear diffusion function. Firstly, under the case of non-degenerate diffusion, it is proved 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 case of degenerate diffusion, we prove that the corresponding problem admits at least one nonnegative global bounded-in-time weak solution. Finally, under some additional conditions, we derive the temporal decay estimate of $w$.
Citation: Pan Zheng, Chunlai Mu, Xiaojun Song. On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1737-1757. doi: 10.3934/dcds.2016.36.1737
References:
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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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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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K. Fujie, M. Winkler and T. Yokota, Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal-dependent sensitivity, Math. Methods Appl. Sci., 38 (2015), 1212-1224. doi: 10.1002/mma.3149.

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K. Fujie and T. Yokota, Boundedness of solutions to parabolic-elliptic chemotaxis-growth systems with signal-dependent sensitivity, Math. Bohem., 139 (2014), 639-647.

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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.

[16]

S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010. doi: 10.1016/j.jde.2014.01.028.

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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.

[19]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005.

[20]

D. Liu and Y. Tao, Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546. doi: 10.1002/mma.3240.

[21]

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

[22]

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

[23]

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

[24]

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.

[25]

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

[26]

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.

[27]

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.

[28]

Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36. doi: 10.1142/S0218202512500443.

[29]

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.

[30]

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.

[31]

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.

[32]

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

[33]

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

[34]

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.

[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, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.

[37]

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.

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]

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.

[5]

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.

[6]

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.

[7]

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

[8]

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.

[9]

K. Fujie, M. Winkler and T. Yokota, Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal-dependent sensitivity, Math. Methods Appl. Sci., 38 (2015), 1212-1224. doi: 10.1002/mma.3149.

[10]

K. Fujie and T. Yokota, Boundedness of solutions to parabolic-elliptic chemotaxis-growth systems with signal-dependent sensitivity, Math. Bohem., 139 (2014), 639-647.

[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]

S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010. doi: 10.1016/j.jde.2014.01.028.

[17]

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.

[18]

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.

[19]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398. doi: 10.1016/j.jmaa.2008.01.005.

[20]

D. Liu and Y. Tao, Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546. doi: 10.1002/mma.3240.

[21]

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

[22]

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

[23]

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

[24]

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.

[25]

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

[26]

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.

[27]

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.

[28]

Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36. doi: 10.1142/S0218202512500443.

[29]

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.

[30]

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.

[31]

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.

[32]

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

[33]

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

[34]

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.

[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, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.

[37]

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.

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