# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 2016, 36 (3) : 1737-1757. doi: 10.3934/dcds.2016.36.1737
##### References:
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Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains,, J. Differential Equations, 256 (2014), 2993. doi: 10.1016/j.jde.2014.01.028. Google Scholar [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. doi: 10.1090/S0002-9947-1992-1046835-6. Google Scholar [18] 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. Google Scholar [19] R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model,, J. Math. Anal. Appl., 343 (2008), 379. doi: 10.1016/j.jmaa.2008.01.005. Google Scholar [20] D. Liu and Y. Tao, Global boundedness in a fully parabolic attraction-repulsion chemotaxis model,, Math. Methods Appl. Sci., 38 (2015), 2537. doi: 10.1002/mma.3240. Google Scholar [21] L. Nirenberg, An extended interpolation inequality,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733. Google Scholar [22] C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311. doi: 10.1007/BF02476407. Google Scholar [23] B. Perthame, Transport Equations in Biology,, Birkhäser-BaselVerlag, (2007). Google Scholar [24] 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. Google Scholar [25] Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system,, , (2014). Google Scholar [26] 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. Google Scholar [27] 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. Google Scholar [28] Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis,, Math. Models Methods Appl. Sci., 23 (2013), 1. doi: 10.1142/S0218202512500443. Google Scholar [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. doi: 10.1137/100802943. Google Scholar [30] 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. Google Scholar [31] 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. Google Scholar [32] J. I. Tello and M. Winkler, A chemotaxis system with logistic source,, Comm. Partial Differential Equations, 32 (2007), 849. doi: 10.1080/03605300701319003. Google Scholar [33] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Stud. Math. Appl., (1977). Google Scholar [34] 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. Google Scholar [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. Google Scholar [36] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008. Google Scholar [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. doi: 10.1016/j.na.2009.07.045. Google Scholar

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. Google Scholar [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. Google Scholar [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. Google Scholar [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. doi: 10.1142/S0218202505000947. Google Scholar [5] 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. Google Scholar [6] 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. Google Scholar [7] A. Friedman, Partial Differential Equations,, Holt, (1969). Google Scholar [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. doi: 10.1016/j.na.2014.06.017. Google Scholar [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. doi: 10.1002/mma.3149. Google Scholar [10] K. Fujie and T. Yokota, Boundedness of solutions to parabolic-elliptic chemotaxis-growth systems with signal-dependent sensitivity,, Math. Bohem., 139 (2014), 639. Google Scholar [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. Google Scholar [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. Google Scholar [13] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I,, Jahresber. Deutsch. Math. -Verein., 105 (2003), 103. Google Scholar [14] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences II,, Jahresber. Deutsch. Math. -Verein., 106 (2004), 51. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.jde.2014.01.028. Google Scholar [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. doi: 10.1090/S0002-9947-1992-1046835-6. Google Scholar [18] 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. Google Scholar [19] R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model,, J. Math. Anal. Appl., 343 (2008), 379. doi: 10.1016/j.jmaa.2008.01.005. Google Scholar [20] D. Liu and Y. Tao, Global boundedness in a fully parabolic attraction-repulsion chemotaxis model,, Math. Methods Appl. Sci., 38 (2015), 2537. doi: 10.1002/mma.3240. Google Scholar [21] L. Nirenberg, An extended interpolation inequality,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733. Google Scholar [22] C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311. doi: 10.1007/BF02476407. Google Scholar [23] B. Perthame, Transport Equations in Biology,, Birkhäser-BaselVerlag, (2007). Google Scholar [24] 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. Google Scholar [25] Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system,, , (2014). Google Scholar [26] 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. Google Scholar [27] 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. Google Scholar [28] Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis,, Math. Models Methods Appl. Sci., 23 (2013), 1. doi: 10.1142/S0218202512500443. Google Scholar [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. doi: 10.1137/100802943. Google Scholar [30] 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. Google Scholar [31] 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. Google Scholar [32] J. I. Tello and M. Winkler, A chemotaxis system with logistic source,, Comm. Partial Differential Equations, 32 (2007), 849. doi: 10.1080/03605300701319003. Google Scholar [33] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Stud. Math. Appl., (1977). Google Scholar [34] 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. Google Scholar [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. Google Scholar [36] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008. Google Scholar [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. doi: 10.1016/j.na.2009.07.045. Google Scholar
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