# American Institute of Mathematical Sciences

June  2018, 23(4): 1675-1688. doi: 10.3934/dcdsb.2018069

## Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion

 School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

* Corresponding author: Chunhua Jin

Received  June 2017 Revised  August 2017 Published  June 2018 Early access  January 2018

Fund Project: The author is supported by NSFC(11471127), Guangdong Natural Science Funds for Distinguished Young Scholar (2015A030306029).

In this paper, we deal with the following coupled chemotaxis-haptotaxis system modeling cancer invasionwith nonlinear diffusion,
 $\left\{ \begin{array}{l}{u_t} = \Delta {u^m} - \chi \nabla \cdot \left( {u \cdot \nabla v} \right) - \xi \nabla \cdot \left( {u \cdot \nabla w} \right) + \mu u\left( {1 - u - w} \right),{\rm{in}}\;\Omega \times {{\mathbb{R}}^ + },\\{v_t} - \nabla v + v = u,\;{\rm{in}}\;\Omega \times {{\mathbb{R}}^ + },\\{w_t} = - vw,\;\;{\rm{in}}\;\Omega \times {{\mathbb{R}}^ + },\end{array} \right.$
where
 $Ω\subset\mathbb R^N$
(
 $N≥ 3$
) is a bounded domain. Under zero-flux boundary conditions, we showed that for any
 $m>0$
, the problem admits a global bounded weak solution for any large initial datum if
 $\frac{χ}{μ}$
is appropriately small. The slow diffusion case (
 $m>1$
) of this problem have been studied by many authors [14,7,19,23], in which, the boundedness and the global in time solution are established for
 $m>\frac{2N}{N+2}$
, but the cases
 $m≤ \frac{2N}{N+2}$
remain open.
Citation: Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1675-1688. doi: 10.3934/dcdsb.2018069
##### References:
 [1] X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys. , 67 (2016), Art. 11, 13 pp. [2] M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Networks and Heterogeneous Media, 1 (2006), 399-439.  doi: 10.3934/nhm.2006.1.399. [3] T. Cieslak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045. [4] D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177. [5] C. Jin, Boundedness and global solvability to a chemotaxis model with nonlinear diffusion, J. Differential Equations, 263 (2017), 5759-5772.  doi: 10.1016/j.jde.2017.06.034. [6] E. Keller and A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. Available from: https://www.researchgate.net/publication/17711401_Initiation_of_Slime_Mold_Aggregation_Viewed_as_an_Instability. doi: 10.1016/0022-5193(70)90092-5. [7] Y. Li and J. Lankeit, Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), 1564-1595.  doi: 10.1088/0951-7715/29/5/1564. [8] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. [9] Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasilinear parabolic sys-tems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180. [10] Z. Szymanska, C. Morales-Rodrigo, M. Lachowicz and M. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models Methods Appl. Sci., 19 (2009), 257-281.  doi: 10.1142/S0218202509003425. [11] T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367.  doi: 10.4310/MAA.2001.v8.n2.a9. [12] C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.  doi: 10.1137/13094058X. [13] Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, arXiv: 1407.7382v1. [14] 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. [15] Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 1901-1914.  doi: 10.3934/dcds.2012.32.1901. [16] Y. Tao and M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115. [17] J. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [18] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020. [19] Y. 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. [20] 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. [21] M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J.Math. Anal. Appl., 48 (2008), 708-729.  doi: 10.1016/j.jmaa.2008.07.071. [22] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426. [23] J. Zheng, Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Discrete Contin. Dyn. Syst., 37 (2017), 627-643.

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##### References:
 [1] X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys. , 67 (2016), Art. 11, 13 pp. [2] M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Networks and Heterogeneous Media, 1 (2006), 399-439.  doi: 10.3934/nhm.2006.1.399. [3] T. Cieslak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045. [4] D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177. [5] C. Jin, Boundedness and global solvability to a chemotaxis model with nonlinear diffusion, J. Differential Equations, 263 (2017), 5759-5772.  doi: 10.1016/j.jde.2017.06.034. [6] E. Keller and A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. Available from: https://www.researchgate.net/publication/17711401_Initiation_of_Slime_Mold_Aggregation_Viewed_as_an_Instability. doi: 10.1016/0022-5193(70)90092-5. [7] Y. Li and J. Lankeit, Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), 1564-1595.  doi: 10.1088/0951-7715/29/5/1564. [8] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. [9] Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasilinear parabolic sys-tems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180. [10] Z. Szymanska, C. Morales-Rodrigo, M. Lachowicz and M. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models Methods Appl. Sci., 19 (2009), 257-281.  doi: 10.1142/S0218202509003425. [11] T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367.  doi: 10.4310/MAA.2001.v8.n2.a9. [12] C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.  doi: 10.1137/13094058X. [13] Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, arXiv: 1407.7382v1. [14] 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. [15] Y. Tao and M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., 32 (2012), 1901-1914.  doi: 10.3934/dcds.2012.32.1901. [16] Y. Tao and M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115. [17] J. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [18] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020. [19] Y. 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. [20] 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. [21] M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J.Math. Anal. Appl., 48 (2008), 708-729.  doi: 10.1016/j.jmaa.2008.07.071. [22] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426. [23] J. Zheng, Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Discrete Contin. Dyn. Syst., 37 (2017), 627-643.
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