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March  2020, 19(3): 1399-1419. doi: 10.3934/cpaa.2020070

Global existence for a chemotaxis-haptotaxis model with $ p $-Laplacian

Changchun Liu and  Pingping Li

Department of Mathematics, Jilin University, Changchun 130012, China

Received  April 2019 Revised  September 2019 Published  November 2019

Fund Project: This work is supported by the Jilin Scientific and Technological Development Program (no. 20170101143JC).

This paper deals with a chemotaxis-haptotaxis model with the slow $ p $-Laplacian diffusion in three-dimensional smooth bounded domains. It is proved that for any $ p>2 $, the chemotaxis-haptotaxis model problem admits a global bounded weak solution if $ \frac{\chi}{\mu} $ is appropriately small.

Keywords: Chemotaxis-haptotaxis, p-Laplacian diffusion, global existence, boundedness.
Mathematics Subject Classification: Primary: 92C17, 35K51; Secondary: 35K65, 35K92.
Citation: Changchun Liu, Pingping Li. Global existence for a chemotaxis-haptotaxis model with $ p $-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1399-1419. doi: 10.3934/cpaa.2020070
References:
[1]

X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys., 67 (2016), 11-13.  doi: 10.1007/s00033-015-0601-3.  Google Scholar

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Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.  doi: 10.1016/j.anihpc.2012.07.002.  Google Scholar

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[21]

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.  Google Scholar

[22]

Y. Wang and Y. Ke, Large time behavior of solution to a fully parabolic chemotaxis-haptotaxis model in higher dimensions, J. Differential Equations, 260 (2016), 6960-6988.  doi: 10.1016/j.jde.2016.01.017.  Google Scholar

[23]

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.  Google Scholar

show all references

References:
[1]

X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys., 67 (2016), 11-13.  doi: 10.1007/s00033-015-0601-3.  Google Scholar

[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.  Google Scholar

[3]

T. Cieślak 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.  Google Scholar

[4]

W. Cong and J. G. Liu, A degenerate p-Laplacian Keller-Segel model, Kinet. Relat. Models, 9 (2016), 687-714.  doi: 10.3934/krm.2016012.  Google Scholar

[5]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.  Google Scholar

[6]

C. Jin, Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1675-1688.  doi: 10.3934/dcdsb.2018069.  Google Scholar

[7]

C. Jin, Global classical solution and boundedness to a chemotaxis-haptotaxis model with re-establishment mechanisms, Bull. London Math. Soc., 50 (2018), 598-618.  doi: 10.1112/blms.12160.  Google Scholar

[8]

E. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[9]

S. Kurima and M. Mizukami, Global weak solutions to a 3-dimensional degenerate and singular chemotaxis-Navier-Stokes system with logistic source, Nonlinear Anal. Real World Appl., 46 (2019), 98-115.  doi: 10.1016/j.nonrwa.2018.09.011.  Google Scholar

[10]

G. Liţcanu and C. Morales-Rodrigo, Asymptotic behavior of global solutions to a model of cell invasion, Math. Models Methods Appl. Sci., 20 (2010), 1721-1758.  doi: 10.1142/S0218202510004775.  Google Scholar

[11]

K. Lin, C. Mu and L. Wang, Large-time behavior of an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124.  doi: 10.1016/j.jmaa.2014.12.052.  Google Scholar

[12]

M. Mei, H. Peng and Z. Wang, Asymptotic profile of a parabolic-hyperbolic system with boundary effect arising from tumor angiogenesis, J. Differential Equations, 259 (2015), 5168-5191.  doi: 10.1016/j.jde.2015.06.022.  Google Scholar

[13]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola. Norm. Sup. Pisa, 13 (1959), 115-162.   Google Scholar

[14]

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.  Google Scholar

[15]

Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, arXiv: 1407.7382, (2014). Google Scholar

[16]

W. Tao and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with slow p-Laplacian diffusion, Nonlinear Anal. Real Word Appl., 45 (2019), 26-52.  doi: 10.1016/j.nonrwa.2018.06.005.  Google Scholar

[17]

Y. Tao and M. Winkler, Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc., 19 (2017), 3641-3678.  doi: 10.4171/JEMS/749.  Google Scholar

[18]

Y. Tao and M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Appl., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115.  Google Scholar

[19]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.  doi: 10.1016/j.anihpc.2012.07.002.  Google Scholar

[20]

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.  Google Scholar

[21]

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.  Google Scholar

[22]

Y. Wang and Y. Ke, Large time behavior of solution to a fully parabolic chemotaxis-haptotaxis model in higher dimensions, J. Differential Equations, 260 (2016), 6960-6988.  doi: 10.1016/j.jde.2016.01.017.  Google Scholar

[23]

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.  Google Scholar

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