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Electromagnetic interior transmission eigenvalue problem for an inhomogeneous medium with a conductive boundary
Global existence for a chemotaxis-haptotaxis model with $ p $-Laplacian
Department of Mathematics, Jilin University, Changchun 130012, China |
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.
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. |
[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. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
L. Nirenberg,
On elliptic partial differential equations, Ann. Scuola. Norm. Sup. Pisa, 13 (1959), 115-162.
|
[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. |
[15] |
Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, arXiv: 1407.7382, (2014). |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
L. Nirenberg,
On elliptic partial differential equations, Ann. Scuola. Norm. Sup. Pisa, 13 (1959), 115-162.
|
[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. |
[15] |
Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, arXiv: 1407.7382, (2014). |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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