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Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension
Global generalized solutions of a haptotaxis model describing cancer cells invasion and metastatic spread
School of Mathematics, Southeast University, Nanjing 211189, China |
$\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{l}}{{u_t} = \Delta u - \chi \nabla \cdot (u\nabla w),}&{x \in \Omega ,\;t > 0,}\\{{v_t} = {d_v}\Delta v - \xi \nabla \cdot (v\nabla w),}&{x \in \Omega ,\;t > 0,}\\{{m_t} = {d_m}\Delta m + u - m,}&{x \in \Omega ,\;t > 0,}\\{{w_t} = - \left( {{\gamma _1}u + m} \right)w,}&{x \in \Omega ,\;t > 0,}\end{array}} \right.}&{(0.1)}\end{array}$ |
$ \Omega\subset \mathbb{R}^3 $ |
$ \chi, \xi, d_{v}, d_{m},\gamma_{1}>0 $ |
$ u $ |
$ v $ |
$ m $ |
$ w $ |
$ (u_0, v_0, m_0, w_0) $ |
References:
[1] |
X. Cao, Boundedness in a three-dimensional chemotaxi–Chaptotaxis model, Z. Angew. Math. Phys., 67 (2016), 13 pp.
doi: 10.1007/s00033-015-0601-3. |
[2] |
M. A. J. Chaplain and G. Lolas,
Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity, Netw. Heterog. Media, 1 (2006), 399-439.
doi: 10.3934/nhm.2006.1.399. |
[3] |
L. C. Franssen, T. Lorenzi, A. E. F. Burgess and M. A. J. Chaplain,
A mathematical framework for modelling the metastatic spread of cancer, Bull. Math. Biol., 81 (2019), 1965-2010.
doi: 10.1007/s11538-019-00597-x. |
[4] |
K. Fujie,
Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.
doi: 10.1016/j.jmaa.2014.11.045. |
[5] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[6] |
C. Jin,
Global classical solutions and convergence to a mathematical model for cancer cells invasion and metastatic spread, J. Differ. Equ., 269 (2020), 3987-4021.
doi: 10.1016/j.jde.2020.03.018. |
[7] |
H. Y. Jin and T. Xiang,
Negligibility of haptotaxis effect in a chemotaxis–haptotaxis model, Math. Models Methods Appl. Sci., 31 (2021), 1373-1417.
doi: 10.1142/S0218202521500287. |
[8] |
J. Lankeit and M. Winkler, Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary, preprint, arXiv: 2103.07232, 2021. |
[9] |
G. Litcanu and C. Morales-Rodrig,
Asymptotic behavior of global solutions to a model of cell invasion, Math. Models Methods Appl. Sci., 20 (2010), 1721-1758.
doi: 10.1142/S0218202510004775. |
[10] |
A. Marciniak-Czochra and M. Ptashnyk,
Boundedness of solutions of a haptotaxis model, Math. Models Methods Appl. Sci., 20 (2010), 449-476.
doi: 10.1142/S0218202510004301. |
[11] |
N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 31 (2014), 851–875.
doi: 10.1016/j.anihpc.2013.07.007. |
[12] |
C. Morales-Rodrigo,
Local existence and uniqueness of regular solutions in a model of tissue invasion by solid tumours, Math. Comput. Model., 47 (2008), 604-613.
doi: 10.1016/j.mcm.2007.02.031. |
[13] |
P. Y. H. Pang and Y. Wang,
Global boundedness of solutions to a chemotaxis-haptotaxis model with tissue remodeling, Math. Models Methods Appl. Sci., 28 (2018), 2211-2235.
doi: 10.1142/S0218202518400134. |
[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 chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.
doi: 10.1016/j.jmaa.2011.02.041. |
[16] |
Y. Tao,
Global existence for a haptotaxis model of cancer invasion with tissue remodeling, Nonlinear Anal. Real World Appl., 12 (2011), 418-435.
doi: 10.1016/j.nonrwa.2010.06.027. |
[17] |
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. |
[18] |
Y. Tao and M. Winkler,
Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equ., 252 (2012), 2520-2543.
doi: 10.1016/j.jde.2011.07.010. |
[19] |
C. Walker and G.F. Webb,
Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Ana., 38 (2006/07), 1694-1713.
doi: 10.1137/060655122. |
[20] |
M. Winkler,
Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., 37 (2012), 319-351.
doi: 10.1080/03605302.2011.591865. |
[21] |
M. Winkler,
The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: global large-data solutions and their relaxation properties, Math. Models Methods Appl. Sci., 26 (2016), 987-1024.
doi: 10.1142/S0218202516500238. |
show all references
References:
[1] |
X. Cao, Boundedness in a three-dimensional chemotaxi–Chaptotaxis model, Z. Angew. Math. Phys., 67 (2016), 13 pp.
doi: 10.1007/s00033-015-0601-3. |
[2] |
M. A. J. Chaplain and G. Lolas,
Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity, Netw. Heterog. Media, 1 (2006), 399-439.
doi: 10.3934/nhm.2006.1.399. |
[3] |
L. C. Franssen, T. Lorenzi, A. E. F. Burgess and M. A. J. Chaplain,
A mathematical framework for modelling the metastatic spread of cancer, Bull. Math. Biol., 81 (2019), 1965-2010.
doi: 10.1007/s11538-019-00597-x. |
[4] |
K. Fujie,
Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.
doi: 10.1016/j.jmaa.2014.11.045. |
[5] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[6] |
C. Jin,
Global classical solutions and convergence to a mathematical model for cancer cells invasion and metastatic spread, J. Differ. Equ., 269 (2020), 3987-4021.
doi: 10.1016/j.jde.2020.03.018. |
[7] |
H. Y. Jin and T. Xiang,
Negligibility of haptotaxis effect in a chemotaxis–haptotaxis model, Math. Models Methods Appl. Sci., 31 (2021), 1373-1417.
doi: 10.1142/S0218202521500287. |
[8] |
J. Lankeit and M. Winkler, Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary, preprint, arXiv: 2103.07232, 2021. |
[9] |
G. Litcanu and C. Morales-Rodrig,
Asymptotic behavior of global solutions to a model of cell invasion, Math. Models Methods Appl. Sci., 20 (2010), 1721-1758.
doi: 10.1142/S0218202510004775. |
[10] |
A. Marciniak-Czochra and M. Ptashnyk,
Boundedness of solutions of a haptotaxis model, Math. Models Methods Appl. Sci., 20 (2010), 449-476.
doi: 10.1142/S0218202510004301. |
[11] |
N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 31 (2014), 851–875.
doi: 10.1016/j.anihpc.2013.07.007. |
[12] |
C. Morales-Rodrigo,
Local existence and uniqueness of regular solutions in a model of tissue invasion by solid tumours, Math. Comput. Model., 47 (2008), 604-613.
doi: 10.1016/j.mcm.2007.02.031. |
[13] |
P. Y. H. Pang and Y. Wang,
Global boundedness of solutions to a chemotaxis-haptotaxis model with tissue remodeling, Math. Models Methods Appl. Sci., 28 (2018), 2211-2235.
doi: 10.1142/S0218202518400134. |
[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 chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.
doi: 10.1016/j.jmaa.2011.02.041. |
[16] |
Y. Tao,
Global existence for a haptotaxis model of cancer invasion with tissue remodeling, Nonlinear Anal. Real World Appl., 12 (2011), 418-435.
doi: 10.1016/j.nonrwa.2010.06.027. |
[17] |
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. |
[18] |
Y. Tao and M. Winkler,
Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equ., 252 (2012), 2520-2543.
doi: 10.1016/j.jde.2011.07.010. |
[19] |
C. Walker and G.F. Webb,
Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Ana., 38 (2006/07), 1694-1713.
doi: 10.1137/060655122. |
[20] |
M. Winkler,
Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., 37 (2012), 319-351.
doi: 10.1080/03605302.2011.591865. |
[21] |
M. Winkler,
The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: global large-data solutions and their relaxation properties, Math. Models Methods Appl. Sci., 26 (2016), 987-1024.
doi: 10.1142/S0218202516500238. |
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