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Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion
School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China |
$\left\{ \begin{gathered} ut = \nabla \cdot \left( {{{\left( {u + 1} \right)}^{m - 1}}\nabla u} \right) - \nabla \cdot \left( {u\nabla v} \right) - \nabla \cdot \left( {u\nabla w} \right) + u\left( {1 - u - w} \right), \hfill \\ ut = \Delta v - v + u, \hfill \\ wt = - vw, \hfill \\ \end{gathered} \right.$ |
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N. D. Alikakos,
$L^p $ bounds of solutions of reaction-diffusion equations, Comm. Partial Diff. Eqns., 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[2] |
N. Bellomo, N. K. Li and P. K. Maini,
On the foundations of cancer modelling: Selected topics, speculations, and perspectives, Math. Models Methods Appl. Sci., 18 (2008), 593-646.
doi: 10.1142/S0218202508002796. |
[3] |
N. Bellomo, A. Belloquid, Y. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[4] |
X. Cao,
Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys., 67 (2016), Art. 11-13 pp.
doi: 10.1007/s00033-015-0601-3. |
[5] |
M. A. J. Chaplain and A. R. A. Anderson,
Mathematical modelling of tissue invasion, in Cancer Modelling and Simulation, L. Preziosi, ed., Chapman Hall/CRC, Boca Raton, FL, (2003), 269-297.
|
[6] |
M. A. J. Chaplain and G. Lolas,
Mathematical modelling of cancer invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 15 (2005), 1685-1734.
doi: 10.1142/S0218202505000947. |
[7] |
M. A. J. Chaplain and G. Lolas,
Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med., 1 (2006), 399-439.
doi: 10.3934/nhm.2006.1.399. |
[8] |
L. Corrias, B. Perthame and H. Zaag,
A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris, Ser. I., 336 (2003), 141-146.
doi: 10.1016/S1631-073X(02)00008-0. |
[9] |
L. Corrias, B. Perthame and H. Zaag,
Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28.
doi: 10.1007/s00032-003-0026-x. |
[10] |
A. Friedman and G. Lolas,
Analysis of a mathematical model of tumor lymphangiogenesis, Math. Models Methods Appl. Sci., 15 (2005), 95-107.
doi: 10.1142/S0218202505003915. |
[11] |
D. D. Haroske and H. Triebel,
Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008. |
[12] |
H. Hajaiej, L. Molinet, T. Ozawa and B. Wang,
Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations, Harmonic Analysis and Nonlinear Partial Differential Equations, in: RIMS Kôkyûroku Bessatsu, B26 (2011), 159-175.
|
[13] |
M. Herrero and J. Velázquez,
A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Super. Pisa Cl. Sci., 24 (1997), 633-683.
|
[14] |
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), 165-198.
doi: 10.1142/S0218202512500480. |
[15] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Diff. Eqns., 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[16] |
J. Liu, J. Zheng and Y. Wang,
Boundedness in a quasilinear chemotaxis-haptotaxis system with logistic source, Z. Angew. Math. Phys., 67 (2016), 1-33.
doi: 10.1007/s00033-016-0620-8. |
[17] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Ser. Int., 40 (1997), 411-433.
|
[18] |
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Diff. Eqns., 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[19] |
E. Keller and L. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[20] |
E. Keller and L. Segel,
Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[21] |
G. Liţanu 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. |
[22] |
G. Meral, C. Stinner and C. Surulescu,
On a multiscale model involving cell contractivity and its effects on tumor invasion, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 189-213.
doi: 10.3934/dcdsb.2015.20.189. |
[23] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxisgrowth system of equations, Nonlinear Anal. TMA., 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[24] |
C. Stinner, C. Surulescu and G. Meral,
A multiscale model for pH-tactic invasion with time-varying carrying capacities, IMA J. Appl. Math., 80 (2015), 1300-1321.
doi: 10.1093/imamat/hxu055. |
[25] |
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. |
[26] |
Y. Tao,
Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source, J. Math. Anal. Appl., 354 (2009), 60-69.
doi: 10.1016/j.jmaa.2008.12.039. |
[27] |
Y. Tao,
Boundedness in a two-dimensional chemotaxis-haptotaxis system, Mathematics, 70 (2014), 165-174.
|
[28] |
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. |
[29] |
Y. Tao and M. Winkler,
A chemotaxis-haptotaxis model: The roles of porous medium diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.
doi: 10.1137/100802943. |
[30] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Diff. Eqns., 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[31] |
Y. Tao and M. Winkler,
Boundedness and stabilization in a multi-dimensional chemotaxis-haptotaxis mode, Proceedings of the Royal Society of Edinburgh, 144 (2014), 1067-1084.
doi: 10.1017/S0308210512000571. |
[32] |
Y. Tao and M. Winkler,
Dominance of chemotaxis in a chemotaxis-haptotaxis model, Nonlinearity, 27 (2014), 1225-1239.
doi: 10.1088/0951-7715/27/6/1225. |
[33] |
Y. Tao and M. Winkler,
Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Diff. Eqns., 257 (2014), 784-815.
doi: 10.1016/j.jde.2014.04.014. |
[34] |
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. |
[35] |
C. Walker and G. F. Webb,
Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Anal., 38 (2007), 1694-1713.
doi: 10.1137/060655122. |
[36] |
L. Wang, Y. Li and C. Mu,
Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. Ser. A., 34 (2014), 789-802.
doi: 10.3934/dcds.2014.34.789. |
[37] |
Y. Wang,
Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Diff. Eqns., 260 (2016), 1975-1989.
doi: 10.1016/j.jde.2015.09.051. |
[38] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Diff. Eqns., 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[39] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Diff. Eqns., 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[40] |
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. |
[41] |
J. Zheng,
Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Diff. Eqns., 259 (2015), 120-140.
doi: 10.1016/j.jde.2015.02.003. |
show all references
References:
[1] |
N. D. Alikakos,
$L^p $ bounds of solutions of reaction-diffusion equations, Comm. Partial Diff. Eqns., 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[2] |
N. Bellomo, N. K. Li and P. K. Maini,
On the foundations of cancer modelling: Selected topics, speculations, and perspectives, Math. Models Methods Appl. Sci., 18 (2008), 593-646.
doi: 10.1142/S0218202508002796. |
[3] |
N. Bellomo, A. Belloquid, Y. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[4] |
X. Cao,
Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys., 67 (2016), Art. 11-13 pp.
doi: 10.1007/s00033-015-0601-3. |
[5] |
M. A. J. Chaplain and A. R. A. Anderson,
Mathematical modelling of tissue invasion, in Cancer Modelling and Simulation, L. Preziosi, ed., Chapman Hall/CRC, Boca Raton, FL, (2003), 269-297.
|
[6] |
M. A. J. Chaplain and G. Lolas,
Mathematical modelling of cancer invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 15 (2005), 1685-1734.
doi: 10.1142/S0218202505000947. |
[7] |
M. A. J. Chaplain and G. Lolas,
Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med., 1 (2006), 399-439.
doi: 10.3934/nhm.2006.1.399. |
[8] |
L. Corrias, B. Perthame and H. Zaag,
A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris, Ser. I., 336 (2003), 141-146.
doi: 10.1016/S1631-073X(02)00008-0. |
[9] |
L. Corrias, B. Perthame and H. Zaag,
Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28.
doi: 10.1007/s00032-003-0026-x. |
[10] |
A. Friedman and G. Lolas,
Analysis of a mathematical model of tumor lymphangiogenesis, Math. Models Methods Appl. Sci., 15 (2005), 95-107.
doi: 10.1142/S0218202505003915. |
[11] |
D. D. Haroske and H. Triebel,
Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008. |
[12] |
H. Hajaiej, L. Molinet, T. Ozawa and B. Wang,
Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations, Harmonic Analysis and Nonlinear Partial Differential Equations, in: RIMS Kôkyûroku Bessatsu, B26 (2011), 159-175.
|
[13] |
M. Herrero and J. Velázquez,
A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Super. Pisa Cl. Sci., 24 (1997), 633-683.
|
[14] |
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), 165-198.
doi: 10.1142/S0218202512500480. |
[15] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Diff. Eqns., 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[16] |
J. Liu, J. Zheng and Y. Wang,
Boundedness in a quasilinear chemotaxis-haptotaxis system with logistic source, Z. Angew. Math. Phys., 67 (2016), 1-33.
doi: 10.1007/s00033-016-0620-8. |
[17] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Ser. Int., 40 (1997), 411-433.
|
[18] |
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Diff. Eqns., 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[19] |
E. Keller and L. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[20] |
E. Keller and L. Segel,
Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[21] |
G. Liţanu 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. |
[22] |
G. Meral, C. Stinner and C. Surulescu,
On a multiscale model involving cell contractivity and its effects on tumor invasion, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 189-213.
doi: 10.3934/dcdsb.2015.20.189. |
[23] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxisgrowth system of equations, Nonlinear Anal. TMA., 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[24] |
C. Stinner, C. Surulescu and G. Meral,
A multiscale model for pH-tactic invasion with time-varying carrying capacities, IMA J. Appl. Math., 80 (2015), 1300-1321.
doi: 10.1093/imamat/hxu055. |
[25] |
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. |
[26] |
Y. Tao,
Global existence of classical solutions to a combined chemotaxis-haptotaxis model with logistic source, J. Math. Anal. Appl., 354 (2009), 60-69.
doi: 10.1016/j.jmaa.2008.12.039. |
[27] |
Y. Tao,
Boundedness in a two-dimensional chemotaxis-haptotaxis system, Mathematics, 70 (2014), 165-174.
|
[28] |
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. |
[29] |
Y. Tao and M. Winkler,
A chemotaxis-haptotaxis model: The roles of porous medium diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.
doi: 10.1137/100802943. |
[30] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Diff. Eqns., 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[31] |
Y. Tao and M. Winkler,
Boundedness and stabilization in a multi-dimensional chemotaxis-haptotaxis mode, Proceedings of the Royal Society of Edinburgh, 144 (2014), 1067-1084.
doi: 10.1017/S0308210512000571. |
[32] |
Y. Tao and M. Winkler,
Dominance of chemotaxis in a chemotaxis-haptotaxis model, Nonlinearity, 27 (2014), 1225-1239.
doi: 10.1088/0951-7715/27/6/1225. |
[33] |
Y. Tao and M. Winkler,
Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Diff. Eqns., 257 (2014), 784-815.
doi: 10.1016/j.jde.2014.04.014. |
[34] |
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. |
[35] |
C. Walker and G. F. Webb,
Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Anal., 38 (2007), 1694-1713.
doi: 10.1137/060655122. |
[36] |
L. Wang, Y. Li and C. Mu,
Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. Ser. A., 34 (2014), 789-802.
doi: 10.3934/dcds.2014.34.789. |
[37] |
Y. Wang,
Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Diff. Eqns., 260 (2016), 1975-1989.
doi: 10.1016/j.jde.2015.09.051. |
[38] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Diff. Eqns., 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[39] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Diff. Eqns., 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[40] |
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. |
[41] |
J. Zheng,
Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Diff. Eqns., 259 (2015), 120-140.
doi: 10.1016/j.jde.2015.02.003. |
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