Figure 1.
A particular example of domain $ \Omega $ .
-
Previous Article
Global asymptotic stability in a chemotaxis-growth model for tumor invasion
- DCDS-S Home
- This Issue
-
Next Article
Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity
February
2020, 13(2): 177-202.
doi: 10.3934/dcdss.2020010
On a chemotaxis model with competitive terms arising in angiogenesis
Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Fac. de Matemáticas, Universidad de Sevilla, Calle Tarfia s/n, 41012, Sevilla, Spain |
In this paper we study an anti-angiogenic therapy model that deactivates the tumor angiogenic factors. The model consists of four parabolic equations and considers the chemotaxis and a logistic law for the endothelial cells and several boundary conditions, some of them are non homogeneous. We study the parabolic problem, proving the existence of a unique global positive solution for positive initial conditions, and the stationary problem, justifying the existence of one real number, an eigenvalue of a certain problem, which determines if the semi-trivial solutions are stable or unstable and the existence of a coexistence state.
Mathematics Subject Classification:
Primary: 35K45, 35K57; Secondary: 92C17.
Citation:
Manuel Delgado, Inmaculada Gayte, Cristian Morales-Rodrigo, Antonio Suárez. On a chemotaxis model with competitive terms arising in angiogenesis. Discrete and Continuous Dynamical Systems - S,
2020, 13
(2)
: 177-202.
doi: 10.3934/dcdss.2020010
![]()
References:
| [1] |
H. Amann, Nonlinear elliptic equations with nonlinear boundary conditions, In New Developments in differential equations (eds. Eckhaus, W.), Math Studies, 21, North-Holland, Amsterdam, (1976), 43-63. |
| [2] |
H. Amann, Maximum principles and principal eigenvalues, In Ten Mathematical Essays on Approximation in Analyis and Topology (eds. J. Ferrera, J. López-Gómez and F. R. Ruíz del Portal), Elsevier, (2005), 1-60.
doi: 10.1016/B978-044451861-3/50001-X. |
| [3] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, In Function Spaces, Differential Operators and Nonlinear Analysis (eds. H. J. Schmeisser, H. Triebel), Teubner, Stuttgart, Leipzig, 133 (1993), 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
| [4] |
H. Amann and J. López-Gómez,
A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374.
doi: 10.1006/jdeq.1998.3440. |
| [5] |
M. A. J. Chaplain,
Avascular growth, angiogenesis and vascular growth in solid tumours: the mathematical modelling of the stages of tumour development, Math. Comput. Modelling, 23 (1996), 47-87.
doi: 10.1016/0895-7177(96)00019-2. |
| [6] |
T. Cieślak and C. Morales-Rodrigo,
Long-time behavior of an angiogenesis model with flux at the tumor boundary, Z. Angew. Math. Phys., 64 (2013), 1625-1641.
doi: 10.1007/s00033-013-0302-8. |
| [7] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation from simple eigenvalues, J. Funt. Anal., 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
| [8] |
M. Delgado, I. Gayte, C. Morales-Rodrigo and A. Suárez,
An angiogenesis model with nonlinear chemotactis response and flux at the tumor bondary, Nonlinear Anal., 72 (2010), 330-347.
doi: 10.1016/j.na.2009.06.057. |
| [9] |
M. Delgado, C. Morales-Rodrigo and A. Suárez,
Anti-angiogenic therapy based on the binding to receptors, Discrete and Continuous Dynamical Systems. Series A, 32 (2012), 3871-3894.
doi: 10.3934/dcds.2012.32.3871. |
| [10] |
J. García-Melián, J. D. Rossi and J. Sabina,
Existence and uniqueness of positive solutions to elliptic problems with sublinear mixed boundary conditions, Comm. Comtemporary Math., 11 (2009), 585-613.
doi: 10.1142/S0219199709003508. |
| [11] |
B. Gidas and J. Spruck,
A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.
doi: 10.1080/03605308108820196. |
| [12] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes Math., 840, Springer-Verlag, 1981.
doi: 10.1007/BFb0089647. |
| [13] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
| [14] |
J. López-Gómez,
Nonlinear eigenvalues and global bifurcations: Applications to the search of positive solutions for general Lotka-Volterra reaction-diffusion systems with two species, Differential Integral Equations, 7 (1994), 1427-1452.
|
| [15] |
J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman & Hall CRC, 2001.
doi: 10.1201/9781420035506. |
| [16] |
J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing, 2013.
doi: 10.1142/9789814440257_0001. |
| [17] |
N. V. Mantzaris, S. Webb and H. G. Othmer,
Mathematical modeling of tumor induced angiogenesis, J. Math. Biol., 49 (2004), 111-187.
doi: 10.1007/s00285-003-0262-2. |
| [18] |
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] |
H. Amann, Nonlinear elliptic equations with nonlinear boundary conditions, In New Developments in differential equations (eds. Eckhaus, W.), Math Studies, 21, North-Holland, Amsterdam, (1976), 43-63. |
| [2] |
H. Amann, Maximum principles and principal eigenvalues, In Ten Mathematical Essays on Approximation in Analyis and Topology (eds. J. Ferrera, J. López-Gómez and F. R. Ruíz del Portal), Elsevier, (2005), 1-60.
doi: 10.1016/B978-044451861-3/50001-X. |
| [3] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, In Function Spaces, Differential Operators and Nonlinear Analysis (eds. H. J. Schmeisser, H. Triebel), Teubner, Stuttgart, Leipzig, 133 (1993), 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
| [4] |
H. Amann and J. López-Gómez,
A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374.
doi: 10.1006/jdeq.1998.3440. |
| [5] |
M. A. J. Chaplain,
Avascular growth, angiogenesis and vascular growth in solid tumours: the mathematical modelling of the stages of tumour development, Math. Comput. Modelling, 23 (1996), 47-87.
doi: 10.1016/0895-7177(96)00019-2. |
| [6] |
T. Cieślak and C. Morales-Rodrigo,
Long-time behavior of an angiogenesis model with flux at the tumor boundary, Z. Angew. Math. Phys., 64 (2013), 1625-1641.
doi: 10.1007/s00033-013-0302-8. |
| [7] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation from simple eigenvalues, J. Funt. Anal., 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
| [8] |
M. Delgado, I. Gayte, C. Morales-Rodrigo and A. Suárez,
An angiogenesis model with nonlinear chemotactis response and flux at the tumor bondary, Nonlinear Anal., 72 (2010), 330-347.
doi: 10.1016/j.na.2009.06.057. |
| [9] |
M. Delgado, C. Morales-Rodrigo and A. Suárez,
Anti-angiogenic therapy based on the binding to receptors, Discrete and Continuous Dynamical Systems. Series A, 32 (2012), 3871-3894.
doi: 10.3934/dcds.2012.32.3871. |
| [10] |
J. García-Melián, J. D. Rossi and J. Sabina,
Existence and uniqueness of positive solutions to elliptic problems with sublinear mixed boundary conditions, Comm. Comtemporary Math., 11 (2009), 585-613.
doi: 10.1142/S0219199709003508. |
| [11] |
B. Gidas and J. Spruck,
A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.
doi: 10.1080/03605308108820196. |
| [12] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes Math., 840, Springer-Verlag, 1981.
doi: 10.1007/BFb0089647. |
| [13] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
| [14] |
J. López-Gómez,
Nonlinear eigenvalues and global bifurcations: Applications to the search of positive solutions for general Lotka-Volterra reaction-diffusion systems with two species, Differential Integral Equations, 7 (1994), 1427-1452.
|
| [15] |
J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman & Hall CRC, 2001.
doi: 10.1201/9781420035506. |
| [16] |
J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing, 2013.
doi: 10.1142/9789814440257_0001. |
| [17] |
N. V. Mantzaris, S. Webb and H. G. Othmer,
Mathematical modeling of tumor induced angiogenesis, J. Math. Biol., 49 (2004), 111-187.
doi: 10.1007/s00285-003-0262-2. |
| [18] |
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. |
| [1] |
Manuel Delgado, Cristian Morales-Rodrigo, Antonio Suárez. Anti-angiogenic therapy based on the binding to receptors. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3871-3894. doi: 10.3934/dcds.2012.32.3871 |
| [2] |
Maciej Leszczyński, Urszula Ledzewicz, Heinz Schättler. Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2315-2334. doi: 10.3934/dcdsb.2019097 |
| [3] |
Filippo Cacace, Valerio Cusimano, Alfredo Germani, Pasquale Palumbo, Federico Papa. Closed-loop control of tumor growth by means of anti-angiogenic administration. Mathematical Biosciences & Engineering, 2018, 15 (4) : 827-839. doi: 10.3934/mbe.2018037 |
| [4] |
Nasser Sweilam, Fathalla Rihan, Seham AL-Mekhlafi. A fractional-order delay differential model with optimal control for cancer treatment based on synergy between anti-angiogenic and immune cell therapies. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2403-2424. doi: 10.3934/dcdss.2020120 |
| [5] |
Robert Stephen Cantrell, Xinru Cao, King-Yeung Lam, Tian Xiang. A PDE model of intraguild predation with cross-diffusion. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3653-3661. doi: 10.3934/dcdsb.2017145 |
| [6] |
Yuan Lou, Wei-Ming Ni, Yaping Wu. On the global existence of a cross-diffusion system. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 193-203. doi: 10.3934/dcds.1998.4.193 |
| [7] |
Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589 |
| [8] |
Michael Winkler, Dariusz Wrzosek. Preface: Analysis of cross-diffusion systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : i-i. doi: 10.3934/dcdss.20202i |
| [9] |
Maxime Breden, Christian Kuehn, Cinzia Soresina. On the influence of cross-diffusion in pattern formation. Journal of Computational Dynamics, 2021, 8 (2) : 213-240. doi: 10.3934/jcd.2021010 |
| [10] |
Yaru Hu, Jinfeng Wang. Dynamics of an SIRS epidemic model with cross-diffusion. Communications on Pure and Applied Analysis, 2022, 21 (1) : 315-336. doi: 10.3934/cpaa.2021179 |
| [11] |
Cristian Morales-Rodrigo. A therapy inactivating the tumor angiogenic factors. Mathematical Biosciences & Engineering, 2013, 10 (1) : 185-198. doi: 10.3934/mbe.2013.10.185 |
| [12] |
Hideki Murakawa. A relation between cross-diffusion and reaction-diffusion. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 147-158. doi: 10.3934/dcdss.2012.5.147 |
| [13] |
Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228 |
| [14] |
Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. On a limiting system in the Lotka--Volterra competition with cross-diffusion. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 435-458. doi: 10.3934/dcds.2004.10.435 |
| [15] |
Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536 |
| [16] |
Kousuke Kuto, Yoshio Yamada. On limit systems for some population models with cross-diffusion. Discrete and Continuous Dynamical Systems - B, 2012, 17 (8) : 2745-2769. doi: 10.3934/dcdsb.2012.17.2745 |
| [17] |
Daniel Ryan, Robert Stephen Cantrell. Avoidance behavior in intraguild predation communities: A cross-diffusion model. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1641-1663. doi: 10.3934/dcds.2015.35.1641 |
| [18] |
Yi Li, Chunshan Zhao. Global existence of solutions to a cross-diffusion system in higher dimensional domains. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 185-192. doi: 10.3934/dcds.2005.12.185 |
| [19] |
F. Berezovskaya, Erika Camacho, Stephen Wirkus, Georgy Karev. "Traveling wave'' solutions of Fitzhugh model with cross-diffusion. Mathematical Biosciences & Engineering, 2008, 5 (2) : 239-260. doi: 10.3934/mbe.2008.5.239 |
| [20] |
Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 957-979. doi: 10.3934/dcdsb.2019198 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]