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

* Corresponding author: C. Morales-Rodrigo

Received  May 2017 Revised  February 2018 Published  January 2019

Fund Project: Supported by MINECO (Spain) grant MTM2015-69875P.

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.

Citation: Manuel Delgado, Inmaculada Gayte, Cristian Morales-Rodrigo, Antonio Suárez. On a chemotaxis model with competitive terms arising in angiogenesis. Discrete & 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. DelgadoI. GayteC. 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.  Google Scholar

[9]

M. DelgadoC. 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.  Google Scholar

[10]

J. García-MeliánJ. 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.  Google Scholar

[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.  Google Scholar

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes Math., 840, Springer-Verlag, 1981. doi: 10.1007/BFb0089647.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[15]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman & Hall CRC, 2001. doi: 10.1201/9781420035506.  Google Scholar

[16]

J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing, 2013. doi: 10.1142/9789814440257_0001.  Google Scholar

[17]

N. V. MantzarisS. 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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. DelgadoI. GayteC. 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.  Google Scholar

[9]

M. DelgadoC. 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.  Google Scholar

[10]

J. García-MeliánJ. 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.  Google Scholar

[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.  Google Scholar

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes Math., 840, Springer-Verlag, 1981. doi: 10.1007/BFb0089647.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[15]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman & Hall CRC, 2001. doi: 10.1201/9781420035506.  Google Scholar

[16]

J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing, 2013. doi: 10.1142/9789814440257_0001.  Google Scholar

[17]

N. V. MantzarisS. 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.  Google Scholar

[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.  Google Scholar

Figure 1.  A particular example of domain $ \Omega $.
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