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Global asymptotic stability in a chemotaxis-growth model for tumor invasion
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Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity
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.
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. |

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