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Entire and ancient solutions of a supercritical semilinear heat equation
Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction
1. | School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China |
2. | Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany |
$ \begin{eqnarray*} \left\{ \begin{array}{lcl} u_t & = & \Delta u - \nabla \cdot (u\nabla v), \\ v_t & = & - (u+w)v, \\ w_t & = & D_w \Delta w - w + uz, \\ z_t & = & D_z \Delta z - z - uz + \beta w, \end{array} \right. \end{eqnarray*} $ |
$ \Omega\subset \mathbb{R}^2 $ |
$ D_w $ |
$ D_z $ |
$ \beta $ |
$ m_c: = \frac{1}{(\beta-1)_+} $ |
$ \frac{1}{|\Omega|} \int_\Omega u(\cdot,0) > m_c $ |
$ m<m_c $ |
$ \frac{1}{|\Omega|} \int_\Omega u(\cdot,0) = m $ |
References:
[1] |
T. Alzahrani, R. Raluca Eftimie and D. Dumitru Trucu,
Multiscale modelling of cancer response to oncolytic viral therapy, Math. Biosci., 310 (2019), 76-95.
doi: 10.1016/j.mbs.2018.12.018. |
[2] |
X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis system, Z. Angew. Math. Phys., 67 (2016), Art. 11, 13pp.
doi: 10.1007/s00033-015-0601-3. |
[3] |
M. A. J. Chaplain and G. Lolas,
Mathematical modelling of cancer cell invasion of tissue: the role of the urokinase plasminogen activation system, Math. Mod. Meth. Appl. Sci., 15 (2005), 1685-1734.
doi: 10.1142/S0218202505000947. |
[4] |
L. Corrias, B. Perthame and H. Zaag,
A chemotaxis model motivated by angiogenesis, C. R. Math. Acad. Sci. Paris, 336 (2003), 141-146.
doi: 10.1016/S1631-073X(02)00008-0. |
[5] |
C. Engwer, A. Hunt and C. Surulescu,
Effective equations for anisotropic glioma spread with proliferation: a multiscale approach and comparisons with previous settings, Math. Med. Biol., 33 (2016), 435-459.
doi: 10.1093/imammb/dqv030. |
[6] |
M. A. Fontelos, A. Friedman and B. Hu,
Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330-1355.
doi: 10.1137/S0036141001385046. |
[7] |
A. Friedman and J. I. Tello,
Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-163.
doi: 10.1016/S0022-247X(02)00147-6. |
[8] |
T. Hillen, K. J. Painter and M. Winkler,
Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Mod. Meth. Appl. Sci., 23 (2013), 165-198.
doi: 10.1142/S0218202512500480. |
[9] |
H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton,
Mathematical modeling of the onset of capillary formation initiating angiogenesis, J. Math. Biol., 42 (2001), 195-238.
doi: 10.1007/s002850000037. |
[10] |
Y. Li and J. Lankeit,
Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), 1564-1595.
doi: 10.1088/0951-7715/29/5/1564. |
[11] |
G. Liţcanu 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. |
[12] |
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. |
[13] |
C. Morales-Rodrigo and J. I. Tello,
Global existence and asymptotic behavior of a tumor angiogenesis model with chemotaxis and haptotaxis, Math. Models Methods Appl. Sci., 24 (2014), 427-464.
doi: 10.1142/S0218202513500553. |
[14] |
P. Y. H. Pang and Y. Wang,
Global boundedness of solutions to a chemotaxis-haptotaxis model with tissue remodeling, Math. Mod. Meth. Appl. Sci., 28 (2018), 2211-2235.
doi: 10.1142/S0218202518400134. |
[15] |
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. |
[16] |
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. |
[17] |
Y. Tao and M. Wang,
A combined chemotaxis-haptotaxis system: The role of logistic source, SIAM J. Math. Anal., 41 (2009), 1533-1558.
doi: 10.1137/090751542. |
[18] |
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. |
[19] |
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. Differential Eq., 257 (2014), 784-815.
doi: 10.1016/j.jde.2014.04.014. |
[20] |
Y. Tao and M. Winkler,
Large time behavior in a mutidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.
doi: 10.1137/15M1014115. |
[21] |
Y. Tao and M. Winkler,
A chemotaxis-haptotaxis system with haptoattractant remodeling: boundedness enforced by mild saturation of signal production, Commun. Pure Appl. Anal., 18 (2019), 2047-2067.
doi: 10.3934/cpaa.2019092. |
[22] |
Y. Tao and M. Winkler,
Global classical solutions to a doubly haptotactic cross-diffusion system modeling oncolytic virotherapy, J. Differential Equations, 268 (2020), 4973-4997.
doi: 10.1016/j.jde.2019.10.046. |
[23] |
Y. Tao and M. Winkler, A critical virus production rate for blow-up suppression in a haptotatxis model for oncolytic virotherapy, Nonlinear Anal., 198 (2020), 111870.
doi: 10.1016/j.na.2020.111870. |
[24] |
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. |
[25] |
Y. Wang,
Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Differential Equations, 260 (2016), 1975-1989.
doi: 10.1016/j.jde.2015.09.051. |
[26] |
M. Winkler and C. Surulescu,
A global weak solutions to a strongly degenerate haptotaxis model, Commun. Math. Sci., 15 (2017), 1581-1616.
doi: 10.4310/CMS.2017.v15.n6.a5. |
[27] |
M. Winkler,
Singular structure formation in a degenerate haptotaxis model involving myopic diffusion, J. Math. Pures Appl., 112 (2018), 118-169.
doi: 10.1016/j.matpur.2017.11.002. |
[28] |
A. Zhigun, C. Surulescu and A. Hunt,
A strongly degenerate diffusion-haptotaxis model of tumour invasion under the go-or-grow dichotomy hypothesis, Math. Methods Appl. Sci., 41 (2018), 2403-2428.
doi: 10.1002/mma.4749. |
[29] |
A. Zhigun, C. Surulescu and A. Uatay, Global existence for a degenerate haptotaxis model of cancer invasion, Z. Angew. Math. Phys., 67 (2016), Art. 146, 29pp.
doi: 10.1007/s00033-016-0741-0. |
show all references
References:
[1] |
T. Alzahrani, R. Raluca Eftimie and D. Dumitru Trucu,
Multiscale modelling of cancer response to oncolytic viral therapy, Math. Biosci., 310 (2019), 76-95.
doi: 10.1016/j.mbs.2018.12.018. |
[2] |
X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis system, Z. Angew. Math. Phys., 67 (2016), Art. 11, 13pp.
doi: 10.1007/s00033-015-0601-3. |
[3] |
M. A. J. Chaplain and G. Lolas,
Mathematical modelling of cancer cell invasion of tissue: the role of the urokinase plasminogen activation system, Math. Mod. Meth. Appl. Sci., 15 (2005), 1685-1734.
doi: 10.1142/S0218202505000947. |
[4] |
L. Corrias, B. Perthame and H. Zaag,
A chemotaxis model motivated by angiogenesis, C. R. Math. Acad. Sci. Paris, 336 (2003), 141-146.
doi: 10.1016/S1631-073X(02)00008-0. |
[5] |
C. Engwer, A. Hunt and C. Surulescu,
Effective equations for anisotropic glioma spread with proliferation: a multiscale approach and comparisons with previous settings, Math. Med. Biol., 33 (2016), 435-459.
doi: 10.1093/imammb/dqv030. |
[6] |
M. A. Fontelos, A. Friedman and B. Hu,
Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330-1355.
doi: 10.1137/S0036141001385046. |
[7] |
A. Friedman and J. I. Tello,
Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-163.
doi: 10.1016/S0022-247X(02)00147-6. |
[8] |
T. Hillen, K. J. Painter and M. Winkler,
Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Mod. Meth. Appl. Sci., 23 (2013), 165-198.
doi: 10.1142/S0218202512500480. |
[9] |
H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton,
Mathematical modeling of the onset of capillary formation initiating angiogenesis, J. Math. Biol., 42 (2001), 195-238.
doi: 10.1007/s002850000037. |
[10] |
Y. Li and J. Lankeit,
Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), 1564-1595.
doi: 10.1088/0951-7715/29/5/1564. |
[11] |
G. Liţcanu 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. |
[12] |
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. |
[13] |
C. Morales-Rodrigo and J. I. Tello,
Global existence and asymptotic behavior of a tumor angiogenesis model with chemotaxis and haptotaxis, Math. Models Methods Appl. Sci., 24 (2014), 427-464.
doi: 10.1142/S0218202513500553. |
[14] |
P. Y. H. Pang and Y. Wang,
Global boundedness of solutions to a chemotaxis-haptotaxis model with tissue remodeling, Math. Mod. Meth. Appl. Sci., 28 (2018), 2211-2235.
doi: 10.1142/S0218202518400134. |
[15] |
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. |
[16] |
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. |
[17] |
Y. Tao and M. Wang,
A combined chemotaxis-haptotaxis system: The role of logistic source, SIAM J. Math. Anal., 41 (2009), 1533-1558.
doi: 10.1137/090751542. |
[18] |
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. |
[19] |
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. Differential Eq., 257 (2014), 784-815.
doi: 10.1016/j.jde.2014.04.014. |
[20] |
Y. Tao and M. Winkler,
Large time behavior in a mutidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.
doi: 10.1137/15M1014115. |
[21] |
Y. Tao and M. Winkler,
A chemotaxis-haptotaxis system with haptoattractant remodeling: boundedness enforced by mild saturation of signal production, Commun. Pure Appl. Anal., 18 (2019), 2047-2067.
doi: 10.3934/cpaa.2019092. |
[22] |
Y. Tao and M. Winkler,
Global classical solutions to a doubly haptotactic cross-diffusion system modeling oncolytic virotherapy, J. Differential Equations, 268 (2020), 4973-4997.
doi: 10.1016/j.jde.2019.10.046. |
[23] |
Y. Tao and M. Winkler, A critical virus production rate for blow-up suppression in a haptotatxis model for oncolytic virotherapy, Nonlinear Anal., 198 (2020), 111870.
doi: 10.1016/j.na.2020.111870. |
[24] |
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. |
[25] |
Y. Wang,
Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Differential Equations, 260 (2016), 1975-1989.
doi: 10.1016/j.jde.2015.09.051. |
[26] |
M. Winkler and C. Surulescu,
A global weak solutions to a strongly degenerate haptotaxis model, Commun. Math. Sci., 15 (2017), 1581-1616.
doi: 10.4310/CMS.2017.v15.n6.a5. |
[27] |
M. Winkler,
Singular structure formation in a degenerate haptotaxis model involving myopic diffusion, J. Math. Pures Appl., 112 (2018), 118-169.
doi: 10.1016/j.matpur.2017.11.002. |
[28] |
A. Zhigun, C. Surulescu and A. Hunt,
A strongly degenerate diffusion-haptotaxis model of tumour invasion under the go-or-grow dichotomy hypothesis, Math. Methods Appl. Sci., 41 (2018), 2403-2428.
doi: 10.1002/mma.4749. |
[29] |
A. Zhigun, C. Surulescu and A. Uatay, Global existence for a degenerate haptotaxis model of cancer invasion, Z. Angew. Math. Phys., 67 (2016), Art. 146, 29pp.
doi: 10.1007/s00033-016-0741-0. |
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