January  2021, 41(1): 439-454. doi: 10.3934/dcds.2020216

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

Received  December 2019 Published  May 2020

We consider the haptotaxis system
$ \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*} $
which arises as a simplified version of a recently proposed model for oncolytic virotherapy. When posed under no-flux boundary conditions in a smoothly bounded domain
$ \Omega\subset \mathbb{R}^2 $
, with positive parameters
$ D_w $
,
$ D_z $
and
$ \beta $
, and along with initial conditions involving suitably regular data, this system is known to admit global classical solutions.
It is shown that with respect to infinite-time blow-up, this system exhibits a critical mass phenomenon related to the quantity
$ m_c: = \frac{1}{(\beta-1)_+} $
: In fact, it is seen that each solution fulfilling
$ \frac{1}{|\Omega|} \int_\Omega u(\cdot,0) > m_c $
must be unbounded, and this is complemented by a boundedness result which inter alia asserts that for any choice of
$ m<m_c $
one can find a nontrivial set of solutions, particularly containing spatially heterogeneous solutions, each of which is bounded though satisfying
$ \frac{1}{|\Omega|} \int_\Omega u(\cdot,0) = m $
.
Citation: Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216
References:
[1]

T. AlzahraniR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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L. CorriasB. 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.  Google Scholar

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C. EngwerA. 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.  Google Scholar

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M. A. FontelosA. 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.  Google Scholar

[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.  Google Scholar

[8]

T. HillenK. 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.  Google Scholar

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H. A. LevineB. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

C. StinnerC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

A. ZhigunC. 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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

T. AlzahraniR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

L. CorriasB. 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.  Google Scholar

[5]

C. EngwerA. 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.  Google Scholar

[6]

M. A. FontelosA. 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.  Google Scholar

[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.  Google Scholar

[8]

T. HillenK. 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.  Google Scholar

[9]

H. A. LevineB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

C. StinnerC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

A. ZhigunC. 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.  Google Scholar

[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.  Google Scholar

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