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  January 2021 Early access  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 and Continuous Dynamical Systems, 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.

[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. 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.

[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.

[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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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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