# American Institute of Mathematical Sciences

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 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. 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. 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. 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. Google Scholar [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. Google Scholar [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. 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. 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. Google Scholar [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. 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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [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. Google Scholar [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. 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. 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. Google Scholar [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. 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. 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. 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. 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. 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  [1] Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. 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