This paper deals with a haptotaxis system modeling oncolytic virotherapy with nonlinear sensitivity
$ \begin{eqnarray*} \left\{ \begin{split}{} &u_t = \Delta u - \nabla\cdot(u\chi(v)\nabla v) + \mu u(1-u) - uz, &(x, t)\in \Omega\times (0, \infty), \\ &v_t = -(u+w)v, &(x, t)\in \Omega\times (0, \infty), \\ &w_t = \Delta w - w + uz, &(x, t)\in \Omega\times (0, \infty), \\ &z_t = \Delta z - z - uz +\beta w, &(x, t)\in \Omega\times (0, \infty), \ \end{split} \right. \end{eqnarray*} $
under homogeneous Neumann boundary conditions in a smoothly bounded domain $ \Omega\subset\mathbb{R}^{3} $, where $ \mu, \beta>0 $ and $ \chi(v) $ is a nonlinear sensitivity. Firstly, we obtain the local well-posedness and global boundedness of solutions for the above system. Moreover, when $ \beta\in(0, 1) $, we can prove that the global solution $ (u, v, w, z) $ exponentially stabilizes to the constant equilibrium $ (1, 0, 0, 0) $ in the topology $ L^{p}(\Omega)\times(L^{\infty}(\Omega))^3 $ with $ p>1 $ as $ t \rightarrow \infty $.
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[1] | M. Agueh, Gagliardo-Nirenberg inequalities involving the gradient $L^2$-norm, C. R. Math., 346 (2008), 757-762. doi: 10.1016/j.crma.2008.05.015. |
[2] | N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. P. Differ. Equ., 4 (1979), 827-868. doi: 10.1080/03605307908820113. |
[3] | T. Alzahrani, R. Eftimie and D. Trucu, Multiscale modelling of cancer response to oncolytic viral therapy, Math. Biosci., 310 (2019), 76-95. doi: 10.1016/j.mbs.2018.12.018. |
[4] | T. Alzahrani, R. Eftimie and D. Trucu, Multiscale moving boundary modelling of cancer interactions with a fusogenic oncolytic virus: The impact of syncytia dynamics, Math. Biosci., 323 (2020), 108296, 22 pp. doi: 10.1016/j.mbs.2019.108296. |
[5] | A. R. A. Anderson, M. A. J. Chaplain, E. L. Newman, R. J. C. Steele and A. M. Thompson, Mathematical modelling of tumour invasion and metastasis, J. Theor. Med., 2 (2000), 129-154. doi: 10.1080/10273660008833042. |
[6] | M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 15 (2005), 1685-1734. doi: 10.1142/S0218202505000947. |
[7] | M. A. J. Chaplain and J. I. Tello, On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Appl. Math. Lett., 57 (2016), 1-6. doi: 10.1016/j.aml.2015.12.001. |
[8] | Z. Chen, Dampening effect of logistic source in a two-dimensional haptotaxis system with nonlinear zero-order interaction, J. Math. Anal. Appl., 492 (2020), 124435, 17 pp. doi: 10.1016/j.jmaa.2020.124435. |
[9] | L. Corrias, B. Perthame and H. Zaag, Global solutions in some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x. |
[10] | K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684. doi: 10.1016/j.jmaa.2014.11.045. |
[11] | F. R. Guarguaglini and R. Natalini, Global existence and uniqueness of solutions for multidimensional weakly parabolic systems arising in chemistry and biology, Commun. Pure Appl. Anal., 6 (2007), 287-309. doi: 10.3934/cpaa.2007.6.287. |
[12] | C. Jin, Global classical solution and boundedness to a chemotaxis-haptotaxis model with re-establishment mechanisms, Bull. Lond. Math. Soc., 50 (2018), 598-618. doi: 10.1112/blms.12160. |
[13] | E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. |
[14] | O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'eva, Linear and Quasi-Linear Equations of Parabolic Type, 1$^{nd}$ edition, American Mathematical Society, Providence, 1968. |
[15] | J. Lankeit and M. Winkler, A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: Global solvability for large nonradial data, Nonlinear Differ. Equ. Appl., 24 (2017), Paper No. 49, 33 pp. doi: 10.1007/s00030-017-0472-8. |
[16] | S. Lawler, M. Speranza, C. Cho and E. Chiocca, Oncolytic viruses in cancer treatment: A review, JAMA Oncol., 3 (2017), 841-849. doi: 10.1001/jamaoncol.2016.2064. |
[17] | J. Li and Y. Wang, Boundedness in a haptotactic cross-diffusion system modeling oncolytic virotherapy, J. Differ. Equ., 270 (2021), 94-113. doi: 10.1016/j.jde.2020.07.032. |
[18] | 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. |
[19] | G. Litcanu and C. Morales-Rodrigo, Asymptotic behavior of global solutions to a model of cell invasion, Math. Mod. Meth. Appl. S., 20 (2010), 1721-1758. doi: 10.1142/S0218202510004775. |
[20] | G. Litcanu and C. Morales-Rodrigo, Global solutions and asymptotic behavior for a parabolic degenerate coupled system arising from biology, Nonlinear Anal., 72 (2010), 77-98. doi: 10.1016/j.na.2009.06.083. |
[21] | C. Morales-Rodrigo, Local existence and uniqueness of regular solutions in a model of tissue invasion by solid tumours, Math. Comput. Model., 47 (2008), 604-613. doi: 10.1016/j.mcm.2007.02.031. |
[22] | P. Pang and Y. Wang, Global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differ. Equ., 263 (2017), 1269-1292. doi: 10.1016/j.jde.2017.03.016. |
[23] | G. Ren and B. Liu, Global classical solvability in a three-dimensional haptotaxis system modeling oncolytic virotherapy, Math. Method. Appl. Sci., 44 (2021), 9275-9291. doi: 10.1002/mma.7354. |
[24] | G. Ren and J. Wei, Analysis of a two-dimensional triply haptotactic model with a fusogenic oncolytic virus and syncytia, Z. Angew. Math. Phys., 72 (2021), Paper No. 134, 23 pp. doi: 10.1007/s00033-021-01572-0. |
[25] | 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. |
[26] | C. Stinner and M. Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity, Nonlinear Anal. Real World Appl., 12 (2011), 3727-3740. doi: 10.1016/j.nonrwa.2011.07.006. |
[27] | X. Tao, Global classical solutions to an oncolytic viral therapy model with triply haptotactic terms, Acta Appl. Math., 171 (2021), Paper No. 5, 11 pp. doi: 10.1007/s10440-020-00375-1. |
[28] | X. Tao, Global weak solutions to an oncolytic viral therapy model with doubly haptotactic terms, Nonlinear Anal. Real World Appl., 60 (2021), Paper No. 103276, 22 pp. doi: 10.1016/j.nonrwa.2020.103276. |
[29] | X. Tao and S. Zhou, Dampening effects on global boundedness and asymptotic behavior in an oncolytic virotherapy model, J. Differ. Equ., 308 (2022), 57-76. doi: 10.1016/j.jde.2021.11.003. |
[30] | Y. Tao, Global existence for a haptotaxis model of cancer invasion with tissue remodeling, Nonlinear Anal. Real World Appl., 12 (2011), 418-435. doi: 10.1016/j.nonrwa.2010.06.027. |
[31] | Y. Tao and M. Winkler, A critical virus production rate for blow-up suppression in a haptotaxis model for oncolytic virotherapy, Nonlinear Anal., 198 (2020), 111870, 17 pp. doi: 10.1016/j.na.2020.111870. |
[32] | 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. Differ. Equ., 257 (2014), 784-815. doi: 10.1016/j.jde.2014.04.014. |
[33] | Y. Tao and M. Winkler, Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction, Discrete Contin. Dyn. Syst., 41 (2021), 439-454. doi: 10.3934/dcds.2020216. |
[34] | Y. Tao and M. Winkler, A critical virus production rate for efficiency of oncolytic virotherapy, Eur. J. Appl. Math., 32 (2021), 301-316. doi: 10.1017/S0956792520000133. |
[35] | Y. Tao and M. Winkler, Global classical solutions to a doubly haptotactic cross-diffusion system modeling oncolytic virotherapy, J. Differ. Equ., 268 (2020), 4973-4997. doi: 10.1016/j.jde.2019.10.046. |
[36] | Y. Tao and M. Winkler, Asymptotic stability of spatial homogeneity in a haptotaxis model for oncolytic virotherapy, P. Roy. Soc. Edinb. A., 152 (2020), 81-101. doi: 10.1017/prm.2020.97. |
[37] | Y. Wang and C. Xu, Asymptotic behaviour of a doubly haptotactic cross-diffusion model for oncolytic virotherapy, P. Roy. Soc. Edinb. A., 153 (2023), 881-906. doi: 10.1017/prm.2022.24. |
[38] | Y. Wang and C. Xu, Asymptotic behavior of a three-dimensional haptotactic cross-diffusion system modeling oncolytic virotherapy, Math. Models Methods Appl. Sci., 33 (2023), 2313-2335. doi: 10.1142/S0218202523400043. |
[39] | M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008. |
[40] | M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673. doi: 10.1002/mana.200810838. |
[41] | M. Winkler, Global existence and slow grow-up in a quasilinear Keller-Segel system with exponentially decaying diffusivity, Nonlinearity, 30 (2017), 735-764. doi: 10.1088/1361-6544/aa565b. |
[42] | M. Winkler, Singular structure formation in a degenerate haptotaxis model involving myopic diffusion, J. Math. Pure Appl., 112 (2018), 118-169. doi: 10.1016/j.matpur.2017.11.002. |
[43] | M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 34 (2011), 176-190. doi: 10.1002/mma.1346. |
[44] | M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: global largedata solutions and their relaxation properties, Math. Models Methods Appl. Sci., 26 (2016), 987-1024. doi: 10.1142/S0218202516500238. |
[45] | M. Winkler, Renormalized radial large-data solutions to the higher-dimensional Keller-Segel system with singular sensitivity and signal absorption, J. Differ. Equ., 264 (2018), 2310-2350. doi: 10.1016/j.jde.2017.10.029. |
[46] | M. Winkler and T. Yokota, Stabilization in the logarithmic Keller-Segel system, Nonlinear Anal., 170 (2018), 123-141. doi: 10.1016/j.na.2018.01.002. |
[47] | J. Zheng and J. Xie, Global classical solutions to a higher-dimensional doubly haptotactic cross-diffusion system modeling oncolytic virotherapy, J. Differ. Equ., 340 (2022), 111-150. doi: 10.1016/j.jde.2022.08.032. |