×

Global solvability and large time behavior to a chemotaxis-haptotaxis model with nonlinear diffusion. (English) Zbl 1415.35054

Summary: In this paper, we deal with the following coupled chemotaxis-haptotaxis system \[ \begin{cases} & u_t = \Delta u^m - \chi \nabla \cdot \left(\frac{u}{(1 + u)^\alpha} \nabla v\right) - \xi \nabla \cdot \left(\frac{u}{(1 + u)^\beta} \nabla \omega\right) + \mu u(1 - u - \omega), \\ & v_t = \Delta v - v + u, \\ & \omega_t = - v \omega \end{cases}. \] in a bounded domain \(\Omega\) of \(\mathbb{R}^3\) corresponding to zero-flux boundary conditions. We showed that for any \(m > 0\), the problem admits a global bounded weak solution. Subsequently, we also discussed the large time behavior of solutions for the fast diffusion case, and showed that when \(0 < m \leq 1\), for appropriately large \(\mu\), for any initial datum, the solution \((u, v, \omega)\) goes to the steady state \((1, 1, 0)\) as \(t \rightarrow \infty\). The global existence result improved the work of J. Liu et al. [Z. Angew. Math. Phys. 67, No. 2, Article ID 21, 33 p. (2016; Zbl 1342.35412) in dimension 3, in which, the global existence is established for \(\min \{\alpha, \beta \} > \frac{4}{3} - m\) with \(m > \frac{4}{3}\) or \(m \leq 1\).

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K51 Initial-boundary value problems for second-order parabolic systems
35K58 Semilinear parabolic equations

Citations:

Zbl 1342.35412
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Patlak, C. S., Random walk with persistence and external bias, Bull. Math. Biophys., 15, 311-338 (1953) · Zbl 1296.82044
[2] Keller, E.; Segel, A., Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26, 399-415 (1970) · Zbl 1170.92306
[3] Corrias, L.; Perthame, B.; Zaag, H., Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72, 1-28 (2004) · Zbl 1115.35136
[4] Dolbeault, J.; Perthame, B., Optimal critical mass in the two-dimensional Keller-Segel model in R2, C. R. Math. Acad. Sci. Paris, 339, 611-616 (2004) · Zbl 1056.35076
[5] Herrero, M. A.; Medina, E.; Velazquez, J. J.L., Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10, 1739-1754 (1997) · Zbl 0909.35071
[6] Nagai, T., Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5, 581-601 (1995) · Zbl 0843.92007
[7] Horstmann, D.; Wang, G., Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12, 159-177 (2001) · Zbl 1017.92006
[8] Lankeit, J., Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26, 2071-2109 (2016) · Zbl 1354.35059
[9] Winkler, M., Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100, 748-767 (2013) · Zbl 1326.35053
[10] Chaplain, M. A.J.; Lolas, G., Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Netw. Heterogen. Media, 1, 399-439 (2006) · Zbl 1108.92023
[11] Tao, Y.; Wang, M., Global solution for a chemotactic-haptotactic model of cancer invasion, Nonlinearity, 21, 2221-2238 (2008) · Zbl 1160.35431
[12] Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, arXiv:1407.7382v1; Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, arXiv:1407.7382v1 · Zbl 1312.35171
[13] Cao, X., Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys., 67, 11 (2016), 13 pp · Zbl 1375.35566
[14] Tao, Y.; Minkler, M., Boundedness and stabilization in a multi-dimensional chemotaxis-haptotaxis model, Proc. Roy. Soc. Edinburgh Sect. A, 144, 1067-1084 (2014) · Zbl 1312.35171
[15] Tao, Y.; Winkler, M., Dominance of chemotaxis in a chemotaxis-haptotaxis model, Nonlinearity, 27, 1225-1239 (2014) · Zbl 1293.35047
[16] Tao, Y.; Winkler, M., A chemotaxis-haptotaxis modle:The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43, 685-704 (2001) · Zbl 1259.35210
[17] Li, Y.; Lankeit, J., Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29, 1564-1595 (2016) · Zbl 1338.35438
[18] Wang, Y., Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Differential Equations, 260, 1975-1989 (2016) · Zbl 1332.35152
[19] Zheng, J., Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, Discrete Contin. Dyn. Syst., 37, 627-643 (2017) · Zbl 1353.92026
[20] Jin, C., Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion, Discrete Contin. Dyn. Syst. Ser. B, 23, 4, 1675-1688 (2018) · Zbl 1396.92008
[21] Liu, J.; Zheng, J.; Wang, Y., Boundedness in a quasilinear chemotaxis-haptotaxis system with logistic source, Z. Angew. Math. Phys., 67, 21 (2016) · Zbl 1342.35412
[22] Tao, Y.; Minkler, M., Large time behavior in a multidimensional chemotaxis-hapotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47, 4229-4250 (2015) · Zbl 1328.35103
[23] C. Jin, Large time behavior of solutions to a chemotaxis model with porous medium diffusion, submitted for publication.; C. Jin, Large time behavior of solutions to a chemotaxis model with porous medium diffusion, submitted for publication.
[24] Stinner, C.; Surulescu, C.; Winkler, M., Global weak solutions in a pde-ode system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46, 1969-2007 (2014) · Zbl 1301.35189
[25] Jin, C., Global classical solution and boundedness to a chemotaxis-haptotaxis model with re-establishment mechanisms, Bull. Lond. Math. Soc., 50, 598-618 (2018) · Zbl 1396.92007
[26] Tao, Y.; Winkler, M., Energy-type estimates and global solvability in a two-dimensional c hemotaxis-hapotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 257, 784-815 (2014) · Zbl 1295.35144
[27] Winker, M., Chemotaxis with logistic source: very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348, 708-729 (2008) · Zbl 1147.92005
[28] Winker, M., Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248, 2889-2905 (2010) · Zbl 1190.92004
[29] Hillen, T.; Painter, K. J.; Winkler, M., Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Methods Appl. Sci., 23, 1, 165-198 (2013) · Zbl 1263.35204
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.