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Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion

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  • This paper deals with a parabolic-elliptic-ODE chemotaxis-haptotaxis system with nonlinear diffusion \begin{eqnarray*}\label{1a} \left\{ \begin{split}{} &u_t=\nabla\cdot(\varphi(u)\nabla u)-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w), \\ &0=\Delta v-v+u, \\ &w_{t}=-vw, \end{split} \right. \end{eqnarray*} under Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{n}$ $(n\geq1)$, where $\chi$, $\xi$ and $\mu$ are positive parameters and $\varphi(u)$ is a nonlinear diffusion. Under the non-degenerate diffusion and some suitable assumptions on positive parameters $\chi,\xi,\mu$, it is shown that the corresponding initial boundary value problem possesses a unique global classical solution that is uniformly bounded in $\Omega\times(0,\infty)$. Moreover, under the degenerate diffusion, it is proved that the corresponding problem admits at least one nonnegative global bounded-in-time weak solution. Finally, for the suitably small initial data $w_{0}$, we give the decay estimate of $w$.
    Mathematics Subject Classification: Primary: 35K55; Secondary: 35B45, 35B33, 35K57, 92C17.

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  • [1]

    N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.doi: 10.1080/03605307908820113.

    [2]

    S. Aznavoorian, M. L. Stracke, H. Krutzsch, E. Schiffmann and L. A. Liotta, Signal transduction for chemotaxis and haptotaxis by matrix molecules in tumor cells, J. Cell Biol., 110 (1990), 1427-1438.doi: 10.1083/jcb.110.4.1427.

    [3]

    D. Besser, P. Verde, Y. Nagamine and F. Blasi, Signal transduction and u-PA/u-PAR system, Fibrinolysis, 10 (1996), 215-237.doi: 10.1016/S0268-9499(96)80018-X.

    [4]

    X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys., 67 (2016), p67, arXiv:1501.05383.doi: 10.1007/s00033-015-0601-3.

    [5]

    X. Cao and S. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, Math. Methods Appl. Sci., 37 (2014), 2326-2330.doi: 10.1002/mma.2992.

    [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 G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med., 1 (2006), 399-439.doi: 10.3934/nhm.2006.1.399.

    [8]

    T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.doi: 10.1088/0951-7715/21/5/009.

    [9]

    A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.

    [10]

    K. Fujie, M. Winkler and T. Yokota, Blow-up prevention by logistic sources in a parabolic-elliptic Keller-Segel system with singular sensitivity, Nonlinear Anal., 109 (2014), 56-71.doi: 10.1016/j.na.2014.06.017.

    [11]

    T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.doi: 10.1007/s00285-008-0201-3.

    [12]

    T. Hillen, K. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Methods Appl. Sci., 23 (2013), 165-198.doi: 10.1142/S0218202512500480.

    [13]

    D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math. -Verein., 105 (2003), 103-165.

    [14]

    D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II. Jahresber. Deutsch. Math. -Verein., 106 (2004), 51-69.

    [15]

    D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.doi: 10.1016/j.jde.2004.10.022.

    [16]

    W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.doi: 10.1090/S0002-9947-1992-1046835-6.

    [17]

    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.

    [18]

    J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.doi: 10.3934/dcdsb.2015.20.1499.

    [19]

    Y. Li and J. Lankeit, Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion, Nonlinearity, 29 (2016), arXiv:1508.05846v1.doi: 10.1088/0951-7715/29/5/1564.

    [20]

    L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733-737.

    [21]

    C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.doi: 10.1007/BF02476407.

    [22]

    B. Perthame, Transport Equations in Biology, Birkhäser-BaselVerlag, Switzerland, 2007.

    [23]

    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.

    [24]

    Y. Tao, Boundedness in a two-dimensional chemotaxis-haptotaxis system, arXiv:1407.7382v1, 2014.

    [25]

    Y. Tao and M. Wang, Global solution for a chemotactic-haptotactic model of cancer invasion, Nonlinearity, 21 (2008), 2221-2238.doi: 10.1088/0951-7715/21/10/002.

    [26]

    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.

    [27]

    Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 1 (2013), 1-36.doi: 10.1142/S0218202512500443.

    [28]

    Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.doi: 10.1137/100802943.

    [29]

    Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.doi: 10.1016/j.jde.2011.08.019.

    [30]

    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.

    [31]

    J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.doi: 10.1080/03605300701319003.

    [32]

    R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Stud. Math. Appl., vol.2, North-Holland, Amsterdam, 1977.

    [33]

    L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.doi: 10.1016/j.jde.2013.12.007.

    [34]

    Y. F. 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.

    [35]

    Z. A. Wang, M. Winkler and D. Wrzosek, Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume-filling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 3502-3525.doi: 10.1137/110853972.

    [36]

    M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.doi: 10.1016/j.jmaa.2008.07.071.

    [37]

    M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.doi: 10.1007/s00332-014-9205-x.

    [38]

    M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.doi: 10.1016/j.na.2009.07.045.

    [39]

    J. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140.doi: 10.1016/j.jde.2015.02.003.

    [40]

    P. Zheng, C. Mu, X. Hu and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl., 424 (2015), 509-522.doi: 10.1016/j.jmaa.2014.11.031.

    [41]

    P. Zheng, C. Mu and X. Song, On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 1737-1757.doi: 10.3934/dcds.2016.36.1737.

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