Advanced Search
Article Contents
Article Contents

Asymptotic behavior in a chemotaxis-growth system with nonlinear production of signals

Abstract Full Text(HTML) Related Papers Cited by
  • We consider the chemotaxis-growth system

    $\left\{ {\begin{array}{*{20}{l}} {{u_t} = \Delta u - \chi \nabla \cdot (u\nabla v) + \mu u(1 - u),}&{x \in \Omega ,{\mkern 1mu} t > 0,}\\ {{v_t} = \Delta v - v + h(u),}&{x \in \Omega ,{\mkern 1mu} t > 0,} \end{array}} \right.$

    under no-flux boundary conditions, in a convex bounded domain $Ω\subset\mathbb{R}^3$ with smooth boundary, where $χ>0$ and $μ>0$ are given parameters, and $h(s)$ is a prescribed function on $[0, ∞)$ .

    It is shown that under the assumption that

    $4|{h}'|<\sqrt{2\mu -7{{\chi }^{2}}},$

    for any given nonnegative $u_0∈ C^0(\bar{Ω})$ and $v_0∈ W^{1, ∞}(Ω)$ the system possesses a global classical solution which is bounded in $Ω× (0, ∞)$ . Moreover, whenever

    $χ |h'| < \sqrt{8μ},$

    any bounded classical solution constructed above stabilizes to the constant stationary solution $(1, h(1))$ as the time goes to infinity.

    Mathematics Subject Classification: Primary:35B40, 35K57, 35Q92, 92C17.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] 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.
    [2] 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.
    [3] S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.
    [4] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instaility, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.
    [5] P. L. Lions, Résolution de problémes elliptiques quasilinéaires, Arch. Ration. Mech. Anal., 74 (1980), 335-353.  doi: 10.1007/BF00249679.
    [6] M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543. 
    [7] N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 31 (2014), 851-875.  doi: 10.1016/j.anihpc.2013.07.007.
    [8] M. R. MyerscoughP. K. Maini and J. Painter, Pattern formation in a generalized chemotactic model, Bull. Math. Biol., 60 (1998), 1-26. 
    [9] E. Nakaguchi and K. Osaki, Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete Contin. Dyn. Syst. B, 18 (2013), 2627-2646.  doi: 10.3934/dcdsb.2013.18.2627.
    [10] E. Nakaguchi and K. Osaki, $L_p$-estimates of solutions to n-dimensional parabolic-parabolic system for chemotaxis with subquadratic degradation, Funkcialaj Ekvacioj, 59 (2016), 51-66.  doi: 10.1619/fesi.59.51.
    [11] E. Nakaguchi and K. Osaki, Global existence of solutions to n-diemnsional parabolic-parabolic system for chemotaxis with subquadratic degradation, Preprint.
    [12] M. E. Orme and M. A. J. Chaplain, A mathematical model of the first steps of tumour-related angiogenesis: Capillary sprout formation and secondary branching, IMA J. Math. Appl. Med. Biol., 13 (1996), 73-98. 
    [13] K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Analysis, 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.
    [14] M. M. Porzio and V. Vespri, Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045.
    [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 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.
    [17] Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.  doi: 10.1007/s00033-015-0541-y.
    [18] Y. Tao and M. Winkler, Boundedness and competitive exclusion in a population model with cross-diffusion for one species, Preprint.
    [19] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.
    [20] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.
  • 加载中

Article Metrics

HTML views(369) PDF downloads(248) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint