\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$

  • * Corresponding author: Rachidi B. Salako

    * Corresponding author: Rachidi B. Salako 
Abstract Full Text(HTML) Related Papers Cited by
  • The current paper is devoted to the study of spreading speeds and traveling wave solutions of the following parabolic-elliptic chemotaxis system,

    where $u(x, t)$ represents the population density of a mobile species and $v(x, t)$ represents the population density of a chemoattractant, and $χ$ represents the chemotaxis sensitivity. We first give a detailed study in the case $N=1$. In this case, it has been shown in an earlier work by the authors of the current paper that, when $0 < χ < 1$, for every nonnegative uniformly continuous and bounded function $u_0(x)$, the system has a unique globally bounded classical solution $(u(x, t;u_0), v(x, t;u_0))$ with initial condition $u(x, 0;u_0)=u_0(x)$. Furthermore, it was shown that, if $0 < χ < \frac{1}{2}$, then the constant steady-state solution $(1, 1)$ is asymptotically stable with respect to strictly positive perturbations. In the current paper, we show that if $0 < χ < 1$, then there are nonnegative constants $c_ - ^*\left( \chi \right) \le c_ + ^*\left( \chi \right)$ such that for every nonnegative initial function $u_0(·)$ with non-empty and compact support ${\rm{supp}}(u_0)$,

    and

    $\mathop {\lim }\limits_{t \to \infty } \mathop {\sup }\limits_{|x| \le ct} [u(x,t;{u_0}) + v(x,t;{u_0})] = 0\quad \forall {\mkern 1mu} {\mkern 1mu} c > c_ + ^*(\chi ).$

    We also show that if $0 < χ < \frac{1}{2}$, there is a positive constant $c^*(χ)$ such that for every $c \ge c^*(χ)$, the system has a traveling wave solution $(u(x, t), v(x, t))$ with speed $c$ and connecting $(1, 1)$ and $(0, 0)$, that is, $(u(x, t), v(x, t))=(U(x-ct), V(x-ct))$ for some functions $U(·)$ and $V(·)$ satisfying $(U(-∞), V(-∞))=(1, 1)$ and $(U(∞), V(∞))=(0, 0)$. Moreover, we show that

    $\mathop {\lim }\limits_{\chi \to 0} {c^*}(\chi ) = \mathop {\lim }\limits_{\chi \to 0} c_ + ^*(\chi ) = \mathop {\lim }\limits_{\chi \to 0} c_ - ^*(\chi ) = 2.$

    We then consider the extensions of the results in the case $N=1$ to the case $N \ge 2$.

    Mathematics Subject Classification: 35B35, 35B40, 35K57, 35Q92, 92C17.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] S. AiW. Huang and Z.-A. Wang, Reaction, diffusion and chemotaxis in wave propagation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1-21.  doi: 10.3934/dcdsb.2015.20.1.
    [2] S. Ai and Z.-A. Wang, Traveling bands for the Keller-Segel model with population growth, Math. Biosci. Eng., 12 (2015), 717-737.  doi: 10.3934/mbe.2015.12.717.
    [3] N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math.Models Methods Appl.Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.
    [4] H. BerestyckiF. Hamel and G. Nadin, Asymptotic spreading in heterogeneous diffusive excita media, Journal of Functional Analysis, 255 (2008), 2146-2189.  doi: 10.1016/j.jfa.2008.06.030.
    [5] H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, Ⅰ -Periodic framework, J. Eur. Math. Soc., 7 (2005), 172-213.  doi: 10.4171/JEMS/26.
    [6] H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems, Ⅱ -General domains, J. Amer. Math. Soc., 23 (2010), 1-34.  doi: 10.1090/S0894-0347-09-00633-X.
    [7] H. Berestycki and G. Nadin, Asymptotic spreading for general heterogeneous Fisher-KPP type, preprint.
    [8] M. Bramson, Convergence of solutions of the Kolmogorov equation to traveling waves, Mem. Amer. Math. Soc., 44 (1983), iv+190 pp. doi: 10.1090/memo/0285.
    [9] J. I. Diaz and T. Nagai, Symmetrization in a parabolic-elliptic system related to chemotaxis, Advances in Mathematical Sciences and Applications, 5 (1995), 659-680. 
    [10] J. I. DiazT. Nagai and J.-M. Rakotoson, Symmetrization techniques on unbounded domains: Application to a chemotaxis system on $\mathbb{R}^{N}$, J. Differential Equations, 145 (1998), 156-183.  doi: 10.1006/jdeq.1997.3389.
    [11] R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.
    [12] M. Freidlin, On wave front propagation in periodic media, In: Stochastic analysis and applications, ed. M. Pinsky, Advances in probablity and related topics, 7 (1984), 147-166. 
    [13] M. Freidlin and J. Gärtner, On the propagation of concentration waves in periodic and ramdom media, Soviet Math. Dokl., 20 (1979), 1282-1286. 
    [14] A. Friedman, Partial Differential Equation of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964.
    [15] M. FunakiM. Mimura and T. Tsujikawa, Travelling front solutions arising in the chemotaxis-growth model, Interfaces Free Bound., 8 (2006), 223-245.  doi: 10.4171/IFB/141.
    [16] E. GalakhovO. Salieva and J. I. Tello, On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.  doi: 10.1016/j.jde.2016.07.008.
    [17] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Berlin Heidelberg New York, 1981.
    [18] D. Horstmann and A. Stevens, A constructive approach to traveling waves in chemotaxis, J. Nonlin. Sci., 14 (2004), 1-25.  doi: 10.1007/s00332-003-0548-y.
    [19] K. Kanga, Angela Steven Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72.  doi: 10.1016/j.na.2016.01.017.
    [20] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.
    [21] E. F. Keller and L. A. Segel, A Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.
    [22] A. KolmogorovI. Petrowsky and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskovskogo Gos. Univ., 1 (1937), 1-26. 
    [23] J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J.Differential Eq., 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.
    [24] J. LiT. Li and Z.-A. Wang, Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819-2849.  doi: 10.1142/S0218202514500389.
    [25] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.
    [26] X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, Journal of Functional Analysis, 259 (2010), 857-903.  doi: 10.1016/j.jfa.2010.04.018.
    [27] B. P. MarchantJ. Norbury and J. A. Sherratt, Travelling wave solutions to a haptotaxis-dominated model of malignant invasion, Nonlinearity, 14 (2001), 1653-1671.  doi: 10.1088/0951-7715/14/6/313.
    [28] G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Anal., 92 (2009), 232-262.  doi: 10.1016/j.matpur.2009.04.002.
    [29] G. NadinB. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with Fisher birth terms, Interfaces Free Bound, 10 (2008), 517-538.  doi: 10.4171/IFB/200.
    [30] T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433. 
    [31] J. NolenM. Rudd and J. Xin, Existence of KPP fronts in spatially-temporally periodic adevction and variational principle for propagation speeds, Dynamics of PDE, 2 (2005), 1-24.  doi: 10.4310/DPDE.2005.v2.n1.a1.
    [32] J. Nolen and J. Xin, Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle, Discrete and Continuous Dynamical Systems, 13 (2005), 1217-1234.  doi: 10.3934/dcds.2005.13.1217.
    [33] R. B. Salako and W. Shen, Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^{N}$, J. Differential Equations, 262 (2017), 5635-5690.  doi: 10.1016/j.jde.2017.02.011.
    [34] D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Advances in Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0.
    [35] W. Shen, Variational principle for spatial spreading speeds and generalized propgating speeds in time almost and space periodic KPP models, Trans. Amer. Math. Soc., 362 (2010), 5125-5168.  doi: 10.1090/S0002-9947-10-04950-0.
    [36] W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69-93. 
    [37] Y. Sugiyama, Global existence in sub-critical cases and finite time blow up in super critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876. 
    [38] Y. Sugiyama and H. Kunii, Global Existence and decay properties for a degenerate keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.  doi: 10.1016/j.jde.2006.03.003.
    [39] J. I. Tello and M. Winkler, A Chemotaxis System with Logistic Source, Communications in Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.
    [40] K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.  doi: 10.1215/kjm/1250522506.
    [41] L. WangC. 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.
    [42] Z.-A. Wang, Mathematics of traveling waves in chemotaxis-review paper, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 601-641.  doi: 10.3934/dcdsb.2013.18.601.
    [43] H. F. Weinberger, Long-time behavior of a class of biology models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.
    [44] H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.
    [45] 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.
    [46] M. Winkler, Aggregation vs.global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.
    [47] M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis and Applications, 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.
    [48] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767, arXiv: 1112.4156v1. doi: 10.1016/j.matpur.2013.01.020.
    [49] 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.
    [50] T. Yokota and N. Yoshino, Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst, (2015), 1125-1133.  doi: 10.3934/proc.2015.1125.
    [51] P. ZhengC. MuX. 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.
    [52] A. Zlatoš, Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl., 98 (2012), 89-102.  doi: 10.1016/j.matpur.2011.11.007.
  • 加载中
SHARE

Article Metrics

HTML views(1331) PDF downloads(228) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return