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Existence of traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source

  • * Corresponding author: Rachidi B. Salako

    * Corresponding author: Rachidi B. Salako 
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  • The current paper is devoted to the study of traveling wave solutions of the following parabolic-elliptic-elliptic chemotaxis systems,

    $\begin{equation}\label{main-eq-abstract}\begin{cases}u_{t} = Δ u- \nabla · ({ χ_1 u} \nabla v_1)+ \nabla · ({ χ_2 u} \nabla v_2) + u(a-bu), \;\;\;\;x∈\mathbb{R}^N, \\0 = Δ v_1-λ_1v_1+μ_1u, \;\;\;\;x∈\mathbb{R}^N, \\0 = Δ v_2-λ_2v_2+μ_2u, \;\;\;\; x∈\mathbb{R}^N, \end{cases}\;\;\;\;\;\;\;\;(0.1)\end{equation}$

    where $a>0, \ b>0, $ $u(x, t)$ represents the population density of a mobile species, $v_1(x, t), $ represents the population density of a chemoattractant, $v_2(x, t)$ represents the population density of a chemorepulsion, the constants $χ_1≥ 0$ and $χ_2≥ 0$ represent the chemotaxis sensitivities, and the positive constants $λ_1, λ_2, μ_1$, and $μ_2$ are related to growth rate of the chemical substances. It was proved in an earlier work by the authors of the current paper that there is a nonnegative constant $K$ depending on the parameters $χ_1, μ_1, λ_1, χ_2, μ_2$, and $λ_2$ such that if $b+χ_2μ_2>χ_1μ_1+K$, then the positive constant steady solution $(\frac{a}{b}, \frac{aμ_1}{bλ_1}, \frac{aμ_2}{bλ_2})$ of (0.1) is asymptotically stable with respect to positive perturbations. In the current paper, we prove that if $b+χ_2μ_2>χ_1μ_1+K$, then there exists a positive number $c^{*}(χ_1, μ_1, λ_1, χ_2, μ_2, λ_2)≥ 2\sqrt{a}$ such that for every $ c∈ ( c^{*}(χ_1, μ_1, λ_1, χ_2, μ_2, λ_2)\ , \ ∞)$ and $ξ∈ S^{N-1}$, the system has a traveling wave solution $(u(x, t), v_1(x, t), v_2(x, t)) = (U(x·ξ-ct), V_1(x·ξ-ct), V_2(x·ξ-ct))$ with speed $c$ connecting the constant solutions $(\frac{a}{b}, \frac{aμ_1}{bλ_1}, \frac{aμ_2}{bλ_2})$ and $(0, 0, 0)$, and it does not have such traveling wave solutions of speed less than 2\sqrt a $. Moreover we prove that

    $\begin{equation*}\lim\limits_{(χ_{1}, χ_2)?(0^+, 0^+)}c^{*}(χ_1, μ_1, λ_1, χ_2, μ_2, λ_2) = \begin{cases}\ 2\sqrt{a} \;\;\text{if}\;\; a≤ \min\{λ_1, λ_2\}\\\frac{a+λ_1}{\sqrt{λ_1}} \;\;\text{if}\;\; λ_1≤ \min\{a, λ_2\}\\\frac{a+λ_2}{\sqrt{λ_2}} \;\;\text{if}\;\; λ_2≤ \min\{a, λ_1\}\end{cases}\end{equation*}$

    for every $ λ_1, λ_2, μ_1, μ_2>0$, and

    $\begin{equation*}\lim\limits_{x?∞}\frac{U(x)}{e^{-\sqrt a μ x}} = 1, \end{equation*}$

    where $μ$ is the only solution of the equation $μ+\frac{1}{μ} = \frac{c}{\sqrt{a}}$ in the interval $(0\ , \ \min\{1, \sqrt{\frac{λ_1}{a}}, \sqrt{\frac{λ_2}{a}}\})$.

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

    Citation:

    \begin{equation} \\ \end{equation}
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  • [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 KellerSegel 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), 173-213.  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, 2015. <hal-01171334v2>.
    [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] T. CieślakP. Laurencot and C. Morales-Rodrigo, Global existence and convergence to steady states in a chemorepulsion system, Parabolic and Navier-Stokes Equations Banach Center Publications, Institute of Mathematics Polish Academy of Sciences Warszawa, 81 (2008), 105-117.  doi: 10.4064/bc81-0-7.
    [10] 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. 
    [11] 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.
    [12] E. Espejoand and T. Suzuki, Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34.  doi: 10.1016/j.aml.2014.04.007.
    [13] 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.
    [14] 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.
    [15] M. Freidlin and J. Gärtner, On the propagation of concentration waves in periodic and ramdom media, Soviet Math. Dokl., 20 (1979), 1282-1286. 
    [16] A. Friedman, Partial Differential Equation of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.
    [17] M. FunakiM. Mimura and T. Tsujikawa, Travelling front solutions arising in the chemotaxisgrowth model, Interfaces Free Bound, 8 (2006), 223-245.  doi: 10.4171/IFB/141.
    [18] 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.
    [19] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag Berlin Heidelberg New York, 1981.
    [20] M. A. Herrero and J. J. L. Velasquez, A blow-up mechanism for a chemotaxis model, Annali Della Scuola Normale Superiore di Pisa, Classe di Scienze, 24 (1997), 633-683. 
    [21] D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multispecies chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlin. Sci., 21 (2011), 231-270.  doi: 10.1007/s00332-010-9082-x.
    [22] 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.
    [23] H. Y. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.  doi: 10.1016/j.jmaa.2014.09.049.
    [24] K. Kanga and A. Steven, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72.  doi: 10.1016/j.na.2016.01.017.
    [25] 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.
    [26] 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.
    [27] 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. 
    [28] 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.
    [29] 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.
    [30] 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.
    [31] K. LinC. Mu and Y. Gao, Boundedness and blow up in the higher-dimensional attractionrepulsion chemotaxis with non-linear diffusion, J. Differential Equations, 261 (2016), 4524-4572.  doi: 10.1016/j.jde.2016.07.002.
    [32] J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31-41.  doi: 10.1080/17513758.2011.571722.
    [33] P. LiuJ. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.  doi: 10.3934/dcdsb.2013.18.2597.
    [34] M. LucaA. Chavez-RossL. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and alzheimers disease senile plaques: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730. 
    [35] B. P. MarchantJ. Norbury and J. A. Sherratt, Travelling wave solutions to a haptotaxisdominated model of malignant invasion, Nonlinearity, 14 (2001), 1653-1671.  doi: 10.1088/0951-7715/14/6/313.
    [36] M. S. Mock, An initial value problem from semiconductor device theory, SIAM J. Math. Anal., 5 (1974), 597-612.  doi: 10.1137/0505061.
    [37] 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.
    [38] 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.
    [39] T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433. 
    [40] 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.
    [41] 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.
    [42] R. B. Salako and W. Shen, Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on ${{\mathbb{R}}^{N}}$. I. Persistence and asymptotic spreading, Math. Models Methods Appl. Sci., 28 (2018), 2237–2273, https://arXiv.org/pdf/1709.05785.pdf. doi: 10.1142/S0218202518400146.
    [43] R. B. Salako and W. Shen, Global classical solutions, stability of constant equilibria, and spreading speeds in attraction-repulsion chemotaxis systems with logistic source on $\mathbb{R}^{N}$, Journal of Dynamics and Differential Equations, (2017), https://doi.org/10.1007/s10884-017-9602-6.
    [44] R. B. Salako and W. Shen, Existence of Traveling wave solution of the full parabolic chemotaxis system, Nonlinear Analysis: Real World Applications, 42 (2018), 93-119.
    [45] R. B. Salako and W. Shen, Spreading Speeds and Traveling waves of a parabolic-elliptic chemotaxis system with logistic source on ${{\mathbb{R}}^{N}}$, Discrete and Continuous Dynamical Systems - Series A, 37 (2017), 6189-6225.  doi: 10.3934/dcds.2017268.
    [46] R. B. Salako and W. Shen, Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on ${{\mathbb{R}}^{N}}$, Journal of Differential Equations, 262 (2017), 5635-5690.  doi: 10.1016/j.jde.2017.02.011.
    [47] 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.
    [48] 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.
    [49] W. Shen, Existence of generalized traveling waves in time recurrent and space periodic monostable equations, J. Appl. Anal. Comput., 1 (2011), 69-93. 
    [50] 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. 
    [51] 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.
    [52] Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.
    [53] 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.
    [54] 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.
    [55] 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.
    [56] Y. Wang, Global bounded weak solutions to a degenerate quasilinear attraction repulsion chemotaxis system with rotation, Computers and Mathematics with Applications, 72 (2016), 2226-2240.  doi: 10.1016/j.camwa.2016.08.024.
    [57] Y. Wang and Z. Xiang, Boundedness in a quasilinear 2D parabolic-parabolic attractionrepulsion chemotaxis system. Discrete Contin, Dyn. Syst. Ser. B, 21 (2016), 1953-1973.  doi: 10.3934/dcdsb.2016031.
    [58] 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.
    [59] H. F. Weinberger, Long–time behavior of a class of biology models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.
    [60] 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.
    [61] 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.
    [62] 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.
    [63] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.
    [64] M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.
    [65] 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.
    [66] T. Yokota and N. Yoshino, Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst., 2015, Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., 1125-1133. doi: 10.3934/proc.2015.1125.
    [67] Q. Zhang and Y. Li, An attraction–repulsion chemotaxis system with logistic source, ZAMM Z. Angew. Math. Mech., 96 (2016), 570-584.  doi: 10.1002/zamm.201400311.
    [68] P. ZhengC. Mu and X. Hu, Boundedness in the higher dimensional attraction-repulsion chemotaxis-growth system, Computers and Mathematics with Applications, 72 (2016), 2194-2202.  doi: 10.1016/j.camwa.2016.08.028.
    [69] 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.
    [70] 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.
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