Global well-posedness of pseudo almost periodic mild solutions for a Keller-Segel system of parabolic-parabolic type
Main Article Content
Abstract
We study the global existence and uniqueness of pseudo almost periodic mild solutions for the parabolic-parabolic Keller-Segel systems on the real hyperbolic spaces Hn(R), where n ⩾ 2. First, we use the dispersive estimates of the scalar heat semigroup to estabilish the wellposedness of bounded mild solutions for the corresponding linear systems. Then, we prove the existence and uniquess of pseudo almost periodic mild solutions by proving a Massara-type principle. Finally, the well-posedness of such solutions for semilinear systems are obtained by employing fixed point arguments.
Article Details
Keywords
Parabolic-Parabolic Keller-Segel systems, Dispersive estimates, Smoothing estimates, Pseudo almost periodic mild solutions, Well-posedness
References
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[17] J. Jiang, Global stability of homogeneous steady states in scaling-invariant spaces for a Keller–Segel–Navier–Stokes system, Journal of Differential Equations, Vol. 267, Iss. 2 (2019), Pages 659-692.
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[20] K. Klaus, Time-Periodic Solutions to Bidomain, Chemotaxis-Fluid, and Q-Tensor Models, Ph.D. Thesis, Technische Universit¨at Darmstadt (2020). doi: 10.25534/tuprints-00013505
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[24] Y. Li and W. Wang, Finite-time blow-up and boundedness in a 2D Keller-Segel system with rotation, Nonlinearity, Vol. 36, No. 1 (2023), pages 287–318.
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[26] A. Montaru, Exponential speed of uniform convergence of the cell density toward equilibrium for subcritical mass in a Patlak-Keller-Segel model, J. Differential Equations Vol. 258, Iss. 3 (2015), 621-668.
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[28] V. Pierfelice, The incompressible Navier-Stokes equations on noncompact manifolds, Journal of Geometric Analysis, 27(1) (2017), 577-617.
[29] T. Nagai, R. Syukuinn and M. Umesako, Decay properties and asymptotic profiles of bounded solutions to a parabolic system of chemotaxis in Rn, Funkcial. Ekvac. 46 (2003), 383–407.
[30] Q. Wang and J. Zhang and L. Zhang, Time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: effect of cellular growth, Discrete Contin. Dyn. Syst. Ser. B, Vol. 22, No. 9 (2017), 3547–3574.
[31] K. Watanabe, Strong time-periodic solutions to chemotaxis–Navier–Stokes equations on bounded domains, Discrete and
Continuous Dynamical Systems, Vol. 42, Iss. 11 (2022), 5577-5590.
[32] Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller–Segel model with a power factor in drift term, J. Diff. Eqns 227 (2006), 333–64.
[33] Van, Nguyen Thi, Nguyet, Tran Minh, Loan, Nguyen Thi and Xuan, Pham Truong. On pseudo almost periodic solutions of the parabolic-elliptic Keller-Segel systems, Mathematica Slovaca, vol. 75, no. 3, (2025), 633-648.
[34] Thi Van, N., The Sac, L. and Truong Xuan, P., Well-posedness and exponential stability of periodic and almost periodic solutions for parabolic–parabolic Keller–Segel systems. Bol. Soc. Mat. Mex. 31, 64 (2025).
[35] P.T. Xuan, N.T. Van and B. Quoc, On Asymptotically Almost Periodic Solution of Parabolic Equations on real hyperbolic Manifolds, J. Math. Anal. Appl., Vol. 517, Iss. 1 (2023), 1-19.
[36] P.T. Xuan and N.T. Van, On asymptotically almost periodic solutions to the Navier–Stokes equations in hyperbolic manifolds. J. Fixed Point Theory Appl. 25, 71 (2023), 32 pages.
[37] P.T. Xuan, V.A. Nguyen Thi, T.V. Thuy, and N.T.Loan, Periodic solutions of the parabolic–elliptic Keller–Segel system on whole spaces, Math. Nachr. (2024), 3003–3023.
[38] P.T. Xuan, N.T. Van and T.V. Thuy, On asymptotically almost periodic solutions of the parabolic-elliptic Keller-Segel system on real hyperbolic manifolds, Dynamical Systems, Vol. 39, No. 4 (2024), 665-682.
[39] P.T. Xuan and N.T. Loan, Almost periodic solutions of the parabolicelliptic Keller–Segel system on the whole space, Arch. Math. 123 (2024), 431–446.
[40] P.T. Xuan, Asymptotically Almost Periodic Solutions of a Chemotaxis Model on Bounded Domains, Siberian Mathematical Journal, Vol. 66 (2025), 857–873.
[41] Z. Zhai, Global well-posedness for nonlocal fractional Keller–Segel systems in critical Besov spaces, Nonlinear Anal. 72 (2010), 3173–89.
[42] M. Winkler, Aggregation vs. global diffusive behavior in the higherdimensional Keller-Segel model, J. Differential Equations 248, (2010), 2889–2905.
[43] M. Winkler, Global large-data solutions in a chemotaxis–(Navier–)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), pp. 319–351.
[44] M. Winkler, Finite-time blow-up in the higher-dimensional parabolicparabolic Keller–Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
[45] M. Winkler, How unstable is spatial homogeneity in Keller-Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic-elliptic cases, Math. Ann. 373 (2019), 1237–1282.
[2] A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional KellerSegel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differ. Eq., 2006 (2006), 1–33.
[3] P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl. 8 (1998), 715–43.
[4] V. Calvez and L. Corrias, The parabolic–parabolic Keller–Segel model in R2, Commun. Math. Sci. 6 (2008), 417–47.
[5] X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete and Continuous Dynamical Systems, 2015, 35(5): 1891–1904.
[6] X. Chen, Well-posedness of the Keller-Segel system in Fourier-BesovMorrey spaces, Z. Anal. Anwend., 37 (2018), 417–433.
[7] L. Corrias and B. Perthame, Critical space for the parabolic–parabolic Keller–Segel model in Rd, C. R. Math. Acad. Sci. Paris 342 (2006), 745–50.
[8] L. Corrias, M. Escobedo and J. Matos, Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller–Segel system in the plane, J. Differential Equations, 257 (2014), 1840–1878.
[9] T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer International Publishing Switzerland (2013).
[10] J. Dolbeault, B. Perthame, Optimal critical mass in the two dimensional Keller-Segel model in R2, C.R.Acad.Sci. (2004).
[11] Lucas C.F. Ferreira and Juliana C. Precioso, Existence and asymptotic behaviour for the parabolic–parabolic Keller–Segel system with singular data, Nonlinearity 24 (2011), 1433.
[12] Lucas C.F. Ferreira, On the uniqueness of mild solutions for the parabolic-elliptic Keller-Segel system in the critical Lp-space, Mathematics in Engineering, Vol. 4 (6) (2021), 1–14.
[13] M. Hieber and C. Stinner, Strong time periodic solutions to Keller-Segel systems: An approach by the quasilinear Arendt-Bu theorem, J. Differential Equations, Vol. 269, Iss. 2 (2020), 1636–1655.
[14] Hao Yu, W. Wang and S. Zheng, Global classical solutions to the Keller–Segel–Navier–Stokes system with matrix-valued sensitivity, Journal of Mathematical Analysis and Applications Vol. 461, Iss. 2 (2018), 1748-
1770.
[15] T. Iwabuchi, Global well-posedness for Keller-Segel system in Besov type spaces, J. Math. Anal. Appl., 379 (2011), 930–948.
[16] J. Jiang, Global stability of Keller–Segel systems in critical Lebesgue spaces, Discrete and Continuous Dynamical Systems, (2020), 40(1): 609-634.
[17] J. Jiang, Global stability of homogeneous steady states in scaling-invariant spaces for a Keller–Segel–Navier–Stokes system, Journal of Differential Equations, Vol. 267, Iss. 2 (2019), Pages 659-692.
[18] E.F. Keller and L.A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (3) (1970), pages 399–415.
[19] H. Kozono and M. Nakao, Periodic solution of the Navier–Stokes equations in unbounded domains, Tˆohoku Math. J. 48, 33–50 (1996).
[20] K. Klaus, Time-Periodic Solutions to Bidomain, Chemotaxis-Fluid, and Q-Tensor Models, Ph.D. Thesis, Technische Universit¨at Darmstadt (2020). doi: 10.25534/tuprints-00013505
[21] H. Kozono and Y. Sugiyama, The Keller–Segel system of parabolic–parabolic type with initial data in weak-Ln2 (Rn) and its application to self-similar solutions, Indiana Univ. Math. J. 57 (2008), pages1467–500.
[22] H. Kozono, Y. Sugiyama and Y. Yahagi, Existence and uniqueness theorem on weak solutions to the parabolic-elliptic Keller-Segel system, J. Differ. Equations, 253 (2012), 2295–2313.
[23] H. Kozono and Y. Sugiyama, Strong solutions to the Keller-Segel system with the weak-Ln/2 initial data and its application to the blow-up rate, Math. Nachr., 283 (2010), 732–751.
[24] Y. Li and W. Wang, Finite-time blow-up and boundedness in a 2D Keller-Segel system with rotation, Nonlinearity, Vol. 36, No. 1 (2023), pages 287–318.
[25] P. Maheux and V. Pierfelice, The Keller-Segel System on the TwoDimensional-Hyperbolic Space, SIAM Journal on Mathematical Analysis, Vol. 52, Iss. 5 (2020), 5036-5065.
[26] A. Montaru, Exponential speed of uniform convergence of the cell density toward equilibrium for subcritical mass in a Patlak-Keller-Segel model, J. Differential Equations Vol. 258, Iss. 3 (2015), 621-668.
[27] D.S. Mitrinovi´c, J.E. Peˇcari´c and A.M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Part of the book series: Mathematics and its Applications (MAEE, volume 53) (1991). doi: 10.1007/978-94-011-3562-7
[28] V. Pierfelice, The incompressible Navier-Stokes equations on noncompact manifolds, Journal of Geometric Analysis, 27(1) (2017), 577-617.
[29] T. Nagai, R. Syukuinn and M. Umesako, Decay properties and asymptotic profiles of bounded solutions to a parabolic system of chemotaxis in Rn, Funkcial. Ekvac. 46 (2003), 383–407.
[30] Q. Wang and J. Zhang and L. Zhang, Time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: effect of cellular growth, Discrete Contin. Dyn. Syst. Ser. B, Vol. 22, No. 9 (2017), 3547–3574.
[31] K. Watanabe, Strong time-periodic solutions to chemotaxis–Navier–Stokes equations on bounded domains, Discrete and
Continuous Dynamical Systems, Vol. 42, Iss. 11 (2022), 5577-5590.
[32] Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller–Segel model with a power factor in drift term, J. Diff. Eqns 227 (2006), 333–64.
[33] Van, Nguyen Thi, Nguyet, Tran Minh, Loan, Nguyen Thi and Xuan, Pham Truong. On pseudo almost periodic solutions of the parabolic-elliptic Keller-Segel systems, Mathematica Slovaca, vol. 75, no. 3, (2025), 633-648.
[34] Thi Van, N., The Sac, L. and Truong Xuan, P., Well-posedness and exponential stability of periodic and almost periodic solutions for parabolic–parabolic Keller–Segel systems. Bol. Soc. Mat. Mex. 31, 64 (2025).
[35] P.T. Xuan, N.T. Van and B. Quoc, On Asymptotically Almost Periodic Solution of Parabolic Equations on real hyperbolic Manifolds, J. Math. Anal. Appl., Vol. 517, Iss. 1 (2023), 1-19.
[36] P.T. Xuan and N.T. Van, On asymptotically almost periodic solutions to the Navier–Stokes equations in hyperbolic manifolds. J. Fixed Point Theory Appl. 25, 71 (2023), 32 pages.
[37] P.T. Xuan, V.A. Nguyen Thi, T.V. Thuy, and N.T.Loan, Periodic solutions of the parabolic–elliptic Keller–Segel system on whole spaces, Math. Nachr. (2024), 3003–3023.
[38] P.T. Xuan, N.T. Van and T.V. Thuy, On asymptotically almost periodic solutions of the parabolic-elliptic Keller-Segel system on real hyperbolic manifolds, Dynamical Systems, Vol. 39, No. 4 (2024), 665-682.
[39] P.T. Xuan and N.T. Loan, Almost periodic solutions of the parabolicelliptic Keller–Segel system on the whole space, Arch. Math. 123 (2024), 431–446.
[40] P.T. Xuan, Asymptotically Almost Periodic Solutions of a Chemotaxis Model on Bounded Domains, Siberian Mathematical Journal, Vol. 66 (2025), 857–873.
[41] Z. Zhai, Global well-posedness for nonlocal fractional Keller–Segel systems in critical Besov spaces, Nonlinear Anal. 72 (2010), 3173–89.
[42] M. Winkler, Aggregation vs. global diffusive behavior in the higherdimensional Keller-Segel model, J. Differential Equations 248, (2010), 2889–2905.
[43] M. Winkler, Global large-data solutions in a chemotaxis–(Navier–)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), pp. 319–351.
[44] M. Winkler, Finite-time blow-up in the higher-dimensional parabolicparabolic Keller–Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
[45] M. Winkler, How unstable is spatial homogeneity in Keller-Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic-elliptic cases, Math. Ann. 373 (2019), 1237–1282.