Global boundedness and asymptotic behavior of the solutions to an attraction–repulsion chemotaxis-growth system

References

• Luca M, Chavez-Ross A, Edelstein-Keshet L, et al. Chemotactic signaling, microglia, and Alzheimer's disease senile plagues: is there a connection? Bull Math Biol. 2003;65:673–730.
   Web of Science ®Google Scholar
• Painter K, Hillen T. Volume-filling and quorum-sensing in models for chemosensitive movement. Canad Appl Math Quart. 2002;10:501–543.
   Google Scholar
• Keller EF, Segel LA. Initiation of slime mold aggregation viewed as an instability. J Theoret Biol. 1970;26:399–415.
   PubMed Web of Science ®Google Scholar
• Bellomo N, Bellouquid A, Tao YS, et al. Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues. Math Models Meth Appl Sci. 2015;25:1663–1763.
   Web of Science ®Google Scholar
• Calvez V, Corrias L, Ebde MA. Blow-up, concentration phenomenon and global existence for the Keller-Segel model in high dimension. Comm Partial Differ Equ. 2012;37:561–584.
   Web of Science ®Google Scholar
• Choi YS, Wang ZA. Prevention of blow up by fast diffusion in chemotaxis. J Math Anal Appl. 2010;362:553–564.
   Web of Science ®Google Scholar
• Herrero MA, Velázquez JJL. A blow-up mechanism for a chemotaxis model. Ann Sc Norm Super Pisa Cl Sci. 1997;24:633–683.
   Google Scholar
• Hittmeir S, Jüngel A. Cross-diffusion preventing blow up in the two-dimensional Keller-Segel model. SIAM J Math Anal. 2011;43:997–1022.
   Web of Science ®Google Scholar
• Horstemann D. From 1970 until present: the Keller-Segel model in chemotaxis and its consequence. I. Jahresber Dtsch Math-Ver. 2003;105:103–165.
   Google Scholar
• Horstemann D, Wang G. Blow-up in a chemotaxis model without symmetry assumptions. European J Appl Math. 2001;12:159–177.
   Web of Science ®Google Scholar
• Horstmann D, Winkler M. Boundedness vs. blow-up in a chemotaxis system. J Differ Equ. 2005;215:52–107.
   Web of Science ®Google Scholar
• Jin HY, Wang ZA. Boundedness, blowup and critical mass phenomenon in competing chemotaxis. J Differ Equ. 2016;260:162–196.
   Web of Science ®Google Scholar
• Nagai T. Blow up of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains. J Inequal Appl. 2001;6:37–55.
   Web of Science ®Google Scholar
• Nagai T. Blow-up of radially symmetric solutions to a chemotaxis system. Adv Math Sci Appl. 1995;5:581–601.
   Google Scholar
• Nagai T, Senba T, Yoshida K. Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis. Funkc Ekvac. 1997;40:411–433.
   Google Scholar
• Osaki K, Yagi A. Finite dimensional attractors for one-dimensional Keller-Segel equations. Funkc Ekvac. 2001;44:441–469.
   Google Scholar
• Tao YS, Winkler M. Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity. J Differ Equ. 2012;252:692–715.
   Web of Science ®Google Scholar
• Lankeit J. Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source. J Differ Equ. 2015;258:1158–1191.
   Web of Science ®Google Scholar
• Osaki K, Yagi A, Tsujikawa T, et al. Exponential attractor for a chemotaxis-growth system of equations. Nonlinear Anal. 2002;51:119–144.
   Web of Science ®Google Scholar
• Tello JI, Winkler M. A chemotaxis system with logistic source. Comm Partial Differ Equ. 2007;32:849–877.
   Web of Science ®Google Scholar
• Winkler M. Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source. Comm Partial Differ Equ. 2010;35:1516–1537.
   Web of Science ®Google Scholar
• Cieślak T, Laurenct P, Morales-Rodrigo C. Global existence and convergence to steady states in a chemorepulsion system. Banach Center Publ Polish Acad Sci. 2008;81:105–117.
   Google Scholar
• Guo Q, Jiang ZX, Zheng SN. Critical mass for an attraction-repulsion chemotaxis system. Appl Anal. 2018;97:2349–2354.
   Web of Science ®Google Scholar
• Li YH, Lin K, Mu CL. Asymptotic behavior for small mass in an attraction-repulsion chemotaxis system. Electron J Differ Equ. 2015;146:1–13.
   Google Scholar
• Tao YS, Wang ZA. Competing effects of attraction vs. repulsion in chemotaxis. J Math Anal Appl. 2013;23:1–36.
   Google Scholar
• Li X, Xiang ZY. On an attraction-repulsion chemotaxis system with a logistic source. IMA J Appl Math. 2016;81:165–198.
   Web of Science ®Google Scholar
• Li Y, Wang W. Boundedness in a four-dimensional attraction-repulsion chemotaxis system with logistic source. Math Meth Appl Sci. 2018;41:4936–4942.
   Web of Science ®Google Scholar
• Xu PP, Zheng SN. Global boundedness in an attraction-repulsion chemotaxis system with logistic source. Appl Math Lett. 2018;83:1–6.
   Web of Science ®Google Scholar
• Zhang QS, Li YX. An attraction-repulsion chemotaxis system with logistic source. ZAMM Z Angew Math Mech. 2016;96:570–584.
   Web of Science ®Google Scholar
• Jin HY, Wang ZA. Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model. Math Meth Appl Sci. 2015;38:444–457.
   Web of Science ®Google Scholar
• Liu J, Wang ZA. Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension. J Biol Dyn. 2012;6:31–41.
   PubMed Web of Science ®Google Scholar
• Liu P, Shi JP, Wang ZA. Pattern formation of the attraction-repulsion Keller-Segel system. Discrete Contin Dyn Syst Ser B. 2013;18:2597–2625.
   Web of Science ®Google Scholar
• Jin HY. Boundedness of the attraction-repulsion Keller-Segel system. J Math Anal Appl. 2015;422:1463–1478.
   Web of Science ®Google Scholar
• Lin K, Mu CL, Wang LC. Large-time behavior of an attraction-repulsion chemotaxis system. J Math Anal Appl. 2015;426:105–124.
   Web of Science ®Google Scholar
• Li D, Mu CL, Lin K, et al. Large time behavior of solution to an attraction-repulsion chemotaxis system with logistic source in three dimensions. J Math Anal Appl. 2017;448:914–936.
   Web of Science ®Google Scholar
• Zheng P, Mu CL, Hu XG. Boundedness in the higher dimensional attraction-repulsion chemotaxis-growth system. J Comput Appl Math. 2016;72:2194–2202.
   Google Scholar
• Xu J, Liu ZR, Shi SJ. Large time behavior of solutions for the attraction-repulsion Keller-Segel system with large initial data. Appl Math Lett. 2019;87:13–19.
   Web of Science ®Google Scholar
• Li HY, Gao JG. Qualitative analysis of a parabolic-elliptic attraction-repulsion chemotaxis model with logistic source. Int J Biomath. 2016;9:1650051. (1–17).
   Google Scholar
• Friedman A. Partial differential equations. Holt, Rinehart and Winston; 1969;270.
   Google Scholar
• Biler P, Hebisch W, Nadzieja T. The Debye system: existence and large time behavior of solutions. Nonlinear Anal Theor Meth Appl. 1994;23:1189–1209.
   Web of Science ®Google Scholar
• Kowalczyk R, Szymańska Z. On the global existence of solutions to an aggregation model. J Math Anal Appl. 2008;343:379–398.
   Web of Science ®Google Scholar
• Ishida S, Seki K, Yokota T. Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains. J Differ Equ. 2014;256:2993–3010.
   Web of Science ®Google Scholar
• Li J, Ke YY, Wang YF. Large time behavior of solutions to a fully parabolic attraction-repulsion chemotaxis system with logistic source. Nonlinear Anal RWA. 2018;39:261–277.
   Google Scholar
• Porzio MM, Vespri V. Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations. J Differ Equ. 1993;103:146–178.
   Web of Science ®Google Scholar
• Tao YS, Winkler M. Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion. SIAM J Math Anal. 2015;47:4229–4250.
   Web of Science ®Google Scholar