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Keywords:
stabilization; degenerate diffusion; Keller-Segel systems
stabilization; degenerate diffusion; Keller-Segel systems
Summary:
This paper presents a stabilization result for weak solutions of degenerate parabolic equations in divergence form. More precisely, the result asserts that the global-in-time weak solution converges to the average of the initial data in some topology as time goes to infinity. It is also shown that the result can be applied to a degenerate parabolic-elliptic Keller-Segel system.
References:
[1] Cao, X.: Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces. Discrete Contin. Dyn. Syst. 35 (2015), 1891–1904. DOI 10.3934/dcds.2015.35.1891 | MR 3294230
[2] Cieślak, T., Stinner, C.: Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2. Acta Appl. Math. 129 (2014), 135–146. DOI 10.1007/s10440-013-9832-5 | MR 3152080 | Zbl 1295.35123
[3] Cieślaka, T., Winkler, M.: Stabilization in a higher-dimensional quasilinear Keller-Segel system with exponentially decaying diffusivity and subcritical sensitivity. Nonlinear Anal. 159 (2017), 129–144. MR 3659827
[4] Hashira, T., Ishida, S., Yokota, T.: Finite-time blow-up for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. J. Differential Equations 264 (2018), 6459–6485. DOI 10.1016/j.jde.2018.01.038 | MR 3770055
[5] Ishida, S., Seki, K., Yokota, T.: 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 | MR 3199754
[6] Ishida, S., Yokota, T.: Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity. Discrete Contin. Dyn. Syst. Ser. S 13 (2020), 212–232. MR 4043690
[7] Ishida, S., Yokota, T.: Weak stabilization in degenerate parabolic equations in divergence form: application to degenerate Keller-Segel systems. Calc. Var. Partial Differential Equations 61 (2022), Paper No. 105. DOI 10.1007/s00526-022-02203-w | MR 4404850
[8] Jiang, J.: Convergence to equilibria of global solutions to a degenerate quasilinear Keller-Segel system. Z. Angew. Math. Phys. 69 (2018), Paper No. 130. DOI 10.1007/s00033-018-1025-7 | MR 3856789
[9] Langlais, M., Phillips, D.: Stabilization of solutions of nonlinear and degenerate evolution equations. Nonlinear Anal. 9 (1985), 321–333. DOI 10.1016/0362-546X(85)90057-4
[10] Lankeit, J.: Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete Contin. Dyn. Syst. Ser. S 13 (2020), 233–255. MR 4043691
[11] Laurençot, P., Mizoguchi, N.: Finite time blowup for the parabolic-parabolic Keller-Segel system with critical diffusion. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), 197–220. DOI 10.1016/j.anihpc.2015.11.002 | MR 3592684
[12] Senba, T., Suzuki, T.: A quasi-linear parabolic system of chemotaxis. Abstr. Appl. Anal. 2006 (2006), 1–21. DOI 10.1155/AAA/2006/23061 | MR 2211660
[13] Sugiyama, Y., Kunii, H.: 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 | MR 2235324
[14] Tao, Y., Winkler, M.: Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity. J. Differential Equations 252 (2012), 692–715. DOI 10.1016/j.jde.2011.08.019 | MR 2852223
[15] Winkler, M.: 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 | MR 2644137 | Zbl 1190.92004
[16] Winkler, M.: 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 | MR 3115832 | Zbl 1326.35053
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