The Keller–Segel system on bounded convex domains in critical spaces

Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Academic Press, Amsterdam (2003)

Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: H.J. Schmeisser, H. Triebel (eds.), Function Spaces, Differential Operators and Nonlinear Analysis, Teubner, pp. 9–126 (1993)

Amann, H.: Linear and Quasilinear Parabolic Problems, Monographs in Mathematics, Vol. 89, Birkhäuser (1995)

Arendt, W., Bu, S.: The operator-valued Marcinkiewicz multiplier theorem and maximal regularity. Math. Z. 240, 311–343 (2002)

Denk, R., Hieber, M., Prüss, J.: $${\cal{R}}$$-Boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem Am. Math. Soc. 788 (2003)

Hieber, M., Prüss, J.: Functional calculi for linear operators in vector-valued $$L^p$$-spaces via the transference principle. Adv. Differ. Equ. 3, 847–872 (1998)

Hieber, M., Robinson, J., Shibata, Y.: Mathematical Analysis of the Navier–Stokes Equations. Lecture Notes in Mathematics, vol. 2254. CIME Found. Subseries, Springer, Cham (2020)

Hieber, M., Stinner, C.: Strong time periodic solutions to Keller–Segel systems: an approach by the quasilinear Arendt–Bu theorem. J. Differ. Equ. 269, 1636–1655 (2020)

Hillen, T., Painter, K.J.: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58, 183–217 (2009)

Horstmann, D.: From 1970 until present: the Keller–Segel model in chemotaxis and its consequences I. Jahresber. Deutsch. Math.-Verein. 105(2003), 103–165

Horstmann, D., Meinlschmidt, H., Rehberg, J.: The full Keller–Segel model is well-posed on nonsmooth domains. Nonlinearity 31, 1560–1592 (2018)

Hytönen, T., van Neerven, J., Veraar, M., Weis, L.: Analysis in Banach Spaces, Vol. II. Springer, New York (2017)

Kozono, H., Sugiyama, Y.: Global strong solution to the semi-linear Keller–Segel system of parabolic–parabolic type with small data in scale invariant spaces. J. Differ. Equ. 247, 1–32 (2009)

Kozono, H., Sugiyama, Y.: Strong solutions to the Keller–Segel system with the weak $$L^{n/2}$$ initial data and its application to the blow-up rate. Math. Nachr. 283, 732–751 (2010)

Kozono, H., Sugiyama, Y., Wachi, T.: Existence and uniqueness theorem on mild solutions to the Keller–Segel system in the scaling invariant space. J. Differ. Equ. 252, 1213–1228 (2012)

Langkeit, J., Winkler, M.: Facing low regularity in chemotaxis systems. Jahresber. Deutsch. Math.-Verein. 122, 35–64 (2020)

Nakaguchi, E., Osaki, K.: $$L_p$$-estimates of solutions to $$n$$-dimensional parabolic–parabolic system for chemotaxis with subquadratic degradation. Funkcial. Ekvac. 59, 51–66 (2016)

Nakaguchi, E., Osaki, K.: Global existence of solutions to an $$n$$-dimensional parabolic–parabolic system for chemotaxis with logistic-type growth and superlinear production. Osaka J. Math. 55, 51–70 (2018)

Prüss, J., Simonett, G.: Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, Vol. 105, Birkhäuser (2016)

Prüss, J., Simonett, G., Wilke, M.: Criticial spaces for quasilinear parabolic evolution equations and applications. J. Differ. Equ. 264, 2028–2074 (2018)

Prüss, J., Wilke, M.: Addendum to the paper “On quasilinear parabolic evolution equations in weighted $$L_p$$-spaces II”. J. Evol. Equ. 17, 1381–1388 (2017)

Tao, Y.: Global dynamics in a higher dimensional repulsion chemotaxis model with nonlinear sensitivity. Discrete Contin. Dyn. Syst. Ser. B 18, 2705–2722 (2013)

Tao, Y., Winkler, M.: Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity. J. Differ. Equ. 252, 692–715 (2012)

Wood, I.: Maximal $$L^p$$-regularity for the Laplacian on Lipschitz domains. Math. Z. 255, 855–875 (2007)