Existence of Solutions for the Keller–Segel Model of Chemotaxis with Measures as Initial Data

• Autorzy: Biler P. , Zienkiewicz J.

Identyfikatory

DOI

10.4064/ba63-1-6

• Języki publikacji: EN

Abstrakty

EN

A simple proof of the existence of solutions for the two-dimensional Keller-Segel model with measures with all the atoms less than 8π as the initial data is given. This result was obtained by Senba and Suzuki (2002) and Bedrossian and Masmoudi (2014) using different arguments. Moreover, we show a uniform bound for the existence time of solutions as well as an optimal hypercontractivity estimate.

Słowa kluczowe

EN

chemotaxis; blow-up of solutions

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2015
• Tom: Vol. 63, no. 1
• Strony: 41--51
• Opis fizyczny: Bibliogr.11 poz.

Twórcy

autor

Biler P.

• Instytut Matematyczny, Uniwersytet Wrocławski, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

autor

Zienkiewicz J.

• Instytut Matematyczny, Uniwersytet Wrocławski, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Bibliografia

• [1] J. Bedrossian and N. Masmoudi, Existence, uniqueness and Lipschitz dependence for Patlak–Keller–Segel and Navier–Stokes in R2 with measure-valued initial data, Arch. Ration. Mech. Anal. 214 (2014), 717–801.
• [2] P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math. 114 (1995), 181–205.
• [3] P. Biler, T. Cieślak, G. Karch and J. Zienkiewicz, Local criteria for blowup of solutions in two-dimensional chemotaxis models, arXiv:1410.7807v2 (2014).
• [4] P. Biler, L. Corrias and J. Dolbeault, Large mass self-similar solutions of the parabolic-parabolic Keller–Segel model, J. Math. Biology 63 (2011), 1–32.
• [5] P. Biler, I. Guerra and G. Karch, Large global-in-time solutions of the parabolic-parabolic Keller–Segel system on the plane, Comm. Pure Appl. Anal., to appear; arXiv:1401.7650 (2014).
• [6] P. Biler, G. Karch and J. Zienkiewicz, Optimal criteria for blowup of radial solutions of chemotaxis systems, arXiv:1407.4501 (2014).
• [7] A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller–Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations 2006, no. 44, 32 pp.
• [8] J. I. Díaz, T. Nagai and J.-M. Rakotoson, Symmetrization techniques on unbounded domains: application to a chemotaxis system on RN, J. Differential Equations 145 (1998), 156–183.
• [9] Y. Giga, T. Miyakawa and H. Osada, Two-dimensional Navier–Stokes flow with measures as initial vorticity, Arch. Ration. Mech. Anal. 104 (1988), 223–250.
• [10] T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations 6 (2001), 21–50.
• [11] T. Senba and T. Suzuki, Weak solutions to a parabolic-elliptic system of chemotaxis, J. Funct. Anal. 191 (2002), 17–51.