Construction of a unique mild solution of one-dimensional Keller-Segel systems with uniformly elliptic operators having variable coefficients

References

[1] Aida, M.—Efendiev, M.—Yagi, A.: Quasilinear abstract parabolic evolution equations and exponential attractors, Osaka J. Math. 42(1) (2005), 101–132.Search in Google Scholar

[2] Albeverio, S.—Bernabei, M. S.—Röckner, M.—Yoshida, M. W.: Homogenization of diffusions on the lattice Z d with periodic drift coefficients, applying a logarithmic Sobolev inequality or a weak Poincare inequality. In: Stoch. Anal. Appl., Abel Symp. 2, 2007, pp. 53–72.10.1007/978-3-540-70847-6_3Search in Google Scholar

[3] Albeverio, S.—Di, P. L.—Mastrogiacomo, E.: Small noise asymptotic expansions for stochastic PDE’s, I. The case of a dissipative polynomially bounded nonlinearity, Tohoku Math. J. 63 (2011), 877–898.10.2748/tmj/1325886292Search in Google Scholar

[4] Albeverio, S.—Röckner, M.—Yoshida, M. W.: A homeomorphism relating path spaces of stochastic processes with values inR^Zrespectively (S¹)^Z, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 17 (2014), Art. ID 1450002, 30 pp.10.1142/S0219025714500027Search in Google Scholar

[5] Albeverio, S.—Yoshida, M. W.: Some abstract considerations on the homogenization problem of infinite dimensional diffusions. In: Applications of Renormalization Group Methods in Mathematical Sciences, RIMS Kokyuroku Bessatsu B21, 2010, pp. 183–192.Search in Google Scholar

[6] Bellomo, N.—Bellouquid, A.—Tao, Y.—Winkler, M.: Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci. 25 (2015), 1663–1763.10.1142/S021820251550044XSearch in Google Scholar

[7] Fukushima, M.: Dirichlet Forms and Markov Processes, Elsevier North-Holland, 1980.Search in Google Scholar

[8] Gross, L.: Logarithmic Sobolev Inequalities and Contractivity Properties of Semigroups. Lecture Notes in Math. 1563, 1993, pp. 54–88.10.1007/BFb0074091Search in Google Scholar

[9] Hillen, T.—Painter, K. J.: A user’s guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009), 183–217.10.1007/s00285-008-0201-3Search in Google Scholar

[10] Keller, E. F.—Segel, L. A.: Initiation of slime mold aggregation viewed as instability, J. Theor. Biol. 26 (1970), 399–415.10.1016/0022-5193(70)90092-5Search in Google Scholar

[11] Kozono, H.-Sugiyama, Y.: The Keller-Segel system of parabolic-parabolic type with initial data in weakLn2(Rⁿ) and its application to self-similar solutions, Indiana Univ. Math. J. 57 (2008), 1468–1500.10.1512/iumj.2008.57.3316Search in Google Scholar

[12] Ma, Z.—Röckner, M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Springer-Verlag, 1992.10.1007/978-3-642-77739-4Search in Google Scholar

[13] Marras, M.—Vernier Piro, S.—Viglialoro, G.: Blow-up phenomena in chemotaxis system with a source term, Math. Methods Appl. Sci. 36(11) (2016), 2787–2798.10.1002/mma.3728Search in Google Scholar

[14] Mizoguchi, N.—Winkler, M.: Blow-up in the two-dimensional parabolic Keller-Segel system, preprint.Search in Google Scholar

[15] Mizohata, S.: The Theory of Partial Differential Equations, Cambridge University Press, 1979.Search in Google Scholar

[16] Nagai, T.: Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl. 5 (1995), 581–601.Search in Google Scholar

[17] Osaki, K.—Yagi, A.: Global existence for a chemotaxis-growth system inR², Adv. Math. Sci. Appl. 12(2) (2002), 587–606.Search in Google Scholar

[18] Reed, M.—Simon, B.: Functional Analysis, Academic Press, Inc., 1972.Search in Google Scholar

[19] Stroock, D. W.: Diffusion semigroups corresponding to uniformly elliptic divergence form operators, Séminaire de Probabilités (Strasbourg) 2 (1988), 316–347.10.1007/BFb0084145Search in Google Scholar

[20] Sugiyama, Y.: Blow-up criterion via scaling invariand quantities with effect on coefficient growth in KellerSegel system, Differential and Integral Equations 23 (2010), 619–634.Search in Google Scholar

[21] Viglialoro, G.: Boundedness proerties of very weak solutions to a fully parabolc chemotaxissystem with logistic source, Nonlinear Anal. Real World Appl. 34 (2017), 520–535.10.1016/j.nonrwa.2016.10.001Search in Google Scholar

[22] Viglialoro, G.: Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl. 439 (2016), 197–212.10.1016/j.jmaa.2016.02.069Search in Google Scholar

[23] Winkler, M.: Global existence and slow grow-up in a quasilinear Keller-Segel system with exponentially decaying diffusivity, Nonlinearity 30 (2017), 735–764.10.1088/1361-6544/aa565bSearch in Google Scholar

[24] Yahagi, Y.: A probabilistic consideration on one dimensional Keller Segel system, Neural Parallel Sci. Compt. 24 (2016), 15–28.Search in Google Scholar

[25] Yahagi, Y.: Asymptotic behavior of solutions to the one-dimensional Keller-Segel system with small chemotaxis, Tokyo J. Math. 41, to appear.10.3836/tjm/1502179267Search in Google Scholar

[26] Yosida, K.: Functional Analysis, Springer-Verlag, 1965.10.1007/978-3-662-25762-3Search in Google Scholar