[1]

Aida, M.—Efendiev, M.—Yagi, A.: *Quasilinear abstract parabolic evolution equations and exponential attractors*, Osaka J. Math. **42**(1) (2005), 101–132.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.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.CrossrefWeb of ScienceGoogle Scholar

[4]

Albeverio, S.—Röckner, M.—Yoshida, M. W.: *A homeomorphism relating path spaces of stochastic processes with values in* **R**^{Z} *respectively* (*S*^{1})^{Z}, Infin. Dimens. Anal. Quantum Probab. Relat. Top. **17** (2014), Art. ID 1450002, 30 pp.Web of ScienceGoogle 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.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.CrossrefWeb of ScienceGoogle Scholar

[7]

Fukushima, M.: *Dirichlet Forms and Markov Processes*, Elsevier North-Holland, 1980.Google Scholar

[8]

Gross, L.: *Logarithmic Sobolev Inequalities and Contractivity Properties of Semigroups*. Lecture Notes in Math. 1563, 1993, pp. 54–88.CrossrefGoogle Scholar

[9]

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

[10]

Keller, E. F.—Segel, L. A.: *Initiation of slime mold aggregation viewed as instability*, J. Theor. Biol. **26** (1970), 399–415.PubMedCrossrefGoogle Scholar

[11]

Kozono, H.-Sugiyama, Y.: *The Keller-Segel system of parabolic*-*parabolic type with initial data in weak* ${L}^{\frac{n}{2}}$(**R**^{n}) *and its application to self*-*similar solutions*, Indiana Univ. Math. J. **57** (2008), 1468–1500.Google Scholar

[12]

Ma, Z.—Röckner, M.: *Introduction to the Theory of (Non*-*Symmetric) Dirichlet Forms*, Springer-Verlag, 1992.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.Google Scholar

[14]

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

[15]

Mizohata, S.: *The Theory of Partial Differential Equations*, Cambridge University Press, 1979.Google Scholar

[16]

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

[17]

Osaki, K.—Yagi, A.: *Global existence for a chemotaxis*-*growth system in* **R**^{2}, Adv. Math. Sci. Appl. **12**(2) (2002), 587–606.Google Scholar

[18]

Reed, M.—Simon, B.: *Functional Analysis*, Academic Press, Inc., 1972.Google Scholar

[19]

Stroock, D. W.: *Diffusion semigroups corresponding to uniformly elliptic divergence form operators*, Séminaire de Probabilités (Strasbourg) **2** (1988), 316–347.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.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.CrossrefGoogle 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.CrossrefWeb of ScienceGoogle 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.Web of ScienceCrossrefGoogle Scholar

[24]

Yahagi, Y.: *A probabilistic consideration on one dimensional Keller Segel system*, Neural Parallel Sci. Compt. **24** (2016), 15–28.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.Web of ScienceGoogle Scholar

[26]

Yosida, K.: *Functional Analysis*, Springer-Verlag, 1965.Google Scholar

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.