On convergence to equilibria for the Keller–Segel chemotaxis model

We show that any global-in-time bounded solution to the Keller–Segel chemotaxis model converges to a single equilibrium as time tends to infinity. The proof is based on a generalized version of the Lojasiewicz–Simon theorem.

[1]  Miguel A. Herrero,et al.  Finite-time aggregation into a single point in a reaction - diffusion system , 1997 .

[2]  Takashi Suzuki,et al.  Chemotactic collapse in a parabolic system of mathematical biology , 2000 .

[3]  H. Duport,et al.  Sur les équations aux dérivées partielles , 1900 .

[4]  A non-smooth version of the Lojasiewicz–Simon theorem with applications to non-local phase-field systems , 2004 .

[5]  L. Simon.  Asymptotics for a class of non-linear evolution equations, with applications to geometric problems , 1983 .

[6]  Takashi Suzuki,et al.  Concentration lemma, Brezis-Merle type inequality, and a parabolic system of chemotaxis , 2001 .

[7]  J. Lions,et al.  Problèmes aux limites non homogènes et applications , 1968 .

[8]  L. Segel,et al.  Initiation of slime mold aggregation viewed as an instability. , 1970, Journal of theoretical biology.

[9]  Dirk Horstmann,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences , 2022 .

[10]  Dirk Horstmann,et al.  Boundedness vs. blow-up in a chemotaxis system , 2005 .

[11]  Herbert Gajewski,et al.  A descent method for the free energy of multicomponent systems , 2006 .

[12]  Atsushi Yagi,et al.  NORM BEHAVIOR OF SOLUTIONS TO A PARABOLIC SYSTEM OF CHEMOTAXIS , 1997 .

[13]  M. A. Herrero,et al.  A blow-up mechanism for a chemotaxis model , 1997 .

[14]  Herbert Gajewski,et al.  On a nonlocal phase separation model , 2003 .

[15]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[16]  Eduard Feireisl,et al.  Convergence for Semilinear Degenerate Parabolic Equations in Several Space Dimensions , 2000 .

[17]  H. Gajewski,et al.  Global Behaviour of a Reaction‐Diffusion System Modelling Chemotaxis , 1998 .

[18]  Piotr Biler,et al.  LOCAL AND GLOBAL SOLVABILITY OF SOME PARABOLIC SYSTEMS MODELLING CHEMOTAXIS , 1998 .

[19]  S. Łojasiewicz,et al.  On the gradient inequality , 1999 .