# American Institute of Mathematical Sciences

## Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity

 1 Institute of Mathematics, Polish Academy of Sciences, Warsaw, 00-656, Poland 2 Department of Mathematics, Tokyo University of Science, Tokyo, 162-8601, Japan

* Corresponding author: Tomasz Cieślak

Received  May 2017 Revised  January 2018 Published  January 2019

The paper should be viewed as complement of an earlier result in [10]. In the paper just mentioned it is shown that 1d case of a quasilinear parabolic-elliptic Keller-Segel system is very special. Namely, unlike in higher dimensions, there is no critical nonlinearity. Indeed, for the nonlinear diffusion of the form $1/u$ all the solutions, independently on the magnitude of initial mass, stay bounded. However, the argument presented in [10] deals with the Jäger-Luckhaus type system. And is very sensitive to this restriction. Namely, the change of variables introduced in [10], being a main step of the method, works only for the Jäger-Luckhaus modification. It does not seem to be applicable in the usual version of the parabolic-elliptic Keller-Segel system. The present paper fulfils this gap and deals with the case of the usual parabolic-elliptic version. To handle it we establish a new Lyapunov-like functional (it is related to what was done in [10]), which leads to global existence of the initial-boundary value problem for any initial mass.

Citation: Tomasz Cieślak, Kentarou Fujie. Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020009
##### References:
 [1] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. Google Scholar [2] B. Bieganowski, T. Cieślak, K. Fujie and T. Senba, Boundedness of solutions to the critical fully parabolic quasilinear one-dimensional Keller-Segel system, Math. Nachr., to appear.Google Scholar [3] P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209. doi: 10.1016/0362-546X(94)90101-5. Google Scholar [4] A. Blanchet, J. A. Carrillo and Ph. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168. doi: 10.1007/s00526-008-0200-7. Google Scholar [5] H. Brézis and W. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590. doi: 10.2969/jmsj/02540565. Google Scholar [6] J. Burczak, T. Cieślak and C. Morales-Rodrigo, Global existence vs. blowup in a fully parabolic quasilinear 1D Keller-Segel system, Nonlinear Anal., 75 (2012), 5215-5228. doi: 10.1016/j.na.2012.04.038. Google Scholar [7] J. Burczak and R. Granero-Belinchon, Critical Keller-Segel meets Burgers on S1:large-time smooth solutions, Nonlinearity, 29 (2016), 3810-3836. doi: 10.1088/0951-7715/29/12/3810. Google Scholar [8] T. Cieślak and K. Fujie, No critical nonlinear diffusion in 1D quasilinear fully parabolic chemotaxis system, Proc. Amer. Math. Soc., 146 (2018), 2529-2540. doi: 10.1090/proc/13939. Google Scholar [9] T. Cieślak and Ph. Laurençot, Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system, C. R. Math. Acad. Sci. Paris, 347 (2009), 237-242. doi: 10.1016/j.crma.2009.01.016. Google Scholar [10] T. Cieślak and Ph. Laurençot, Looking for critical nonlinearity in the one-dimensional quasilinear Smoluchowski-Poisson system, Discrete Contin. Dyn. Syst., 26 (2010), 417-430. doi: 10.3934/dcds.2010.26.417. Google Scholar [11] T. Cieślak and Ph. Laurençot, Global existence vs. blowup for the one dimensional quasilinear Smoluchowski-Poisson system, Progress in Nonlinear Differential Equations and Their Applications, 80 (2011), 95-109. doi: 10.1007/978-3-0348-0075-4_6. Google Scholar [12] T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076. doi: 10.1088/0951-7715/21/5/009. Google Scholar [13] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6. Google Scholar [14] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar [15] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Ser. Int., 40 (1997), 411-433. Google Scholar [16] E. Nasreddine, Global existence of solutions to a parabolic-elliptic Keller-Segel system with critical degenerate diffusion, J. Math. Anal. Appl., 417 (2014), 144-163. doi: 10.1016/j.jmaa.2014.02.069. Google Scholar [17] Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876. Google Scholar [18] Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. Differential Equations, 12 (2007), 121-144. Google Scholar

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##### References:
 [1] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. Google Scholar [2] B. Bieganowski, T. Cieślak, K. Fujie and T. Senba, Boundedness of solutions to the critical fully parabolic quasilinear one-dimensional Keller-Segel system, Math. Nachr., to appear.Google Scholar [3] P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209. doi: 10.1016/0362-546X(94)90101-5. Google Scholar [4] A. Blanchet, J. A. Carrillo and Ph. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168. doi: 10.1007/s00526-008-0200-7. Google Scholar [5] H. Brézis and W. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan, 25 (1973), 565-590. doi: 10.2969/jmsj/02540565. Google Scholar [6] J. Burczak, T. Cieślak and C. Morales-Rodrigo, Global existence vs. blowup in a fully parabolic quasilinear 1D Keller-Segel system, Nonlinear Anal., 75 (2012), 5215-5228. doi: 10.1016/j.na.2012.04.038. Google Scholar [7] J. Burczak and R. Granero-Belinchon, Critical Keller-Segel meets Burgers on S1:large-time smooth solutions, Nonlinearity, 29 (2016), 3810-3836. doi: 10.1088/0951-7715/29/12/3810. Google Scholar [8] T. Cieślak and K. Fujie, No critical nonlinear diffusion in 1D quasilinear fully parabolic chemotaxis system, Proc. Amer. Math. Soc., 146 (2018), 2529-2540. doi: 10.1090/proc/13939. Google Scholar [9] T. Cieślak and Ph. Laurençot, Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system, C. R. Math. Acad. Sci. Paris, 347 (2009), 237-242. doi: 10.1016/j.crma.2009.01.016. Google Scholar [10] T. Cieślak and Ph. Laurençot, Looking for critical nonlinearity in the one-dimensional quasilinear Smoluchowski-Poisson system, Discrete Contin. Dyn. Syst., 26 (2010), 417-430. doi: 10.3934/dcds.2010.26.417. Google Scholar [11] T. Cieślak and Ph. Laurençot, Global existence vs. blowup for the one dimensional quasilinear Smoluchowski-Poisson system, Progress in Nonlinear Differential Equations and Their Applications, 80 (2011), 95-109. doi: 10.1007/978-3-0348-0075-4_6. Google Scholar [12] T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076. doi: 10.1088/0951-7715/21/5/009. Google Scholar [13] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6. Google Scholar [14] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar [15] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Ser. Int., 40 (1997), 411-433. Google Scholar [16] E. Nasreddine, Global existence of solutions to a parabolic-elliptic Keller-Segel system with critical degenerate diffusion, J. Math. Anal. Appl., 417 (2014), 144-163. doi: 10.1016/j.jmaa.2014.02.069. Google Scholar [17] Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876. Google Scholar [18] Y. Sugiyama, Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models, Adv. Differential Equations, 12 (2007), 121-144. Google Scholar
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