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February  2020, 13(2): 119-137. doi: 10.3934/dcdss.2020007

Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity

Tobias Black 1,

Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany

Received  April 2017 Revised  November 2017 Published  January 2019

We investigate the parabolic-elliptic Keller-Segel model
$ \begin{align*} \left\{ \begin{array}{r@{\, }l@{\quad}l@{\quad}l@{\, }c} u_{t}& = \Delta u-\, \chi\nabla\!\cdot(\frac{u}{v}\nabla v), \ &x\in\Omega, & t>0, \\ 0& = \Delta v-\, v+u, \ &x\in\Omega, & t>0, \\ \frac{\partial u}{\partial\nu}& = \frac{\partial v}{\partial\nu} = 0, &x\in\partial \Omega, & t>0, \\ u(&x, 0) = u_0(x), \ &x\in\Omega, & \end{array}\right. \end{align*} $
in a bounded domain
$ \Omega\subset\mathbb{R}^n $
$ (n\geq2) $
 with smooth boundary.
We introduce a notion of generalized solvability which is consistent with the classical solution concept, and we show that whenever
$ 0<\chi<\frac{n}{n-2} $
 and the initial data satisfy only certain requirements on regularity and on positivity, one can find at least one global generalized solution.
Keywords: Chemotaxis, global existence, logarithmic sensitivity, generalized solution.
Mathematics Subject Classification: Primary: 35K55, 35D99; Secondary: 35A01, 35Q92, 92C17.
Citation: Tobias Black. Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 119-137. doi: 10.3934/dcdss.2020007
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]

P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359.   Google Scholar

[3]

T. Black, Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D, J. Differential Equations, 265 (2018), 2296–2339. doi: 10.1016/j.jde.2018.04.035.  Google Scholar

[4]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.  Google Scholar

[5]

H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in L1, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.  Google Scholar

[6]

X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differential Equations, 55 (2016), Art. 107, 39 pp. doi: 10.1007/s00526-016-1027-2.  Google Scholar

[7]

K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.  doi: 10.1016/j.jmaa.2014.11.045.  Google Scholar

[8]

K. Fujie and T. Senba, Global existence and boundedness of radial solutions to a two dimensional fully parabolic chemotaxis system with general sensitivity, Nonlinearity, 29 (2016), 2417-2450.  doi: 10.1088/0951-7715/29/8/2417.  Google Scholar

[9]

K. Fujie and T. Senba, Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 81-102.  doi: 10.3934/dcdsb.2016.21.81.  Google Scholar

[10]

K. Fujie, M. Winkler and T. Yokota, Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal-dependent sensitivity, Math. Methods Appl. Sci., 38 (2015), 1212-1224.  doi: 10.1002/mma.3149.  Google Scholar

[11]

T. Hillen and K. J. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[12]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.   Google Scholar

[13]

S. Itô, Diffusion Equations, volume 114 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1992.  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]

J. Lankeit, A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 39 (2016), 394-404.  doi: 10.1002/mma.3489.  Google Scholar

[16]

J. Lankeit and M. Winkler, A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: Global solvability for large nonradial data, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 49, 33pp. doi: 10.1007/s00030-017-0472-8.  Google Scholar

[17]

M. Mizukami and T. Yokota, A unified method for boundedness in fully parabolic chemotaxis systems with signal-dependent sensitivity, Math. Nachr., 290 (2017), 2648-2660.  doi: 10.1002/mana.201600399.  Google Scholar

[18]

T. Nagai and T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 145-156.   Google Scholar

[19]

G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen, Bull. Math. Biol., 40 (1978), 671-674.  doi: 10.1007/BF02460738.  Google Scholar

[20]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65–96. doi: 10.1007/BF01762360.  Google Scholar

[21]

C. Stinner and M. Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity, Nonlinear Anal. Real World Appl., 12 (2011), 3727-3740.  doi: 10.1016/j.nonrwa.2011.07.006.  Google Scholar

[22]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[23]

Y. Tao and M. Winkler, Persistence of mass in a chemotaxis system with logistic source, J. Differential Equations, 259 (2015), 6142-6161.  doi: 10.1016/j.jde.2015.07.019.  Google Scholar

[24]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[25]

M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 34 (2011), 176-190.  doi: 10.1002/mma.1346.  Google Scholar

[26]

M. Winkler, Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3092-3115.  doi: 10.1137/140979708.  Google Scholar

[27]

M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: global large-data solutions and their relaxation properties, Math. Models Methods Appl. Sci., 26 (2016), 987-1024.  doi: 10.1142/S0218202516500238.  Google Scholar

show all references

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]

P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359.   Google Scholar

[3]

T. Black, Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D, J. Differential Equations, 265 (2018), 2296–2339. doi: 10.1016/j.jde.2018.04.035.  Google Scholar

[4]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.  Google Scholar

[5]

H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in L1, J. Math. Soc. Japan, 25 (1973), 565-590.  doi: 10.2969/jmsj/02540565.  Google Scholar

[6]

X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differential Equations, 55 (2016), Art. 107, 39 pp. doi: 10.1007/s00526-016-1027-2.  Google Scholar

[7]

K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.  doi: 10.1016/j.jmaa.2014.11.045.  Google Scholar

[8]

K. Fujie and T. Senba, Global existence and boundedness of radial solutions to a two dimensional fully parabolic chemotaxis system with general sensitivity, Nonlinearity, 29 (2016), 2417-2450.  doi: 10.1088/0951-7715/29/8/2417.  Google Scholar

[9]

K. Fujie and T. Senba, Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 81-102.  doi: 10.3934/dcdsb.2016.21.81.  Google Scholar

[10]

K. Fujie, M. Winkler and T. Yokota, Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal-dependent sensitivity, Math. Methods Appl. Sci., 38 (2015), 1212-1224.  doi: 10.1002/mma.3149.  Google Scholar

[11]

T. Hillen and K. J. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[12]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.   Google Scholar

[13]

S. Itô, Diffusion Equations, volume 114 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1992.  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]

J. Lankeit, A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 39 (2016), 394-404.  doi: 10.1002/mma.3489.  Google Scholar

[16]

J. Lankeit and M. Winkler, A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: Global solvability for large nonradial data, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 49, 33pp. doi: 10.1007/s00030-017-0472-8.  Google Scholar

[17]

M. Mizukami and T. Yokota, A unified method for boundedness in fully parabolic chemotaxis systems with signal-dependent sensitivity, Math. Nachr., 290 (2017), 2648-2660.  doi: 10.1002/mana.201600399.  Google Scholar

[18]

T. Nagai and T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 145-156.   Google Scholar

[19]

G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen, Bull. Math. Biol., 40 (1978), 671-674.  doi: 10.1007/BF02460738.  Google Scholar

[20]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65–96. doi: 10.1007/BF01762360.  Google Scholar

[21]

C. Stinner and M. Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity, Nonlinear Anal. Real World Appl., 12 (2011), 3727-3740.  doi: 10.1016/j.nonrwa.2011.07.006.  Google Scholar

[22]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[23]

Y. Tao and M. Winkler, Persistence of mass in a chemotaxis system with logistic source, J. Differential Equations, 259 (2015), 6142-6161.  doi: 10.1016/j.jde.2015.07.019.  Google Scholar

[24]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[25]

M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 34 (2011), 176-190.  doi: 10.1002/mma.1346.  Google Scholar

[26]

M. Winkler, Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3092-3115.  doi: 10.1137/140979708.  Google Scholar

[27]

M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: global large-data solutions and their relaxation properties, Math. Models Methods Appl. Sci., 26 (2016), 987-1024.  doi: 10.1142/S0218202516500238.  Google Scholar

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