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

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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 and 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.
[2]	P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359.
[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.
[4]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	S. Itô, Diffusion Equations, volume 114 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1992.
[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.
[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.
[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.
[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.
[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.
[19]	G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen, Bull. Math. Biol., 40 (1978), 671-674. doi: 10.1007/BF02460738.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[2]	P. Biler, Global solutions to some parabolic-elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359.
[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.
[4]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	S. Itô, Diffusion Equations, volume 114 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1992.
[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.
[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.
[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.
[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.
[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.
[19]	G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen, Bull. Math. Biol., 40 (1978), 671-674. doi: 10.1007/BF02460738.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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