September  2019, 24(9): 4665-4684. doi: 10.3934/dcdsb.2018328

Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production

1. 

School of Mathematical Sciences, Peking University, Beijing, 100871, China

2. 

School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

* Corresponding author: Wei Wang

Received  May 2018 Published  September 2019 Early access  January 2019

Fund Project: This work is supported by the National Natural Science Foundation of China (11671066, 11571020, 11671021).

In this paper we develop a new and convenient technique, with fractional Gagliardo-Nirenberg type inequalities inter alia involved, to treat the quasilinear fully parabolic chemotaxis system with indirect signal production: $ u_t = \nabla\cdot(D(u)\nabla u-S(u)\nabla v) $, $ \tau_1v_t = \Delta v-a_1v+b_1w $, $ \tau_2w_t = \Delta w-a_2w+b_2u $, under homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset\Bbb{R}^{n} $ ($ n\geq 1 $), where $ \tau_i,a_i,b_i>0 $ ($ i = 1,2 $) are constants, and the diffusivity $ D $ and the density-dependent sensitivity $ S $ satisfy $ D(s)\geq a_0(s+1)^{-\alpha} $ and $ 0\leq S(s)\leq b_0(s+1)^{\beta} $ for all $ s\geq 0 $ with $ a_0,b_0>0 $ and $ \alpha,\beta\in\Bbb R $. It is proved that if $ \alpha+\beta<3 $ and $ n = 1 $, or $ \alpha+\beta<4/n $ with $ n\geq 2 $, for any properly regular initial data, this problem has a globally bounded and classical solution. Furthermore, consider the quasilinear attraction-repulsion chemotaxis model: $ u_t = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla z)+\xi\nabla\cdot(u\nabla w) $, $ z_t = \Delta z-\rho z+\mu u $, $ w_t = \Delta w-\delta w+\gamma u $, where $ \chi,\mu,\xi,\gamma,\rho,\delta>0 $, and the diffusivity $ D $ fulfills $ D(s)\geq c_0(s+1)^{M-1} $ for any $ s\geq 0 $ with $ c_0>0 $ and $ M\in\Bbb R $. As a corollary of the aforementioned assertion, it is shown that when the repulsion cancels the attraction (i.e. $ \chi\mu = \xi\gamma $), the solution is globally bounded if $ M>-1 $ and $ n = 1 $, or $ M>2-4/n $ with $ n\geq 2 $. This seems to be the first result for this quasilinear fully parabolic problem that genuinely concerns the contribution of repulsion.

Citation: Mengyao Ding, Wei Wang. Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4665-4684. doi: 10.3934/dcdsb.2018328
References:
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H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (1984), 593-676. 

[3]

J. Bergh and J. Löfström, Interpolation Spaces. An introduction, Springer-Verlag, Berlin, 1976.

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E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

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E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

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A. Friedman, Partial Differential Equations, Holt, Rinehard and Winston, New York, 1969.

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K. Fujie and T. Senba, Application of an Adams type inequality to a two-chemical substances chemotaxis system, J. Differential Equations, 263 (2017), 88-148.  doi: 10.1016/j.jde.2017.02.031.

[8]

H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations, RIMS Kôkyûroku Bessatsu B26: Harmonic Analysis and Nonlinear Partial Differential Equations, 26 (2011), 159-175.

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.

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M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super Pisa Cl. Sci., 24 (1997), 633-683. 

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M. Hieber and J. Prüss, Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), 1647-1669.  doi: 10.1080/03605309708821314.

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D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[13]

B. Hu and Y. Tao, To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111-2128.  doi: 10.1142/S0218202516400091.

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S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.

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H. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.  doi: 10.1016/j.jmaa.2014.09.049.

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H. Jin and Z. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Methods Appl. Sci., 38 (2015), 444-457.  doi: 10.1002/mma.3080.

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H. Jin and T. Xiang, Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions, Discrete Continuous Dynam. Systems - B, 23 (2018), 3071-3085.  doi: 10.3934/dcdsb.2017197.

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E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. 

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O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Amer. Math. Soc., Providence, R.I., 1968.

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J. Lankeit, Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion, J. Differential Equations, 262 (2017), 4052-4084.  doi: 10.1016/j.jde.2016.12.007.

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G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996. doi: 10.1142/3302.

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K. LinC. Mu and L. Wang, Large time behavior for an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124.  doi: 10.1016/j.jmaa.2014.12.052.

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D. Liu and Y. Tao, Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546.  doi: 10.1002/mma.3240.

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J. Liu and Z. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dynam., 6 (2012), 31-41.  doi: 10.1080/17513758.2011.571722.

[25]

M. LucaA. Chavez-RossL. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, microglia, and Alzheimer's disease senile plague: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730. 

[26]

N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system, preprint.

[27]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. 

[28]

L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Sup. Pisa, 20 (1966), 733-737. 

[29]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.

[30]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. 

[31]

K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. 

[32]

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.

[33]

Y. Tao and M. Winkler, Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc., 19 (2017), 3641-3678.  doi: 10.4171/JEMS/749.

[34]

J. I. Tello and D. Wrzosek, Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.  doi: 10.1142/S0218202516400108.

[35]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North Holland, Amsterdam, 1978.

[36]

H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[37]

M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.  doi: 10.1002/mma.319.

[38]

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.

[39]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.

[40]

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.

[41]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. 
[2]

H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (1984), 593-676. 

[3]

J. Bergh and J. Löfström, Interpolation Spaces. An introduction, Springer-Verlag, Berlin, 1976.

[4]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[5]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[6]

A. Friedman, Partial Differential Equations, Holt, Rinehard and Winston, New York, 1969.

[7]

K. Fujie and T. Senba, Application of an Adams type inequality to a two-chemical substances chemotaxis system, J. Differential Equations, 263 (2017), 88-148.  doi: 10.1016/j.jde.2017.02.031.

[8]

H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations, RIMS Kôkyûroku Bessatsu B26: Harmonic Analysis and Nonlinear Partial Differential Equations, 26 (2011), 159-175.

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.

[10]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Sc. Norm. Super Pisa Cl. Sci., 24 (1997), 633-683. 

[11]

M. Hieber and J. Prüss, Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), 1647-1669.  doi: 10.1080/03605309708821314.

[12]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[13]

B. Hu and Y. Tao, To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111-2128.  doi: 10.1142/S0218202516400091.

[14]

S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.

[15]

H. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.  doi: 10.1016/j.jmaa.2014.09.049.

[16]

H. Jin and Z. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Methods Appl. Sci., 38 (2015), 444-457.  doi: 10.1002/mma.3080.

[17]

H. Jin and T. Xiang, Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions, Discrete Continuous Dynam. Systems - B, 23 (2018), 3071-3085.  doi: 10.3934/dcdsb.2017197.

[18]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. 

[19]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Amer. Math. Soc., Providence, R.I., 1968.

[20]

J. Lankeit, Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion, J. Differential Equations, 262 (2017), 4052-4084.  doi: 10.1016/j.jde.2016.12.007.

[21]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996. doi: 10.1142/3302.

[22]

K. LinC. Mu and L. Wang, Large time behavior for an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124.  doi: 10.1016/j.jmaa.2014.12.052.

[23]

D. Liu and Y. Tao, Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546.  doi: 10.1002/mma.3240.

[24]

J. Liu and Z. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dynam., 6 (2012), 31-41.  doi: 10.1080/17513758.2011.571722.

[25]

M. LucaA. Chavez-RossL. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, microglia, and Alzheimer's disease senile plague: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730. 

[26]

N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system, preprint.

[27]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. 

[28]

L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Sup. Pisa, 20 (1966), 733-737. 

[29]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.

[30]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. 

[31]

K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. 

[32]

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.

[33]

Y. Tao and M. Winkler, Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc., 19 (2017), 3641-3678.  doi: 10.4171/JEMS/749.

[34]

J. I. Tello and D. Wrzosek, Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.  doi: 10.1142/S0218202516400108.

[35]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North Holland, Amsterdam, 1978.

[36]

H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[37]

M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.  doi: 10.1002/mma.319.

[38]

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.

[39]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.

[40]

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.

[41]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.

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