# American Institute of Mathematical Sciences

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  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 & Continuous Dynamical Systems - B, 2019, 24 (9) : 4665-4684. doi: 10.3934/dcdsb.2018328
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Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. Google Scholar [2] H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (1984), 593-676. Google Scholar [3] J. Bergh and J. Löfström, Interpolation Spaces. An introduction, Springer-Verlag, Berlin, 1976. Google Scholar [4] E. Di Nezza, G. 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. Google Scholar [5] E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2. Google Scholar [6] A. Friedman, Partial Differential Equations, Holt, Rinehard and Winston, New York, 1969. Google Scholar [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. Google Scholar [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. Google Scholar [9] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [14] S. Ishida, K. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. Google Scholar [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. Google Scholar [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. Google Scholar [21] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996. doi: 10.1142/3302. Google Scholar [22] K. Lin, C. 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. Google Scholar [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. Google Scholar [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. Google Scholar [25] M. Luca, A. Chavez-Ross, L. 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. Google Scholar [26] N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system, preprint.Google Scholar [27] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. Google Scholar [28] L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Sup. Pisa, 20 (1966), 733-737. Google Scholar [29] K. Osaki, T. Tsujikawa, A. 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. Google Scholar [30] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. Google Scholar [31] K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [35] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North Holland, Amsterdam, 1978. Google Scholar [36] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983. doi: 10.1007/978-3-0346-0416-1. Google Scholar [37] M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925. doi: 10.1002/mma.319. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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