# American Institute of Mathematical Sciences

February  2014, 34(2): 789-802. doi: 10.3934/dcds.2014.34.789

## Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source

 1 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China 2 Department of Mathematics, Sichuan Normal University, Chengdu, 610066

Received  October 2012 Revised  March 2013 Published  August 2013

This paper deals with the global existence and boundedness of the solutions for the chemotaxis system with logistic source \begin{eqnarray*} \left\{ \begin{array}{llll} u_t=\nabla\cdot(\phi(u)\nabla u)-\nabla\cdot(\varphi(u)\nabla v)+f(u),\quad &x\in \Omega,\quad t>0,\\ v_t=\Delta v-v+u,\quad &x\in\Omega,\quad t>0,\\ \end{array} \right. \end{eqnarray*} under homogeneous Neumann boundary conditions in a convex smooth bounded domain $\Omega\subset \mathbb{R}^n (n\geq2),$ with non-negative initial data $u_0\in C^0(\overline{\Omega})$ and $v_0\in W^{1,\theta}{(\Omega)}$ (with some $\theta>n$). The nonlinearities $\phi$ and $\varphi$ are assumed to generalize the prototypes \begin{eqnarray*} \phi(u)=(u+1)^{-\alpha},\,\,\,\,\,\, \varphi(u)=u(u+1)^{\beta-1} \end{eqnarray*} with $\alpha\in \mathbb{R}$ and $\beta\in \mathbb{R}$. $f(u)$ is a smooth function generalizing the logistic function \begin{eqnarray*} f(u)=ru-bu^\gamma,\,\,\,\,\,\, u\geq0,\,\,\text{with}\,\, r\geq0,\,\,b>0\,\,\text{and}\,\,\gamma>1. \end{eqnarray*} It is proved that the corresponding initial-boundary value problem possesses a unique global classical solution that is uniformly bounded provided that some technical conditions are fulfilled.
Citation: Liangchen Wang, Yuhuan Li, Chunlai Mu. Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 789-802. doi: 10.3934/dcds.2014.34.789
##### References:
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Super. Pisa (3), 20 (1966), 733. Google Scholar [18] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations,, Nonlinear Anal., 51 (2002), 119. doi: 10.1016/S0362-546X(01)00815-X. Google Scholar [19] K. Osaki and A. Yagi, Global existence for a chemotaxis-growth system in $\mathbbR^2$,, Adv. Math. Sci. Appl., 12 (2002), 587. Google Scholar [20] K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations,, Funkcial. Ekvac., 44 (2001), 441. Google Scholar [21] K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Q., 10 (2002), 501. 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. doi: 10.1016/j.jde.2011.08.019. Google Scholar [23] J. I. Tello and M. Winkler, A chemotaxis system with logistic source,, Comm. Partial Differential Equations, 32 (2007), 849. doi: 10.1080/03605300701319003. Google Scholar [24] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008. Google Scholar [25] M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity,, Math. Nachr., 283 (2010), 1664. doi: 10.1002/mana.200810838. Google Scholar [26] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source,, Comm. Partial Differential Equations, 35 (2010), 1516. doi: 10.1080/03605300903473426. Google Scholar [27] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, in press, (). doi: 10.1016/j.matpur.2013.01.020. Google Scholar [28] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?,, Math. Methods Appl. Sci., 33 (2010), 12. doi: 10.1002/mma.1146. Google Scholar [29] M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect,, Nonlinear Anal., 72 (2010), 1044. doi: 10.1016/j.na.2009.07.045. Google Scholar [30] M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties,, J. Math. Anal. Appl., 348 (2008), 708. doi: 10.1016/j.jmaa.2008.07.071. Google Scholar [31] D. Wrzosek, Long time behaviour of solutions to a chemotaxis model with volume-filling effect,, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 431. doi: 10.1017/S0308210500004649. Google Scholar

show all references

##### References:
 [1] J. Burczak, T. Cieślak and C. Morales-Rodrigo, Global existence vs. blow-up in a fully parabolic quasilinear 1D Keller-Segel system,, Nonlinear Anal., 75 (2012), 5215. doi: 10.1016/j.na.2012.04.038. Google Scholar [2] Y.-S. Choi and Z.-A. Wang, Prevention of blow-up by fast diffusion in chemotaxis,, J. Math. Anal. Appl., 362 (2010), 553. doi: 10.1016/j.jmaa.2009.08.012. Google Scholar [3] T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions,, J. Differential Equations, 252 (2012), 5832. doi: 10.1016/j.jde.2012.01.045. Google Scholar [4] T. Cieślak, Quasilinear nonuniformly parabolic system modelling chemotaxis,, J. Math. Anal. Appl., 326 (2007), 1410. doi: 10.1016/j.jmaa.2006.03.080. Google Scholar [5] T. Cieślak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2,, , (). Google Scholar [6] T. Cieślak and P. Laurençot, Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system,, Ann. I. H. Poincaré Anal. Non Linéaire, 27 (2010), 437. doi: 10.1016/j.anihpc.2009.11.016. Google Scholar [7] T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis,, Nonlinearity, 21 (2008), 1057. doi: 10.1088/0951-7715/21/5/009. Google Scholar [8] A. Friedman, "Partial Differential Equations,", Holt, (1969). Google Scholar [9] M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 633. Google Scholar [10] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52. doi: 10.1016/j.jde.2004.10.022. Google Scholar [11] D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions,, European J. Appl. Math., 12 (2001), 159. doi: 10.1017/S0956792501004363. Google Scholar [12] T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding,, Adv. Appl. Math., 26 (2001), 280. doi: 10.1006/aama.2001.0721. Google Scholar [13] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar [14] 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. doi: 10.2307/2153966. Google Scholar [15] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theoret. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar [16] R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model,, J. Math. Anal. Appl., 343 (2008), 379. doi: 10.1016/j.jmaa.2008.01.005. Google Scholar [17] L. Nirenberg, An extended interpolation inequality,, Ann. Sc. Norm. Super. Pisa (3), 20 (1966), 733. Google Scholar [18] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations,, Nonlinear Anal., 51 (2002), 119. doi: 10.1016/S0362-546X(01)00815-X. Google Scholar [19] K. Osaki and A. Yagi, Global existence for a chemotaxis-growth system in $\mathbbR^2$,, Adv. Math. Sci. Appl., 12 (2002), 587. Google Scholar [20] K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations,, Funkcial. Ekvac., 44 (2001), 441. Google Scholar [21] K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Q., 10 (2002), 501. 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. doi: 10.1016/j.jde.2011.08.019. Google Scholar [23] J. I. Tello and M. Winkler, A chemotaxis system with logistic source,, Comm. Partial Differential Equations, 32 (2007), 849. doi: 10.1080/03605300701319003. Google Scholar [24] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008. Google Scholar [25] M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity,, Math. Nachr., 283 (2010), 1664. doi: 10.1002/mana.200810838. Google Scholar [26] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source,, Comm. Partial Differential Equations, 35 (2010), 1516. doi: 10.1080/03605300903473426. Google Scholar [27] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, in press, (). doi: 10.1016/j.matpur.2013.01.020. Google Scholar [28] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?,, Math. Methods Appl. Sci., 33 (2010), 12. doi: 10.1002/mma.1146. Google Scholar [29] M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect,, Nonlinear Anal., 72 (2010), 1044. doi: 10.1016/j.na.2009.07.045. Google Scholar [30] M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties,, J. Math. Anal. Appl., 348 (2008), 708. doi: 10.1016/j.jmaa.2008.07.071. Google Scholar [31] D. Wrzosek, Long time behaviour of solutions to a chemotaxis model with volume-filling effect,, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 431. doi: 10.1017/S0308210500004649. Google Scholar
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