In this paper, we study a chemotaxis system with nonlinear indirect signal production
$ \left\{ {\begin{array}{*{20}{l}} {{u_t} = \Delta \left( {\gamma \left( v \right) u } \right)}+ru-\mu u^l, \quad &x\in \Omega, t>0, \\ {{v_t} = \Delta v - v + w^{\beta}}, \quad &x\in \Omega, t>0, \\ {{w_t} = - \delta w + u}, \quad &x\in \Omega, t>0, \end{array}} \right. $
under homogeneous Neumann boundary conditions in a smooth bounded domain $ \Omega \subset {\mathbb{R}^n}(n\geq2) $, where the parameters $ r $, $ \mu $, $ \beta $, $ \delta > 0 $, and $ l > 1 $, the motility function $ \gamma\in C^{3}([0, \infty)) $, $ \gamma(v) > 0 $ is bounded, $ \gamma^{'}(v) < 0 $, and $ \frac{\gamma^{'}(v)}{\gamma(v)} $ is bounded. We show that if $ {\frac{l}{\beta}} > {\frac{n}{2}} $, the system has a unique global classical solution. Moreover, the solution exponentially converges to $ ((\frac{r} {\mu})^{\frac{1}{l-1}}, (\frac{1}{\delta})^{\beta}(\frac{r}{\mu})^{\frac{\beta}{l-1}}, \frac{1}{\delta}(\frac{r}{\mu})^{\frac{1}{l-1}})) $ in the large time limit under some extra hypotheses.
Citation: Ya Tian, Jing Luo. Boundedness and large time behavior of a signal-dependent motility system with nonlinear indirect signal production[J]. Electronic Research Archive, 2024, 32(11): 6301-6319. doi: 10.3934/era.2024293
In this paper, we study a chemotaxis system with nonlinear indirect signal production
$ \left\{ {\begin{array}{*{20}{l}} {{u_t} = \Delta \left( {\gamma \left( v \right) u } \right)}+ru-\mu u^l, \quad &x\in \Omega, t>0, \\ {{v_t} = \Delta v - v + w^{\beta}}, \quad &x\in \Omega, t>0, \\ {{w_t} = - \delta w + u}, \quad &x\in \Omega, t>0, \end{array}} \right. $
under homogeneous Neumann boundary conditions in a smooth bounded domain $ \Omega \subset {\mathbb{R}^n}(n\geq2) $, where the parameters $ r $, $ \mu $, $ \beta $, $ \delta > 0 $, and $ l > 1 $, the motility function $ \gamma\in C^{3}([0, \infty)) $, $ \gamma(v) > 0 $ is bounded, $ \gamma^{'}(v) < 0 $, and $ \frac{\gamma^{'}(v)}{\gamma(v)} $ is bounded. We show that if $ {\frac{l}{\beta}} > {\frac{n}{2}} $, the system has a unique global classical solution. Moreover, the solution exponentially converges to $ ((\frac{r} {\mu})^{\frac{1}{l-1}}, (\frac{1}{\delta})^{\beta}(\frac{r}{\mu})^{\frac{\beta}{l-1}}, \frac{1}{\delta}(\frac{r}{\mu})^{\frac{1}{l-1}})) $ in the large time limit under some extra hypotheses.
| [1] |
E. Keller, L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 25 (1970), 399–415. https://doi.org/10.1016/0022-5193(70)90092-5 doi: 10.1016/0022-5193(70)90092-5
|
| [2] |
K. Osaki, A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2001), 441–470. https://doi.org/10.1016/0022-2364(85)90127-1 doi: 10.1016/0022-2364(85)90127-1
|
| [3] |
T. Nagai, T. Senba, K. Yoshid, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, 4 (1997), 411–433. https://doi.org/10.1142/S1664360722500126 doi: 10.1142/S1664360722500126
|
| [4] |
D. Horstmann, G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math., 12 (2001), 159–177. https://doi.org/10.1017/s0956792501004363 doi: 10.1017/s0956792501004363
|
| [5] |
M. Winkler, Aggregation versus global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equations, 248 (2010), 2889–2905. https://doi.org/10.1016/j.jde.2010.02.008 doi: 10.1016/j.jde.2010.02.008
|
| [6] |
M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Part Differ. Equations, 35 (2010), 1516–1537. https://doi.org/10.1080/03605300903473426 doi: 10.1080/03605300903473426
|
| [7] |
J. I. Tello, M. Winkler, A chemotaxis system with logistic source, Commun. Part. Differ. Equations, 32 (2007), 849–877. https://doi.org/10.1080/03605300701319003 doi: 10.1080/03605300701319003
|
| [8] |
X. Fu, L. Tang, C. Liu, J. Huang, T. Hwa, P. Lenz, Stripe formation in bacterial systems with density-suppressed motility, Phys. Rev. Lett., 108 (2012), 198102. https://doi.org/10.1103/physrevlett.108.198102 doi: 10.1103/physrevlett.108.198102
|
| [9] |
C. Liu, X. Fu, L. Liu, X. Ren, C. K. L. Chau, S. Li, et al., Sequential establishment of stripe patterns in an expanding cell population, Science, 334 (2011), 238–241. https://doi.org/10.3410/f.13985959.15441064 doi: 10.3410/f.13985959.15441064
|
| [10] |
Y. Tao, M. Winkler, Effects of signal-dependent motilities in a Keller-Segel-type reaction diffusion system, Math. Models Methods Appl. Sci., 27 (2017), 1645–1683. https://doi.org/10.1142/s0218202517500282 doi: 10.1142/s0218202517500282
|
| [11] |
C. Yoon, Y. Kim, Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Appl. Math., 149 (2017), 101–123. https://doi.org/10.1007/s10440-016-0089-7 doi: 10.1007/s10440-016-0089-7
|
| [12] |
K. Fujie, J. Jiang, Comparison methods for a Keller-Segel-type model of pattern formations with density-suppressed motilities, Calculus Var. Partial Differ. Equations, 60 (2021), 92. https://doi.org/10.1007/s00526-021-01943-5 doi: 10.1007/s00526-021-01943-5
|
| [13] |
H. Jin, Z. Wang, Critical mass on the Keller-Segel system with signal-dependent motility, Proc. Am. Math. Soc., 148 (2020), 4855–4873. https://doi.org/10.1090/proc/15124 doi: 10.1090/proc/15124
|
| [14] |
L. Desvillettes, Y. Kim, A. Trescases, C. Yoon, A logarithmic chemotaxis model featuring global existence and aggregation, Nonlinear Anal. Real World Appl., 50 (2019), 562–582. https://doi.org/10.1016/j.nonrwa.2019.05.010 doi: 10.1016/j.nonrwa.2019.05.010
|
| [15] |
J. Jiang, P. Laurençot, Global existence and uniform boundedness in a chemotaxis model with signal-dependent motility, J. Differ. Equations, 299 (2021), 513–541. https://doi.org/10.1016/j.jde.2021.07.029 doi: 10.1016/j.jde.2021.07.029
|
| [16] |
Z. Wang, On the parabolic-elliptic Keller-Segel system with signal-dependent motilities: A paradigm for global boundedness and steady states, Math. Methods Appl. Sci., 44 (2021), 10881–10898. https://doi.org/10.22541/au.159317660.09415314 doi: 10.22541/au.159317660.09415314
|
| [17] |
H. Jin, Y. Kim, Z. Wang, Boundedness, stabilization and pattern formation driven by density suppressed motility, SIAM J. Appl. Math., 78 (2018), 1632–1657. https://doi.org/10.1137/17m1144647 doi: 10.1137/17m1144647
|
| [18] |
W. Lv, Q. Wang, Global existence for a class of Keller-Segel models with signal-dependent motility and general logistic term, Evol. Equations Control. Theory, 10 (2021), 25–36. https://doi.org/10.1016/j.nonrwa.2020.103160 doi: 10.1016/j.nonrwa.2020.103160
|
| [19] |
W. Lv, Q. Wang, An n-dimensional chemotaxis system with signal- dependent motility and generalized logistic source: Global existence and asymp- totic stabilization, Proc. R. Soc. Edinburgh Sect., 151 (2021), 821–841. https://doi.org/10.1017/prm.2020.38 doi: 10.1017/prm.2020.38
|
| [20] |
W. Lv, Global existence for a class of chemotaxis-consumption systems with signal dependent motility and generalized logistic source, Nonlinear. Anal. Real. World. Appl., 56 (2020), 103160. https://doi.org/10.1016/j.nonrwa.2020.103160 doi: 10.1016/j.nonrwa.2020.103160
|
| [21] |
X. Tao, Z. Fang, Global boundedness and stability in a density-suppressed motility model with generalized logistic source and nonlinear signal production, ZAMP, 73 (2022), 1–19. https://doi.org/10.1007/s00033-022-01775-z doi: 10.1007/s00033-022-01775-z
|
| [22] |
Y. Tian, G. Xie, Global boundedness and large time behavior in a signal-dependent motility system with nonlinear signal consumption, ZAMP, 75 (2024), 7. https://doi.org/10.21203/rs.3.rs-3147707/v1 doi: 10.21203/rs.3.rs-3147707/v1
|
| [23] |
S. Strohm, R. Tyson, J. Powell, Pattern formation in a model for mountain pine beetle dispersal: Linking model predictions to data, Bull. Math. Biol., 75 (2013), 1778–1797. https://doi.org/10.1007/s11538-013-9868-8 doi: 10.1007/s11538-013-9868-8
|
| [24] |
B. Hu, 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. https://doi.org/10.1142/s0218202516400091 doi: 10.1142/s0218202516400091
|
| [25] |
S. Qiu, C. Mu, L. Wang, Boundedness in the higher-dimensional quasilinear chemotaxis-growth system with indirect attractant production, Comput. Math. Appl., 75 (2018), 3213–3223. https://doi.org/10.1016/j.camwa.2018.01.042 doi: 10.1016/j.camwa.2018.01.042
|
| [26] |
H. Li, Y. Tao, Boundedness in a chemotaxis system with indirect signal production and generalized logistic source, Appl. Math. Lett., 77 (2018), 108–113. https://doi.org/10.1016/j.aml.2017.10.006 doi: 10.1016/j.aml.2017.10.006
|
| [27] |
G. Ren, B. Liu, Boundedness in a chemotaxis system under a critical parameter condition, Bull. Braz. Math. Soc., 52 (2021), 281–289. https://doi.org/10.1007/s00574-020-00202-z doi: 10.1007/s00574-020-00202-z
|
| [28] |
Y. Tao, M. Winkler, Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc., 19 (2017), 3641–3678. https://doi.org/10.4171/JEMS/749 doi: 10.4171/JEMS/749
|
| [29] |
Y. Tao, M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAMJ. Math. Anal., 47 (2015), 4229–4250. https://doi.org/10.1137/15M1014115 doi: 10.1137/15M1014115
|
| [30] |
C. Stinner, M. Winkler, A critical exponent in a quasilinear Keller-Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects, J. Evol. Equations, 24 (2024), 26. https://doi.org/10.1007/s00028-024-00954-x doi: 10.1007/s00028-024-00954-x
|
| [31] |
W. Lv, Q. Wang, Global existence for a class of chemotaxis systems with signal-dependent motility, indirect signal production and generalized logistic source, ZAMP, 71 (2020), 53. https://doi.org/10.1007/s00033-020-1276-y doi: 10.1007/s00033-020-1276-y
|
| [32] |
H. Amann, Dynamic theory of quasilinear parabolic equations. Ⅱ. Reaction-diffusion systems, Differ. Integr. Equations, 3 (1990), 13–75. https://doi.org/10.57262/die/1371586185 doi: 10.57262/die/1371586185
|
| [33] |
C. Stinner, C. Surulrscu, M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, STAM J. Math. Anal., 46 (2014), 1969–2007. https://doi.org/10.1137/13094058X doi: 10.1137/13094058X
|
| [34] |
Y. Tao, M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equations, 252 (2012), 692–715. https://doi.org/10.1016/j.jde.2011.08.019 doi: 10.1016/j.jde.2011.08.019
|
| [35] | O. A. Lady'zhenskaya, V. Solonnikov, N. N. Ural'ceva, Linear and Quasilinesr Equations of Parabolic Type, American Mathematical Soc., 1968. https://doi.org/10.1007/978-3-663-13911-9_1 |
| [36] |
Y. Tao, M. Winkler, Large time behavior in a multi-dimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229–4250. doilinkhttps://doi.org/10.1016/j.aml.2016.03.019 doi: 10.1016/j.aml.2016.03.019
|
| [37] |
T. B. Issa, W. Shen, Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources, SIAM J. Appl. Dyn. Syst., 16 (2017), 926–973. https://doi.org/10.48550/arXiv.1609.00794 doi: 10.48550/arXiv.1609.00794
|
Ya Tian, Jing Luo. Boundedness and large time behavior of a signal-dependent motility system with nonlinear indirect signal production[J]. Electronic Research Archive, 2024, 32(11): 6301-6319. doi: 10.3934/era.2024293
DownLoad: