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Orbital stability and spectral properties of solitary waves of Klein–Gordon equation with concentrated nonlinearity
Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production
School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China |
$ \begin{equation*} \begin{cases} \partial_{t} u_1 = \nabla \cdot (D_1(u_{1})\nabla u_{1} - S_1(u_{1})\nabla v) + f_{1}(u_{1}),\quad &x\in\Omega,\quad t>0,\\ \partial_{t} u_2 = \nabla \cdot (D_2(u_{2})\nabla u_{2} - S_2(u_{2})\nabla v) + f_{2}(u_{2}),\quad &x\in\Omega,\quad t>0,\\ \partial_{t} v = \Delta v-v+g_1(u_{1})+g_2(u_{2}),\quad &x\in\Omega,\quad t>0 \end{cases} \end{equation*} $ |
$ \Omega\subset \mathbb{R}^{n} $ |
$ (n\geq2) $ |
$ D_{i}(s) \geq C_{d_{i}} (1+s)^{-\alpha_i} $ |
$ S_{i}(s) \leq C_{s_{i}} s (1+s)^{\beta_{i}-1} $ |
$ s\geq0 $ |
$ C_{d_{i}},C_{s_{i}}>0 $ |
$ \alpha_i,\beta_{i} \in \mathbb{R} $ |
$ f_{i}(s) \leq r_{i}s - \mu_{i} s^{k_{i}} $ |
$ g_{i}(s)\leq s^{\gamma_{i}} $ |
$ s\geq0 $ |
$ r_{i} \in \mathbb{R} $ |
$ \mu_{i},\gamma_{i} > 0 $ |
$ k_{i} > 1 $ |
$ (i = 1,2) $ |
$ f_{i}(s) $ |
$ r_{i}>0 $ |
$ \mu_{i} $ |
$ ((\frac{r_{1}}{\mu_{1}})^{\frac{1}{k_{1}-1}}, (\frac{r_{2}}{\mu_{2}})^{\frac{1}{k_{2}-1}}, (\frac{r_{1}}{\mu_{1}})^{\frac{\gamma_{1}}{k_{1}-1}} + (\frac{r_{2}}{\mu_{2}})^{\frac{\gamma_{2}}{k_{2}-1}}) $ |
$ t\rightarrow \infty $ |
References:
[1] |
X. Bai and M. Winkler,
Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.
doi: 10.1512/iumj.2016.65.5776. |
[2] |
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. |
[3] |
T. Black,
Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 1253-1272.
doi: 10.3934/dcdsb.2017061. |
[4] |
T. Black, J. Lankeit and M. Mizukami,
On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876.
|
[5] |
M. Ding, W. Wang, S. Zhou and S. Zheng,
Asymptotic stability in a fully parabolic quasilinear chemotaxis model with general logistic source and signal production, J. Diff. Equations., 268 (2020), 6729-6777.
doi: 10.1016/j.jde.2019.11.052. |
[6] |
E. Espejo, K. Vilches and C. Conca,
A simultaneous blow-up problem arising in tumor modeling, J. Math. Biol., 79 (2019), 1357-1399.
|
[7] |
D. D. Haroske, H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008. |
[8] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[9] |
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differ. Equ., 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[10] |
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. |
[11] |
R. Kowalczyk and Z. Szymańska,
On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.
doi: 10.1016/j.jmaa.2008.01.005. |
[12] |
D. Liu and Y. Tao,
Boundedness in a chemotaxis system with nonlinear signal production, Appl. Math. J. Chin. Univ. Ser. B., 31 (2016), 379-388.
doi: 10.1007/s11766-016-3386-z. |
[13] |
N. Mizoguchi and P. Souplet,
Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire., 31 (2014), 851-875.
doi: 10.1016/j.anihpc.2013.07.007. |
[14] |
M. Mizukami,
Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. S., 13 (2020), 269-278.
doi: 10.3934/dcdss.2020015. |
[15] |
M. Negreanu and J. I. Tello,
Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differ. Equ., 258 (2015), 1592-1617.
doi: 10.1016/j.jde.2014.11.009. |
[16] |
K. Osaki and A. Yagi,
Global existence for a chemotaxis-growth system in $R^2$, Adv. Math. Sci. Appl., 12 (2002), 587-606.
|
[17] |
X. Pan and L. Wang, Boundedness in a two-species chemotaxis system with nonlinear sensitivity and signal secretion, J. Math. Anal. Appl., (2021), 125078. |
[18] |
X. Pan and L. Wang,
Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production, C. R. Math., 359 (2021), 161-168.
doi: 10.5802/crmath.148. |
[19] |
X. Pan and L. Wang, On a quasilinear fully parabolic two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., (2021).
doi: 10.3934/dcdsb.2021047. |
[20] |
X. Pan, L. Wang and J. Zhang,
Boundedness in a three-dimensional two-species and two-stimuli chemotaxis system with chemical signalling loop, Math. Method. Appl. Sci., 43 (2020), 9529-9542.
doi: 10.1002/mma.6621. |
[21] |
X. Pan, L. Wang, J Zhang and J Wang, Boundedness in a three-dimensional two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 71 (2020), 15pp.
doi: 10.1007/s00033-020-1248-2. |
[22] |
C. Stinner, J.I. Tello and M. Winkler,
Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[23] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equ., 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[24] |
Y. Tao and M. Winkler,
Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 20 (2015), 3165-3183.
doi: 10.3934/dcdsb.2015.20.3165. |
[25] |
M. Tian and S. Zheng,
Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species, Commun. Pure Appl. Anal., 15 (2016), 243-260.
|
[26] |
J. I. Tello and M. Winkler,
Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[27] |
L. Wang, Improvement of conditions for boundedness in a two-species chemotaxis competition system of parabolic-parabolic-elliptic type, J. Math. Anal. Appl., 481 (2020), 123705.
doi: 10.1016/j.jmaa.2019.123705. |
[28] |
L. Wang, Y. Li and C. Mu,
Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802.
|
[29] |
L. Wang and C. Mu,
A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 4585-4601.
doi: 10.3934/dcdsb.2020114. |
[30] |
L. Wang, J. Zhang, C. Mu and X. Hu,
Boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 191-221.
doi: 10.3934/dcdsb.2019178. |
[31] |
M. Winkler,
A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056.
doi: 10.1088/1361-6544/aaaa0e. |
[32] |
T. Xiang,
How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?, J. Math. Anal. Appl., 459 (2018), 1172-1200.
doi: 10.1016/j.jmaa.2017.11.022. |
[33] |
L. Xie and Y. Wang,
On a fully parabolic chemotaxis system with Lotka-Volterra competitive kinetics, J. Math. Anal. Appl., 471 (2019), 584-598.
|
show all references
References:
[1] |
X. Bai and M. Winkler,
Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.
doi: 10.1512/iumj.2016.65.5776. |
[2] |
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. |
[3] |
T. Black,
Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 1253-1272.
doi: 10.3934/dcdsb.2017061. |
[4] |
T. Black, J. Lankeit and M. Mizukami,
On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876.
|
[5] |
M. Ding, W. Wang, S. Zhou and S. Zheng,
Asymptotic stability in a fully parabolic quasilinear chemotaxis model with general logistic source and signal production, J. Diff. Equations., 268 (2020), 6729-6777.
doi: 10.1016/j.jde.2019.11.052. |
[6] |
E. Espejo, K. Vilches and C. Conca,
A simultaneous blow-up problem arising in tumor modeling, J. Math. Biol., 79 (2019), 1357-1399.
|
[7] |
D. D. Haroske, H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008. |
[8] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[9] |
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differ. Equ., 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[10] |
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. |
[11] |
R. Kowalczyk and Z. Szymańska,
On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.
doi: 10.1016/j.jmaa.2008.01.005. |
[12] |
D. Liu and Y. Tao,
Boundedness in a chemotaxis system with nonlinear signal production, Appl. Math. J. Chin. Univ. Ser. B., 31 (2016), 379-388.
doi: 10.1007/s11766-016-3386-z. |
[13] |
N. Mizoguchi and P. Souplet,
Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire., 31 (2014), 851-875.
doi: 10.1016/j.anihpc.2013.07.007. |
[14] |
M. Mizukami,
Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. S., 13 (2020), 269-278.
doi: 10.3934/dcdss.2020015. |
[15] |
M. Negreanu and J. I. Tello,
Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differ. Equ., 258 (2015), 1592-1617.
doi: 10.1016/j.jde.2014.11.009. |
[16] |
K. Osaki and A. Yagi,
Global existence for a chemotaxis-growth system in $R^2$, Adv. Math. Sci. Appl., 12 (2002), 587-606.
|
[17] |
X. Pan and L. Wang, Boundedness in a two-species chemotaxis system with nonlinear sensitivity and signal secretion, J. Math. Anal. Appl., (2021), 125078. |
[18] |
X. Pan and L. Wang,
Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production, C. R. Math., 359 (2021), 161-168.
doi: 10.5802/crmath.148. |
[19] |
X. Pan and L. Wang, On a quasilinear fully parabolic two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., (2021).
doi: 10.3934/dcdsb.2021047. |
[20] |
X. Pan, L. Wang and J. Zhang,
Boundedness in a three-dimensional two-species and two-stimuli chemotaxis system with chemical signalling loop, Math. Method. Appl. Sci., 43 (2020), 9529-9542.
doi: 10.1002/mma.6621. |
[21] |
X. Pan, L. Wang, J Zhang and J Wang, Boundedness in a three-dimensional two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 71 (2020), 15pp.
doi: 10.1007/s00033-020-1248-2. |
[22] |
C. Stinner, J.I. Tello and M. Winkler,
Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[23] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equ., 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[24] |
Y. Tao and M. Winkler,
Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 20 (2015), 3165-3183.
doi: 10.3934/dcdsb.2015.20.3165. |
[25] |
M. Tian and S. Zheng,
Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species, Commun. Pure Appl. Anal., 15 (2016), 243-260.
|
[26] |
J. I. Tello and M. Winkler,
Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[27] |
L. Wang, Improvement of conditions for boundedness in a two-species chemotaxis competition system of parabolic-parabolic-elliptic type, J. Math. Anal. Appl., 481 (2020), 123705.
doi: 10.1016/j.jmaa.2019.123705. |
[28] |
L. Wang, Y. Li and C. Mu,
Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802.
|
[29] |
L. Wang and C. Mu,
A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 4585-4601.
doi: 10.3934/dcdsb.2020114. |
[30] |
L. Wang, J. Zhang, C. Mu and X. Hu,
Boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 191-221.
doi: 10.3934/dcdsb.2019178. |
[31] |
M. Winkler,
A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056.
doi: 10.1088/1361-6544/aaaa0e. |
[32] |
T. Xiang,
How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?, J. Math. Anal. Appl., 459 (2018), 1172-1200.
doi: 10.1016/j.jmaa.2017.11.022. |
[33] |
L. Xie and Y. Wang,
On a fully parabolic chemotaxis system with Lotka-Volterra competitive kinetics, J. Math. Anal. Appl., 471 (2019), 584-598.
|
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