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Critical system involving fractional Laplacian
Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion
Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China |
$\left\{ \begin{array}{lll}&u_t = \nabla\cdot(\nabla u-\chi u\nabla v)+au-bu^\theta, &x\in \Omega, t>0, \\&0 = \Delta v -v+u^\kappa, & x\in \Omega, t>0 \end{array}\right.$ |
$u$ |
$Ω\subset \mathbb{R}^n(n≥ 2)$ |
$χ, b, κ>0$ |
$a∈ \mathbb{R}$ |
$θ>1$ |
$κ+1<\max\{θ, 1+\frac{2}{n}\}$ |
$θ = κ+1, \ \ b≥ \frac{(κ n-2)}{κ n}χ.$ |
$χ$ |
$ L^θ(Ω)$ |
$χ \to ∞$ |
$θ = 2$ |
$κ = 1$ |
$θ = κ+1$ |
$((a/b)^{\frac{1}{κ}}, a/b)$ |
$(0,0)$ |
$χ$ |
$b$ |
$θ = 2$ |
$κ = 1$ |
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. S. 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] |
X. Cao and S. Zheng,
Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, Math. Methods Appl. Sci., 37 (2014), 2326-2330.
doi: 10.1002/mma.2992. |
[4] |
A. Friedman,
Partial Differential Equations, Holt, Rinehart Winston, New York, 1969. |
[5] |
E. Galakhova, O. Salievab and J. Tello,
On a Parabolic-Elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.
doi: 10.1016/j.jde.2016.07.008. |
[6] |
X. He and S. Zheng,
Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source, J. Math. Anal. Appl., 436 (2016), 970-982.
doi: 10.1016/j.jmaa.2015.12.058. |
[7] |
T. Hillen and K. Painter,
A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[8] |
D. Horstmann,
From 1970 until now: the Keller-Segal model in chaemotaxis and its consequence Ⅰ, Jahresber DMV, 105 (2003), 103-165.
|
[9] |
B. Hu and Y. Tao,
Boundedness in a parabolic-elliptic chemotaxis-growth system under a critical parameter condition, Appl. Math. Lett., 64 (2017), 1-7.
doi: 10.1016/j.aml.2016.08.003. |
[10] |
W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis,
Trans. Amer. Math. Soc., 329 (1992), 819-824.
doi: 10.2307/2153966. |
[11] |
K. Kang and A. Stevens,
Blowup and global solutions in a chemotaxis-growth system, Nonlinear Anal., 135 (2016), 57-72.
doi: 10.1016/j.na.2016.01.017. |
[12] |
E. Keller and L. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theoret Biol., 26 (1970), 399-415.
|
[13] |
K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa,
Spatial pattern formation in a chemotaxis-diffusion-growth model, Phys. D, 241 (2012), 1629-1639.
doi: 10.1016/j.physd.2012.06.009. |
[14] |
O. Ladyzhenskaya, V. Solonnikov and N. Uralceva,
Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968. |
[15] |
D. Liu and Y. Tao,
Boundedness in a chemotaxis system with nonlinear signal production, Appl. Math. J. Chinese Univ. Ser. B, 31 (2016), 379-388.
doi: 10.1007/s11766-016-3386-z. |
[16] |
Y. Lou and W. M. Ni,
Diffusion vs cross-diffusion: an elliptic approach, J. Differential Equations, 154 (1999), 157-190.
doi: 10.1006/jdeq.1998.3559. |
[17] |
L. Nirenberg,
Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Sciences, New York University, New York, 1974. |
[18] |
P. Rabinowitz,
Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.
|
[19] |
Y. Tao and M. Winkler,
Dominance of chemotaxis in a chemotaxis-haptotaxis model, Nonlinearity, 27 (2014), 1225-1239.
doi: 10.1088/0951-7715/27/6/1225. |
[20] |
Y. Tao and M. Winkler,
Persistence of mass in a chemotaxis system with logistic source, J. Differential Equations, 259 (2015), 6142-6161.
doi: 10.1016/j.jde.2015.07.019. |
[21] |
J. Tello and M. Winkler,
chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[22] |
L. Wang, C. Mu and P. Zheng,
On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.
doi: 10.1016/j.jde.2013.12.007. |
[23] |
Z. Wang and T. Xiang, A class of chemotaxis systems with growth source and nonlinear secretion, arXiv: 1510.07204, 2015. |
[24] |
M. Winkler, Chemotaxis with logistic source: very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 38 (2008), 708-729.
doi: 10.1016/j.jmaa.2008.07.071. |
[25] |
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. |
[26] |
M. Winkler,
Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[27] |
M. Winkler,
Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-279.
doi: 10.3934/dcdsb.2017135. |
[28] |
T. Xiang,
Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differential Equations, 258 (2015), 4275-4323.
doi: 10.1016/j.jde.2015.01.032. |
[29] |
T. Xiang,
On a class of Keller-Segel chemotaxis systems with cross-diffusion, J. Differential Equations, 259 (2015), 4273-4326.
doi: 10.1016/j.jde.2015.05.021. |
[30] |
J. Zheng,
Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140.
doi: 10.1016/j.jde.2015.02.003. |
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. S. 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] |
X. Cao and S. Zheng,
Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, Math. Methods Appl. Sci., 37 (2014), 2326-2330.
doi: 10.1002/mma.2992. |
[4] |
A. Friedman,
Partial Differential Equations, Holt, Rinehart Winston, New York, 1969. |
[5] |
E. Galakhova, O. Salievab and J. Tello,
On a Parabolic-Elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.
doi: 10.1016/j.jde.2016.07.008. |
[6] |
X. He and S. Zheng,
Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source, J. Math. Anal. Appl., 436 (2016), 970-982.
doi: 10.1016/j.jmaa.2015.12.058. |
[7] |
T. Hillen and K. Painter,
A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[8] |
D. Horstmann,
From 1970 until now: the Keller-Segal model in chaemotaxis and its consequence Ⅰ, Jahresber DMV, 105 (2003), 103-165.
|
[9] |
B. Hu and Y. Tao,
Boundedness in a parabolic-elliptic chemotaxis-growth system under a critical parameter condition, Appl. Math. Lett., 64 (2017), 1-7.
doi: 10.1016/j.aml.2016.08.003. |
[10] |
W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis,
Trans. Amer. Math. Soc., 329 (1992), 819-824.
doi: 10.2307/2153966. |
[11] |
K. Kang and A. Stevens,
Blowup and global solutions in a chemotaxis-growth system, Nonlinear Anal., 135 (2016), 57-72.
doi: 10.1016/j.na.2016.01.017. |
[12] |
E. Keller and L. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theoret Biol., 26 (1970), 399-415.
|
[13] |
K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa,
Spatial pattern formation in a chemotaxis-diffusion-growth model, Phys. D, 241 (2012), 1629-1639.
doi: 10.1016/j.physd.2012.06.009. |
[14] |
O. Ladyzhenskaya, V. Solonnikov and N. Uralceva,
Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968. |
[15] |
D. Liu and Y. Tao,
Boundedness in a chemotaxis system with nonlinear signal production, Appl. Math. J. Chinese Univ. Ser. B, 31 (2016), 379-388.
doi: 10.1007/s11766-016-3386-z. |
[16] |
Y. Lou and W. M. Ni,
Diffusion vs cross-diffusion: an elliptic approach, J. Differential Equations, 154 (1999), 157-190.
doi: 10.1006/jdeq.1998.3559. |
[17] |
L. Nirenberg,
Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Sciences, New York University, New York, 1974. |
[18] |
P. Rabinowitz,
Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.
|
[19] |
Y. Tao and M. Winkler,
Dominance of chemotaxis in a chemotaxis-haptotaxis model, Nonlinearity, 27 (2014), 1225-1239.
doi: 10.1088/0951-7715/27/6/1225. |
[20] |
Y. Tao and M. Winkler,
Persistence of mass in a chemotaxis system with logistic source, J. Differential Equations, 259 (2015), 6142-6161.
doi: 10.1016/j.jde.2015.07.019. |
[21] |
J. Tello and M. Winkler,
chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[22] |
L. Wang, C. Mu and P. Zheng,
On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.
doi: 10.1016/j.jde.2013.12.007. |
[23] |
Z. Wang and T. Xiang, A class of chemotaxis systems with growth source and nonlinear secretion, arXiv: 1510.07204, 2015. |
[24] |
M. Winkler, Chemotaxis with logistic source: very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 38 (2008), 708-729.
doi: 10.1016/j.jmaa.2008.07.071. |
[25] |
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. |
[26] |
M. Winkler,
Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[27] |
M. Winkler,
Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-279.
doi: 10.3934/dcdsb.2017135. |
[28] |
T. Xiang,
Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differential Equations, 258 (2015), 4275-4323.
doi: 10.1016/j.jde.2015.01.032. |
[29] |
T. Xiang,
On a class of Keller-Segel chemotaxis systems with cross-diffusion, J. Differential Equations, 259 (2015), 4273-4326.
doi: 10.1016/j.jde.2015.05.021. |
[30] |
J. Zheng,
Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140.
doi: 10.1016/j.jde.2015.02.003. |
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