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Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms
Global stabilization of the full attraction-repulsion Keller-Segel system
1. | Department of Mathematics, South China University of Technology, Guangzhou 510640, China |
2. | Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China |
$\left\{ {\begin{array}{*{20}{l}}{{u_t} = \Delta u - \nabla \cdot (\chi u\nabla v) + \nabla \cdot (\xi u\nabla w),}&{x \in \Omega ,t > 0,}\\{{v_t} = {D_1}\Delta v + \alpha u - \beta v,}&{x \in \Omega ,t > 0,}\\{{w_t} = {D_2}\Delta w + \gamma u - \delta w,}&{x \in \Omega ,t > 0,}\\{u(x,0) = {u_0}(x),v(x,0) = {v_0}(x),w(x,0) = {w_0}(x)}&{x \in \Omega ,}\end{array}} \right.\;\;\;\;\left( * \right)$ |
$ \Omega\subset \mathbb{R}^2 $ |
$ (u_0,v_0,w_0) \in [W^{1,\infty}(\Omega)]^3 $ |
$ \frac{\xi\gamma}{\chi\alpha}\geq \max\Big\{\frac{D_1}{D_2},\frac{D_2}{D_1},\frac{\beta}{\delta},\frac{\delta}{\beta}\Big\} $ |
$ D_1,D_2,\chi,\xi,\alpha,\beta,\gamma $ |
$ \delta $ |
$ (u,v,w) $ |
$ (\bar{u}_0,\frac{\alpha}{\beta}\bar{u}_0,\frac{\gamma}{\delta}\bar{u}_0) $ |
$ t\to+\infty $ |
$ \bar{u}_0 = \frac{1}{|\Omega|}\int_\Omega u_0dx $ |
$ \frac{\xi\gamma}{\chi\alpha}> \max\Big\{\frac{\beta}{\delta},\frac{\delta}{\beta}\Big\} $ |
$ D_1\ne D_2 $ |
References:
[1] |
N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler,
Towards 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. |
[2] |
J. P. Bourguignon and H. Brezis,
Remarks on Euler Equation, J. Functional Analysis, 15 (1974), 341-363.
doi: 10.1016/0022-1236(74)90027-5. |
[3] |
J. A. Carrillo, A. Juöngle, P. A. Markowich, G. Toscani and A. Unterreiter,
Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82.
doi: 10.1007/s006050170032. |
[4] |
T. Cieślak, Ph. Laurenc |
[5] |
E. Espejo and T. Suzuki,
Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34.
doi: 10.1016/j.aml.2014.04.007. |
[6] |
D. Horstemann,
From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I. Jahresber. Deutsch. Math. Verien., 105 (2003), 103-165.
|
[7] |
D. Horstmann and G. Wang,
Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[8] |
H. Y. 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. |
[9] |
H. Y. Jin and Z. Liu,
Large time behavior of the full attraction-repulsion Keller-Segel system in the whole space, Appl. Math. Lett., 47 (2015), 13-20.
doi: 10.1016/j.aml.2015.03.004. |
[10] |
H. Y. Jin and Z. A. 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. |
[11] |
H. Y. Jin and Z. A. Wang,
Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 260 (2016), 162-196.
doi: 10.1016/j.jde.2015.08.040. |
[12] |
R. Kowalczyk and Z. Szyma$\acute{n}$ska,
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. |
[13] |
O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, American Mathematical Society, Providence, R.I., 1968. |
[14] |
Y. Li and Y. X. Li,
Blow-up of nonradial solutions to attraction-repulsion chemotaxis system in two dimensions, Nonlinear Anal. Real Word Appl., 30 (2016), 170-183.
doi: 10.1016/j.nonrwa.2015.12.003. |
[15] |
K. Lin and C. Mu,
Global existence and convergence to steady states for an attraction-repulsion chemotaxis system, Nonlinear Anal. Real Word Appl., 31 (2016), 630-642.
doi: 10.1016/j.nonrwa.2016.03.012. |
[16] |
K. Lin, C. Mu and L. Wang,
Large-time behavior of an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124.
doi: 10.1016/j.jmaa.2014.12.052. |
[17] |
K. Lin, C. Mu and D. Zhou,
Stabilization in a higher-dimensional attraction-repulsion chemotaxis system if repulsion dominates over attraction, Math. Models Methods Appl. Sci., 28 (2018), 1105-1134.
doi: 10.1142/S021820251850029X. |
[18] |
D. Liu and Y. S. Tao,
Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546.
doi: 10.1002/mma.3240. |
[19] |
P. Liu, J. P. Shi and Z. A. Wang,
Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.
doi: 10.3934/dcdsb.2013.18.2597. |
[20] |
J. Liu and Z. A. Wang,
Classical solutions and steady states of an attraction-repulsion chemotaxis model in one dimension, J. Biol. Dyn., 6 (2012), 31-41.
doi: 10.1080/17513758.2011.571722. |
[21] |
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, Microglia, and Alzheimer's disease senile plagues: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730. Google Scholar |
[22] |
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.
|
[23] |
K. J. Painter and T. Hillen,
Volume-filling quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.
|
[24] |
M. M. Porzio and V. Vespri,
Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.
doi: 10.1006/jdeq.1993.1045. |
[25] |
R. Shi and W. Wang,
Well-posedness for a model derived from an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 423 (2015), 497-520.
doi: 10.1016/j.jmaa.2014.10.006. |
[26] |
Y. S. Tao,
Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2705-2722.
doi: 10.3934/dcdsb.2013.18.2705. |
[27] |
Y. S. Tao and Z. A. Wang,
Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[28] |
Y. S. Tao and M. Winkler,
Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.
doi: 10.1137/15M1014115. |
[29] |
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. |
[30] |
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. |
show all references
References:
[1] |
N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler,
Towards 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. |
[2] |
J. P. Bourguignon and H. Brezis,
Remarks on Euler Equation, J. Functional Analysis, 15 (1974), 341-363.
doi: 10.1016/0022-1236(74)90027-5. |
[3] |
J. A. Carrillo, A. Juöngle, P. A. Markowich, G. Toscani and A. Unterreiter,
Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82.
doi: 10.1007/s006050170032. |
[4] |
T. Cieślak, Ph. Laurenc |
[5] |
E. Espejo and T. Suzuki,
Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34.
doi: 10.1016/j.aml.2014.04.007. |
[6] |
D. Horstemann,
From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I. Jahresber. Deutsch. Math. Verien., 105 (2003), 103-165.
|
[7] |
D. Horstmann and G. Wang,
Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[8] |
H. Y. 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. |
[9] |
H. Y. Jin and Z. Liu,
Large time behavior of the full attraction-repulsion Keller-Segel system in the whole space, Appl. Math. Lett., 47 (2015), 13-20.
doi: 10.1016/j.aml.2015.03.004. |
[10] |
H. Y. Jin and Z. A. 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. |
[11] |
H. Y. Jin and Z. A. Wang,
Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 260 (2016), 162-196.
doi: 10.1016/j.jde.2015.08.040. |
[12] |
R. Kowalczyk and Z. Szyma$\acute{n}$ska,
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. |
[13] |
O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, American Mathematical Society, Providence, R.I., 1968. |
[14] |
Y. Li and Y. X. Li,
Blow-up of nonradial solutions to attraction-repulsion chemotaxis system in two dimensions, Nonlinear Anal. Real Word Appl., 30 (2016), 170-183.
doi: 10.1016/j.nonrwa.2015.12.003. |
[15] |
K. Lin and C. Mu,
Global existence and convergence to steady states for an attraction-repulsion chemotaxis system, Nonlinear Anal. Real Word Appl., 31 (2016), 630-642.
doi: 10.1016/j.nonrwa.2016.03.012. |
[16] |
K. Lin, C. Mu and L. Wang,
Large-time behavior of an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124.
doi: 10.1016/j.jmaa.2014.12.052. |
[17] |
K. Lin, C. Mu and D. Zhou,
Stabilization in a higher-dimensional attraction-repulsion chemotaxis system if repulsion dominates over attraction, Math. Models Methods Appl. Sci., 28 (2018), 1105-1134.
doi: 10.1142/S021820251850029X. |
[18] |
D. Liu and Y. S. Tao,
Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546.
doi: 10.1002/mma.3240. |
[19] |
P. Liu, J. P. Shi and Z. A. Wang,
Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.
doi: 10.3934/dcdsb.2013.18.2597. |
[20] |
J. Liu and Z. A. Wang,
Classical solutions and steady states of an attraction-repulsion chemotaxis model in one dimension, J. Biol. Dyn., 6 (2012), 31-41.
doi: 10.1080/17513758.2011.571722. |
[21] |
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, Microglia, and Alzheimer's disease senile plagues: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730. Google Scholar |
[22] |
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.
|
[23] |
K. J. Painter and T. Hillen,
Volume-filling quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.
|
[24] |
M. M. Porzio and V. Vespri,
Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.
doi: 10.1006/jdeq.1993.1045. |
[25] |
R. Shi and W. Wang,
Well-posedness for a model derived from an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 423 (2015), 497-520.
doi: 10.1016/j.jmaa.2014.10.006. |
[26] |
Y. S. Tao,
Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2705-2722.
doi: 10.3934/dcdsb.2013.18.2705. |
[27] |
Y. S. Tao and Z. A. Wang,
Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[28] |
Y. S. Tao and M. Winkler,
Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.
doi: 10.1137/15M1014115. |
[29] |
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. |
[30] |
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. |
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