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Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation
Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source
1. | Department of Basic Science, Jilin Jianzhu University, Changchun 130118, China |
2. | School of Information, Renmin University of China, Beijing, 100872, China |
3. | School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China |
$ \left\{\begin{array}{ll} u_t = \Delta u-\chi\nabla\cdot(u\nabla v)- \xi\nabla\cdot(u\nabla w)+u(a-\mu u^{r-1}-w), \\ {v_t = \Delta v- v +u}, \quad \\ {w_t = - vw}, \quad\\ {\frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = 0}, \quad x\in \partial\Omega, t>0, \\ {u(x, 0) = u_0(x)}, v(x, 0) = v_0(x), w(x, 0) = w_0(x), \quad x\in \Omega, \ \end{array}\right. ~~~~~~~~~~~~~~~~~(0.1)$ |
$ \mathbb{R}^N(N\geq1) $ |
$ r>1 $ |
$ a\in \mathbb{R}, \mu>0, \chi>0 $ |
$ r>2 $ |
$ \begin{equation*} \mu>\mu^{*} = \begin{array}{ll} \frac{(N-2)_{+}}{N}(\chi+C_{\beta}) C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1}, if r = 2, \ \end{array} \end{equation*} $ |
$ C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1} $ |
$ C_{\beta} $ |
$ \xi $ |
$ \|u_0\|_{C(\bar{\Omega})}, \ \|v_0\|_{W^{1, \infty}(\Omega)} $ |
$ \|w_0\|_{L^\infty(\Omega)} $ |
References:
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N. Bellomo, A. Belloquid, 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. |
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X. Cao, Boundedness in a three-dimensional chemotaxis–haptotaxis model, Z. Angew. Math. Phys., 67 (2015), Art. 11, 13 pp.
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M. A. J. Chaplain and G. Lolas,
Mathematical modelling of cancer invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 11 (2005), 1685-1734.
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M. A. J. Chaplain and G. Lolas,
Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med., 1 (2006), 399-439.
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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. Diff. Eqns., 252 (2012), 5832-5851.
|
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A use's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
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T. Hillen, K. J. Painter and M. Winkler,
Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Methods Appl. Sci., 23 (2013), 165-198.
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From 1970 until present: the Keller–Segel model in chemotaxis and its consequences, I. Jahresberichte der Deutschen Mathematiker-Vereinigung, 105 (2003), 103-165.
|
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D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Diff. Eqns, 215 (2005), 52-107.
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S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains, J. Diff. Eqns., 256 (2014), 2993-3010.
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On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Am. Math. Soc., 329 (1992), 819-824.
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E. Keller and L. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
|
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X. Li and Z. Xiang,
Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 3503-3531.
doi: 10.3934/dcds.2015.35.3503. |
[15] |
G. Liţcanu and C. Morales-Rodrigo,
Asymptotic behavior of global solutions to a model of cell invasion, Math. Models Methods Appl. Sci., 20 (2010), 1721-1758.
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J. Liu, J. Zheng and Y. Wang, Boundedness in a quasilinear chemotaxis-haptotaxis system with logistic source, Z. Angew. Math. Phys., 67 (2016), Art. 21, 33 pp.
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A. Marciniak-Czochra and M. Ptashnyk,
Boundedness of solutions of a haptotaxis model, Math. Models Methods Appl. Sci., 20 (2010), 449-476.
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V. Nanjundiah,
Chemotaxis, signal relaying and aggregation morpholog, J. Theor. Biol., 42 (1973), 63-105.
|
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K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis growth system of equations, Nonlinear Anal. TMA., 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
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K. J. Painter and T. Hillen,
Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.
|
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J. Simon,
Compact sets in the space Lp(O, T; B), Annali di Matematica Pura ed Applicata, 146 (1986), 65-96.
doi: 10.1007/BF01762360. |
[22] |
Y. Tao,
Global existence of classical solutions to a combined chemotaxis–haptotaxis model with logistic source, J. Math. Anal. Appl., 354 (2009), 60-69.
doi: 10.1016/j.jmaa.2008.12.039. |
[23] |
Y. Tao,
Boundedness in a two-dimensional chemotaxis–haptotaxis system, Journal of Oceanography, 70 (2014), 165-174.
|
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Y. Tao and M. Wang,
Global solution for a chemotactic–haptotactic model of cancer invasion, Nonlinearity, 21 (2008), 2221-2238.
doi: 10.1088/0951-7715/21/10/002. |
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Y. Tao and M. Wang,
A combined chemotaxis–haptotaxis system: The role of logistic source, SIAM J. Math. Anal., 41 (2009), 1533-1558.
doi: 10.1137/090751542. |
[26] |
Y. Tao and M. Winkler,
A chemotaxis–haptotaxis model: The roles of porous medium diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.
doi: 10.1137/100802943. |
[27] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity, J. Diff. Eqns., 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[28] |
Y. Tao and M. Winkler,
Boundedness and stabilization in a multi-dimensional chemotaxis–haptotaxis model, Proceedings of the Royal Society of Edinburgh, 144 (2014), 1067-1084.
doi: 10.1017/S0308210512000571. |
[29] |
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. |
[30] |
Y. Tao and M. Winkler,
Energy-type estimates and global solvability in a two-dimensional chemotaxis–haptotaxis model with remodeling of non-diffusible attractant, J. Diff. Eqns., 257 (2014), 784-815.
doi: 10.1016/j.jde.2014.04.014. |
[31] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Diff. Eqns., 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
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C. Walker and G. F. Webb,
Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Anal., 38 (2007), 1694-1713.
doi: 10.1137/060655122. |
[33] |
L. Wang, C. Mu and P. Zheng,
On a quasilinear parabolic–elliptic chemotaxis system with logistic source, J. Diff. Eqns., 256 (2014), 1847-1872.
doi: 10.1016/j.jde.2013.12.007. |
[34] |
Y. Wang,
Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Diff. Eqns., 260 (2016), 1975-1989.
doi: 10.1016/j.jde.2015.09.051. |
[35] |
Y. Wang and Y. Ke,
Large time behavior of solution to a fully parabolic chemotaxis-haptotaxis model in higher dimensions, J. Diff. Eqns., 260 (2016), 6960-6988.
doi: 10.1016/j.jde.2016.01.017. |
[36] |
M. Winkler,
Does a volume-filling effect always prevent chemotactic collapse, Math. Methods Appl. Sci., 33 (2010), 12-24.
doi: 10.1002/mma.1146. |
[37] |
M. Winkler,
Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source, Comm. Partial Diff. Eqns., 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[38] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Diff. Eqns., 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[39] |
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. |
[40] |
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. |
[41] |
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. |
[42] |
M. Winkler, Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), Art. 69, 40 pp.
doi: 10.1007/s00033-018-0935-8. |
[43] |
M. Winkler and K. C. Djie,
Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. TMA., 72 (2010), 1044-1064.
doi: 10.1016/j.na.2009.07.045. |
[44] |
T. Xiang,
Boundedness and global existence in the higher-dimensional parabolic–parabolic chemotaxis system with/without growth source, J. Diff. Eqns., 258 (2015), 4275-4323.
doi: 10.1016/j.jde.2015.01.032. |
[45] |
Q. Zhang and Y. Li,
Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.
doi: 10.1007/s00033-013-0383-4. |
[46] |
J. Zheng,
Optimal controls of multi-dimensional modified Swift-Hohenberg equation, International Journal of Control, 88 (2015), 2117-2125.
doi: 10.1080/00207179.2015.1038587. |
[47] |
J. Zheng,
Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source, J. Diff. Eqns., 259 (2015), 120-140.
doi: 10.1016/j.jde.2015.02.003. |
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J. Zheng,
Boundedness of solutions to a quasilinear parabolic–parabolic Keller–Segel system with logistic source, J. Math. Anal. Appl., 431 (2015), 867-888.
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Boundedness and global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with nonlinear a logistic source, J. Math. Anal. Appl., 450 (2017), 1047-1061.
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J. Zheng,
Boundedness in a two-species quasi-linear chemotaxis system with two chemicals, Topological Methods in Nonlinear Analysis, 49 (2017), 463-480.
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J. Zheng,
Boundedness of solution of a higher-dimensional parabolic–ODE–parabolic chemotaxis–haptotaxis model with generalized logistic source, Nonlinearity, 30 (2017), 1987-2009.
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J. Zheng,
Boundedness of solutions to a quasilinear higher-dimensional chemotaxis–haptotaxis model with nonlinear diffusion, Discrete and Continuous Dynamical Systems, 37 (2017), 627-643.
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J. Zheng,
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J. Zheng, A new result for global existence and boundedness of solutions to a parabolic–parabolic Keller–Segel system with logistic source, J. Math. Anal. Appl., 431 (2015), 867–888, arXiv: 1712.00906, 2017.
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J. Zheng and Y. Wang,
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show all references
References:
[1] |
N. Bellomo, A. Belloquid, 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. |
[2] |
X. Cao, Boundedness in a three-dimensional chemotaxis–haptotaxis model, Z. Angew. Math. Phys., 67 (2015), Art. 11, 13 pp.
doi: 10.1007/s00033-015-0601-3. |
[3] |
M. A. J. Chaplain and G. Lolas,
Mathematical modelling of cancer invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 11 (2005), 1685-1734.
doi: 10.1142/S0218202505000947. |
[4] |
M. A. J. Chaplain and G. Lolas,
Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med., 1 (2006), 399-439.
doi: 10.3934/nhm.2006.1.399. |
[5] |
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. Diff. Eqns., 252 (2012), 5832-5851.
|
[6] |
H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo–Nirenberg inequalities and applications to Navier–Stokes and generalized boson equations, in: Harmonic Analysis and Nonlinear Partial Differential Equations, in: RIMS Kôkyûroku Bessatsu, Res. Inst. Math. Sci. (RIMS), Kyoto, 26 (2011), 159–175. |
[7] |
T. Hillen and K. J. Painter,
A use's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[8] |
T. Hillen, K. J. Painter and M. Winkler,
Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Methods Appl. Sci., 23 (2013), 165-198.
doi: 10.1142/S0218202512500480. |
[9] |
D. Horstmann,
From 1970 until present: the Keller–Segel model in chemotaxis and its consequences, I. Jahresberichte der Deutschen Mathematiker-Vereinigung, 105 (2003), 103-165.
|
[10] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Diff. Eqns, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[11] |
S. Ishida, K. Seki and T. Yokota,
Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains, J. Diff. Eqns., 256 (2014), 2993-3010.
doi: 10.1016/j.jde.2014.01.028. |
[12] |
W. Jäger and S. Luckhaus,
On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Am. Math. Soc., 329 (1992), 819-824.
doi: 10.1090/S0002-9947-1992-1046835-6. |
[13] |
E. Keller and L. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
|
[14] |
X. Li and Z. Xiang,
Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 3503-3531.
doi: 10.3934/dcds.2015.35.3503. |
[15] |
G. Liţcanu and C. Morales-Rodrigo,
Asymptotic behavior of global solutions to a model of cell invasion, Math. Models Methods Appl. Sci., 20 (2010), 1721-1758.
doi: 10.1142/S0218202510004775. |
[16] |
J. Liu, J. Zheng and Y. Wang, Boundedness in a quasilinear chemotaxis-haptotaxis system with logistic source, Z. Angew. Math. Phys., 67 (2016), Art. 21, 33 pp.
doi: 10.1007/s00033-016-0620-8. |
[17] |
A. Marciniak-Czochra and M. Ptashnyk,
Boundedness of solutions of a haptotaxis model, Math. Models Methods Appl. Sci., 20 (2010), 449-476.
doi: 10.1142/S0218202510004301. |
[18] |
V. Nanjundiah,
Chemotaxis, signal relaying and aggregation morpholog, J. Theor. Biol., 42 (1973), 63-105.
|
[19] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis growth system of equations, Nonlinear Anal. TMA., 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[20] |
K. J. Painter and T. Hillen,
Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.
|
[21] |
J. Simon,
Compact sets in the space Lp(O, T; B), Annali di Matematica Pura ed Applicata, 146 (1986), 65-96.
doi: 10.1007/BF01762360. |
[22] |
Y. Tao,
Global existence of classical solutions to a combined chemotaxis–haptotaxis model with logistic source, J. Math. Anal. Appl., 354 (2009), 60-69.
doi: 10.1016/j.jmaa.2008.12.039. |
[23] |
Y. Tao,
Boundedness in a two-dimensional chemotaxis–haptotaxis system, Journal of Oceanography, 70 (2014), 165-174.
|
[24] |
Y. Tao and M. Wang,
Global solution for a chemotactic–haptotactic model of cancer invasion, Nonlinearity, 21 (2008), 2221-2238.
doi: 10.1088/0951-7715/21/10/002. |
[25] |
Y. Tao and M. Wang,
A combined chemotaxis–haptotaxis system: The role of logistic source, SIAM J. Math. Anal., 41 (2009), 1533-1558.
doi: 10.1137/090751542. |
[26] |
Y. Tao and M. Winkler,
A chemotaxis–haptotaxis model: The roles of porous medium diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.
doi: 10.1137/100802943. |
[27] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity, J. Diff. Eqns., 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[28] |
Y. Tao and M. Winkler,
Boundedness and stabilization in a multi-dimensional chemotaxis–haptotaxis model, Proceedings of the Royal Society of Edinburgh, 144 (2014), 1067-1084.
doi: 10.1017/S0308210512000571. |
[29] |
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. |
[30] |
Y. Tao and M. Winkler,
Energy-type estimates and global solvability in a two-dimensional chemotaxis–haptotaxis model with remodeling of non-diffusible attractant, J. Diff. Eqns., 257 (2014), 784-815.
doi: 10.1016/j.jde.2014.04.014. |
[31] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Diff. Eqns., 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[32] |
C. Walker and G. F. Webb,
Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Anal., 38 (2007), 1694-1713.
doi: 10.1137/060655122. |
[33] |
L. Wang, C. Mu and P. Zheng,
On a quasilinear parabolic–elliptic chemotaxis system with logistic source, J. Diff. Eqns., 256 (2014), 1847-1872.
doi: 10.1016/j.jde.2013.12.007. |
[34] |
Y. Wang,
Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion, J. Diff. Eqns., 260 (2016), 1975-1989.
doi: 10.1016/j.jde.2015.09.051. |
[35] |
Y. Wang and Y. Ke,
Large time behavior of solution to a fully parabolic chemotaxis-haptotaxis model in higher dimensions, J. Diff. Eqns., 260 (2016), 6960-6988.
doi: 10.1016/j.jde.2016.01.017. |
[36] |
M. Winkler,
Does a volume-filling effect always prevent chemotactic collapse, Math. Methods Appl. Sci., 33 (2010), 12-24.
doi: 10.1002/mma.1146. |
[37] |
M. Winkler,
Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source, Comm. Partial Diff. Eqns., 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[38] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Diff. Eqns., 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[39] |
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. |
[40] |
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. |
[41] |
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. |
[42] |
M. Winkler, Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), Art. 69, 40 pp.
doi: 10.1007/s00033-018-0935-8. |
[43] |
M. Winkler and K. C. Djie,
Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. TMA., 72 (2010), 1044-1064.
doi: 10.1016/j.na.2009.07.045. |
[44] |
T. Xiang,
Boundedness and global existence in the higher-dimensional parabolic–parabolic chemotaxis system with/without growth source, J. Diff. Eqns., 258 (2015), 4275-4323.
doi: 10.1016/j.jde.2015.01.032. |
[45] |
Q. Zhang and Y. Li,
Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.
doi: 10.1007/s00033-013-0383-4. |
[46] |
J. Zheng,
Optimal controls of multi-dimensional modified Swift-Hohenberg equation, International Journal of Control, 88 (2015), 2117-2125.
doi: 10.1080/00207179.2015.1038587. |
[47] |
J. Zheng,
Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source, J. Diff. Eqns., 259 (2015), 120-140.
doi: 10.1016/j.jde.2015.02.003. |
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