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Optimality conditions for a controlled sweeping process with applications to the crowd motion model
Uniform $L^{∞}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model
1. | School of Mathematics, Jilin University, Changchun 130012, China |
2. | Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708, USA |
This paper investigates the existence of a uniform in time $L^{∞}$ bounded weak entropy solution for the quasilinear parabolic-parabolic Keller-Segel model with the supercritical diffusion exponent $0<m<2-\frac{2}{d}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(2-m)}{2}}$ norm of initial data is smaller than a universal constant. Moreover, the weak entropy solution $u(x,t)$ satisfies mass conservation when $m>1-\frac{2}{d}$. We also prove the local existence of weak entropy solutions and a blow-up criterion for general $L^1\cap L^{∞}$ initial data.
References:
[1] |
S. Bian and J.-G. Liu,
Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $m>0$, Comm. Math. Phys., 323 (2013), 1017-1070.
doi: 10.1007/s00220-013-1777-z. |
[2] |
S. Bian, J.-G. Liu and C. Zou,
Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$, Kinet. Relat. Models, 7 (2014), 9-28.
doi: 10.3934/krm.2014.7.9. |
[3] |
A. Blanchet, J. Carrillo and N. Masmoudi,
Infinite time aggregation for the critical Patlak-Keller-Segel model in ${\mathbb{R}}^2$, Comm. Pure Appl. Math., 61 (2008), 1449-1481.
doi: 10.1002/cpa.20225. |
[4] |
M. Brenner, P. Constantin, L. Kadanoff, A. Schenkel and S. Venkataramani,
Diffusion, attraction and collapse, Nonlinearity, 12 (1999), 1071-1098.
doi: 10.1088/0951-7715/12/4/320. |
[5] |
V. Calvez and L. Corrias,
The parabolic-parabolic Keller-Segel model in ${\mathbb{R}}^2$, Commun. Math. Sci., 6 (2008), 417-447.
doi: 10.4310/CMS.2008.v6.n2.a8. |
[6] |
X. Chen, A. Jüngel and J.-G. Liu,
A note on Aubin-Lions-Dubinskiĭ lemmas, Acta Appl. Math., 133 (2014), 33-43.
doi: 10.1007/s10440-013-9858-8. |
[7] |
L. Corrias and B. Perthame,
Critical space for the parabolic-parabolic Keller-Segel model in ${\mathbb{R}}^d$, C. R. Math. Acad. Sci. Paris, 342 (2006), 745-750.
doi: 10.1016/j.crma.2006.03.008. |
[8] |
L. Evans,
Partial Differential Equations Second edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. |
[9] |
M. A. Herrero, E. Medina and J. J. L. Velázquez,
Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), 1739-1754.
doi: 10.1088/0951-7715/10/6/016. |
[10] |
M. Hieber and J. Prüss,
Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), 1647-1669.
doi: 10.1080/03605309708821314. |
[11] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.
|
[12] |
S. Ishida and T. Yokota,
Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, J. Differential Equations, 252 (2012), 2469-2491.
doi: 10.1016/j.jde.2011.08.047. |
[13] |
S. Ishida and T. Yokota,
Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569-2596.
doi: 10.3934/dcdsb.2013.18.2569. |
[14] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[15] |
P. Kunstmann and L. Weis,
Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^{∞}$-functional calculus, Lecture Notes in Math., 1855 (2004), 65-311.
doi: 10.1007/978-3-540-44653-8_2. |
[16] |
J.-G. Liu and J. Wang,
A note on $L^{∞}$-bound and uniqueness to a degenerate Keller-Segel model, Acta Appl. Math., 142 (2016), 173-188.
doi: 10.1007/s10440-015-0022-5. |
[17] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. |
[18] |
B. Perthame,
Transport Equations in Biology Birkhäuser Verlag, Basel, 2007. |
[19] |
Y. Sugiyama,
Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180.
|
[20] |
Y. Sugiyama and H. Kunii,
Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.
doi: 10.1016/j.jde.2006.03.003. |
[21] |
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] |
S. Bian and J.-G. Liu,
Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $m>0$, Comm. Math. Phys., 323 (2013), 1017-1070.
doi: 10.1007/s00220-013-1777-z. |
[2] |
S. Bian, J.-G. Liu and C. Zou,
Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$, Kinet. Relat. Models, 7 (2014), 9-28.
doi: 10.3934/krm.2014.7.9. |
[3] |
A. Blanchet, J. Carrillo and N. Masmoudi,
Infinite time aggregation for the critical Patlak-Keller-Segel model in ${\mathbb{R}}^2$, Comm. Pure Appl. Math., 61 (2008), 1449-1481.
doi: 10.1002/cpa.20225. |
[4] |
M. Brenner, P. Constantin, L. Kadanoff, A. Schenkel and S. Venkataramani,
Diffusion, attraction and collapse, Nonlinearity, 12 (1999), 1071-1098.
doi: 10.1088/0951-7715/12/4/320. |
[5] |
V. Calvez and L. Corrias,
The parabolic-parabolic Keller-Segel model in ${\mathbb{R}}^2$, Commun. Math. Sci., 6 (2008), 417-447.
doi: 10.4310/CMS.2008.v6.n2.a8. |
[6] |
X. Chen, A. Jüngel and J.-G. Liu,
A note on Aubin-Lions-Dubinskiĭ lemmas, Acta Appl. Math., 133 (2014), 33-43.
doi: 10.1007/s10440-013-9858-8. |
[7] |
L. Corrias and B. Perthame,
Critical space for the parabolic-parabolic Keller-Segel model in ${\mathbb{R}}^d$, C. R. Math. Acad. Sci. Paris, 342 (2006), 745-750.
doi: 10.1016/j.crma.2006.03.008. |
[8] |
L. Evans,
Partial Differential Equations Second edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. |
[9] |
M. A. Herrero, E. Medina and J. J. L. Velázquez,
Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), 1739-1754.
doi: 10.1088/0951-7715/10/6/016. |
[10] |
M. Hieber and J. Prüss,
Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), 1647-1669.
doi: 10.1080/03605309708821314. |
[11] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.
|
[12] |
S. Ishida and T. Yokota,
Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, J. Differential Equations, 252 (2012), 2469-2491.
doi: 10.1016/j.jde.2011.08.047. |
[13] |
S. Ishida and T. Yokota,
Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569-2596.
doi: 10.3934/dcdsb.2013.18.2569. |
[14] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[15] |
P. Kunstmann and L. Weis,
Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^{∞}$-functional calculus, Lecture Notes in Math., 1855 (2004), 65-311.
doi: 10.1007/978-3-540-44653-8_2. |
[16] |
J.-G. Liu and J. Wang,
A note on $L^{∞}$-bound and uniqueness to a degenerate Keller-Segel model, Acta Appl. Math., 142 (2016), 173-188.
doi: 10.1007/s10440-015-0022-5. |
[17] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. |
[18] |
B. Perthame,
Transport Equations in Biology Birkhäuser Verlag, Basel, 2007. |
[19] |
Y. Sugiyama,
Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differential Integral Equations, 20 (2007), 133-180.
|
[20] |
Y. Sugiyama and H. Kunii,
Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.
doi: 10.1016/j.jde.2006.03.003. |
[21] |
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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