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Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials
Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces
1. | Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada |
References:
[1] |
J. Bergh and J. L$\ddoto$fstr$\ddoto$m, "Interpolation Spaces: An Introduction," Springer, Heidelberg, 1976. |
[2] |
P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743. |
[3] |
P. Biler and G. Karch, Blow up of solutions to generalized Keller-Segel model, J. Evol. Equ., 2 (2010), 247-262.
doi: doi:10.1007/s00028-009-0048-0. |
[4] |
P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quatratic evolution problems, SIAM J. APPL. MATH., 59 (1998), 845-869.
doi: doi:10.1137/S0036139996313447. |
[5] |
P. Biler and G. Wu, Two-dimensional chemotaxis models with fractional diffusion, Math. Meth. Appl. Sci., 32 (2009), 112-126.
doi: doi:10.1002/mma.1036. |
[6] |
V. Calvez, B. Perthame and M. Sharifi Tabar, Modified Keller-Segel system and critical mass for the log interaction kernel, Contemporary Math., 429 (2007), 45-62 Stochastic analysis and pde; Chen, Gui-Qiang (ed.) et al., AMS. |
[7] |
L. Corrias and B. Perthame, Critical space for the parabolic-parabolic Keller-Segel model in $\mathbbR^n$, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 745-750. |
[8] |
C. Escudero, The fractional Keller-Segel model, Nonlinearity, 19 (2006), 2909-2918.
doi: doi:10.1088/0951-7715/19/12/010. |
[9] |
D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences, Jahresber. Deutch. Math.-Verein., 105 (2003), 103-165. |
[10] |
E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.
doi: doi:10.1016/0022-5193(71)90050-6. |
[11] |
H. Kozono and Y. Sugiyama, The Keller-Segel system of parabolic-parabolic type with initial data in weak $L^{n/2}(R^n)$ and its application to self-similar solutions, Inidana Univ. Math. J., 57 (2008), 1467-1500.
doi: doi:10.1512/iumj.2008.57.3316. |
[12] |
H. Kozono and Y. Sugiyama, Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces, J. Differ. Equations, 247 (2009), 1-32.
doi: doi:10.1016/j.jde.2009.03.027. |
[13] |
H. Kozono, T. Ogawa and Y. Taniuchi, Navier-Stokes equations in the Besov space near $L^\infty$ and BMO, Kyushu J. Math., 57 (2003), 303-324.
doi: doi:10.2206/kyushujm.57.303. |
[14] |
P. G. Lemarié-Rieusset, "Recent Development in the Navier-Stokes Problem," Chapman & Hall/CRC Press, Boca Raton, 2002.
doi: doi:10.1201/9781420035674. |
[15] |
D. Li, J. Rodrigo and X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem, to appear in Rev.Mat. Iberoam., 1 (2010), 295-332. |
[16] |
Y. Meyer, Wavelets, paraproducts and Navier-Stokes equations, Current developments in mathematics, 1996 (Cambridge, MA), 105-212, Internat. Press, MA, 1997. |
[17] |
C. Miao, B. Yuan and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Anal. TMA, 68 (2008), 461-484.
doi: doi:10.1016/j.na.2006.11.011. |
[18] |
J. D. Murray, "Spatial Models and Biomedical Applications, in: Mathematical Biology," II, Third ed., in: Interdiscip. Appl. Math., vol. 18, Springer- Verlag, New York, 2003. |
[19] |
T. Runst and W. Sickel, "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations," de Gruyter Series in Nonlinear Analysis and Applications, vol. 3 (Berlin: Walter de Gruyter, 1996). |
[20] |
E. M. Stein, "Harmonic Analysis: Real-Varible Methods, Orthogonality, and Oscillatory Integrals," Princeton University Press, Princeton, New Jersey, 1993. |
[21] |
Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with power factor in drift term, J. Differ. Equations, 227 (2006), 333-364.
doi: doi:10.1016/j.jde.2006.03.003. |
[22] |
Y. Sugiyama, Time global existence and asympotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differetial Integral Equations, 20 (2007), 133-180. |
[23] |
H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, 1978. |
[24] |
G. Wu and J. Yuan, Well-posedness of the Cauchy problem for the fractional power dissipative equation in critical Besov spaces, J. Math. Anal. Appl., 340 (2008), 1326-1335.
doi: doi:10.1016/j.jmaa.2007.09.060. |
[25] |
Z. Zhai, Strichartz type estimates for fractional heat equations, J. Math. Anal. Appl., 356 (2009), 642-658. |
[26] |
Z. Zhai, Global well-posedness for nonlocal fractional Keller-Segel systems in critical Besov spaces, Nonlinear Analysis TMA, 72 (2010), 3173-3189.
doi: doi:10.1016/j.na.2009.12.011. |
show all references
References:
[1] |
J. Bergh and J. L$\ddoto$fstr$\ddoto$m, "Interpolation Spaces: An Introduction," Springer, Heidelberg, 1976. |
[2] |
P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743. |
[3] |
P. Biler and G. Karch, Blow up of solutions to generalized Keller-Segel model, J. Evol. Equ., 2 (2010), 247-262.
doi: doi:10.1007/s00028-009-0048-0. |
[4] |
P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quatratic evolution problems, SIAM J. APPL. MATH., 59 (1998), 845-869.
doi: doi:10.1137/S0036139996313447. |
[5] |
P. Biler and G. Wu, Two-dimensional chemotaxis models with fractional diffusion, Math. Meth. Appl. Sci., 32 (2009), 112-126.
doi: doi:10.1002/mma.1036. |
[6] |
V. Calvez, B. Perthame and M. Sharifi Tabar, Modified Keller-Segel system and critical mass for the log interaction kernel, Contemporary Math., 429 (2007), 45-62 Stochastic analysis and pde; Chen, Gui-Qiang (ed.) et al., AMS. |
[7] |
L. Corrias and B. Perthame, Critical space for the parabolic-parabolic Keller-Segel model in $\mathbbR^n$, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 745-750. |
[8] |
C. Escudero, The fractional Keller-Segel model, Nonlinearity, 19 (2006), 2909-2918.
doi: doi:10.1088/0951-7715/19/12/010. |
[9] |
D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences, Jahresber. Deutch. Math.-Verein., 105 (2003), 103-165. |
[10] |
E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.
doi: doi:10.1016/0022-5193(71)90050-6. |
[11] |
H. Kozono and Y. Sugiyama, The Keller-Segel system of parabolic-parabolic type with initial data in weak $L^{n/2}(R^n)$ and its application to self-similar solutions, Inidana Univ. Math. J., 57 (2008), 1467-1500.
doi: doi:10.1512/iumj.2008.57.3316. |
[12] |
H. Kozono and Y. Sugiyama, Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces, J. Differ. Equations, 247 (2009), 1-32.
doi: doi:10.1016/j.jde.2009.03.027. |
[13] |
H. Kozono, T. Ogawa and Y. Taniuchi, Navier-Stokes equations in the Besov space near $L^\infty$ and BMO, Kyushu J. Math., 57 (2003), 303-324.
doi: doi:10.2206/kyushujm.57.303. |
[14] |
P. G. Lemarié-Rieusset, "Recent Development in the Navier-Stokes Problem," Chapman & Hall/CRC Press, Boca Raton, 2002.
doi: doi:10.1201/9781420035674. |
[15] |
D. Li, J. Rodrigo and X. Zhang, Exploding solutions for a nonlocal quadratic evolution problem, to appear in Rev.Mat. Iberoam., 1 (2010), 295-332. |
[16] |
Y. Meyer, Wavelets, paraproducts and Navier-Stokes equations, Current developments in mathematics, 1996 (Cambridge, MA), 105-212, Internat. Press, MA, 1997. |
[17] |
C. Miao, B. Yuan and B. Zhang, Well-posedness of the Cauchy problem for the fractional power dissipative equations, Nonlinear Anal. TMA, 68 (2008), 461-484.
doi: doi:10.1016/j.na.2006.11.011. |
[18] |
J. D. Murray, "Spatial Models and Biomedical Applications, in: Mathematical Biology," II, Third ed., in: Interdiscip. Appl. Math., vol. 18, Springer- Verlag, New York, 2003. |
[19] |
T. Runst and W. Sickel, "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations," de Gruyter Series in Nonlinear Analysis and Applications, vol. 3 (Berlin: Walter de Gruyter, 1996). |
[20] |
E. M. Stein, "Harmonic Analysis: Real-Varible Methods, Orthogonality, and Oscillatory Integrals," Princeton University Press, Princeton, New Jersey, 1993. |
[21] |
Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with power factor in drift term, J. Differ. Equations, 227 (2006), 333-364.
doi: doi:10.1016/j.jde.2006.03.003. |
[22] |
Y. Sugiyama, Time global existence and asympotic behavior of solutions to degenerate quasi-linear parabolic systems of chemotaxis, Differetial Integral Equations, 20 (2007), 133-180. |
[23] |
H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, 1978. |
[24] |
G. Wu and J. Yuan, Well-posedness of the Cauchy problem for the fractional power dissipative equation in critical Besov spaces, J. Math. Anal. Appl., 340 (2008), 1326-1335.
doi: doi:10.1016/j.jmaa.2007.09.060. |
[25] |
Z. Zhai, Strichartz type estimates for fractional heat equations, J. Math. Anal. Appl., 356 (2009), 642-658. |
[26] |
Z. Zhai, Global well-posedness for nonlocal fractional Keller-Segel systems in critical Besov spaces, Nonlinear Analysis TMA, 72 (2010), 3173-3189.
doi: doi:10.1016/j.na.2009.12.011. |
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