Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces

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January 2011, 10(1): 287-308. doi: 10.3934/cpaa.2011.10.287

Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces

Zhichun Zhai ^1,

1.	Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Received January 2010 Revised May 2010 Published November 2010

Abstract
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In this paper, we study two types of generalized Keller-Segel system of chemotaxis. We establish the global existence and uniqueness of solutions to the semilinear Keller-Segel system of doubly parabolic type and the nonlinear nonlocal type Keller-Segel system with data in Besov spaces. Moreover, we prove the stability of solution to the first type. Our main tools are the $L^p-L^q$ estimates for $e^{-t(-\triangle)^{\theta/2}}$ in Besov spaces and the perturbation of linearization.

Keywords: Besov spaces., well-posedness, Keller-Segel system, Chemotaxis.

Mathematics Subject Classification: Primary: 35K55, 35K15; Secondary: 92C17, 92B0.

Citation: Zhichun Zhai. Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces. Communications on Pure and Applied Analysis, 2011, 10 (1) : 287-308. doi: 10.3934/cpaa.2011.10.287

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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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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