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

1. 

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

Received  January 2010 Revised  May 2010 Published  November 2010

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.
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.

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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