## Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems.(English)Zbl 1113.35028

Global existence and large time asymptotics of non-negative and integrable solutions are studied for the parabolic-elliptic system
\begin{align*}{ \partial_t u & = \nabla\cdot \left( \nabla u^m - u^{q-1} \nabla v \right) \;\;\text{ in }\;\; (0,\infty)\times {\Bbb R}^N\,, \cr 0 & = \Delta v - v + u \;\;\text{ in }\;\; (0,\infty)\times{\Bbb R}^N\,. }\end{align*}
This system is an extended version of the simplified Keller-Segel model for chemotaxis which corresponds to the parameters $$m=1$$ and $$q=2$$. Assuming that $$m>1$$, $$q\geq 2$$, and $$q>m+(2/N)$$, the global existence of a solution $$(u,v)$$ is established provided the $$L^{N(q-m)/2}$$-norm of the initial condition is sufficiently small. The solution $$(u,v)$$ thus obtained decays in $$L^p({\mathbb R}^N)$$ at the same rate as the solution $$w$$ to the porous medium equation $$\partial_t w - \Delta w^m=0$$ in $$(0,\infty)\times{\mathbb R}^N$$ for $$p\in [1,\infty)$$. In addition, the drift term becomes negligible for large times and $$u$$ behaves as the solution $$w$$ to $$\partial_t w - \Delta w^m=0$$ in $$(0,\infty)\times {\mathbb R}^N$$. However, the estimates of the distance between $$u$$ and $$w$$ are not optimal and have been subsequently improved by the same authors [S. Luckhaus and Y. Sugiyama, “Asymptotic profile with the optimal convergence rate for a parabolic equation of chemotaxis in super-critical cases”, to appear in Indiana Univ. J. Math.].

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35K45 Initial value problems for second-order parabolic systems 35K57 Reaction-diffusion equations 35K65 Degenerate parabolic equations 92C17 Cell movement (chemotaxis, etc.)
Full Text:

### References:

 [1] N.D. Alikakos, Lp bounds of solutions of reaction-diffusion equations. Comm. Partial Diff. Equ.4 (1979) 827-868. Zbl0421.35009 · Zbl 0421.35009 · doi:10.1080/03605307908820113 [2] G.I. Barenblatt, On some unsteady motions of a fluid and a gas in a porous medium. Prikl. Mat. Mekh.16 (1952) 67-78. · Zbl 0049.41902 [3] P.H. Bénilan, Opérateurs accrétifs et semi-groupes dans les espaces Lp (1 \leq p \leq \infty ). France-Japan Seminar, Tokyo (1976). [4] P. Biler, T. Nadzieja and R. Stanczy, Nonisothermal systems of self-attracting Fermi-Dirac particles. Banach Center Pulb.66 (2004) 61-78. Zbl1146.35415 · Zbl 1146.35415 [5] P. Biler, M. Cannone, I.A. Guerra and G. Karch, Global regular and singular solutions for a model of gravitating particles. Math. Ann.330 (2004) 693-708. Zbl1078.35097 · Zbl 1078.35097 · doi:10.1007/s00208-004-0565-7 [6] H. Brezis, Analyse fonctionnelle, Theorie et applications. Masson (1983). Zbl0511.46001 · Zbl 0511.46001 [7] S. Childress and J.K. Percus, Nonlinear aspects of chemotaxis. Math. Biosci.56 (1981) 217-237. · Zbl 0481.92010 · doi:10.1016/0025-5564(81)90055-9 [8] J.I. Diaz, T. Nagai and J.M. Rakotoson, Symmetrization techniques on unbounded domains: Application to a chemotaxis system on \Bbb R N . J. Diffierential Equations145 (1998) 156-183. Zbl0908.35016 · Zbl 0908.35016 · doi:10.1006/jdeq.1997.3389 [9] J. Duoandikoetxea, Fourier Analysis, Graduate studies Mathematics29 AMS, Providence, Rhode Island (2000). · Zbl 0969.42001 [10] A. Friedman and S. Kamin, The asymptotic behavior of gas in an N-dimensional porous medium. Trans. Amer. Math. Soc.262 (1980) 551-563. Zbl0447.76076 · Zbl 0447.76076 · doi:10.2307/1999846 [11] H. Fujita, On the blowing up of solutions of the Cauchy problem for u t = \Delta u + u 1 + \alpha . J. Fac. Sci. Univ. Tokyo Sect. I13 (1966) 109-124. · Zbl 0163.34002 [12] V.A. Galaktionov, Blow-up for quasilinear heat equations with critical Fujita’s exponents. Proc. Roy. Soc. Edinburgh Sect. A124 (1994) 517-525. · Zbl 0808.35053 · doi:10.1017/S0308210500028766 [13] V.A. Galaktionov and S.P. Kurdyumov, A.P. Mikhailov and A.A. Samarskiin, On unbounded solutions of the Cauchy problem for a parabolic equation u t = \nabla \cdot ( u \sigma \nabla u ) + u \beta . Sov. Phys., Dokl.25 (1980) 458-459. · Zbl 0515.35045 [14] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin-New York (1983). Zbl0599.35040 · Zbl 0599.35040 [15] M.A. Herrero and Juan J.L. Velázquez, Chemotactic collapse for the Keller-Segel model. J. Math. Biol.35 (1996) 177-194. Zbl0866.92009 · Zbl 0866.92009 · doi:10.1007/s002850050049 [16] M.A. Herrero and Juan J.L. Velázquez, Singularity patterns in a chemotaxis model. Math. Ann.306 (1996) 583-623. Zbl0864.35008 · Zbl 0864.35008 · doi:10.1007/BF01445268 [17] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math.-Verein.105 (2003) 103-165. · Zbl 1071.35001 [18] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. II. Jahresber. Deutsch. Math.-Verein.106 (2004) 51-69. · Zbl 1072.35007 [19] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc.329 (1992) 819-824. · Zbl 0746.35002 · doi:10.2307/2153966 [20] S. Kamin, Similar solutions and the asymptotics of filtration equations. Arch. Rational Mech. Anal.60 (1976) 171-183. · Zbl 0336.76036 [21] S. Kamin and J.L. Vazquez, Fundamental solutions and asymptotic behaviour for the p-Laplacian equation. Rev. Mat. Iberoamericana4 (1988) 339-354. · Zbl 0699.35158 · doi:10.4171/RMI/77 [22] T. Kawanago, Existence and behavior of solutions for ut = \Delta (um) + u l. Adv. Math. Sci. Appl.7 (1997) 367-400. · Zbl 0876.35061 [23] E.F. Keller and L.A. Segel, Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol.26 (1970) 399-415. · Zbl 1170.92306 · doi:10.1016/0022-5193(70)90092-5 [24] K. Mochizuki and R. Suzuki, Critical exponent and critical blow-up for quasilinear parabolic equations. Israel J. Math.98 (1997) 141-156. Zbl0880.35057 · Zbl 0880.35057 · doi:10.1007/BF02937331 [25] T. Nagai, T. Senba and K. Yoshida, Application of the Moser-Trudinger inequality to a parabolic system of chemotaxis. Funkc. Ekvacioj40 (1997) 411-433. Zbl0901.35104 · Zbl 0901.35104 [26] T. Nagai, R. Syukuinn and M. Umesako, Decay Properties and Asymptotic Profiles of Bounded Solutions to a Parabolic System of Chemotaxis in \Bbb R N . Funkc. Ekvacioj46 (2003) 383-407. · Zbl 1330.35476 [27] M. Nakao, Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations. Nonlinear Analysis, Theory, Method Appl.10 (1986) 299-314. · Zbl 0595.35058 · doi:10.1016/0362-546X(86)90005-2 [28] T. Senba and T. Suzuki, Local and norm behavior of blowup solutions to a parabolic system of chemotaxis. J. Korean Math. Soc.37 (2000) 929-941. · Zbl 0967.35011 [29] E.M. Stein, Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. (1970). Zbl0207.13501 · Zbl 0207.13501 [30] Y. Sugiyama, Global existence and decay properties of solutions for some degenerate quasilinear parabolic systems modelling chemotaxis. Nonlinear Anal.63 (2005) 1051-1062. Zbl1224.35207 · Zbl 1224.35207 · doi:10.1016/j.na.2005.03.020 [31] Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems for chemotaxis-growth models, (submitted). Zbl1212.35241 · Zbl 1212.35241 [32] 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 (in press). Zbl1102.35046 · Zbl 1102.35046 · doi:10.1016/j.jde.2006.03.003 [33] Y. Sugiyama, Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems. Diff. Integral Equations, (to appear). · Zbl 1212.35240 [34] J.L. Vazquez, Asymptotic behaviour for the porous medium equation posed in the whole space. J. Evol. Equ.3 (2003) 67-118. Zbl1036.35108 · Zbl 1036.35108 · doi:10.1007/s000280300004 [35] L. Véron, Coercivité et propriétés régularisantes des semi-groupes nonlinéaires dans les espaces de Banach. Ann. Fac. Sci. Toulouse1 (1979) 171-200. · Zbl 0426.35052 · doi:10.5802/afst.535
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.