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Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems

Published online by Cambridge University Press:  22 July 2006

Stephan Luckhaus  and
Yoshie Sugiyama
Stephan Luckhaus
Affiliation:
Departement of mathematics and computer science, Universität Leipzig, Leipzig, 04109, Germany. luckhaus@mis.mpg.de
Yoshie Sugiyama
Affiliation:
Department of Mathematics and Computer Science, Tsuda College, 2-1-1, Tsuda-chou, Kodaira-shi, Tokyo, 187-8577, Japan. sugiyama@tsuda.ac.jp
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Abstract

We consider the following reaction-diffusion equation:

$$ {\rm (KS)}\left\{\begin{array}{llll}u_t=\nabla \cdot \Big( \nabla u^m - u^{q-1} \nabla v \Big),& x \in \mathbb{R}^N, \ 0<t<\infty, \nonumber0 = \Delta v - v + u, & x \in \mathbb{R}^N, \ 0<t<\infty, \nonumberu(x,0) = u_0(x), & x \in \mathbb{R}^N,\end{array}\right.$$

where $N \ge 1, \ m > 1, \ q \ge \max\{m+\frac{2}{N},2\}$ .
In [Sugiyama, Nonlinear Anal.63 (2005) 1051–1062; Submitted; J. Differential Equations (in press)]it was shown that in the case of $q \ge \max\{m+\frac{2}{N},2\}$ , the above problem (KS) is solvable globally in time for “small $L^{\frac{N(q-m)}{2}}$ data”.Moreover, the decay of the solution (u,v) in $L^p(\mathbb{R}^N)$ was proved.In this paper, we consider the case of “ $q \ge \max\{m+\frac{2}{N},2\}$ and small $L^{\ell}$ data” with any fixed $\ell \ge \frac{N(q-m)}{2}$ and show that (i) there exists a time global solution (u,v) of (KS) andit decays to 0 as t tends to ∞ and(ii) a solution u of the first equation in (KS)behaveslike the Barenblatt solution asymptotically as t tends to ∞,where the Barenblatt solution is the exact solution (with self-similarity) of the porous medium equation $u_t = \Delta u^m$ with m>1.

Keywords

Degenerate parabolic systemchemotaxisKeller-Segel modeldrift termdecay propertyasymptotic behaviorFujita exponentporous medium equationBarenblatt solution.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 3 , May 2006 , pp. 597 - 621
Copyright
© EDP Sciences, SMAI, 2006

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References

Alikakos, N.D., Lp bounds of solutions of reaction-diffusion equations. Comm. Partial Diff. Equ. 4 (1979) 827–868. CrossRef
Barenblatt, G.I., On some unsteady motions of a fluid and a gas in a porous medium. Prikl. Mat. Mekh. 16 (1952) 67–78.
P.H. Bénilan, Opérateurs accrétifs et semi-groupes dans les espaces Lp (1 ≤ p ≤ ∞). France-Japan Seminar, Tokyo (1976).
Biler, P., Nadzieja, T. and Stanczy, R., Nonisothermal systems of self-attracting Fermi-Dirac particles. Banach Center Pulb. 66 (2004) 61–78. CrossRef
Biler, P., Cannone, M., Guerra, I.A. and Karch, G., Global regular and singular solutions for a model of gravitating particles. Math. Ann. 330 (2004) 693–708. CrossRef
H. Brezis, Analyse fonctionnelle, Theorie et applications. Masson (1983).
Childress, S. and Percus, J.K., Nonlinear aspects of chemotaxis. Math. Biosci. 56 (1981) 217–237. CrossRef
Diaz, J.I., Nagai, T. and Rakotoson, J.M., Symmetrization techniques on unbounded domains: Application to a chemotaxis system on $\mathbb{R}^N$ . J. Diffierential Equations 145 (1998) 156–183. CrossRef
J. Duoandikoetxea, Fourier Analysis, Graduate studies Mathematics 29 AMS, Providence, Rhode Island (2000).
Friedman, A. and Kamin, S., The asymptotic behavior of gas in an N-dimensional porous medium. Trans. Amer. Math. Soc. 262 (1980) 551–563.
Fujita, H., On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha }$ . J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966) 109–124.
Galaktionov, V.A., Blow-up for quasilinear heat equations with critical Fujita's exponents. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994) 517–525. CrossRef
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.
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin-New York (1983).
Herrero, M.A. and Juan J.L. Velázquez, Chemotactic collapse for the Keller-Segel model. J. Math. Biol. 35 (1996) 177–194. CrossRef
Herrero, M.A. and Juan J.L. Velázquez, Singularity patterns in a chemotaxis model. Math. Ann. 306 (1996) 583–623. CrossRef
Horstmann, D., From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math.-Verein. 105 (2003) 103–165.
Horstmann, D., From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. II. Jahresber. Deutsch. Math.-Verein. 106 (2004) 51–69.
Jäger, W. and Luckhaus, S., On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc. 329 (1992) 819–824. CrossRef
Kamin, S., Similar solutions and the asymptotics of filtration equations. Arch. Rational Mech. Anal. 60 (1976) 171–183.
Kamin, S. and Vazquez, J.L., Fundamental solutions and asymptotic behaviour for the p-Laplacian equation. Rev. Mat. Iberoamericana 4 (1988) 339–354. CrossRef
Kawanago, T., Existence and behavior of solutions for ut = Δ(um) + u l . Adv. Math. Sci. Appl. 7 (1997) 367–400.
Keller, E.F. and Segel, L.A., Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970) 399–415. CrossRef
Mochizuki, K. and Suzuki, R., Critical exponent and critical blow-up for quasilinear parabolic equations. Israel J. Math. 98 (1997) 141–156. CrossRef
Nagai, T., Senba, T. and Yoshida, K., Application of the Moser-Trudinger inequality to a parabolic system of chemotaxis. Funkc. Ekvacioj 40 (1997) 411–433.
Nagai, T., Syukuinn, R. and Umesako, M., Decay Properties and Asymptotic Profiles of Bounded Solutions to a Parabolic System of Chemotaxis in $\mathbb{R}^N$ . Funkc. Ekvacioj 46 (2003) 383–407. CrossRef
M. Nakao, Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations. Nonlinear Analysis, Theory, Method Appl. 10 (1986) 299–314.
Senba, T. and Suzuki, T., Local and norm behavior of blowup solutions to a parabolic system of chemotaxis. J. Korean Math. Soc. 37 (2000) 929–941.
E.M. Stein, Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. (1970).
Sugiyama, Y., Global existence and decay properties of solutions for some degenerate quasilinear parabolic systems modelling chemotaxis. Nonlinear Anal. 63 (2005) 1051–1062. CrossRef
Y. Sugiyama, Time global existence and asymptotic behavior of solutions to degenerate quasi-linear parabolic systems for chemotaxis-growth models, (submitted).
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).
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).
Vazquez, J.L., Asymptotic behaviour for the porous medium equation posed in the whole space. J. Evol. Equ. 3 (2003) 67–118. CrossRef
Véron, L., Coercivité et propriétés régularisantes des semi-groupes nonlinéaires dans les espaces de Banach. Ann. Fac. Sci. Toulouse 1 (1979) 171–200. CrossRef