Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems
ESAIM: Modélisation mathématique et analyse numérique, Volume 40 (2006) no. 3, pp. 597-621.

We consider the following reaction-diffusion equation:

( KS )u t =· u m - u q-1 v,x N ,0<t<,0=Δv-v+u,x N ,0<t<,u(x,0)=u 0 (x),x N ,
where N1,m>1,qmax{m+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 qmax{m+2 N,2}, the above problem (KS) is solvable globally in time for “small L N(q-m) 2 data”. Moreover, the decay of the solution (u,v) in L p ( N ) was proved. In this paper, we consider the case of “qmax{m+2 N,2} and small L data” with any fixed N(q-m) 2 and show that (i) there exists a time global solution (u,v) of (KS) and it decays to 0 as t tends to and (ii) a solution u of the first equation in (KS) behaves like 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 =Δu m with m>1.

DOI: 10.1051/m2an:2006025
Classification: 35B40, 35K45, 35K55, 35k65
Keywords: degenerate parabolic system, chemotaxis, Keller-Segel model, drift term, decay property, asymptotic behavior, Fujita exponent, porous medium equation, Barenblatt solution
@article{M2AN_2006__40_3_597_0,
     author = {Luckhaus, Stephan and Sugiyama, Yoshie},
     title = {Large time behavior of solutions in super-critical cases to degenerate {Keller-Segel} systems},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {597--621},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {3},
     year = {2006},
     doi = {10.1051/m2an:2006025},
     mrnumber = {2245322},
     zbl = {1113.35028},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an:2006025/}
}
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Luckhaus, Stephan; Sugiyama, Yoshie. Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems. ESAIM: Modélisation mathématique et analyse numérique, Volume 40 (2006) no. 3, pp. 597-621. doi : 10.1051/m2an:2006025. http://archive.numdam.org/articles/10.1051/m2an:2006025/

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