Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 3, p. 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 : https://doi.org/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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {3},
     year = {2006},
     pages = {597-621},
     doi = {10.1051/m2an:2006025},
     zbl = {1113.35028},
     mrnumber = {2245322},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2006__40_3_597_0}
}
Luckhaus, Stephan; Sugiyama, Yoshie. Large time behavior of solutions in super-critical cases to degenerate Keller-Segel systems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 3, pp. 597-621. doi : 10.1051/m2an:2006025. http://www.numdam.org/item/M2AN_2006__40_3_597_0/