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

Stephan Luckhaus, Yoshie Sugiyama

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23 Citations (Scopus)

Abstract

We consider the following reaction-diffusion equation: (KS){ ut = Δ · (Δum - uq-1 Δv), x ∈ ℝN, 0 < t < ∞, 0 = Δv - v + u, x ∈ ℝ, 0 < t < ∞, (u(x,0}=u0(x), x ∈ ℝN, where N ≥ 1, m > 1, q ≥ max{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 q ≥ max{m + 2/N, 2}, the above problem (KS) is solvable globally in time for "small LN(q-m)/2 data". Moreover, the decay of the solution (u, v) in Lp(ℝN) was proved. In this paper, we consider the case of "q ≥ max{m+ 2/N, 2} and small L data" with any fixed ℓ ≥ 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 ut = Δum with m > 1.

Original languageEnglish
Pages (from-to)597-621
Number of pages25
JournalMathematical Modelling and Numerical Analysis
Volume40
Issue number3
DOIs
Publication statusPublished - Jul 28 2006

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All Science Journal Classification (ASJC) codes

  • Analysis
  • Numerical Analysis
  • Modelling and Simulation
  • Computational Mathematics
  • Applied Mathematics

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