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Asymptotic stability of a decaying solution to the Keller-Segel system of degenerate type. (English) Zbl 1224.35229

Summary: We discuss the global behavior of the weak solution of the Keller-Segel system of degenerate type. Asymptotic stability of the Barenblatt-Pattle solution and its convergence rate for the decaying weak solution in \(L^1(\mathbb {R}^{n})\) is shown for the degenerated case \(1 <\alpha <2-2/n\). The method is based on the techniques applied to the Fokker-Planck equation due to J. A. Carrillo and G. Toscani [Indiana Univ. Math. J. 49, No. 1, 113–142 (2000; Zbl 0963.35098)] deriving from the explicit time decay of the free energy functional and some new estimates for the nonlinear interaction involving the critical type Sobolev inequality. We give the rigorous justification of those procedures via some approximating procedures.

MSC:

35K65 Degenerate parabolic equations
35M31 Initial value problems for mixed-type systems of PDEs
35B35 Stability in context of PDEs
35K59 Quasilinear parabolic equations

Citations:

Zbl 0963.35098
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