Global existence and decay properties for a degenerate Keller–Segel model with a power factor in drift term

Abstract

The following degenerate parabolic system modelling chemotaxis is considered:where m 1, q 2, τ = 0 or 1, and N 1. The aim of this paper is to prove the existence of a time global weak solution (u, v) of (KS) with the L ∞ (0, ∞; L ∞ (R N )) bound. Such a global bound is obtained in the case of (i) m > q − 2 N for large initial data and (ii) 1 m q − 2 N for small initial data. In the case of (ii), the decay properties of the solution (u, v) are also discussed.

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

“…Therefore, we obtain the the same decay estimate as (4.16) for v ε . Furthermore, by the similar argument to that in Section 5 in [32], we can prove that there exists a subsequence {u εn } such that…”

“…) In the following section, we shall prepare several lemmas which will be used in the sequent sections. In Section 3, we introduce the results obtained in [30][31][32] concerning the existence of a time global strong solution of the approximated problem of (KS). In Section 4, we organize the proof of the decay of a solution (u, v).…”

“…This exponent q = m + 2 N is called the Fujita exponent [11]. For (KS) with N ≥ 1, m > 1, q ≥ 2, in [30][31][32]] the Fujita's exponent was found. Specifically, in [32] it was shown that (i) when q < m + 2 N , the problem (KS) is globally solvable without any restriction on the size of the initial data; and (ii) when m > 1 and…”

“…In [30][31][32], the following proposition concerning the existence of the strong solution was proved:…”

Stability of Stationary Solutions and Existence of Oscillating Solutions to a Chemotaxis System in High Dimensional Spaces

“…If U I A L 1 ðR n Þ V L n=2 ðR n Þ and kU I k L n=2 is su‰ciently small, the solution ðU; V Þ exists globally in time and decays algebraically (see [11,9]). Here, k Á k L p is the standard L p ðR n Þ norm for p A ½1; y .…”