Extinction, decay and blow-up for Keller–Segel systems of fast diffusion type

Abstract

We consider the quasi-linear Keller-Segel system of singular type, where the principal part u m represents a fast diffusion like 0 < m < 1. We first construct a global weak solution with small initial data in the scaling invariant norm L N(q−m) 2 for all dimensions N 2 and all exponents q 2. As for the large initial data, we show that there exists a blow-up solution in the case of N = 2. In the second part, the decay property in L r with 1 < r < ∞ for 1 − 2 N m < 1 with the mass conservation is shown. On the other hand, in the case of 0 < m < 1 − 2 N , the extinction phenomenon of solution is proved. It is clarified that the case of m = 1 − 2 N exhibits the borderline in the sense that the decay and extinction occur when the diffusion power m changes across 1 − 2 N . For the borderline case of m = 1 − 2 N , our solution decays in L r exponentially as t → ∞.

Learning interacting particle systems: Diffusion parameter estimation for aggregation equations

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Extinction behavior of solutions for a quasilinear parabolic system with nonlocal sources

“…The first result concerned the extinction behavior of solutions for the heat equation with absorption was established in [18]. In recent years, there are many works on the extinction properties of solutions for different kinds of evolution equations (see [3,5,6,13,23,24,26,31] and the references therein). In particular, Li and Wu [20] studied the extinction properties of solutions to the following problem where 0 < m < 1 and k; p > 0, and showed that if p > m, the unique solution with small initial data vanishes in finite time, and if p < m, the maximal solution is positive in X for all t > 0 by using the supersolution and subsolution methods.…”

Extinction and decay estimates of solutions for a polytropic filtration equation with the nonlocal source and interior absorption

“…The first result concerning the extinction of a solution for the general heat equation with absorption was established in. In recent years, there were many works on the extinction properties of solutions for different kinds of evolution equations (see and the references therein ). In particular, Liu studied the extinction properties of solutions to the following problem with the nonlocal source and absorption where q , p ∈ (0,1), d , k > 0, , and obtained the critical extinction exponent q = p .…”

“…The first result concerning the extinction of a solution for the general heat equation with absorption was established in [11]. In recent years, there were many works on the extinction properties of solutions for different kinds of evolution equations (see [12][13][14][15][16][17][18][19][20][21][22][23][24][25] and the references therein ). In particular, Liu [26] studied the extinction properties of solutions to the following problem with the nonlocal source and absorption 8 <…”