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2018 Semigroup and Blow-Up Dynamics of Attraction Keller-Segel Equations in Scale of Banach Spaces
David S. I. Iiyambo, Robert Willie
Afr. Diaspora J. Math. (N.S.) 21(1): 1-31 (2018).

Abstract

In this paper, we study the asymptotic and blow-up dynamics of the attraction Keller-Segel chemotaxis system of equations in scale of Banach spaces $E^\alpha_q = H^{2\alpha,q}(\Omega), −1 \le \alpha \le 1,1 \lt q \lt \infty$, where $\Omega \subset \mathbb{R}^N$ is a bounded spatial domain. We show that the system of equations is well-posed for a perturbed analytic semigroup, whenever $2\chi + a \lt \left( \frac{Ne\pi}{2} \right)^{\beta+\frac{\gamma}{2}-\frac{1}{2}}$, where $\chi$ is the chemical attractivity coefficient, $a$ is the rate of production of chemical, and $q, \beta, \gamma$ are of the scale spaces. Thus, as $t\nearrow\infty$, the asymptotic dynamics are captured in the limit set $\mathcal{M}\cup \{0\}$, where $\mathcal{M} = |\Omega|L^1 -$spatial average solutions. The constants for the sharp space embedding $E^\alpha_q \subset L ^\Theta(\Omega) (1\lt\Theta\le\infty)$ indicate that for either the application of Banach fixed point theorem, or the global existence of solutions, no need of either the time for a contraction mapping, nor the initial data of the system of equations, to be small, respectively. In blow-up dynamics, we prove that the solutions blow-up at the borderline scale spaces $E^\alpha_q, \alpha = \frac{N}{2q}$, independent of time $t > 0$, if the chemo-attractivity coefficient dominates the Moser-Trudinger threshold value. An analysis of the finite time bounds for blow-up of solutions in norm of $L^{2p}(\Omega),1 \le p \le 6$ and $\Omega \subset \mathbb{R}^N,N = 2, 3$, is also furnished.

Citation

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David S. I. Iiyambo. Robert Willie. "Semigroup and Blow-Up Dynamics of Attraction Keller-Segel Equations in Scale of Banach Spaces." Afr. Diaspora J. Math. (N.S.) 21 (1) 1 - 31, 2018.

Information

Published: 2018
First available in Project Euclid: 7 March 2018

zbMATH: 06873705
MathSciNet: MR3763735

Subjects:
Primary: 35B40-4, 35K50, 35K57

Rights: Copyright © 2018 Mathematical Research Publishers

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Vol.21 • No. 1 • 2018
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