Semigroup and Blow-Up Dynamics of Attraction Keller-Segel Equations in Scale of Banach Spaces

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.