Approaching optimality in blow-up results for Keller–Segel systems with logistic-type dampening
نویسندگان
چکیده
Nonnegative solutions of the Neumann initial-boundary value problem for chemotaxis system \begin{align}\label{prob:star}\tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u v) + \lambda \mu u^\kappa, \\\\ 0 v \overline m(t) u, \quad \frac1{|\Omega|} \int_\Omega u(\cdot, t) \end{cases} \end{align} in smooth bounded domains $\Omega \subset \mathbb R^n$, $n \ge 1$, are known to be global-in-time if $\lambda \geq 0$, $\mu > 0$ and $\kappa 2$. In present work, we show that exponent 2$ is actually critical four- higher dimensional setting. More precisely, \begin{alignat*}{3} \qquad n &\geq 4, &&\quad \kappa \in (1, 2) &&\text{and} \text{or}\qquad 5, 2 \left(0, \frac{n-4}{n}\right), \end{alignat*} balls R^n$ parameters $m_0 construct a nonnegative initial datum $u_0 C^0(\overline \Omega)$ with $\int_\Omega u_0 m_0$ which corresponding solution $(u, v)$ \eqref{prob:star} blows up finite time. Moreover, 3D, obtain finite-time blow-up \frac32)$ (and 0$). As corner stone our analysis, certain data, prove mass accumulation function $w(s, \int_0^{\sqrt[n]{s}} \rho^{n-1} u(\rho, \,\mathrm d\rho$ fulfills estimate $w_s \le \frac{w}{s}$. Using this information, then $u$ by showing suitably chosen $s_0$ $\gamma$, $\phi(t) \int_0^{s_0} s^{-\gamma} (s_0 s) w(s, t)$ cannot exist globally.
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ژورنال
عنوان ژورنال: Nonlinear Differential Equations And Applications Nodea
سال: 2021
ISSN: ['1420-9004', '1021-9722']
DOI: https://doi.org/10.1007/s00030-021-00677-9