Publication Date:
2018-03-07
Description:
We consider radially symmetric solutions of the Keller–Segel system with generalized logistic source given by $$\begin{aligned} \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v) + \lambda u - \mu u^\kappa , \\ 0 = \Delta v - v + u, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$ under homogeneous Neumann boundary conditions in the ball \(\Omega =B_R(0) \subset \mathbb {R}^n\) for \(n\ge 3\) and \(R〉0\) , where \(\lambda \in \mathbb {R}, \mu 〉0\) and \(\kappa 〉1\) . Under the assumption that $$\begin{aligned} \kappa 〈 \left\{ \begin{array}{ll} \frac{7}{6} &{}\quad \text {if } n\in \{3,4\}, \\ 1+ \frac{1}{2(n-1)} &{}\quad \text {if } n \ge 5, \end{array} \right. \end{aligned}$$ a condition on the initial data is derived which is seen to be sufficient to ensure the occurrence of finite-time blow-up for the corresponding solution of ( \(\star \) ). Moreover, this criterion is shown to be mild enough so as to allow for the conclusion that in fact any positive continuous radial function on \(\overline{\Omega }\) is the limit in \(L^1(\Omega )\) of a sequence \((u_{0k})_{k\in \mathbb {N}}\) of continuous radial initial data which are such that for each \(k\in \mathbb {N}\) the associated initial-boundary value problem for ( \(\star \) ) exhibits a finite-time explosion phenomenon in the above sense. In particular, this apparently provides the first rigorous detection of blow-up in a superlinearly dampened but otherwise essentially original Keller–Segel system in the physically relevant three-dimensional case.
Print ISSN:
0044-2275
Electronic ISSN:
1420-9039
Topics:
Mathematics
,
Physics
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