Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system

By Michael Winkler. 2011

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We study the Neumann initial-boundary value problem for the fully parabolic Keller-Segel system u_t=\Delta u - \nabla \cdot (u\nabla v), \qquad x\in\Omega, \ t>0, [1mm] v_t=\Delta v-v+u, \qquad x\in\Omega, \ t>0, where $\Omega$ is a ball in ${\mathbb{R}}^n$ with $n\ge 3$. It is proved that for any prescribed $m>0$ there exist radially symmetric positive initial data $(u_0,v_0) \in C^0(\bar\Omega) \times W^{1,\infty}(\Omega)$ with $\int_\Omega u_0=m$ such that the corresponding solution blows up in finite time. Moreover, by providing an essentially explicit blow-up criterion it is shown that within the space of all radial functions, the set of such blow-up enforcing initial data indeed is large in an appropriate sense; in particular, this set is dense with respect to the topology of $L^p(\Omega) \times W^{1,2}(\Omega)$ for any $p \in (1,\frac{2n}{n+2})$.

Subfields: Cell Biology, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/18018021973959eec6daea1c6c77ff78ea858148

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Citations in the corpus

Effects of signal-dependent motilities in a Keller–Segel-type reaction–diffusion system (2017, background): “[9], [36] and also [20]), a large variety of results in the literature underlines the crucial role of the interplay between the chemotactic sensitivity S and the diffusion rate D therein.”
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis–fluid system (2015, background): “1 detects an explicit parameter condition μ ≥ 23 under which the suitable quadratic degradation of bacteria is sufficient to exclude any blow-up phenomenon, as known to occur when μ = 0 even in the case of the corresponding Keller-Segel system obtained on letting u ≡ 0 ([29]).”
An attraction‐repulsion chemotaxis system with logistic source (2016, background): “A large amount of efforts were devoted to making clear whether solutions to the so-called minimal model (f (u) = 0) are blowup or bounded [4, 13, 16, 24, 27].”
How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system? (2015, background): “Indeed, intense analysis on the latter has confirmed the conjecture that spontaneous formation of aggregates, in the extreme mathematical sense of finite-time blow-up of solutions, may arise even in the simple two-component framework containing a population of cells moving chemotactically upward gra”
Global classical solution and boundedness to a chemotaxis‐haptotaxis model with re‐establishment mechanisms (2018, background): “When μ = 0, we refer to [3, 6, 10, 13, 22, 24] for the study of the global solutions or blow up property.”
Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^{N}$ (2016, background): “The reader is referred to [2], [7], [14], [40], [43], [45], [46], [47], [49], [50], [52], and references therein for other studies of (1.”
Chemotaxis can prevent thresholds on population density (2014, background): “Moreover, blow-up of solutions with large enough initial mass has been shown to be a generic phenomenon of the equation in some sense even for the parabolic-parabolic version of the system [13, 24].”