English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Our main result shows that the mass $2\pi$ is critical for the minimalKeller-Segel system \begin{align}\label{prob:abstract}\tag{$\star$}\begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v), \\ v_t = \Delta v - v+ u, \end{cases} \end{align} considered in a quarter disc $\Omega = \{\,(x_1,x_2) \in \mathbb R : x_1 > 0, x_2 > 0, x_1^2 + x_2^2 < R^2\,\}$, $R > 0$, inthe following sense: For all reasonably smooth nonnegative initial data $u_0,v_0$ with $\int_\Omega u_0 < 2\pi$, there exists a global classical solution tothe Neumann initial boundary value problem associated to \eqref{prob:abstract},while for all $m > 2 \pi$ there exist nonnegative initial data $u_0, v_0$ with$\int_\Omega u_0 = m$ so that the corresponding classical solution of thisproblem blows up in finite time. At the same time, this gives an example ofboundary blow-up in \eqref{prob:abstract}. Up to now, precise values ofcritical masses had been observed in spaces of radially symmetric functions orfor parabolic-elliptic simplifications of \eqref{prob:abstract} only.

Corners and collapse: Some simple observations concerning critical masses and boundary blow-up in the fully parabolic Keller-Segel system