Absence of critical mass phenomena in one-dimensional critical quasilinear Keller-Segel systems

Mathematics > Analysis of PDEs [Submitted on 16 Jun 2026] Title:Absence of critical mass phenomena in one-dimensional critical quasilinear Keller-Segel systems View PDF HTML (experimental)Abstract:We consider the Neumann initial boundary value problem associated to the chemotaxis system \begin{align}\label{prob:abstract}\tag{$\star$} \begin{cases} u_t = \big((u+1)^{m-1} u_x - u(u+1)^m v_x\big)_x & \text{in $(0, 1) \times (0, \infty)$}, \\ v_t = v_{xx} - v + u, &\text{in $(0, 1) \times (0, \infty)$}, \end{cases} \end{align} where $m \in \mathbb R$ is a given parameter. The relation between diffusion and taxis sensitivity is critical since the ratio $u(u+1)^m/(u+1)^{m-1}$ grows like $u^{2/n}$ for large $u$ with $n = \dim((0, 1)) = 1$. Nonetheless, we show that there is no critical mass phenomenon if $m \le -1$; that is, in that case all solutions emanating from suitably regular initial data are globally bounded. For certain parabolic-elliptic simplifications of \eqref{prob:abstract}, we obtain the same conclusion for all $m \in (-\infty, -1] \cup (0, \infty)$ and even for all $m \in \mathbb R$ if the initial datum is additionally assumed to be monotone. This stands in contrast to critical mass phenomena known to occur for critical quasilinear Keller-Segel systems considered in higher-dimensional domains. Accordingly, we make use of several special features of the one-dimensional setting such as the boundedness of the energy functional from below, the embedding $W^{1, n} \hookrightarrow L^\infty$, and the fact that the mass accumulation function solves a spatially non-degenerate parabolic equation. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.