A double critical mass phenomenon in a no-flux-Dirichlet Keller-Segel system

TL;DR

This paper investigates a Keller-Segel system with Dirichlet boundary conditions, revealing a double critical mass phenomenon that determines whether solutions remain bounded, blow up, or exhibit both behaviors, with implications for cell biophysics.

Contribution

It identifies a novel secondary critical mass level in a Keller-Segel model with Dirichlet conditions, expanding understanding of singular structure formation.

Findings

Existence of two critical mass levels in radial solutions.

Different regimes of global boundedness, blow-up, or both.

Numerical illustrations and biophysical interpretations for planar cells.

Abstract

Derived from a biophysical model for the motion of a crawling cell, the system \[(*)~\begin{cases}u_t=\Delta u-\nabla\cdot(u\nabla v)\\0=\Delta v-kv+u\end{cases}\] is investigated in a finite domain $\Omega\subset\mathbb{R}^n$, $n\geq2$, with $k\geq0$. While a comprehensive literature is available for cases with $(*)$ describing chemotaxis systems and hence being accompanied by homogeneous Neumann-type boundary conditions, the presently considered modeling context, besides yet requiring the flux $\partial_\nu u-u\partial_\nu v$ to vanish on $\partial\Omega$, inherently involves homogeneous Dirichlet conditions for the attractant $v$, which in the current setting corresponds to the cell's cytoskeleton being free of pressure at the boundary. This modification in the boundary setting is shown to go along with a substantial change with respect to the potential to support the emergence of singular structures: It is, inter alia, revealed that in contexts of radial solutions in balls there exist two critical mass levels, distinct from each other whenever $k>0$ or $n\geq3$, that separate ranges within which (i) all solutions are global in time and remain bounded, (ii) both global bounded and exploding solutions exist, or (iii) all nontrivial solutions blow up in finite time. While critical mass phenomena distinguishing between regimes of type (i) and (ii) belong to the well-understood characteristics of $(*)$ when posed under classical no-flux boundary conditions in planar domains, the discovery of a distinct secondary critical mass level related to the occurrence of (iii) seems to have no nearby precedent. In the planar case with the domain being a disk, the analytical results are supplemented with some numerical illustrations, and it is discussed how the findings can be interpreted biophysically for the situation of a cell on a flat substrate.