A double critical mass phenomenon in a no-flux-Dirichlet Keller-Segel system (Q2136294)

In this paper, the authors considered the following cross-diffusion system \[ u_t=\Delta u-\nabla\cdot(u\nabla v),\quad x\in\Omega,t>0, \] \[ 0=\Delta v-kv+u,\quad x\in\Omega,t>0, \] \[ u(x,0)=u_0(x),\quad x\in\Omega, \] \[ \frac{\partial u}{\partial \nu}-u\frac{\partial v}{\partial\nu}=v=0,\quad x\in\partial\Omega, \] in a finite domain \(\Omega\subset\mathbb{R}^N\), \(N\ge 2,\) with \(k\ge0\). 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 \(N\ge 3\), 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. While critical mass phenomena distinguishing between regimes of type (i) and (ii) belong to the well-understood characteristics of the above system 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.