TY - JOUR

T1 - A double critical mass phenomenon in a no-flux-Dirichlet Keller-Segel system

AU - Fuhrmann, Jan

AU - Lankeit, Johannes

AU - Winkler, Michael

N1 - Funding Information: The third author acknowledges support of the Deutsche Forschungsgemeinschaft within the project Emergence of structures and advantages in cross-diffusion systems [project number 411007140 ].

PY - 2022/6

Y1 - 2022/6

N2 - 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.

AB - 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.

KW - Blow-up

KW - Critical mass

KW - Keller-Segel

UR - http://www.scopus.com/inward/record.url?scp=85129026007&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2101.06748

DO - 10.48550/arXiv.2101.06748

M3 - Article

VL - 162

SP - 124

EP - 151

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

SN - 0021-7824

ER -