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arXiv · 2206.01144

Regular solutions of chemotaxis-consumption systems involving tensor-valued sensitivities and Robin type boundary conditions

Jaewook Ahn·Kyungkeun Kang·Jihoon Lee
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Abstract

This paper deals with a parabolic-elliptic chemotaxis-consumption system with tensor-valued sensitivity $S(x,n,c)$ under no-flux boundary conditions for $n$ and Robin-type boundary conditions for $c$. The global existence of bounded classical solutions is established in dimension two under general assumptions on tensor-valued sensitivity $S$. One of main steps is to show that $\nabla c(\cdot,t)$ becomes tiny in $L^{2}(B_{r}(x)\cap \Omega)$ for every $x\in\overline{\Omega}$ and $t$ when $r$ is sufficiently small, which seems to be of independent interest. On the other hand, in the case of scalar-valued sensitivity $S=\chi(x,n,c)\mathbb{I}$, there exists a bounded classical solution globally in time for two and higher dimensions provided the domain is a ball with radius $R$ and all given data are radial. The result of the radial case covers scalar-valued sensitivity $\chi$ that can be singular at $c=0$.

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Jaewook Ahn, Kyungkeun Kang, Jihoon Lee. 2022-06-02. Regular solutions of chemotaxis-consumption systems involving tensor-valued sensitivities and Robin type boundary conditions. https://arxiv.org/abs/2206.01144

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