Instantaneous regularization of measure-valued population densities in a Keller--Segel system with flux limitation
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| Format: | Preprint |
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2024
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| author | Kohatsu, Shohei |
| author_facet | Kohatsu, Shohei |
| contents | This paper is concerned with the Keller--Segel system with flux limitation,
\begin{align} \tag{$\ast$} \begin{cases} u_t=Δu - \nabla \cdot (uf(|\nabla v|^{2})\nabla v), \\ v_t=Δv - v + u \end{cases} \end{align} in bounded $n$-dimensional domains with homogeneous Neumann boundary conditions, where $f$ generalizes the prototype obtained on letting \[
f(ξ) = k_f(1 + ξ)^{-α}, \quad ξ\ge 0, \] with $k_f > 0$ and $α> 0$. In this framework, it is shown that if either $n = 1$ and $α> 0$ is arbitrary, or $n \ge 2$ and $α> \frac{n-2}{2(n-1)}$, then for any nonnegative initial data belonging to the space of Radon measures for the population density and to $W^{1,q}$ with $q \in (\max\{1, (1-2α)n\}, \frac{n}{n-1})$ for the signal density, there exists a global classical solution of the Neumann problem for $(\ast)$, which is continuous at $t = 0$ in an appropriate sense. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_17955 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Instantaneous regularization of measure-valued population densities in a Keller--Segel system with flux limitation Kohatsu, Shohei Analysis of PDEs Primary: 35B65, Secondary: 35Q92, 35A09, 92C17 This paper is concerned with the Keller--Segel system with flux limitation, \begin{align} \tag{$\ast$} \begin{cases} u_t=Δu - \nabla \cdot (uf(|\nabla v|^{2})\nabla v), \\ v_t=Δv - v + u \end{cases} \end{align} in bounded $n$-dimensional domains with homogeneous Neumann boundary conditions, where $f$ generalizes the prototype obtained on letting \[ f(ξ) = k_f(1 + ξ)^{-α}, \quad ξ\ge 0, \] with $k_f > 0$ and $α> 0$. In this framework, it is shown that if either $n = 1$ and $α> 0$ is arbitrary, or $n \ge 2$ and $α> \frac{n-2}{2(n-1)}$, then for any nonnegative initial data belonging to the space of Radon measures for the population density and to $W^{1,q}$ with $q \in (\max\{1, (1-2α)n\}, \frac{n}{n-1})$ for the signal density, there exists a global classical solution of the Neumann problem for $(\ast)$, which is continuous at $t = 0$ in an appropriate sense. |
| title | Instantaneous regularization of measure-valued population densities in a Keller--Segel system with flux limitation |
| topic | Analysis of PDEs Primary: 35B65, Secondary: 35Q92, 35A09, 92C17 |
| url | https://arxiv.org/abs/2402.17955 |