arXiv subjects

Shohei Kohatsu

Publications and source records attributed to Shohei Kohatsu.

4 recordsLinked to original sources

arXiv2025

Global boundedness of weak solutions to a flux-limited Keller--Segel system with superlinear production

The flux-limited Keller--Segel system \begin{align*} \begin{cases} u_t = Δu - χ\nabla \cdot (u|\nabla v|^{p-2}\nabla v), \\[] v_t = Δv - v + u^θ \end{cases} \end{align*} is considered under homogeneous Neumann boundary conditions in a bounded domain $Ω\subset \mathbb{R}^n$ $(n \in \mathbb{N})$. In the case that $θ\le 1$, existence of global bounded weak solutions was established in the previous work (arXiv:2501.04370 ; to be appear in Proceedings of the conference "Critical Phenomena in Nonlinear Partial Differential Equations, Harmonic Analysis, and Functional Inequalities."). The purpose of this paper is to prove that global bounded weak solutions can also be constructed in the case $θ> 1$ with a smallness condition on $p$.

math.AP↗

arXiv2025

Stability of constant equilibria in a Keller--Segel system with gradient dependent chemotactic sensitivity and sublinear signal production

This paper deals with the homogeneous Neumann boundary-value problem for the Keller--Segel system \begin{align*} \begin{cases} u_t=Δu - χ\nabla \cdot (u|\nabla v|^{p-2}\nabla v),\\[] v_t=Δv - v + u^θ \end{cases} \end{align*} in $n$-dimensional bounded smooth domains for suitably regular nonnegative initial data, where $χ> 0$, $p \in (1, \infty)$ and $θ\in (0,1]$. Under smallness conditions on $p$ and $θ$, we prove that the spatially homogeneous equilibrium solution is stable. This generalizes the result in Kohatsu--Yokota (Le Matematiche, 2023; 78; 213--237) from the case $θ= 1$ to more general values of $θ$.

math.AP↗

arXiv2024

Instantaneous regularization of measure-valued population densities in a Keller--Segel system with flux limitation

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.

math.AP↗

arXiv2024

Large densities in a competitive two-species chemotaxis system in the non-symmetric case

This paper deals with the two-species chemotaxis system with Lotka-Volterra competitive kinetics, \begin{align*} \begin{cases} u_t = d_1 Δu - χ_1 \nabla \cdot (u \nabla w) + μ_1 u (1 - u - a_1 v), & x\inΩ,\ t>0,\\ v_t = d_2 Δv - χ_2 \nabla \cdot (v \nabla w) + μ_2 v (1 - a_2 u - v), & x\inΩ,\ t>0,\\ 0 = d_3 Δw + αu + βv - γw, & x\inΩ,\ t>0, \end{cases} \end{align*} under homogeneous Neumann boundary conditions and suitable initial conditions, where $Ω\subset \mathbb{R}^n$ $(n \in \mathbb{N})$ is a bounded domain with smooth boundary, $d_1, d_2, d_3, χ_1, χ_2, μ_1, μ_2 > 0$, $a_1, a_2 \ge 0$ and $α, β, γ> 0$. Under largeness conditions on $χ_1$ and $χ_2$, we show that for suitably regular initial data, any thresholds of the population density can be surpassed, which extends the previous results to the non-symmetric case. The paper contains a well-posedness result for the hyperbolic-elliptic limit system with $d_1=d_2=0$.

math.AP↗