Shohei Kohatsu

This paper deals with the chemotaxis-haptotaxis system with flux limitation and nonlinear signal production, $ \begin{align*} \begin{cases} u_t = \Delta u -\chi \nabla \cdot \left( \dfrac{u\nabla v}{\left(1 + |\nabla v |^2 \right)^{\alpha}} \right) -\xi \nabla \cdot (u\nabla w) + \mu u(1-u-w), \\ 0 = \Delta v - v + u^{\beta}, \\ w_t = -vw \end{cases} \end{align*} $ in a bounded domain $ \Omega \subset \mathbb{R}^n $ $ (n \geq 2) $ with smooth boundary, where $ \alpha \in \mathbb{R} $ and $ \chi, \xi, \mu, \beta>0 $ are given constants. The first result of this paper asserts global existence and boundedness of classical solutions $ (u, v, w) $ provided that either $ \beta n > 1, \quad \frac{1}{2} - \frac{1}{2(\beta n - 1)} < \alpha $ or $ \beta n \le 1. $ The second result shows that when $ \alpha \ge 0 $, under a smallness condition on the initial datum for $ w $, the function $ t \mapsto \|w(\cdot, t)\|_{W^{1, \infty}(\Omega)} $ decays exponentially as $ t \to \infty $.

We consider the flux-limited Keller–Segel system with nonlinear diffusion, $\begin{cases} u_t = \nabla \cdot \big( (u+1)^{m-1}\nabla u \big) - \nabla \cdot \left( \dfrac{u}{(1+|\nabla v|^2)^{\alpha}} \nabla v \right), \\ v_t = \Delta v - v + u \end{cases} \quad\quad\quad\quad (*)$ in smoothly bounded domains $ \Omega \subset \mathbb{R}^n $ $ (n \in \mathbb{N}) $, and show that if $ m $ and $ \alpha $ satisfy suitable assumptions, then for any nonnegative initial data $ u_0 $, $ v_0 $ belonging to the space $ \mathcal{M}(\overline{\Omega}) $ of Radon measures on $ \overline{\Omega} $, which can be identified with $ (C^0(\overline{\Omega}))^{*} $, for the former and to a first-order Sobolev space $ W^{1,q}(\Omega) $ for the latter, there exists a global smooth solution $ (u,v) \in \big( C^{\infty}(\overline{\Omega} \times (0, \infty)) \big)^2 $ of the Neumann problem for $ (\ast) $, which is continuous at $ t = 0 $ in the sense that $ u(\cdot, t) \stackrel{\star}{\rightharpoonup} u_0 \quad \text{ in }\ \mathcal{M}(\overline{\Omega}) \quad \text{ and } \quad v(\cdot, t) \to v_0 \quad \text{ in }\ W^{1,q}(\Omega) $ as $ t \searrow 0 $. This extends previous results of [ 13 ] to the case $ m \neq 1 $, where such a smoothing property is derived using an argument based on standard $ L^p $-$ L^q $ estimates for the Neumann heat semigroup, which are applicable only in the case $ m = 1 $.

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 $p$.

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 $θ$.