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arXiv2026

Existence and uniqueness of global weak solutions to degenerate volume-filling chemotaxis systems with source terms

This paper is concerned with a no-flux initial-boundary value problem for the degenerate volume-filling chemotaxis system with source terms, \begin{align*} u_t = \nabla \cdot (D(u,v) \nabla u - h(u,v) \nabla v) + f(x, u, v), \quad v_t = Δv + g(u,v), \quad x\in Ω, \ t>0 \end{align*} in a smoothly bounded domain $Ω\subset \mathbb{R}^N$ $(N \in \mathbb{N})$. It is shown that when $D$, $h$, $f$ and $g$ satisfy suitable assumptions involving $D(1,\cdot)=h(0,\cdot)=h(1,\cdot)=f(\cdot,0,\cdot) = f(\cdot,1,\cdot) = 0$, for nonnegative initial data $u_0$ and $v_0$ with $u_0\le 1$, there exists a global weak solution $(u, v)$ with $u\le 1$. In addition, uniqueness of global weak solutions is established when $D(r,s) = D(r)$ for all $r\in[0,1]$ and $s\in[0,\infty)$, and $D$, $h$, $f$, $g$ and $v_0$ are supposed to satisfy additional conditions.

math.AP↗

arXiv2026

Global well-posedness and flat-hump-shaped stationary solutions for degenerate chemotaxis systems with threshold density

In a smoothly bounded domain $Ω\subset \mathbb{R}^N$ $(N\in \mathbb{N})$, a no-flux initial-boundary value problem for the degenerate chemotaxis system with volume-filling effects, \begin{align*} u_t = \nabla \cdot (D(u,v) \nabla u - h(u,v) \nabla v), \quad v_t = Δv + g(u,v), \quad x\in Ω, \ t>0, \end{align*} is considered under the assumptions that $D(1,s)=0$ and that $h(0,s)=h(1,s)=0$. Here, initial data $u_0$ and $v_0$ have suitable regularity and satisfy $0\le u_0\le 1$ and $v_0\ge 0$ with $\nabla v_0 \cdot ν|_{\partial Ω} = 0$. It is proved that there exists a global weak solution such that $0\le u\le 1$ and $v\ge 0$. Moreover, when $D(r,s) = D(r)$ for all $r\in[0,1]$ and $s\in[0,\infty)$ and additional conditions on $D$, $h$ and $g$ are assumed, uniqueness of global weak solutions with the mass conservation law $\int_Ωu(x,t) \, dx = \int_Ωu_0(x) \, dx$ is shown. Also, a flat-hump-shaped stationary solution is constructed in the one-dimensional setting

math.AP↗