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

Abstract: 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 $Ω\subset \mathbb{R}<sup>N$ $(N \in \mathbb{N})$. 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.