Global weak solutions to fully cross-diffusive systems with nonlinear diffusion and saturated taxis sensitivity

Abstract: Systems of the type $$\begin{cases} u_t = \nabla \cdot (D_1(u) \nabla u - S_1(u) \nabla v) + f_1(u, v),\ v_t = \nabla \cdot (D_2(v) \nabla v + S_2(v) \nabla u) + f_2(u, v) \end{cases} \qquad (\star)$$ can be used to model pursuit-evasion relationships between predators and prey. Apart from local kinetics given by $f_1$ and $f_2$, the key components in this system are the taxis terms $-\nabla \cdot (S_1(u) \nabla v)$ and $+\nabla \cdot (S_2(v) \nabla u)$; that is, the species are not only assumed to move around randomly in space but are also able to partially direct their movement depending on the nearby presence of the other species. In the present article, we construct global weak solutions of ($\star$) for certain prototypical nonlinear functions $D_i$, $S_i$ and $f_i$, $i \in {1, 2}$. To that end, we first make use of a fourth-order regularization to obtain global solutions to approximate systems and then rely on an entropy-like identity associated with ($\star$) for obtaining various a~priori estimates.