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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 modelpursuit-evasion relationships between predators and prey. Apart from localkinetics given by $f_1$ and $f_2$, the key components in this system are thetaxis terms $-\nabla \cdot (S_1(u) \nabla v)$ and $+\nabla \cdot (S_2(v) \nablau)$; that is, the species are not only assumed to move around randomly in spacebut are also able to partially direct their movement depending on the nearbypresence of the other species. In the present article, we construct global weaksolutions 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 afourth-order regularization to obtain global solutions to approximate systemsand then rely on an entropy-like identity associated with ($\star$) forobtaining various a~priori estimates.

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