Strong convergence of weighted gradients in parabolic equations and applications to global generalized solvability of cross-diffusive systems

In the first part of the present paper, we show that strong convergence of $(v_{0\epsilon }{)}_{\epsilon \in (0,1)}$ in $L^1(\mathrm{\Omega })$ and weak convergence of $(f_{\epsilon }{)}_{\epsilon \in (0,1)}$ in $L_{\text{loc}}^1(\bar{\mathrm{\Omega }}\times [0,\mathrm{\infty }))$ not only suffice to conclude that solutions to the initial boundary value problem \begin{align*} \begin{cases} v_{\varepsilon t} = \Delta v_\varepsilon + f_\varepsilon(x, t) & \text{in $\mathrm{\Omega }\times (0,\mathrm{\infty })$}, \ \partial_\nu v_\varepsilon = 0 & \text{on $\partial \Omega \times (0, \infty)$}, \ v_\varepsilon(\cdot, 0) = v_{0 \varepsilon} & $\text{in }\mathrm{\Omega }$, \end{cases} \end{align*} which we consider in smooth, bounded domains $\mathrm{\Omega }$, converge to the unique weak solution of the limit problem, but that also certain weighted gradients of $v_{\epsilon }$ converge strongly in $L_{\text{loc}}^2(\bar{\mathrm{\Omega }}\times [0,\mathrm{\infty }))$ along a subsequence. We then make use of these findings to obtain global generalized solutions to various cross-diffusive systems. Inter alia, we establish global generalized solvability of the system \begin{align*} \begin{cases} u_t = \Delta u - \chi \nabla \cdot (\frac{u}{v} \nabla v) + g(u), \ v_t = \Delta v - uv, \end{cases} \end{align*} where $\chi >0$ and $g\in C^1([0,\mathrm{\infty }))$ are given, merely provided that ($g(0)\ge 0$ and) $-g$ grows superlinearily. This result holds in all space dimensions and does neither require any symmetry assumptions nor the smallness of certain parameters. Thereby, we expand on a corresponding result for quadratically growing $-g$ proved by Lankeit and Lankeit (Nonlinearity, 32(5):1569–1596, 2019).