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

TL;DR

This paper proves strong convergence of weighted gradients in parabolic equations and applies these results to establish the global existence of solutions for certain cross-diffusive systems in all space dimensions.

Contribution

It introduces new convergence results for weighted gradients in parabolic equations and uses them to prove global solvability of complex cross-diffusive systems without symmetry or smallness constraints.

Findings

Weighted gradients converge strongly in $L^2_{loc}$

Global generalized solutions exist for systems with superlinear growth in $g$

Results hold in all space dimensions without symmetry assumptions

Abstract

In the first part of the present paper, we show that strong convergence of $(v_{0 \varepsilon})_{\varepsilon \in (0, 1)}$ in $L^1(\Omega)$ and weak convergence of $(f_{\varepsilon})_{\varepsilon \in (0, 1)}$ in $L_{\textrm{loc}}^1(\overline \Omega \times [0, \infty))$ 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 $\Omega \times (0, \infty)$}, \\ \partial_\nu v_\varepsilon = 0 & \text{on $\partial \Omega \times (0, \infty)$}, \\ v_\varepsilon(\cdot, 0) = v_{0 \varepsilon} & \text{in $\Omega$}, \end{cases} \end{align*} which we consider in smooth, bounded domains $\Omega$, converge to the unique weak solution of the limit problem, but that also certain weighted gradients of $v_\varepsilon$ converge strongly in $L_{\textrm{loc}}^2(\overline \Omega \times [0, \infty))$ 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 $g \in C^1([0, \infty))$ are given, merely provided that ($g(0) \geq 0$ 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).