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

Abstract

In the first part of the present paper, we show that strong convergence of $(v_{0\epsilon }{)}_{\epsilon \in (0,1)}$ in $L^1(\Omega )$ and weak convergence of $(f_{\epsilon }{)}_{\epsilon \in (0,1)}$ in $L_{\text{loc}}^1(\overline{\Omega }\times [0,\infty ))$ not only suffice to conclude that solutions to the initial boundary value problem \begin{align*} \begin{cases} v_{\varepsilon t} = Δv_\varepsilon + f_\varepsilon(x, t) & \text{in $\Omega \times (0,\infty )$}, \\ \partial_νv_\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 $v_{\epsilon }$ converge strongly in $L_{\text{loc}}^2(\overline{\Omega }\times [0,\infty ))$ 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 = Δu - χ\nabla \cdot (\frac{u}{v} \nabla v) + g(u), \\ v_t = Δv - uv, \end{cases} \end{align*} where $\chi >0$ and $g\in C^1([0,\infty ))$ 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).

43 pages

References in corpus (1)

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

Cited by in corpus (2)

• Classical and generalized solutions of an alarm-taxis model
• Strong convergence of weighted gradients in parabolic equations and applications to global generalized solvability of cross-diffusive systems