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 (v0ε)ε∈(0,1) in L¹(Ω) and weak convergence of (fε)ε∈(0,1) in Lloc1(Ω¯×[0,∞)) not only suffice to conclude that solutions to the initial boundary value problem {vεt=Δvε+fε(x,t)inΩ×(0,∞),∂νvε=0on∂Ω×(0,∞),vε(·,0)=v0εinΩ, which we consider in smooth, bounded domains Ω , converge to the unique weak solution of the limit problem, but that also certain weighted gradients of v_ε converge strongly in Lloc2(Ω¯×[0,∞)) 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 {ut=Δu-χ∇·(uv∇v)+g(u),vt=Δv-uv, where χ> 0 and g∈ C¹([0 , ∞)) are given, merely provided that (g(0) ≥ 0 and) - g grows superlinearly. 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).