TY - JOUR

T1 - Strong convergence of weighted gradients in parabolic equations and applications to global generalized solvability of cross-diffusive systems

AU - Fuest, Mario

N1 - Funding Information: The author is grateful for various insightful comments by both the reviewer and the editor, which helped to improve the article. In particular, he thanks the reviewer and the editor for pointing him to the references [7 , 8] and [30 , 31 , 35], respectively.

PY - 2023/9

Y1 - 2023/9

N2 - 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 vε 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).

AB - 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 vε 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).

KW - Chemotaxis

KW - Generalized solutions

KW - Global existence

KW - Strong convergence of approximations

UR - http://www.scopus.com/inward/record.url?scp=85163017078&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2202.00317

DO - 10.48550/arXiv.2202.00317

M3 - Article

AN - SCOPUS:85163017078

VL - 23

JO - Journal of evolution equations

JF - Journal of evolution equations

SN - 1424-3199

IS - 3

M1 - 49

ER -