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 varepsilon })_{varepsilon in (0, 1)} in L^1(Omega ) and weak convergence of (f_{varepsilon })_{varepsilon 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 vεt=Δvε+fε(x,t)inΩ×(0,∞),∂νvε=0on∂Ω×(0,∞),vε(·,0)=v0εinΩ,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} v_{\\varepsilon t} = \\Delta v_\\varepsilon + f_\\varepsilon (x, t) &{} \ ext {in }\\Omega \ imes (0, \\infty ), \\\\ \\partial _\ u v_\\varepsilon = 0 &{} \ ext {on }\\partial \\Omega \ imes (0, \\infty ), \\\\ v_\\varepsilon (\\cdot , 0) = v_{0 \\varepsilon } &{} \ ext {in }\\Omega , \\end{array}\\right. } \\end{aligned}$$\\end{document}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_{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 ut=Δu-χ∇·(uv∇v)+g(u),vt=Δv-uv,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u_t = \\Delta u - \\chi \ abla \\cdot (\\frac{u}{v} \ abla v) + g(u), \\\\ v_t = \\Delta v - uv, \\end{array}\\right. } \\end{aligned}$$\\end{document}where chi > 0 and g in C^1([0, infty )) are given, merely provided that (g(0) ge 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).