---
_id: '63337'
abstract:
- lang: eng
  text: <jats:title>Abstract</jats:title><jats:p>In bounded<jats:italic>n</jats:italic>-dimensional
    domains<jats:italic>Ω</jats:italic>, the Neumann problem for the parabolic equation</jats:p><jats:p><jats:disp-formula
    id="j_anona-2020-0013_eq_001"><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink"
    xlink:href="graphic/j_anona-2020-0013_eq_001.png" position="float" orientation="portrait"
    /><jats:tex-math>$$\begin{array}{} \displaystyle u_t = \nabla \cdot \Big( A(x,t)\cdot\nabla
    u\Big) + \nabla \cdot \Big(b(x,t)u\Big) - f(x,t,u)+g(x,t) \end{array}$$</jats:tex-math></jats:alternatives><jats:label>(*)</jats:label></jats:disp-formula></jats:p><jats:p>is
    considered for sufficiently regular matrix-valued<jats:italic>A</jats:italic>,
    vector-valued<jats:italic>b</jats:italic>and real valued<jats:italic>g</jats:italic>,
    and with<jats:italic>f</jats:italic>representing superlinear absorption in generalizing
    the prototypical choice given by<jats:italic>f</jats:italic>(⋅, ⋅,<jats:italic>s</jats:italic>)
    =<jats:italic>s<jats:sup>α</jats:sup></jats:italic>with<jats:italic>α</jats:italic>&gt;
    1. Problems of this form arise in a natural manner as sub-problems in several
    applications such as cross-diffusion systems either of Keller-Segel or of Shigesada-Kawasaki-Teramoto
    type in mathematical biology, and accordingly a natural space for initial data
    appears to be<jats:italic>L</jats:italic><jats:sup>1</jats:sup>(<jats:italic>Ω</jats:italic>).</jats:p><jats:p>The
    main objective thus consists in examining how far solutions can be constructed
    for initial data merely assumed to be integrable, with major challenges potentially
    resulting from the interplay between nonlinear degradation on the one hand, and
    the possibly destabilizing drift-type action on the other in such contexts. Especially,
    the applicability of well-established methods such as techniques relying on entropy-like
    structures available in some particular cases, for instance, seems quite limited
    in the present setting, as these typically rely on higher initial regularity properties.</jats:p><jats:p>The
    first of the main results shows that in the general framework of (*), nevertheless
    certain global very weak solutions can be constructed through a limit process
    involving smooth solutions to approximate variants thereof, provided that the
    ingredients of the latter satisfy appropriate assumptions with regard to their
    stabilization behavior.</jats:p><jats:p>The second and seemingly most substantial
    part of the paper develops a method by which it can be shown, under suitably stregthened
    hypotheses on the integrability of<jats:italic>b</jats:italic>and the degradation
    parameter<jats:italic>α</jats:italic>, that the solutions obtained above in fact
    form genuine weak solutions in a naturally defined sense. This is achieved by
    properly exploiting a weak integral inequality, as satisfied by the very weak
    solution at hand, through a testing procedure that appears to be novel and of
    potentially independent interest.</jats:p><jats:p>To underline the strength of
    this approach, both these general results are thereafter applied to two specific
    cross-diffusion systems. Inter alia, this leads to a statement on global solvability
    in a logistic Keller-Segel system under the assumption<jats:italic>α</jats:italic>&gt;<jats:inline-formula><jats:alternatives><jats:inline-graphic
    xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_anona-2020-0013_eq_002.png"
    /><jats:tex-math>$\begin{array}{} \frac{2n+4}{n+4} \end{array}$</jats:tex-math></jats:alternatives></jats:inline-formula>on
    the respective degradation rate which seems substantially milder than any previously
    found condition in the literature. Apart from that, for a Shigesada-Kawasaki-Teramoto
    system some apparently first results on global solvability for<jats:italic>L</jats:italic><jats:sup>1</jats:sup>initial
    data are derived.</jats:p>
author:
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Winkler M. The role of superlinear damping in the construction of solutions
    to drift-diffusion problems with initial data in L1. <i>Advances in Nonlinear
    Analysis</i>. 2019;9(1):526-566. doi:<a href="https://doi.org/10.1515/anona-2020-0013">10.1515/anona-2020-0013</a>
  apa: Winkler, M. (2019). The role of superlinear damping in the construction of
    solutions to drift-diffusion problems with initial data in L1. <i>Advances in
    Nonlinear Analysis</i>, <i>9</i>(1), 526–566. <a href="https://doi.org/10.1515/anona-2020-0013">https://doi.org/10.1515/anona-2020-0013</a>
  bibtex: '@article{Winkler_2019, title={The role of superlinear damping in the construction
    of solutions to drift-diffusion problems with initial data in L1}, volume={9},
    DOI={<a href="https://doi.org/10.1515/anona-2020-0013">10.1515/anona-2020-0013</a>},
    number={1}, journal={Advances in Nonlinear Analysis}, publisher={Walter de Gruyter
    GmbH}, author={Winkler, Michael}, year={2019}, pages={526–566} }'
  chicago: 'Winkler, Michael. “The Role of Superlinear Damping in the Construction
    of Solutions to Drift-Diffusion Problems with Initial Data in L1.” <i>Advances
    in Nonlinear Analysis</i> 9, no. 1 (2019): 526–66. <a href="https://doi.org/10.1515/anona-2020-0013">https://doi.org/10.1515/anona-2020-0013</a>.'
  ieee: 'M. Winkler, “The role of superlinear damping in the construction of solutions
    to drift-diffusion problems with initial data in L1,” <i>Advances in Nonlinear
    Analysis</i>, vol. 9, no. 1, pp. 526–566, 2019, doi: <a href="https://doi.org/10.1515/anona-2020-0013">10.1515/anona-2020-0013</a>.'
  mla: Winkler, Michael. “The Role of Superlinear Damping in the Construction of Solutions
    to Drift-Diffusion Problems with Initial Data in L1.” <i>Advances in Nonlinear
    Analysis</i>, vol. 9, no. 1, Walter de Gruyter GmbH, 2019, pp. 526–66, doi:<a
    href="https://doi.org/10.1515/anona-2020-0013">10.1515/anona-2020-0013</a>.
  short: M. Winkler, Advances in Nonlinear Analysis 9 (2019) 526–566.
date_created: 2025-12-18T19:45:09Z
date_updated: 2025-12-18T20:01:54Z
doi: 10.1515/anona-2020-0013
intvolume: '         9'
issue: '1'
language:
- iso: eng
page: 526-566
publication: Advances in Nonlinear Analysis
publication_identifier:
  issn:
  - 2191-950X
publication_status: published
publisher: Walter de Gruyter GmbH
status: public
title: The role of superlinear damping in the construction of solutions to drift-diffusion
  problems with initial data in L1
type: journal_article
user_id: '31496'
volume: 9
year: '2019'
...