---
_id: '10005'
abstract:
- lang: eng
  text: We study systems of nonlinear partial differential equations of parabolic
    type, in which the elliptic operator is replaced by the first-order divergence
    operator acting on a flux function, which is related to the spatial gradient of
    the unknown through an additional implicit equation. This setting, broad enough
    in terms of applications, significantly expands the paradigm of nonlinear parabolic
    problems. Formulating four conditions concerning the form of the implicit equation,
    we first show that these conditions describe a maximal monotone p-coercive graph.
    We then establish the global-in-time and large-data existence of a (weak) solution
    and its uniqueness. To this end, we adopt and significantly generalize Minty’s
    method of monotone mappings. A unified theory, containing several novel tools,
    is developed in a way to be tractable from the point of view of numerical approximations.
acknowledgement: "M. Bulíček and J. Málek acknowledge the support of the project No.
  18-12719S financed by the Czech\r\nScience foundation (GAČR). E. Maringová acknowledges
  support from Charles University Research program \r\nUNCE/SCI/023, the grant SVV-2020-260583
  by the Ministry of Education, Youth and Sports, Czech Republic\r\nand from the Austrian
  Science Fund (FWF), grants P30000, W1245, and F65. M. Bulíček and J. Málek are\r\nmembers
  of the Nečas Center for Mathematical Modelling.\r\n"
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Miroslav
  full_name: Bulíček, Miroslav
  last_name: Bulíček
- first_name: Erika
  full_name: Maringová, Erika
  id: dbabca31-66eb-11eb-963a-fb9c22c880b4
  last_name: Maringová
- first_name: Josef
  full_name: Málek, Josef
  last_name: Málek
citation:
  ama: Bulíček M, Maringová E, Málek J. On nonlinear problems of parabolic type with
    implicit constitutive equations involving flux. <i>Mathematical Models and Methods
    in Applied Sciences</i>. 2021;31(09). doi:<a href="https://doi.org/10.1142/S0218202521500457">10.1142/S0218202521500457</a>
  apa: Bulíček, M., Maringová, E., &#38; Málek, J. (2021). On nonlinear problems of
    parabolic type with implicit constitutive equations involving flux. <i>Mathematical
    Models and Methods in Applied Sciences</i>. World Scientific. <a href="https://doi.org/10.1142/S0218202521500457">https://doi.org/10.1142/S0218202521500457</a>
  chicago: Bulíček, Miroslav, Erika Maringová, and Josef Málek. “On Nonlinear Problems
    of Parabolic Type with Implicit Constitutive Equations Involving Flux.” <i>Mathematical
    Models and Methods in Applied Sciences</i>. World Scientific, 2021. <a href="https://doi.org/10.1142/S0218202521500457">https://doi.org/10.1142/S0218202521500457</a>.
  ieee: M. Bulíček, E. Maringová, and J. Málek, “On nonlinear problems of parabolic
    type with implicit constitutive equations involving flux,” <i>Mathematical Models
    and Methods in Applied Sciences</i>, vol. 31, no. 09. World Scientific, 2021.
  ista: Bulíček M, Maringová E, Málek J. 2021. On nonlinear problems of parabolic
    type with implicit constitutive equations involving flux. Mathematical Models
    and Methods in Applied Sciences. 31(09).
  mla: Bulíček, Miroslav, et al. “On Nonlinear Problems of Parabolic Type with Implicit
    Constitutive Equations Involving Flux.” <i>Mathematical Models and Methods in
    Applied Sciences</i>, vol. 31, no. 09, World Scientific, 2021, doi:<a href="https://doi.org/10.1142/S0218202521500457">10.1142/S0218202521500457</a>.
  short: M. Bulíček, E. Maringová, J. Málek, Mathematical Models and Methods in Applied
    Sciences 31 (2021).
date_created: 2021-09-12T22:01:25Z
date_published: 2021-08-25T00:00:00Z
date_updated: 2023-09-04T11:43:45Z
day: '25'
department:
- _id: JuFi
doi: 10.1142/S0218202521500457
external_id:
  arxiv:
  - '2009.06917'
  isi:
  - '000722222900004'
intvolume: '        31'
isi: 1
issue: '09'
keyword:
- Nonlinear parabolic systems
- implicit constitutive theory
- weak solutions
- existence
- uniqueness
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2009.06917
month: '08'
oa: 1
oa_version: Preprint
project:
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
  grant_number: F6504
  name: Taming Complexity in Partial Differential Systems
publication: Mathematical Models and Methods in Applied Sciences
publication_identifier:
  eissn:
  - 1793-6314
  issn:
  - 0218-2025
publication_status: published
publisher: World Scientific
quality_controlled: '1'
scopus_import: '1'
status: public
title: On nonlinear problems of parabolic type with implicit constitutive equations
  involving flux
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 31
year: '2021'
...