{"scopus_import":"1","publisher":"World Scientific","date_created":"2021-09-12T22:01:25Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Preprint","oa":1,"status":"public","title":"On nonlinear problems of parabolic type with implicit constitutive equations involving flux","quality_controlled":"1","date_updated":"2023-09-04T11:43:45Z","month":"08","article_processing_charge":"No","day":"25","type":"journal_article","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2009.06917"}],"abstract":[{"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.","lang":"eng"}],"department":[{"_id":"JuFi"}],"issue":"09","date_published":"2021-08-25T00:00:00Z","publication_status":"published","year":"2021","volume":31,"intvolume":"        31","project":[{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems","grant_number":"F6504"}],"keyword":["Nonlinear parabolic systems","implicit constitutive theory","weak solutions","existence","uniqueness"],"isi":1,"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","publication_identifier":{"eissn":["1793-6314"],"issn":["0218-2025"]},"external_id":{"isi":["000722222900004"],"arxiv":["2009.06917"]},"author":[{"full_name":"Bulíček, Miroslav","last_name":"Bulíček","first_name":"Miroslav"},{"first_name":"Erika","last_name":"Maringová","full_name":"Maringová, Erika","id":"dbabca31-66eb-11eb-963a-fb9c22c880b4"},{"full_name":"Málek, Josef","first_name":"Josef","last_name":"Málek"}],"publication":"Mathematical Models and Methods in Applied Sciences","doi":"10.1142/S0218202521500457","_id":"10005","language":[{"iso":"eng"}],"article_type":"original","citation":{"short":"M. Bulíček, E. Maringová, J. Málek, Mathematical Models and Methods in Applied Sciences 31 (2021).","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>","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>.","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).","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>","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.","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>."}}