Energy classification in a nonstandard fourth‐order parabolic equation with a Navier boundary condition
This paper deals with a fourth‐order parabolic equation involving variable exponents, subject to Navier boundary conditions. We firstly give an optimal classification on the initial energy for blow‐up or global solutions, which is characterized by the sign of the Nehari energy and the difference bet...
Saved in:
| Published in | Mathematical methods in the applied sciences Vol. 46; no. 4; pp. 3703 - 3720 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Freiburg
Wiley Subscription Services, Inc
15.03.2023
|
| Subjects | |
| Online Access | Get full text |
| ISSN | 0170-4214 1099-1476 |
| DOI | 10.1002/mma.8717 |
Cover
Loading…
| Abstract | This paper deals with a fourth‐order parabolic equation involving variable exponents, subject to Navier boundary conditions. We firstly give an optimal classification on the initial energy for blow‐up or global solutions, which is characterized by the sign of the Nehari energy and the difference between the initial energy and the depth of the potential well. Then, large time estimates of solutions are obtained. The results of the paper extend and complete the ones in “Applied Mathematics Letters 78(2018)141–146.” |
|---|---|
| AbstractList | This paper deals with a fourth‐order parabolic equation involving variable exponents, subject to Navier boundary conditions. We firstly give an optimal classification on the initial energy for blow‐up or global solutions, which is characterized by the sign of the Nehari energy and the difference between the initial energy and the depth of the potential well. Then, large time estimates of solutions are obtained. The results of the paper extend and complete the ones in “Applied Mathematics Letters 78(2018)141–146.” This paper deals with a fourth‐order parabolic equation involving variable exponents, subject to Navier boundary conditions. We firstly give an optimal classification on the initial energy for blow‐up or global solutions, which is characterized by the sign of the Nehari energy and the difference between the initial energy and the depth of the potential well. Then, large time estimates of solutions are obtained. The results of the paper extend and complete the ones in “ Applied Mathematics Letters 78(2018)141–146.” |
| Author | Sun, Xizheng Liu, Bingchen Wang, Yiming |
| Author_xml | – sequence: 1 givenname: Bingchen orcidid: 0000-0003-4136-5375 surname: Liu fullname: Liu, Bingchen email: bcliu@upc.edu.cn organization: China University of Petroleum – sequence: 2 givenname: Xizheng surname: Sun fullname: Sun, Xizheng organization: China University of Petroleum – sequence: 3 givenname: Yiming surname: Wang fullname: Wang, Yiming organization: China University of Petroleum |
| BookMark | eNp1kL1OwzAURi0EEm1B4hEisbCk2I5T22NVlR-phQVmy3Ec6iq1Wzuh6sYj8Iw8CU7DhGCydHXOd32_ITi1zmoArhAcIwjx7WYjx4wiegIGCHKeIkInp2AAEYUpwYicg2EIawghQwgPgJlb7d8OiaplCKYySjbG2cTYRCYxOTTSltKXSeVa36y-Pj6dL7VPttLLwtVGJXrX9sreNKsoPcl3E4HCtZ0Yg50tTQdcgLNK1kFf_rwj8Ho3f5k9pIvn-8fZdJEqzDOaSi4p01QrxpQsSZFnOOeMQEYV1xDnsKgmWKOcVogjQijkE4qkJqTiTMdRNgLXfe7Wu12rQyPW8e82rhSY0owzigmM1LinlHcheF0JZZrjIY2XphYIiq5OEesUXZ1RuPklbL3ZxAv_QtMe3ZtaH_7lxHI5PfLfoYeHdA |
| CitedBy_id | crossref_primary_10_11650_tjm_250301 |
| Cites_doi | 10.1016/j.jde.2015.08.022 10.1109/TIP.2003.819229 10.2991/978-94-6239-112-3 10.1016/j.nonrwa.2018.03.009 10.1016/j.aml.2012.12.017 10.1016/j.camwa.2018.01.047 10.1016/S0022-247X(03)00474-8 10.1515/anona-2020-0141 10.1016/j.aml.2016.08.013 10.1016/j.nonrwa.2020.103155 10.1007/s00205-002-0208-7 10.1016/j.aml.2013.07.012 10.1016/j.jfa.2013.03.010 10.1016/j.jde.2020.03.047 10.1016/j.na.2008.11.030 10.1007/s00033-011-0148-x 10.1007/s11425-017-9280-x 10.1016/j.aml.2017.11.015 10.1016/0022-0248(95)01048-3 10.1016/j.nonrwa.2019.01.001 10.1017/S0308210510000399 10.1109/83.869184 10.1016/j.jmaa.2017.08.047 10.1080/00036811.2011.632767 10.1016/j.aml.2020.106664 10.1007/s11565-006-0002-9 10.1016/j.nonrwa.2019.01.020 10.3934/dcds.2016.36.715 10.1016/j.jmaa.2013.10.040 10.1201/b17415 10.1090/S0033-569X-2010-01197-0 10.1137/050624522 |
| ContentType | Journal Article |
| Copyright | 2022 John Wiley & Sons, Ltd. 2023 John Wiley & Sons, Ltd. |
| Copyright_xml | – notice: 2022 John Wiley & Sons, Ltd. – notice: 2023 John Wiley & Sons, Ltd. |
| DBID | AAYXX CITATION 7TB 8FD FR3 JQ2 KR7 |
| DOI | 10.1002/mma.8717 |
| DatabaseName | CrossRef Mechanical & Transportation Engineering Abstracts Technology Research Database Engineering Research Database ProQuest Computer Science Collection Civil Engineering Abstracts |
| DatabaseTitle | CrossRef Civil Engineering Abstracts Engineering Research Database Technology Research Database Mechanical & Transportation Engineering Abstracts ProQuest Computer Science Collection |
| DatabaseTitleList | CrossRef Civil Engineering Abstracts |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Sciences (General) Mathematics |
| EISSN | 1099-1476 |
| EndPage | 3720 |
| ExternalDocumentID | 10_1002_mma_8717 MMA8717 |
| Genre | article |
| GrantInformation_xml | – fundername: Shandong Provincial Natural Science Foundation of China funderid: ZR2020MA020; ZR2021MA003 – fundername: Fundamental Research Funds for the Central Universities funderid: 202111016 |
| GroupedDBID | -~X .3N .GA 05W 0R~ 10A 1L6 1OB 1OC 1ZS 33P 3SF 3WU 4.4 50Y 50Z 51W 51X 52M 52N 52O 52P 52S 52T 52U 52W 52X 5GY 5VS 66C 702 7PT 8-0 8-1 8-3 8-4 8-5 8UM 930 A03 AAESR AAEVG AAHHS AAHQN AAMNL AANLZ AAONW AAXRX AAYCA AAZKR ABCQN ABCUV ABIJN ABJNI ABPVW ACAHQ ACCFJ ACCZN ACGFS ACIWK ACPOU ACXBN ACXQS ADBBV ADEOM ADIZJ ADKYN ADMGS ADOZA ADXAS ADZMN AEEZP AEIGN AEIMD AENEX AEQDE AEUQT AEUYR AFBPY AFFPM AFGKR AFPWT AFWVQ AHBTC AITYG AIURR AIWBW AJBDE AJXKR ALAGY ALMA_UNASSIGNED_HOLDINGS ALUQN ALVPJ AMBMR AMYDB ATUGU AUFTA AZBYB AZVAB BAFTC BFHJK BHBCM BMNLL BMXJE BNHUX BROTX BRXPI BY8 CO8 CS3 D-E D-F DCZOG DPXWK DR2 DRFUL DRSTM DU5 EBS F00 F01 F04 F5P G-S G.N GNP GODZA H.T H.X HBH HGLYW HHY HZ~ IX1 J0M JPC KQQ LATKE LAW LC2 LC3 LEEKS LH4 LITHE LOXES LP6 LP7 LUTES LYRES MEWTI MK4 MRFUL MRSTM MSFUL MSSTM MXFUL MXSTM N04 N05 NF~ O66 O9- OIG P2P P2W P2X P4D Q.N Q11 QB0 QRW R.K ROL RWI RWS RX1 RYL SUPJJ UB1 V2E W8V W99 WBKPD WH7 WIB WIH WIK WOHZO WQJ WRC WXSBR WYISQ XBAML XG1 XPP XV2 ZZTAW ~02 ~IA ~WT AAYXX AEYWJ AGHNM AGYGG AMVHM CITATION 7TB 8FD AAMMB AEFGJ AGXDD AIDQK AIDYY FR3 JQ2 KR7 |
| ID | FETCH-LOGICAL-c2937-a9a78e7ec88cad4b5325984087c9e0250bf62e157f19144709671ae44f98e1913 |
| IEDL.DBID | DR2 |
| ISSN | 0170-4214 |
| IngestDate | Fri Jul 25 12:12:02 EDT 2025 Tue Jul 01 03:02:47 EDT 2025 Thu Apr 24 23:08:26 EDT 2025 Wed Jan 22 16:24:07 EST 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 4 |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c2937-a9a78e7ec88cad4b5325984087c9e0250bf62e157f19144709671ae44f98e1913 |
| Notes | Funding information Shandong Provincial Natural Science Foundation of China (ZR2021MA003 and ZR2020MA020) and the Fundamental Research Funds for the Central Universities (202111016). ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ORCID | 0000-0003-4136-5375 |
| PQID | 2773987240 |
| PQPubID | 1016386 |
| PageCount | 18 |
| ParticipantIDs | proquest_journals_2773987240 crossref_citationtrail_10_1002_mma_8717 crossref_primary_10_1002_mma_8717 wiley_primary_10_1002_mma_8717_MMA8717 |
| ProviderPackageCode | CITATION AAYXX |
| PublicationCentury | 2000 |
| PublicationDate | 15 March 2023 |
| PublicationDateYYYYMMDD | 2023-03-15 |
| PublicationDate_xml | – month: 03 year: 2023 text: 15 March 2023 day: 15 |
| PublicationDecade | 2020 |
| PublicationPlace | Freiburg |
| PublicationPlace_xml | – name: Freiburg |
| PublicationTitle | Mathematical methods in the applied sciences |
| PublicationYear | 2023 |
| Publisher | Wiley Subscription Services, Inc |
| Publisher_xml | – name: Wiley Subscription Services, Inc |
| References | 2017; 64 2012; 142 2006; 52 2013; 26 2020; 63 2000; 9 2011; 62 2014; 27 2009 2020; 269 2013; 92 2020; 56 2013; 264 1996; 163 2018; 43 2016; 36 2014; 412 2003; 12 2021; 10 2010; 68 2015; 259 2018; 458 2002; 164 2006; 66 2009; 71 2019; 48 2017 2015 2021; 111 2014 2018; 78 2005; 18 2003; 286 e_1_2_8_28_1 e_1_2_8_29_1 e_1_2_8_24_1 e_1_2_8_25_1 e_1_2_8_26_1 e_1_2_8_27_1 Gazzola F (e_1_2_8_32_1) 2005; 18 e_1_2_8_3_1 e_1_2_8_2_1 e_1_2_8_5_1 e_1_2_8_4_1 e_1_2_8_7_1 e_1_2_8_6_1 e_1_2_8_9_1 e_1_2_8_8_1 e_1_2_8_20_1 e_1_2_8_21_1 e_1_2_8_22_1 e_1_2_8_23_1 e_1_2_8_17_1 e_1_2_8_18_1 e_1_2_8_19_1 e_1_2_8_13_1 e_1_2_8_14_1 e_1_2_8_35_1 e_1_2_8_15_1 e_1_2_8_16_1 e_1_2_8_10_1 e_1_2_8_31_1 e_1_2_8_11_1 e_1_2_8_34_1 e_1_2_8_12_1 e_1_2_8_33_1 e_1_2_8_30_1 |
| References_xml | – volume: 43 start-page: 451 year: 2018 end-page: 466 article-title: A class of fourth‐order parabolic equation with arbitrary initial energy publication-title: Nonlinear Anal Real World Appl – year: 2009 – volume: 264 start-page: 2732 year: 2013 end-page: 2763 article-title: Global existence and finite time blow‐up for a class of semilinear pseudo‐parabolic equations publication-title: J Funct Anal – volume: 68 start-page: 459 year: 2010 end-page: 468 article-title: Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data publication-title: Q Appl Math – volume: 36 start-page: 715 year: 2016 end-page: 730 article-title: Finite‐time blow‐up and extinction rates of solutions to an initial Neumann problem involving the ‐Laplace operator and a non‐local term publication-title: Disc Cont Dyn Syst – volume: 412 start-page: 326 year: 2014 end-page: 333 article-title: Blow‐up versus extinction in a nonlocal ‐Laplace equation with Neumann boundary conditions publication-title: J Math Anal Appl – volume: 52 start-page: 19 year: 2006 end-page: 36 article-title: On stationary thermo‐rheological viscous flows publication-title: Ann Univ Ferrara Sez VII Sci Mat – volume: 164 start-page: 213 year: 2002 end-page: 259 article-title: Regularity results for stationary electro‐rheological fluids publication-title: Arch Ration Mech Anal – volume: 56 start-page: 103155 year: 2020 article-title: Potential well method for ‐Laplacian equations with variable exponent sources publication-title: Nonlinear Anal Real World Appl – volume: 10 start-page: 261 year: 2021 end-page: 288 article-title: Global existence and finite time blowup for a nonlocal semilinear pseudo‐parabolic equation publication-title: Ad Nonlinear Anal – volume: 62 start-page: 909 year: 2011 end-page: 926 article-title: Study of weak solutions for a fourth‐order parabolic equation with variable exponent of nonlinearity publication-title: Z Angew Math Phys – volume: 142 start-page: 1027 year: 2012 end-page: 1042 article-title: Critical exponents for a semilinear parabolic equation with variable reaction publication-title: Proc Roy Soc Edinburgh Sect A – volume: 64 start-page: 67 year: 2017 end-page: 73 article-title: Blow‐up phenomena for a pseudo‐parabolic equation with variable exponents publication-title: Appl Math Lett – volume: 48 start-page: 54 year: 2019 end-page: 70 article-title: Global asymptotical behavior of solutions to a class of fourth order parabolic equation modeling epitaxial growth publication-title: Nonlinear Anal Real World Appl – year: 2014 – volume: 9 start-page: 1723 year: 2000 end-page: 1730 article-title: Fourth‐order partial differential equations for noise removal publication-title: IEEE Trans Image Process – volume: 163 start-page: 8 year: 1996 end-page: 21 article-title: Some causes and a consequence of epitaxial roughening publication-title: J Cryst Growth – volume: 92 start-page: 651 year: 2013 end-page: 664 article-title: Asymptotic analysis for blow‐up solutions in parabolic equations involving variable exponents publication-title: Appl Anal – volume: 71 start-page: 1094 year: 2009 end-page: 1099 article-title: Blow‐up for parabolic and hyperbolic problems with variable exponents publication-title: Nonlinear Anal – volume: 63 start-page: 321 year: 2020 end-page: 356 article-title: Global well‐posedness of coupled parabolic systems publication-title: Science China Math – volume: 458 start-page: 9 year: 2018 end-page: 20 article-title: Finite time blow‐up for a thin‐film equation with initial data at arbitrary energy level publication-title: J Math Anal Appl – volume: 48 start-page: 383 year: 2019 end-page: 409 article-title: A nonlinear diffusion problem with convection and anisotropic nonstandard growth conditions publication-title: Nonlinear Anal Real World Appl – volume: 66 start-page: 1383 year: 2006 end-page: 1406 article-title: Variable exponent, linear growth functionals in image restoration publication-title: SIAM J Math Appl – volume: 269 start-page: 4914 year: 2020 end-page: 4959 article-title: Global existence and blow up of solutions for pseudo‐parabolic equation with singular potential publication-title: J Differ Equ – volume: 26 start-page: 539 year: 2013 end-page: 543 article-title: Blow‐up of solutions for a semilinear parabolic equation involving variable source and positive initial energy publication-title: Appl Math Lett – volume: 286 start-page: 459 year: 2003 end-page: 490 article-title: A fourth‐order parabolic equation modeling epitaxial thin‐film growth publication-title: J Math Anal Appl – volume: 111 start-page: 106664 year: 2021 article-title: A complete classification of initial energy in a ‐Laplace pseudo‐parabolic equation publication-title: Appl Math Lett – volume: 18 start-page: 961 year: 2005 end-page: 990 article-title: Finite time blow up and global solutions for semilinear parabolic equations with initial data at high energy level publication-title: Differ Integr Equ – year: 2017 – volume: 27 start-page: 49 year: 2014 end-page: 52 article-title: Lower bounds for the blow‐up time in a semilinear parabolic problem involving a variable source publication-title: Appl Math Lett – volume: 78 start-page: 141 year: 2018 end-page: 146 article-title: Asymptotic behavior for a fourth‐order parabolic equation modeling thin film growth publication-title: Appl Math Lett – volume: 259 start-page: 7260 year: 2015 end-page: 7283 article-title: Solutions of fourth‐order parabolic equation modeling thin film growth publication-title: J Differ Equ – year: 2015 – volume: 12 start-page: 1579 year: 2003 end-page: 1590 article-title: Noise removal using fourth‐order partial differential equation with applications to medical magnetic resonance images in space and time publication-title: IEEE Trans Image Process – ident: e_1_2_8_8_1 doi: 10.1016/j.jde.2015.08.022 – ident: e_1_2_8_3_1 doi: 10.1109/TIP.2003.819229 – volume: 18 start-page: 961 year: 2005 ident: e_1_2_8_32_1 article-title: Finite time blow up and global solutions for semilinear parabolic equations with initial data at high energy level publication-title: Differ Integr Equ – ident: e_1_2_8_35_1 doi: 10.2991/978-94-6239-112-3 – ident: e_1_2_8_5_1 doi: 10.1016/j.nonrwa.2018.03.009 – ident: e_1_2_8_17_1 doi: 10.1016/j.aml.2012.12.017 – ident: e_1_2_8_24_1 doi: 10.1016/j.camwa.2018.01.047 – ident: e_1_2_8_6_1 doi: 10.1016/S0022-247X(03)00474-8 – ident: e_1_2_8_29_1 doi: 10.1515/anona-2020-0141 – ident: e_1_2_8_20_1 doi: 10.1016/j.aml.2016.08.013 – ident: e_1_2_8_26_1 doi: 10.1016/j.nonrwa.2020.103155 – ident: e_1_2_8_12_1 doi: 10.1007/s00205-002-0208-7 – ident: e_1_2_8_18_1 doi: 10.1016/j.aml.2013.07.012 – ident: e_1_2_8_33_1 doi: 10.1016/j.jfa.2013.03.010 – ident: e_1_2_8_28_1 doi: 10.1016/j.jde.2020.03.047 – ident: e_1_2_8_15_1 doi: 10.1016/j.na.2008.11.030 – ident: e_1_2_8_22_1 doi: 10.1007/s00033-011-0148-x – ident: e_1_2_8_30_1 doi: 10.1007/s11425-017-9280-x – ident: e_1_2_8_2_1 doi: 10.1016/j.aml.2017.11.015 – ident: e_1_2_8_10_1 doi: 10.1016/0022-0248(95)01048-3 – ident: e_1_2_8_11_1 doi: 10.1016/j.nonrwa.2019.01.001 – ident: e_1_2_8_16_1 doi: 10.1017/S0308210510000399 – ident: e_1_2_8_4_1 doi: 10.1109/83.869184 – ident: e_1_2_8_9_1 doi: 10.1016/j.jmaa.2017.08.047 – ident: e_1_2_8_19_1 doi: 10.1080/00036811.2011.632767 – ident: e_1_2_8_27_1 doi: 10.1016/j.aml.2020.106664 – ident: e_1_2_8_13_1 doi: 10.1007/s11565-006-0002-9 – ident: e_1_2_8_25_1 doi: 10.1016/j.nonrwa.2019.01.020 – ident: e_1_2_8_21_1 doi: 10.3934/dcds.2016.36.715 – ident: e_1_2_8_7_1 doi: 10.1016/j.jmaa.2013.10.040 – ident: e_1_2_8_34_1 doi: 10.1201/b17415 – ident: e_1_2_8_23_1 – ident: e_1_2_8_31_1 doi: 10.1090/S0033-569X-2010-01197-0 – ident: e_1_2_8_14_1 doi: 10.1137/050624522 |
| SSID | ssj0008112 |
| Score | 2.3148582 |
| Snippet | This paper deals with a fourth‐order parabolic equation involving variable exponents, subject to Navier boundary conditions. We firstly give an optimal... |
| SourceID | proquest crossref wiley |
| SourceType | Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 3703 |
| SubjectTerms | Applications of mathematics asymptotic behavior blow‐up Boundary conditions Classification extinction Fluid flow higher order parabolic equation Mathematical analysis potential well method |
| Title | Energy classification in a nonstandard fourth‐order parabolic equation with a Navier boundary condition |
| URI | https://onlinelibrary.wiley.com/doi/abs/10.1002%2Fmma.8717 https://www.proquest.com/docview/2773987240 |
| Volume | 46 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnZ3PS8MwFMeD7KQHdVNxOiWC-OPQbWnTJT0O2RhCdxAHAw8l6VIU3dStu3jyT_Bv9C_xvbTdVBTEU6EkbZqX13yTvHxCyHHCWMs1mjuoHRyeNMGlWlI7gWDGE0nCPRtEE_ZbvQG_HPrDPKoS98JkfIjFhBt6hv1fo4MrPWssoaHjsaqD2seN5BiqhXroakmOkswudCIdxuEu4wV3tuk2ioxfe6KlvPwsUm0v090gN0X5suCS-_o81fX45Ru68X8fsEnWc_FJ21lrKZMVM6mQtXBBbp1VSDl39hk9y4nU51vkrmN3CNIYpTbGFllz0rsJVXSSCUycjqAJvCC9fX99s0BPilhxjdxhap4zojjFaV_I1FfYHVNtz3SawoMfceUcEmyTQbdzfdFz8jManBiEgnBUoIQ0wsRSxmrEte_BeAoGjVLEgUF9pRNoC8wXCYLkuIARk2DKcJ4E0sAtb4eUoKBml9BAqliOAiNxtdaIplQBtzwzDqKWjfwqOS3sFcU5wBzP0XiIMvSyG0GNRlijVXK0SPmUQTt-SFMrTB7lbjuLXCG8QApQOVVyYm33a_4oDNt43ftrwn2yikfVY_wa82uklE7n5gAETaoPbdP9AAhL8gs |
| linkProvider | Wiley-Blackwell |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnZ3JTsMwEIZHUA7AgR1RViMhlkPaOnFqR5wQApUlPSCQOCBFTuoIBBRoy4UTj8Az8iTMOEkLCCTEKVJkJ47tif_x8g3ARsp53TWxcEg7OCKtoUnVVewEkhtPpqnw7CaasFlvXIjjS_9yCHaLszAZH6I_4UaWYf_XZOA0IV0dUEPv73UF5b4chhEK6G39qbMBO0pxu9RJfBhHuFwU5NmaWy1yfh2LBgLzs0y148zhJFwVJcy2l9xWnntxJXn5Bm_85ydMwUSuP9le1mGmYci0Z2A87MNbuzMwndt7l23nUOqdWbg5sIcEWUJqm7YX2RZlN22mWTvTmDQjwVJ8Qe_6_fXNMj0ZkcVjQg8z85RBxRnN_GKmpqYRmcU2rFMHH_xAi-eYYA4uDg_O9xtOHqbBSVArSEcHWiojTaJUolsi9j10qdBvVDIJDEmsOMXuwH2ZEktOSHSaJNdGiDRQBm9581DCgpoFYIHSiWoFRtGCrZE1pQNhkWYCdS1v-WXYKhosSnKGOYXSuIsy-rIbYY1GVKNlWO-nfMy4HT-kWS7aPMottxu5UnqBkih0yrBpG-_X_FEY7tF18a8J12C0cR6eRqdHzZMlGKPI9bSdjfvLUOp1ns0K6ptevGr78QchSPYm |
| linkToPdf | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnZ1JS8QwFMcfLiB6cBfHNYK4HDpO2nSSHkUd3GYQURA8lLSToKjjMuPFkx_Bz-gn8b20nVFREE-FkrRpXl7zT_LyC8Cq5bzqm0R4pB08YSvoUlWVeJHkJpDWisAF0dQb1f1zcXgRXuRRlbQXJuNDdCfcyDPc_5oc_KFpt3rQ0Ls7XUa1L_thUFQrilr07mkPHaW4W-kkPIwnfC4K8GzF3ypyfu2Kevrys0p13UxtDC6LAmbRJTfl505STl--sRv_9wXjMJqrT7adNZcJ6DOtSRipd9Gt7UmYyL29zTZyJPXmFFzvuS2CLCWtTcFFzp7susU0a2UKk-YjmMUXdK7eX98c0ZMRVzwh8DAzjxlSnNG8L2ZqaOqPWeIOdXrCB9_T0jkmmIbz2t7Zzr6XH9LgpagUpKcjLZWRJlUq1U2RhAEOqHDUqGQaGRJYicXGwENpiSQnJA6ZJNdGCBspg7eCGRjAgppZYJHSqWpGRtFyrZEVpSPhgGYCVS1vhiVYL-wVpznBnA7SuI0z9rIfY43GVKMlWOmmfMioHT-kWShMHud-2459KYNISZQ5JVhztvs1f1yvb9N17q8Jl2HoZLcWHx80juZhmI6tp1g2Hi7AQOfp2SyiuOkkS64VfwBoHfTe |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Energy+classification+in+a+nonstandard+fourth%E2%80%90order+parabolic+equation+with+a+Navier+boundary+condition&rft.jtitle=Mathematical+methods+in+the+applied+sciences&rft.au=Liu%2C+Bingchen&rft.au=Sun%2C+Xizheng&rft.au=Wang%2C+Yiming&rft.date=2023-03-15&rft.issn=0170-4214&rft.eissn=1099-1476&rft.volume=46&rft.issue=4&rft.spage=3703&rft.epage=3720&rft_id=info:doi/10.1002%2Fmma.8717&rft.externalDBID=10.1002%252Fmma.8717&rft.externalDocID=MMA8717 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0170-4214&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0170-4214&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0170-4214&client=summon |