Pub Date : 2023-09-12DOI: 10.1007/s00030-023-00876-6
Leah Schätzler, Jarkko Siltakoski
Abstract In this paper, we study the Cauchy–Dirichlet problem $$begin{aligned} left{ begin{array}{ll} partial _t u - {text {div}} left( D_xi f(t, Du)right) = 0 &{} quad hbox {in} Omega _T, u = u_o &{} quad hbox { on} partial _{mathcal {P}} Omega _T, end{array} right. end{aligned}$$ ∂tu-divDξf(t,Du)=0inΩT,u=uoon∂PΩT, where $$Omega subset mathbb {R}^n$$ Ω⊂Rn is a convex and bounded domain, $$f:[0,T]times {mathbb {R}}^n rightarrow {mathbb {R}}$$ f:[0,T]×Rn→R is $$L^1$$ L1 -integrable in time and convex in the second variable. Assuming that the initial and boundary datum $$u_o:{overline{Omega }}rightarrow {mathbb {R}}$$ uo:Ω¯→R satisfies the bounded slope condition, we prove the exis
摘要本文研究了柯西-狄利克雷问题$$begin{aligned} left{ begin{array}{ll} partial _t u - {text {div}} left( D_xi f(t, Du)right) = 0 &{} quad hbox {in} Omega _T, u = u_o &{} quad hbox { on} partial _{mathcal {P}} Omega _T, end{array} right. end{aligned}$$∂tu - div D ξ f (t, du)在Ω t上= 0，在∂P Ω t上u = u o，其中$$Omega subset mathbb {R}^n$$ Ω∧R n是一个凸有界定域，$$f:[0,T]times {mathbb {R}}^n rightarrow {mathbb {R}}$$ f: [0, t] × R n→R是$$L^1$$ L 1 -在时间上可积，在第二变量上是凸的。假设初始和边界基准$$u_o:{overline{Omega }}rightarrow {mathbb {R}}$$ uo: Ω¯→R满足有界斜率条件，证明了在空间变量上存在唯一的Lipschitz连续变分解。
{"title":"The bounded slope condition for parabolic equations with time-dependent integrands","authors":"Leah Schätzler, Jarkko Siltakoski","doi":"10.1007/s00030-023-00876-6","DOIUrl":"https://doi.org/10.1007/s00030-023-00876-6","url":null,"abstract":"Abstract In this paper, we study the Cauchy–Dirichlet problem $$begin{aligned} left{ begin{array}{ll} partial _t u - {text {div}} left( D_xi f(t, Du)right) = 0 &{} quad hbox {in} Omega _T, u = u_o &{} quad hbox { on} partial _{mathcal {P}} Omega _T, end{array} right. end{aligned}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mfenced> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>-</mml:mo> <mml:mtext>div</mml:mtext> <mml:mfenced> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>ξ</mml:mi> </mml:msub> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mfenced> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:mtd> <mml:mtd> <mml:mrow> <mml:mrow /> <mml:mspace /> <mml:mtext>in</mml:mtext> <mml:mspace /> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>T</mml:mi> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mrow /> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>o</mml:mi> </mml:msub> </mml:mrow> </mml:mtd> <mml:mtd> <mml:mrow> <mml:mrow /> <mml:mspace /> <mml:mspace /> <mml:mtext>on</mml:mtext> <mml:mspace /> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>P</mml:mi> </mml:msub> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>T</mml:mi> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:mfenced> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> where $$Omega subset mathbb {R}^n$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> is a convex and bounded domain, $$f:[0,T]times {mathbb {R}}^n rightarrow {mathbb {R}}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>→</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> is $$L^1$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> -integrable in time and convex in the second variable. Assuming that the initial and boundary datum $$u_o:{overline{Omega }}rightarrow {mathbb {R}}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>o</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mover> <mml:mi>Ω</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>→</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> satisfies the bounded slope condition, we prove the exis","PeriodicalId":49747,"journal":{"name":"Nodea-Nonlinear Differential Equations and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135825657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-08DOI: 10.1007/s00030-023-00882-8
Burton Brown, Mathew Gluck, Vince Guingona, Thomas Hammons, Miriam Parnes, Sean Pooley, Avery Schweitzer
{"title":"The Brezis–Nirenberg problem for systems involving divergence-form operators","authors":"Burton Brown, Mathew Gluck, Vince Guingona, Thomas Hammons, Miriam Parnes, Sean Pooley, Avery Schweitzer","doi":"10.1007/s00030-023-00882-8","DOIUrl":"https://doi.org/10.1007/s00030-023-00882-8","url":null,"abstract":"","PeriodicalId":49747,"journal":{"name":"Nodea-Nonlinear Differential Equations and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79380522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-07DOI: 10.1007/s00030-023-00885-5
Wei Cheng, Jiahui Hong
{"title":"Correction to: Local strict singular characteristics II: existence for stationary equations on R2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin","authors":"Wei Cheng, Jiahui Hong","doi":"10.1007/s00030-023-00885-5","DOIUrl":"https://doi.org/10.1007/s00030-023-00885-5","url":null,"abstract":"","PeriodicalId":49747,"journal":{"name":"Nodea-Nonlinear Differential Equations and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87940520","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-29DOI: 10.1007/s00030-023-00873-9
M. Aouadi, Souad Guerine
{"title":"Observability and attractors of nonlinear Von Kármán beams","authors":"M. Aouadi, Souad Guerine","doi":"10.1007/s00030-023-00873-9","DOIUrl":"https://doi.org/10.1007/s00030-023-00873-9","url":null,"abstract":"","PeriodicalId":49747,"journal":{"name":"Nodea-Nonlinear Differential Equations and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82354290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-29DOI: 10.1007/s00030-023-00883-7
S. Hsu
{"title":"Asymptotic behaviour of blow-up solutions of the fast diffusion equation","authors":"S. Hsu","doi":"10.1007/s00030-023-00883-7","DOIUrl":"https://doi.org/10.1007/s00030-023-00883-7","url":null,"abstract":"","PeriodicalId":49747,"journal":{"name":"Nodea-Nonlinear Differential Equations and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78423887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-26DOI: 10.1007/s00030-023-00881-9
Chunpeng Wang, Yanan Zhou
{"title":"Carleman estimate and null controllability for a degenerate parabolic equation with a slightly superlinear reaction term","authors":"Chunpeng Wang, Yanan Zhou","doi":"10.1007/s00030-023-00881-9","DOIUrl":"https://doi.org/10.1007/s00030-023-00881-9","url":null,"abstract":"","PeriodicalId":49747,"journal":{"name":"Nodea-Nonlinear Differential Equations and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82953755","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-31DOI: 10.1007/s00030-023-00875-7
R. Molle, D. Passaseo
{"title":"Nonexistence results for elliptic problems with supercritical growth in thin planar domains","authors":"R. Molle, D. Passaseo","doi":"10.1007/s00030-023-00875-7","DOIUrl":"https://doi.org/10.1007/s00030-023-00875-7","url":null,"abstract":"","PeriodicalId":49747,"journal":{"name":"Nodea-Nonlinear Differential Equations and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86908639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-12DOI: 10.1007/s00030-023-00868-6
A. Buică
{"title":"Bifurcations from a normally degenerate cycle in forced planar differential equations","authors":"A. Buică","doi":"10.1007/s00030-023-00868-6","DOIUrl":"https://doi.org/10.1007/s00030-023-00868-6","url":null,"abstract":"","PeriodicalId":49747,"journal":{"name":"Nodea-Nonlinear Differential Equations and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89643724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-11DOI: 10.1007/s00030-023-00869-5
Yanyan Guo, B. Ruf
{"title":"A system of superlinear elliptic equations in a cylinder","authors":"Yanyan Guo, B. Ruf","doi":"10.1007/s00030-023-00869-5","DOIUrl":"https://doi.org/10.1007/s00030-023-00869-5","url":null,"abstract":"","PeriodicalId":49747,"journal":{"name":"Nodea-Nonlinear Differential Equations and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79546356","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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