Pub Date : 2023-10-17DOI: 10.1007/s00028-023-00917-8
J. M. Mazón, A. Molino, J. Toledo
Abstract This paper is concerned with the Neumann problem for a class of doubly nonlinear equations for the 1-Laplacian, $$begin{aligned} frac{partial v}{partial t} - Delta _1 u ni 0 hbox { in } (0, infty ) times Omega , quad vin gamma (u), end{aligned}$$ ∂v∂t-Δ1u∋0in(0,∞)×Ω,v∈γ(u), and initial data in $$L^1(Omega )$$ L1(Ω) , where $$Omega $$ Ω is a bounded smooth domain in $${mathbb {R}}^N$$ RN and $$gamma $$ γ is a maximal monotone graph in $${mathbb {R}}times {mathbb {R}}$$ R×R . We prove that, under certain assumptions on the graph $$gamma $$ γ , there is existence and uniqueness of solutions. Moreover, we proof that these solutions coincide with the ones of the Neumann problem for the total variational flow. We show that such assumptions are necessary.
研究一类双非线性1-拉普拉斯方程的Neumann问题。 $$begin{aligned} frac{partial v}{partial t} - Delta _1 u ni 0 hbox { in } (0, infty ) times Omega , quad vin gamma (u), end{aligned}$$ ∂v∂t - Δ 1 u∈0 in(0，∞)× Ω， v∈γ (u)，初始数据in $$L^1(Omega )$$ l1 (Ω)，其中 $$Omega $$ 中的有界光滑域Ω $${mathbb {R}}^N$$ R N和 $$gamma $$ γ是中的极大单调图 $${mathbb {R}}times {mathbb {R}}$$ R × R。我们在图上的某些假设下证明了这一点 $$gamma $$ γ，解存在唯一性。此外，我们证明了这些解与总变分流的诺伊曼问题的解是一致的。我们证明这些假设是必要的。
{"title":"Doubly nonlinear equations for the 1-Laplacian","authors":"J. M. Mazón, A. Molino, J. Toledo","doi":"10.1007/s00028-023-00917-8","DOIUrl":"https://doi.org/10.1007/s00028-023-00917-8","url":null,"abstract":"Abstract This paper is concerned with the Neumann problem for a class of doubly nonlinear equations for the 1-Laplacian, $$begin{aligned} frac{partial v}{partial t} - Delta _1 u ni 0 hbox { in } (0, infty ) times Omega , quad vin gamma (u), end{aligned}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>v</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>∋</mml:mo> <mml:mn>0</mml:mn> <mml:mspace /> <mml:mtext>in</mml:mtext> <mml:mspace /> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mi>v</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>γ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> and initial data in $$L^1(Omega )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , where $$Omega $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>Ω</mml:mi> </mml:math> is a bounded smooth domain in $${mathbb {R}}^N$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:math> and $$gamma $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>γ</mml:mi> </mml:math> is a maximal monotone graph in $${mathbb {R}}times {mathbb {R}}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>×</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> . We prove that, under certain assumptions on the graph $$gamma $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>γ</mml:mi> </mml:math> , there is existence and uniqueness of solutions. Moreover, we proof that these solutions coincide with the ones of the Neumann problem for the total variational flow. We show that such assumptions are necessary.","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"22 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135994235","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-14DOI: 10.1007/s00028-023-00915-w
Mikhail Anikushin
{"title":"Frequency theorem and inertial manifolds for neutral delay equations","authors":"Mikhail Anikushin","doi":"10.1007/s00028-023-00915-w","DOIUrl":"https://doi.org/10.1007/s00028-023-00915-w","url":null,"abstract":"","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"117 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135800300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-06DOI: 10.1007/s00028-023-00918-7
Ning-An Lai, Nico Michele Schiavone
{"title":"Lifespan estimates for the compressible Euler equations with damping via Orlicz spaces techniques","authors":"Ning-An Lai, Nico Michele Schiavone","doi":"10.1007/s00028-023-00918-7","DOIUrl":"https://doi.org/10.1007/s00028-023-00918-7","url":null,"abstract":"","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135352549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-30DOI: 10.1007/s00028-023-00913-y
Kexin Yan, Hao Wang, Ying Fu
{"title":"Ill-posedness of the Cauchy problem for the $$mu $$-Camassa–Holm-type equations","authors":"Kexin Yan, Hao Wang, Ying Fu","doi":"10.1007/s00028-023-00913-y","DOIUrl":"https://doi.org/10.1007/s00028-023-00913-y","url":null,"abstract":"","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"89 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136342605","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-27DOI: 10.1007/s00028-023-00911-0
M. S. Alves, R. N. Monteiro
{"title":"Stability for a composite of Timoshenko laminated beams","authors":"M. S. Alves, R. N. Monteiro","doi":"10.1007/s00028-023-00911-0","DOIUrl":"https://doi.org/10.1007/s00028-023-00911-0","url":null,"abstract":"","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135535712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-15DOI: 10.1007/s00028-023-00912-z
O. El Mennaoui, S. Hadd, Y. Kharou
{"title":"On the admissibility of observation operators in the context of maximal regularity","authors":"O. El Mennaoui, S. Hadd, Y. Kharou","doi":"10.1007/s00028-023-00912-z","DOIUrl":"https://doi.org/10.1007/s00028-023-00912-z","url":null,"abstract":"","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135397030","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-21DOI: 10.1007/s00028-023-00910-1
N. Hayashi, P. Naumkin
{"title":"Modified scattering for the fractional mKdV equation","authors":"N. Hayashi, P. Naumkin","doi":"10.1007/s00028-023-00910-1","DOIUrl":"https://doi.org/10.1007/s00028-023-00910-1","url":null,"abstract":"","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"10 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75359371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-14DOI: 10.1007/s00028-023-00908-9
M. Sobajima
{"title":"Weighted energy method for semilinear wave equations with time-dependent damping","authors":"M. Sobajima","doi":"10.1007/s00028-023-00908-9","DOIUrl":"https://doi.org/10.1007/s00028-023-00908-9","url":null,"abstract":"","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"74 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74567847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-05DOI: 10.1007/s00028-023-00899-7
A. Audrito
{"title":"On the existence and Hölder regularity of solutions to some nonlinear Cauchy–Neumann problems","authors":"A. Audrito","doi":"10.1007/s00028-023-00899-7","DOIUrl":"https://doi.org/10.1007/s00028-023-00899-7","url":null,"abstract":"","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"1 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82324536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-25DOI: 10.1007/s00028-023-00907-w
N. Carreño, Cristóbal Loyola
{"title":"An explicit time for the uniform null controllability of a linear Korteweg-de Vries equation","authors":"N. Carreño, Cristóbal Loyola","doi":"10.1007/s00028-023-00907-w","DOIUrl":"https://doi.org/10.1007/s00028-023-00907-w","url":null,"abstract":"","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"24 5","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72371708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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