In this paper, we discuss the Hyers–Ulam stability of closable (unbounded) operators with some examples. We also present results pertaining to the Hyers–Ulam stability of the sum and product of closable operators to have the Hyers–Ulam stability and the necessary and sufficient conditions of the Schur complement and the quadratic complement of block matrix in order to have the Hyers–Ulam stability.
{"title":"Hyers–Ulam stability of unbounded closable operators in Hilbert spaces","authors":"Arup Majumdar, P. Sam Johnson, Ram N. Mohapatra","doi":"10.1002/mana.202300484","DOIUrl":"10.1002/mana.202300484","url":null,"abstract":"<p>In this paper, we discuss the Hyers–Ulam stability of closable (unbounded) operators with some examples. We also present results pertaining to the Hyers–Ulam stability of the sum and product of closable operators to have the Hyers–Ulam stability and the necessary and sufficient conditions of the Schur complement and the quadratic complement of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mo>×</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$2 times 2$</annotation>\u0000 </semantics></math> block matrix <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$mathcal {A}$</annotation>\u0000 </semantics></math> in order to have the Hyers–Ulam stability.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 10","pages":"3887-3903"},"PeriodicalIF":0.8,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142176981","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}
{"title":"Planar Choquard equations with critical exponential reaction and Neumann boundary condition","authors":"Sushmita Rawat, Vicenţiu D. Rădulescu, K. Sreenadh","doi":"10.1002/mana.202400095","DOIUrl":"10.1002/mana.202400095","url":null,"abstract":"<p>We study the existence of positive weak solutions for the following problem:\u0000\u0000 </p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 10","pages":"3847-3869"},"PeriodicalIF":0.8,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202400095","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141922265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we investigate the stabilization of the transmission problem of the degenerate wave equation and the heat equation under the Coleman–Gurtin heat conduction law or Gurtin–Pipkin law with the memory effect. We investigate the polynomial stability of this system when employing the Coleman–Gurtin heat conduction, establishing a decay rate of type . Next, we demonstrate exponential stability in the case when Gurtin–Pipkin heat conduction is applied.
{"title":"The energy decay rate of a transmission system governed by the degenerate wave equation with drift and under heat conduction with the memory effect","authors":"Mohammad Akil, Genni Fragnelli, Ibtissam Issa","doi":"10.1002/mana.202300571","DOIUrl":"10.1002/mana.202300571","url":null,"abstract":"<p>In this paper, we investigate the stabilization of the transmission problem of the degenerate wave equation and the heat equation under the Coleman–Gurtin heat conduction law or Gurtin–Pipkin law with the memory effect. We investigate the polynomial stability of this system when employing the Coleman–Gurtin heat conduction, establishing a decay rate of type <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>t</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$t^{-4}$</annotation>\u0000 </semantics></math>. Next, we demonstrate exponential stability in the case when Gurtin–Pipkin heat conduction is applied.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 10","pages":"3766-3796"},"PeriodicalIF":0.8,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141871436","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}
The work deals with the Ericksen–Leslie system for nematic liquid crystals on the space with . In our work, we suppose the initial condition stays on an arc connecting two fixed orthogonal vectors on the unit sphere. Thanks to this geometric assumption, we prove through energy a priori estimates the local existence and the global existence for small initial data of a solution
{"title":"Local and global solutions on arcs for the Ericksen–Leslie problem in \u0000 \u0000 \u0000 R\u0000 N\u0000 \u0000 $mathbb {R}^N$","authors":"Daniele Barbera, Vladimir Georgiev","doi":"10.1002/mana.202300253","DOIUrl":"10.1002/mana.202300253","url":null,"abstract":"<p>The work deals with the Ericksen–Leslie system for nematic liquid crystals on the space <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>N</mi>\u0000 </msup>\u0000 <annotation>$mathbb {R}^N$</annotation>\u0000 </semantics></math> with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 <mo>≥</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$Nge 3$</annotation>\u0000 </semantics></math>. In our work, we suppose the initial condition <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>v</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <annotation>$v_0$</annotation>\u0000 </semantics></math> stays on an arc connecting two fixed orthogonal vectors on the unit sphere. Thanks to this geometric assumption, we prove through energy a priori estimates the local existence and the global existence for small initial data of a solution\u0000\u0000 </p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 10","pages":"3584-3624"},"PeriodicalIF":0.8,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141871440","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}
<p>In this paper, we focus on the well-posedness, blow-up phenomena, and continuity of the data-to-solution map of the Cauchy problem for a two-component higher order Camassa–Holm (CH) system. The local well-posedness is established in Besov spaces <span></span><math> <semantics> <mrow> <msubsup> <mi>B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mn>1</mn> </mrow> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </msubsup> <mo>×</mo> <msubsup> <mi>B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mo>+</mo> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> </msubsup> </mrow> <annotation>$B_{p,1}^{frac{1}{p}} times B_{p,1}^{2+frac{1}{p}}$</annotation> </semantics></math> with <span></span><math> <semantics> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> <annotation>$1 le p < infty$</annotation> </semantics></math>, which improves the local well-posedness result proved before in Tang and Liu [Z. Angew. Math. Phys. 66 (2015), 1559–1580], Ye and Yin [arXiv preprint arXiv:2109.00948 (2021)], Zhang and Li [Nonlinear Anal. Real World Appl. 35 (2017), 414–440], and Zhou [Math. Nachr. 291 (2018), no. 10, 1595–1619]. Next, we consider the continuity of the solution-to-data map, that is, the ill-posedness is derived in Besov space <span></span><math> <semantics> <mrow> <msubsup> <mi>B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>∞</mi> </mrow> <mrow> <mi>s</mi> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> <mo>×</mo> <msubsup> <mi>B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>∞</mi> </mrow> <mi>s</mi> </msubsup> </mrow> <annotation>$B_{p,infty }^{s
{"title":"On the Cauchy problem for a two-component higher order Camassa–Holm system","authors":"Shouming Zhou, Luhang Zhou, Rong Chen","doi":"10.1002/mana.202300382","DOIUrl":"10.1002/mana.202300382","url":null,"abstract":"<p>In this paper, we focus on the well-posedness, blow-up phenomena, and continuity of the data-to-solution map of the Cauchy problem for a two-component higher order Camassa–Holm (CH) system. The local well-posedness is established in Besov spaces <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>B</mi>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <mfrac>\u0000 <mn>1</mn>\u0000 <mi>p</mi>\u0000 </mfrac>\u0000 </msubsup>\u0000 <mo>×</mo>\u0000 <msubsup>\u0000 <mi>B</mi>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mo>+</mo>\u0000 <mfrac>\u0000 <mn>1</mn>\u0000 <mi>p</mi>\u0000 </mfrac>\u0000 </mrow>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation>$B_{p,1}^{frac{1}{p}} times B_{p,1}^{2+frac{1}{p}}$</annotation>\u0000 </semantics></math> with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>≤</mo>\u0000 <mi>p</mi>\u0000 <mo><</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation>$1 le p &lt; infty$</annotation>\u0000 </semantics></math>, which improves the local well-posedness result proved before in Tang and Liu [Z. Angew. Math. Phys. 66 (2015), 1559–1580], Ye and Yin [arXiv preprint arXiv:2109.00948 (2021)], Zhang and Li [Nonlinear Anal. Real World Appl. 35 (2017), 414–440], and Zhou [Math. Nachr. 291 (2018), no. 10, 1595–1619]. Next, we consider the continuity of the solution-to-data map, that is, the ill-posedness is derived in Besov space <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>B</mi>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>,</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 <mo>−</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msubsup>\u0000 <mo>×</mo>\u0000 <msubsup>\u0000 <mi>B</mi>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>,</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <mi>s</mi>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation>$B_{p,infty }^{s","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 10","pages":"3797-3834"},"PeriodicalIF":0.8,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141871439","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}
{"title":"Boundedness in a quasilinear forager–exploiter model","authors":"Jianping Wang, Qianying Zhang","doi":"10.1002/mana.202300507","DOIUrl":"10.1002/mana.202300507","url":null,"abstract":"<p>We study a forager–exploiter model with nonlinear diffusions:\u0000\u0000 </p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 10","pages":"3741-3765"},"PeriodicalIF":0.8,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141871437","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}
We obtain a complete description of the spectrum of the Fréchet algebra of symmetric analytic functions bounded on balls on the sequence space . This is achieved after proving that on the analogous algebra for , , the radius function of any evaluation homomorphism , coincides with the norm of .
{"title":"On the spectrum of the algebra of bounded-type symmetric analytic functions on \u0000 \u0000 \u0000 ℓ\u0000 1\u0000 \u0000 $ell _1$","authors":"Iryna Chernega, Pablo Galindo, Andriy Zagorodnyuk","doi":"10.1002/mana.202300415","DOIUrl":"10.1002/mana.202300415","url":null,"abstract":"<p>We obtain a complete description of the spectrum of the Fréchet algebra of symmetric analytic functions bounded on balls on the sequence space <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℓ</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <annotation>$ell _1$</annotation>\u0000 </semantics></math>. This is achieved after proving that on the analogous algebra for <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℓ</mi>\u0000 <mi>p</mi>\u0000 </msub>\u0000 <annotation>$ell _p$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>≤</mo>\u0000 <mi>p</mi>\u0000 <mo><</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation>$1le p &lt;infty$</annotation>\u0000 </semantics></math>, the radius function of any evaluation homomorphism <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>δ</mi>\u0000 <mi>x</mi>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <mi>x</mi>\u0000 <mo>∈</mo>\u0000 <msub>\u0000 <mi>ℓ</mi>\u0000 <mi>p</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$delta _x, nobreakspace x in ell _p$</annotation>\u0000 </semantics></math>, coincides with the norm of <span></span><math>\u0000 <semantics>\u0000 <mi>x</mi>\u0000 <annotation>$x$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 10","pages":"3835-3846"},"PeriodicalIF":0.8,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141871438","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}
In this paper, we establish a weighted Kato–Ponce inequality for factors in the endpoint case. Furthermore, we extend the validity of the Kato–Ponce inequality from the class of Schwartz functions to the broader class of functions living in a (weighted) fractional Sobolev space.
{"title":"Weighted Kato–Ponce inequalities for multiple factors","authors":"Sean Douglas, Loukas Grafakos","doi":"10.1002/mana.202300443","DOIUrl":"10.1002/mana.202300443","url":null,"abstract":"<p>In this paper, we establish a weighted Kato–Ponce inequality for <span></span><math>\u0000 <semantics>\u0000 <mi>m</mi>\u0000 <annotation>$m$</annotation>\u0000 </semantics></math> factors in the endpoint case. Furthermore, we extend the validity of the Kato–Ponce inequality from the class of Schwartz functions to the broader class of functions living in a (weighted) fractional Sobolev space.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 10","pages":"3700-3722"},"PeriodicalIF":0.8,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780450","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}
Vladimir Gol'dshtein, Paz Hashash, Alexander Ukhlov
We study connections between the -differentiability and the -differentiability of Sobolev functions. We prove that -differentiability implies the -differentiability, but the opposite implication is not valid. The notion of approximate differentiability is discussed as well. In addition, we consider the -differentiability of Sobolev functions -almost everywhere.
{"title":"On differentiability of Sobolev functions with respect to the Sobolev norm","authors":"Vladimir Gol'dshtein, Paz Hashash, Alexander Ukhlov","doi":"10.1002/mana.202300545","DOIUrl":"10.1002/mana.202300545","url":null,"abstract":"<p>We study connections between the <span></span><math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>W</mi>\u0000 <mi>p</mi>\u0000 <mn>1</mn>\u0000 </msubsup>\u0000 <annotation>$W^1_p$</annotation>\u0000 </semantics></math>-differentiability and the <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <mi>p</mi>\u0000 </msub>\u0000 <annotation>$L_p$</annotation>\u0000 </semantics></math>-differentiability of Sobolev functions. We prove that <span></span><math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>W</mi>\u0000 <mi>p</mi>\u0000 <mn>1</mn>\u0000 </msubsup>\u0000 <annotation>$W^1_p$</annotation>\u0000 </semantics></math>-differentiability implies the <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <mi>p</mi>\u0000 </msub>\u0000 <annotation>$L_p$</annotation>\u0000 </semantics></math>-differentiability, but the opposite implication is not valid. The notion of approximate differentiability is discussed as well. In addition, we consider the <span></span><math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>W</mi>\u0000 <mi>p</mi>\u0000 <mn>1</mn>\u0000 </msubsup>\u0000 <annotation>$W^1_p$</annotation>\u0000 </semantics></math>-differentiability of Sobolev functions <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mo>cap</mo>\u0000 <mi>p</mi>\u0000 </msub>\u0000 <annotation>$operatorname{cap}_p$</annotation>\u0000 </semantics></math>-almost everywhere.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 10","pages":"3681-3699"},"PeriodicalIF":0.8,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202300545","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：