Advanced Nonlinear Studies最新文献

{"title":"Solutions to the coupled Schrödinger systems with steep potential well and critical exponent","authors":"Zongyan Lv, Zhongwei Tang","doi":"10.1515/ans-2023-0149","DOIUrl":"https://doi.org/10.1515/ans-2023-0149","url":null,"abstract":"In the present paper, we consider the coupled Schrödinger systems with critical exponent:<jats:disp-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\" overflow=\"scroll\"> <m:mfenced close=\"\" open=\"{\"> <m:mrow> <m:mtable> <m:mtr> <m:mtd columnalign=\"left\"> <m:mo>−</m:mo> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mfenced close=\")\" open=\"(\"> <m:mrow> <m:mi>λ</m:mi> <m:msub> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfenced> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:munderover accent=\"true\" accentunder=\"false\"> <m:mrow> <m:mo>∑</m:mo> </m:mrow> <m:mrow> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:munderover> <m:msub> <m:mrow> <m:mi>β</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mfenced close=\"|\" open=\"|\"> <m:mrow> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mfenced close=\"|\" open=\"|\"> <m:mrow> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfenced> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mtext> </m:mtext> <m:mtext> in </m:mtext> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width=\"1em\"/> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=\"left\"> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width=\"1em\"/> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1,2</m:mn> <m:mo>,</m:mo> <m:mo>…</m:mo> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo>,</m:mo> <m:mspace width=\"1em\"/> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> <jats:tex-math>$$begin{cases}-{Delta}{u}_{i}+left(lambda {V}_{i}left(xright)+{a}_{i}right){u}_{i}=sum _{j=1}^{d}{beta }_{ij}{leftvert {u}_{j}rightvert }^{3}leftvert {u}_{i}rightvert {u}_{i}quad ,text{in},{mathbb{R}}^{3},quad hfill {u}_{i}in {H}^{1}left({mathbb{R}}^{N}right),quad i=1,2,dots ,d,quad hfill end{cases}$$</jats:tex-math> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xli","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"18 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Solitons to the Willmore flow","authors":"Pak Tung Ho, Juncheol Pyo","doi":"10.1515/ans-2023-0150","DOIUrl":"https://doi.org/10.1515/ans-2023-0150","url":null,"abstract":"The Willmore flow is the negative gradient flow of the Willmore energy. In this paper, we consider a special kind of solutions to Willmore flow, which we call solitons, and investigate their geometric properties.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"11 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142184674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Remarks on analytical solutions to compressible Navier–Stokes equations with free boundaries","authors":"Jianwei Dong, Manwai Yuen","doi":"10.1515/ans-2023-0146","DOIUrl":"https://doi.org/10.1515/ans-2023-0146","url":null,"abstract":"In this paper, we consider the free boundary problem of the radially symmetric compressible Navier–Stokes equations with viscosity coefficients of the form <jats:italic>μ</jats:italic>(<jats:italic>ρ</jats:italic>) = <jats:italic>ρ</jats:italic> <jats:sup> <jats:italic>θ</jats:italic> </jats:sup>, <jats:italic>λ</jats:italic>(<jats:italic>ρ</jats:italic>) = (<jats:italic>θ</jats:italic> − 1)<jats:italic>ρ</jats:italic> <jats:sup> <jats:italic>θ</jats:italic> </jats:sup> in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math> ${mathbb{R}}^{N}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0146_ineq_001.png\"/> </jats:alternatives> </jats:inline-formula>. Under the continuous density boundary condition, we correct some errors in (Z. H. Guo and Z. P. Xin, “Analytical solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients and free boundaries,” <jats:italic>J. Differ. Equ.</jats:italic>, vol. 253, no. 1, pp. 1–19, 2012) for <jats:italic>N</jats:italic> = 3, <jats:italic>θ</jats:italic> = <jats:italic>γ</jats:italic> &gt; 1 and improve the spreading rate of the free boundary, where <jats:italic>γ</jats:italic> is the adiabatic exponent. Moreover, we construct an analytical solution for <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mi>θ</m:mi> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:mfrac> </m:math> <jats:tex-math> $theta =frac{2}{3}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0146_ineq_002.png\"/> </jats:alternatives> </jats:inline-formula>, <jats:italic>N</jats:italic> = 3 and <jats:italic>γ</jats:italic> &gt; 1, and we prove that the free boundary grows linearly in time by using some new techniques. When <jats:italic>θ</jats:italic> = 1, under the stress free boundary condition, we construct some analytical solutions for <jats:italic>N</jats:italic> = 2, <jats:italic>γ</jats:italic> = 2 and <jats:italic>N</jats:italic> = 3, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mi>γ</m:mi> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>5</m:mn> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:mfrac> </m:math> <jats:tex-math> $gamma =frac{5}{3}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0146_ineq_003.png\"/> </jats:alternatives> </jats:inline-formula>, respectively.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"28 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507103","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Homogenization of Smoluchowski-type equations with transmission boundary conditions","authors":"Bruno Franchi, Silvia Lorenzani","doi":"10.1515/ans-2023-0143","DOIUrl":"https://doi.org/10.1515/ans-2023-0143","url":null,"abstract":"In this work, we prove a two-scale homogenization result for a set of diffusion-coagulation Smoluchowski-type equations with transmission boundary conditions. This system is meant to describe the aggregation and diffusion of pathological tau proteins in the cerebral tissue, a process associated with the onset and evolution of a large variety of tauopathies (such as Alzheimer’s disease). We prove the existence, uniqueness, positivity and boundedness of solutions to the model equations derived at the microscale (that is the scale of single neurons). Then, we study the convergence of the homogenization process to the solution of a macro-model asymptotically consistent with the microscopic one.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"17 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Periodic solutions for a coupled system of wave equations with x-dependent coefficients","authors":"Jiayu Deng, Shuguan Ji","doi":"10.1515/ans-2023-0144","DOIUrl":"https://doi.org/10.1515/ans-2023-0144","url":null,"abstract":"This paper is concerned with the periodic solutions for a coupled system of wave equations with <jats:italic>x</jats:italic>-dependent coefficients. Such a model arises naturally when two waves propagate simultaneously in the nonisotrpic media. In this paper, for the periods having the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mi>T</m:mi> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mn>2</m:mn> <m:mi>a</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>b</m:mi> </m:mrow> </m:mfrac> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>b</m:mi> <m:mspace width=\"0.3333em\"/> <m:mspace width=\"0.28em\"/> <m:mtext>are positive integers</m:mtext> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:tex-math>$T=frac{2a-1}{b}left(a,b text{are,positive,integers}right)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0144_ineq_001.png\"/> </jats:alternatives> </jats:inline-formula> and some types of boundary conditions, we obtain the existence of the time periodic solutions and analyze the asymptotic behaviors as the coupled parameter goes to zero, when the nonlinearities are superlinear and monotone, by using the variational method. In particular, the condition ess inf <jats:italic>η</jats:italic> <jats:sub> <jats:italic>ϱ</jats:italic> </jats:sub>(<jats:italic>x</jats:italic>) &gt; 0 is not required.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"31 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Regularity of center-outward distribution functions in non-convex domains","authors":"Eustasio del Barrio, Alberto González-Sanz","doi":"10.1515/ans-2023-0140","DOIUrl":"https://doi.org/10.1515/ans-2023-0140","url":null,"abstract":"For a probability <jats:italic>P</jats:italic> in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mi>d</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>${mathbb{R}}^{d}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0140_ineq_001.png\"/> </jats:alternatives> </jats:inline-formula> its center outward distribution function F <jats:sub>±</jats:sub>, introduced in V. Chernozhukov, A. Galichon, M. Hallin, and M. Henry (“Monge–Kantorovich depth, quantiles, ranks and signs,” <jats:italic>Ann. Stat.</jats:italic>, vol. 45, no. 1, pp. 223–256, 2017) and M. Hallin, E. del Barrio, J. Cuesta-Albertos, and C. Matrán (“Distribution and quantile functions, ranks and signs in dimension d: a measure transportation approach,” <jats:italic>Ann. Stat.</jats:italic>, vol. 49, no. 2, pp. 1139–1165, 2021), is a new and successful concept of multivariate distribution function based on mass transportation theory. This work proves, for a probability <jats:italic>P</jats:italic> with density locally bounded away from zero and infinity in its support, the continuity of the center-outward map on the interior of the support of <jats:italic>P</jats:italic> and the continuity of its inverse, the quantile, Q <jats:sub>±</jats:sub>. This relaxes the convexity assumption in E. del Barrio, A. González-Sanz, and M. Hallin (“A note on the regularity of optimal-transport-based center-outward distribution and quantile functions,” <jats:italic>J. Multivariate Anal.</jats:italic>, vol. 180, p. 104671, 2020). Some important consequences of this continuity are Glivenko–Cantelli type theorems and characterisation of weak convergence by the stability of the center-outward map.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"86 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507179","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An example related to Whitney’s extension problem for L 2,p (R2) when 1 < p < 2","authors":"Jacob Carruth, Arie Israel","doi":"10.1515/ans-2023-0126","DOIUrl":"https://doi.org/10.1515/ans-2023-0126","url":null,"abstract":"In this paper, we prove the existence of a bounded linear extension operator <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mi>T</m:mi> <m:mo>:</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>→</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:tex-math>$T:{L}^{2,p}left(Eright)to {L}^{2,p}left({mathbb{R}}^{2}right)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0126_ineq_002.png\"/> </jats:alternatives> </jats:inline-formula> when 1 &lt; <jats:italic>p</jats:italic> &lt; 2, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mi>E</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math>$Esubset {mathbb{R}}^{2}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0126_ineq_003.png\"/> </jats:alternatives> </jats:inline-formula> is a certain discrete set with fractal structure. Our proof makes use of a theorem of Fefferman–Klartag (“Linear extension operators for Sobolev spaces on radially symmetric binary trees,” <jats:italic>Adv. Nonlinear Stud.</jats:italic>, vol. 23, no. 1, p. 20220075, 2023) on the existence of linear extension operators for radially symmetric binary trees.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"24 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141196839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sobolev extension in a simple case","authors":"Marjorie Drake, Charles Fefferman, Kevin Ren, Anna Skorobogatova","doi":"10.1515/ans-2023-0132","DOIUrl":"https://doi.org/10.1515/ans-2023-0132","url":null,"abstract":"In this paper, we establish the existence of a bounded, linear extension operator <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>T</m:mi> <m:mspace width=\"0.17em\"/> <m:mo>:</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>→</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:tex-math> $T :{L}^{2,p}left(Eright)to {L}^{2,p}left({mathbb{R}}^{2}right)$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0132_ineq_001.png\"/> </jats:alternatives> </jats:inline-formula> when 1 &lt; <jats:italic>p</jats:italic> &lt; 2 and <jats:italic>E</jats:italic> is a finite subset of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math> ${mathbb{R}}^{2}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0132_ineq_002.png\"/> </jats:alternatives> </jats:inline-formula> contained in a line.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"83 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141150512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Limiting behavior of quasilinear wave equations with fractional-type dissipation","authors":"Barbara Kaltenbacher, Mostafa Meliani, Vanja Nikolić","doi":"10.1515/ans-2023-0139","DOIUrl":"https://doi.org/10.1515/ans-2023-0139","url":null,"abstract":"In this work, we investigate a class of quasilinear wave equations of Westervelt type with, in general, nonlocal-in-time dissipation. They arise as models of nonlinear sound propagation through complex media with anomalous diffusion of Gurtin–Pipkin type. Aiming at minimal assumptions on the involved memory kernels – which we allow to be weakly singular – we prove the well-posedness of such wave equations in a general theoretical framework. In particular, the Abel fractional kernels, as well as Mittag-Leffler-type kernels, are covered by our results. The analysis is carried out uniformly with respect to the small involved parameter on which the kernels depend and which can be physically interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the behavior of solutions as this parameter vanishes, and in this way relate the equations to their limiting counterparts. To establish the limiting problems, we distinguish among different classes of kernels and analyze and discuss all ensuing cases.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"106 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Two solutions for Dirichlet double phase problems with variable exponents","authors":"Eleonora Amoroso, Gabriele Bonanno, Giuseppina D’Aguì, Patrick Winkert","doi":"10.1515/ans-2023-0134","DOIUrl":"https://doi.org/10.1515/ans-2023-0134","url":null,"abstract":"This paper is devoted to the study of a double phase problem with variable exponents and Dirichlet boundary condition. Based on an abstract critical point theorem, we establish existence results under very general assumptions on the nonlinear term, such as a subcritical growth and a superlinear condition. In particular, we prove the existence of two bounded weak solutions with opposite energy sign and we state some special cases in which they turn out to be nonnegative.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"41 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140933590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}