Pub Date : 2026-01-21DOI: 10.1016/j.jmaa.2026.130452
Eduardo H. Gomes Tavares , Anderson J.A. Ramos , Marcio A. Jorge Silva , JinYun Yuan
We provide a complete analytical and numerical characterization of the stability behavior for a shearing Cattaneo-Timoshenko system governed by a recently proposed stability number, denoted herein by . Unlike previous results restricted to energy-based approaches or specific parameter configurations, we prove that is not only sufficient but also necessary for exponential stability. Otherwise, the system exhibits optimal algebraic decay rate for regular initial data. The theoretical results are aligned with a proper numerical analysis where a finite difference scheme confirms the characterization of stability in terms of and its effectiveness in predicting the complete system's asymptotic behavior.
{"title":"Full stability characterization of shearing Cattaneo-Timoshenko systems","authors":"Eduardo H. Gomes Tavares , Anderson J.A. Ramos , Marcio A. Jorge Silva , JinYun Yuan","doi":"10.1016/j.jmaa.2026.130452","DOIUrl":"10.1016/j.jmaa.2026.130452","url":null,"abstract":"<div><div>We provide a complete analytical and numerical characterization of the stability behavior for a shearing Cattaneo-Timoshenko system governed by a recently proposed stability number, denoted herein by <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>τ</mi></mrow></msub></math></span>. Unlike previous results restricted to energy-based approaches or specific parameter configurations, we prove that <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo>=</mo><mn>0</mn></math></span> is not only sufficient but also necessary for exponential stability. Otherwise, the system exhibits optimal algebraic decay rate <span><math><msup><mrow><mi>t</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span> for regular initial data. The theoretical results are aligned with a proper numerical analysis where a finite difference scheme confirms the characterization of stability in terms of <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>τ</mi></mrow></msub></math></span> and its effectiveness in predicting the complete system's asymptotic behavior.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130452"},"PeriodicalIF":1.2,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078314","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 : 2026-01-20DOI: 10.1016/j.jmaa.2026.130445
Mohammad Saeid Abolhassanifar , Reza Saadati , Mohammad Bagher Ghaemi , Donal O'Regan
<div><div>We investigate a class of nonlinear nonlocal problems that integrate two complex mechanisms: Kirchhoff-type nonlocal diffusion and Choquard-type critical convolution nonlinearity involving the Hardy–Littlewood–Sobolev (HLS) critical exponent. Specifically, we consider the following equation in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>:<span><span><span><math><mo>−</mo><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><msup><mrow><mo>(</mo><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></munder><mo>|</mo><mi>∇</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mspace></mspace><mi>d</mi><mi>x</mi><mo>)</mo></mrow><mrow><mi>θ</mi></mrow></msup><mo>)</mo></mrow><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>u</mi><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>⁎</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>q</mi></mrow></msup><mo>)</mo></mrow><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>,</mo><mspace></mspace><mi>u</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo><mo>,</mo><mspace></mspace><mi>u</mi><mo>></mo><mn>0</mn><mo>,</mo></math></span></span></span> where <span><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>></mo><mn>0</mn></math></span>, <span><math><mn>0</mn><mo><</mo><mi>θ</mi><mo><</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mn>3</mn></mrow></mfrac></math></span>, <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>3</mn></math></span>, <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></mrow><mrow><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>3</mn><mo>−</mo><mi>α</mi></mrow></msup></mrow></mfrac></math></span> is the Riesz potential, and <span><math><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is a positive, asymptotically constant potential. This formulation simultaneously captures <em>Kirchhoff-type nonlocality</em> through the energy-dependent coefficient of the Laplacian, and <em>Choquard criticality</em> via a convolution nonlinearity with critical exponent <span><math><mi>q</mi><mo>=</mo><mn>3</mn><mo>+</mo><mi>α</mi></math></span>.</div><div>By combining variational methods, Pohožaev-type identities, and global compactness techniques adapted to this doubly nonlocal setting, we prove the existence of positive ground state solutions in both subcritical and critical cases. Moreover, we analyze the asymptotic behavior as the nonlocal parameters approach their critical limits, <span><math><mi>θ</mi><mo>→</mo><msup><mrow><mn>1</mn></mrow><mrow><mo>−</mo></mrow></msup></math></span> and <span><math><mi>α</
{"title":"Ground state solutions and asymptotic behavior for a nonlocal Kirchhoff–Choquard equation with variable potential in R3","authors":"Mohammad Saeid Abolhassanifar , Reza Saadati , Mohammad Bagher Ghaemi , Donal O'Regan","doi":"10.1016/j.jmaa.2026.130445","DOIUrl":"10.1016/j.jmaa.2026.130445","url":null,"abstract":"<div><div>We investigate a class of nonlinear nonlocal problems that integrate two complex mechanisms: Kirchhoff-type nonlocal diffusion and Choquard-type critical convolution nonlinearity involving the Hardy–Littlewood–Sobolev (HLS) critical exponent. Specifically, we consider the following equation in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>:<span><span><span><math><mo>−</mo><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><msup><mrow><mo>(</mo><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></munder><mo>|</mo><mi>∇</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mspace></mspace><mi>d</mi><mi>x</mi><mo>)</mo></mrow><mrow><mi>θ</mi></mrow></msup><mo>)</mo></mrow><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>u</mi><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>⁎</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>q</mi></mrow></msup><mo>)</mo></mrow><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>,</mo><mspace></mspace><mi>u</mi><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo><mo>,</mo><mspace></mspace><mi>u</mi><mo>></mo><mn>0</mn><mo>,</mo></math></span></span></span> where <span><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>></mo><mn>0</mn></math></span>, <span><math><mn>0</mn><mo><</mo><mi>θ</mi><mo><</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mn>3</mn></mrow></mfrac></math></span>, <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>3</mn></math></span>, <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></mrow><mrow><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>3</mn><mo>−</mo><mi>α</mi></mrow></msup></mrow></mfrac></math></span> is the Riesz potential, and <span><math><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is a positive, asymptotically constant potential. This formulation simultaneously captures <em>Kirchhoff-type nonlocality</em> through the energy-dependent coefficient of the Laplacian, and <em>Choquard criticality</em> via a convolution nonlinearity with critical exponent <span><math><mi>q</mi><mo>=</mo><mn>3</mn><mo>+</mo><mi>α</mi></math></span>.</div><div>By combining variational methods, Pohožaev-type identities, and global compactness techniques adapted to this doubly nonlocal setting, we prove the existence of positive ground state solutions in both subcritical and critical cases. Moreover, we analyze the asymptotic behavior as the nonlocal parameters approach their critical limits, <span><math><mi>θ</mi><mo>→</mo><msup><mrow><mn>1</mn></mrow><mrow><mo>−</mo></mrow></msup></math></span> and <span><math><mi>α</","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"559 2","pages":"Article 130445"},"PeriodicalIF":1.2,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081414","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 work, we study the asymptotic behaviour of solutions to the heat equation in exterior domains, i.e., domains which are the complement of a smooth compact set in . Different homogeneous boundary conditions are considered, including Dirichlet, Robin, and Neumann ones. In this second part of our work, for , we consider the case of bounded initial data and prove that, after some correction term, the solutions become close to the solutions in the whole space and show how complex behaviours appear. We also analyse the case of initial data in with where all solutions essentially decay to 0 and the convergence rate could be arbitrarily slow.
{"title":"Asymptotic behaviour of the heat equation in an exterior domain with general boundary conditions II. The case of bounded and of Lp data","authors":"Joaquín Domínguez-de-Tena , Aníbal Rodríguez-Bernal","doi":"10.1016/j.jmaa.2026.130448","DOIUrl":"10.1016/j.jmaa.2026.130448","url":null,"abstract":"<div><div>In this work, we study the asymptotic behaviour of solutions to the heat equation in exterior domains, i.e., domains which are the complement of a smooth compact set in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>. Different homogeneous boundary conditions are considered, including Dirichlet, Robin, and Neumann ones. In this second part of our work, for <span><math><mi>N</mi><mo>≥</mo><mn>3</mn></math></span>, we consider the case of bounded initial data and prove that, after some correction term, the solutions become close to the solutions in the whole space and show how complex behaviours appear. We also analyse the case of initial data in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> with <span><math><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mo>∞</mo></math></span> where all solutions essentially decay to 0 and the convergence rate could be arbitrarily slow.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 1","pages":"Article 130448"},"PeriodicalIF":1.2,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038212","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 : 2026-01-20DOI: 10.1016/j.jmaa.2026.130451
Yuan Chen
We investigate the mass-preserving -gradient flow associated with a generalized Cahn–Hilliard equation. Our focus is on the sharp interface regime, where the interface width parameter is small. For well-prepared initial data, we rigorously prove that, as , solutions of the diffuse-interface model converge to the volume-preserving Willmore flow in arbitrary spatial dimensions . The proof incorporates matched asymptotic expansions, spectrum and energy estimates to establish the convergence of the order parameter away from the interface, alongside a precise motion law derivation for the limiting interface. This result extends the analysis of Fei and Liu [16] from two- and three-dimensional settings to general N-dimensional domains, and it applies to a broad class of symmetric double-well potentials beyond the classical quartic form.
{"title":"Volume preserving Willmore flow in a generalized Cahn-Hilliard flow","authors":"Yuan Chen","doi":"10.1016/j.jmaa.2026.130451","DOIUrl":"10.1016/j.jmaa.2026.130451","url":null,"abstract":"<div><div>We investigate the mass-preserving <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-gradient flow associated with a generalized Cahn–Hilliard equation. Our focus is on the sharp interface regime, where the interface width parameter <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span> is small. For well-prepared initial data, we rigorously prove that, as <span><math><mi>ε</mi><mo>→</mo><mn>0</mn></math></span>, solutions of the diffuse-interface model converge to the <em>volume-preserving Willmore flow</em> in arbitrary spatial dimensions <span><math><mi>N</mi><mo>≥</mo><mn>2</mn></math></span>. The proof incorporates matched asymptotic expansions, spectrum and energy estimates to establish the convergence of the order parameter away from the interface, alongside a precise motion law derivation for the limiting interface. This result extends the analysis of Fei and Liu <span><span>[16]</span></span> from two- and three-dimensional settings to general <em>N</em>-dimensional domains, and it applies to a broad class of symmetric double-well potentials beyond the classical quartic form.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130451"},"PeriodicalIF":1.2,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023451","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 : 2026-01-20DOI: 10.1016/j.jmaa.2026.130450
Howard S. Cohl , Loyal Durand
By starting with Durand's double integral representation for a product of two Jacobi functions of the second kind, we derive an integral representation for a product of two Jacobi functions of the second kind in kernel form. We also derive a Bateman-type sum for a product of two Jacobi functions of the second kind. From this integral representation we derive integral representations for the Jacobi function of the first kind in both the hyperbolic and trigonometric contexts. From the integral representations for Jacobi functions, we also derive integral representations for products of limiting functions such as associated Legendre functions of the first and second kind, Ferrers functions and also Gegenbauer functions of the first and second kind. By examining the behavior of one of these products near singularities of the relevant functions, we also derive integral representations for single functions, including a Laplace-type integral representation for the Jacobi function of the second kind. Finally, we use the product formulas for the functions of the second kind to derive Nicholson-type integral relations for the sums of squares of Jacobi functions of the first and second kinds, and in a confluent limit, Laguerre functions of the first and second kinds, which generalize the relation to those functions.
{"title":"Integral representation for a product of two Jacobi functions of the second kind","authors":"Howard S. Cohl , Loyal Durand","doi":"10.1016/j.jmaa.2026.130450","DOIUrl":"10.1016/j.jmaa.2026.130450","url":null,"abstract":"<div><div>By starting with Durand's double integral representation for a product of two Jacobi functions of the second kind, we derive an integral representation for a product of two Jacobi functions of the second kind in kernel form. We also derive a Bateman-type sum for a product of two Jacobi functions of the second kind. From this integral representation we derive integral representations for the Jacobi function of the first kind in both the hyperbolic and trigonometric contexts. From the integral representations for Jacobi functions, we also derive integral representations for products of limiting functions such as associated Legendre functions of the first and second kind, Ferrers functions and also Gegenbauer functions of the first and second kind. By examining the behavior of one of these products near singularities of the relevant functions, we also derive integral representations for single functions, including a Laplace-type integral representation for the Jacobi function of the second kind. Finally, we use the product formulas for the functions of the second kind to derive Nicholson-type integral relations for the sums of squares of Jacobi functions of the first and second kinds, and in a confluent limit, Laguerre functions of the first and second kinds, which generalize the relation <span><math><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>x</mi></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>i</mi><mi>x</mi></mrow></msup><mo>=</mo><mn>1</mn></math></span> to those functions.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"559 1","pages":"Article 130450"},"PeriodicalIF":1.2,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146024433","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 : 2026-01-20DOI: 10.1016/j.jmaa.2026.130449
Lisbeth Carrero, Pedro Hernández-Llanos
<div><div>In this paper, we study the existence and nonexistence of solutions for the following Kirchhoff-type fractional <span><math><mo>(</mo><mi>p</mi><mtext>-</mtext><mi>q</mi><mo>)</mo></math></span>-Laplacian problem:<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mi>M</mi><mrow><mo>(</mo><msubsup><mrow><mo>[</mo><mi>u</mi><mo>]</mo></mrow><mrow><mi>p</mi><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>p</mi></mrow></msubsup><mo>)</mo></mrow><msubsup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mi>u</mi><mo>+</mo><mi>M</mi><mrow><mo>(</mo><msubsup><mrow><mo>[</mo><mi>u</mi><mo>]</mo></mrow><mrow><mi>q</mi><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mi>q</mi></mrow></msubsup><mo>)</mo></mrow><msubsup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mi>u</mi></mtd><mtd></mtd></mtr><mtr><mtd><mspace></mspace><mo>=</mo><mi>λ</mi><mo>[</mo><mi>a</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><mi>b</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>]</mo><mo>+</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo></mtd><mtd><mtext>in </mtext><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd><mtext>on </mtext><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>∖</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> (<span><math><mi>N</mi><mo>≥</mo><mn>1</mn></math></span>) is a bounded domain with smooth boundary, <span><math><mn>0</mn><mo><</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mn>1</mn></math></span>, and <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>p</mi><mo><</mo><mi>N</mi></math></span>. We assume <span><math><mn>1</mn><mo><</mo><mi>q</mi><mo>≤</mo><mi>p</mi><mo><</mo><mi>θ</mi><mi>p</mi><mo><</mo><msubsup><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>:</mo><mo>=</mo><mfrac><mrow><mi>N</mi><mi>p</mi></mrow><mrow><mi>N</mi><mo>−</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>p</mi></mrow></mfrac></math></span>, and <span><math><mi>λ</mi><mo>∈</mo><mi>R</mi></math></span>. The functions <span><math><mi>a</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>b</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>
{"title":"Kirchhoff-type equations involving the fractional (p,q)-Laplacian","authors":"Lisbeth Carrero, Pedro Hernández-Llanos","doi":"10.1016/j.jmaa.2026.130449","DOIUrl":"10.1016/j.jmaa.2026.130449","url":null,"abstract":"<div><div>In this paper, we study the existence and nonexistence of solutions for the following Kirchhoff-type fractional <span><math><mo>(</mo><mi>p</mi><mtext>-</mtext><mi>q</mi><mo>)</mo></math></span>-Laplacian problem:<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mi>M</mi><mrow><mo>(</mo><msubsup><mrow><mo>[</mo><mi>u</mi><mo>]</mo></mrow><mrow><mi>p</mi><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>p</mi></mrow></msubsup><mo>)</mo></mrow><msubsup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mi>u</mi><mo>+</mo><mi>M</mi><mrow><mo>(</mo><msubsup><mrow><mo>[</mo><mi>u</mi><mo>]</mo></mrow><mrow><mi>q</mi><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mi>q</mi></mrow></msubsup><mo>)</mo></mrow><msubsup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mi>u</mi></mtd><mtd></mtd></mtr><mtr><mtd><mspace></mspace><mo>=</mo><mi>λ</mi><mo>[</mo><mi>a</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><mi>b</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>]</mo><mo>+</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo></mtd><mtd><mtext>in </mtext><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo></mtd><mtd><mtext>on </mtext><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>∖</mo><mi>Ω</mi><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> (<span><math><mi>N</mi><mo>≥</mo><mn>1</mn></math></span>) is a bounded domain with smooth boundary, <span><math><mn>0</mn><mo><</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mn>1</mn></math></span>, and <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>p</mi><mo><</mo><mi>N</mi></math></span>. We assume <span><math><mn>1</mn><mo><</mo><mi>q</mi><mo>≤</mo><mi>p</mi><mo><</mo><mi>θ</mi><mi>p</mi><mo><</mo><msubsup><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>:</mo><mo>=</mo><mfrac><mrow><mi>N</mi><mi>p</mi></mrow><mrow><mi>N</mi><mo>−</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>p</mi></mrow></mfrac></math></span>, and <span><math><mi>λ</mi><mo>∈</mo><mi>R</mi></math></span>. The functions <span><math><mi>a</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>b</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130449"},"PeriodicalIF":1.2,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078213","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 : 2026-01-20DOI: 10.1016/j.jmaa.2026.130447
Yanping Chen , Teng Wang , Huoxiong Wu
Let be a homogeneous function of degree zero and have mean value zero. Consider the rough singular integral We prove that is of weak type if the rough kernel function Ω belongs to the block space for some . This result substantially extends a classical theorem of Seeger (1996) [19].
{"title":"Weak type (1,1) bounds for singular integrals with rough kernels on block spaces","authors":"Yanping Chen , Teng Wang , Huoxiong Wu","doi":"10.1016/j.jmaa.2026.130447","DOIUrl":"10.1016/j.jmaa.2026.130447","url":null,"abstract":"<div><div>Let <span><math><mi>Ω</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> be a homogeneous function of degree zero and have mean value zero. Consider the rough singular integral<span><span><span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>Ω</mi></mrow></msub><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mrow><mi>p</mi><mo>.</mo><mi>v</mi><mo>.</mo></mrow><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></munder><mfrac><mrow><mi>Ω</mi><mo>(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo>)</mo></mrow><mrow><mo>|</mo><mi>x</mi><mo>−</mo><mi>y</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfrac><mi>f</mi><mo>(</mo><mi>y</mi><mo>)</mo><mi>d</mi><mi>y</mi><mo>.</mo></math></span></span></span> We prove that <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>Ω</mi></mrow></msub></math></span> is of weak type <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> if the rough kernel function Ω belongs to the block space <span><math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>q</mi></mrow><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow></msubsup><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> for some <span><math><mi>q</mi><mo>></mo><mn>1</mn></math></span>. This result substantially extends a classical theorem of Seeger (1996) <span><span>[19]</span></span>.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"559 1","pages":"Article 130447"},"PeriodicalIF":1.2,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146024431","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 : 2026-01-19DOI: 10.1016/j.jmaa.2026.130443
Zuzana Došlá , Mauro Marini , Serena Matucci
A nonlinear differential equation with inhomogeneous differential operator Φ, which is regularly varying at zero, is considered. The operator Φ can be viewed as an extension of the p-Laplacian operator and arises in many physical problems, as illustrated by several examples. In particular, the existence of global positive bounded solutions on the half-line with the Neumann type boundary conditions is studied by means of an abstract fixed point theorem and certain properties of an associated half-linear equation. The results do not require the explicit form of the inverse operator of Φ, and are completed by an asymptotic analysis of these solutions near infinity.
{"title":"Global positive bounded solutions for equations with regularly varying operator","authors":"Zuzana Došlá , Mauro Marini , Serena Matucci","doi":"10.1016/j.jmaa.2026.130443","DOIUrl":"10.1016/j.jmaa.2026.130443","url":null,"abstract":"<div><div>A nonlinear differential equation with inhomogeneous differential operator Φ, which is regularly varying at zero, is considered. The operator Φ can be viewed as an extension of the <em>p</em>-Laplacian operator and arises in many physical problems, as illustrated by several examples. In particular, the existence of global positive bounded solutions on the half-line with the Neumann type boundary conditions is studied by means of an abstract fixed point theorem and certain properties of an associated half-linear equation. The results do not require the explicit form of the inverse operator of Φ, and are completed by an asymptotic analysis of these solutions near infinity.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"559 1","pages":"Article 130443"},"PeriodicalIF":1.2,"publicationDate":"2026-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146024508","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 : 2026-01-19DOI: 10.1016/j.jmaa.2026.130444
Razvan Gabriel Iagar , Diana-Rodica Munteanu
<div><div>We study the large time behavior of solutions to the Cauchy problem for the quasilinear absorption-diffusion equation<span><span><span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>=</mo><mi>Δ</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>σ</mi></mrow></msup><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>,</mo><mspace></mspace><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>×</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo><mo>,</mo></math></span></span></span> with exponents <span><math><mi>p</mi><mo>></mo><mi>m</mi><mo>></mo><mn>1</mn></math></span> and <span><math><mi>σ</mi><mo>></mo><mn>0</mn></math></span> and with initial conditions either satisfying<span><span><span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo><mo>∩</mo><mi>C</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo><mo>,</mo><mspace></mspace><munder><mi>lim</mi><mrow><mo>|</mo><mi>x</mi><mo>|</mo><mo>→</mo><mo>∞</mo></mrow></munder><mo></mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>θ</mi></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>A</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span></span></span> for some <span><math><mi>θ</mi><mo>≥</mo><mn>0</mn></math></span>. A number of different asymptotic profiles are identified, and uniform convergence on time-expanding sets towards them is established, according to the position of both <em>p</em> and <em>θ</em> with respect to the following critical exponents<span><span><span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>σ</mi><mo>)</mo><mo>=</mo><mi>m</mi><mo>+</mo><mfrac><mrow><mi>σ</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>N</mi></mrow></mfrac><mo>,</mo><mspace></mspace><msub><mrow><mi>θ</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>=</mo><mfrac><mrow><mi>σ</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>p</mi><mo>−</mo><mi>m</mi></mrow></mfrac><mo>,</mo><mspace></mspace><msup><mrow><mi>θ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>=</mo><mi>N</mi><mo>.</mo></math></span></span></span> More precisely, solutions in radially symmetric self-similar form decaying as <span><math><mo>|</mo><mi>x</mi><mo>|</mo><mo>→</mo><mo>∞</mo></math></span> with the rates<span><span><span><math><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∼</mo><mi>A</mi><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><msub><mrow><mi>θ</mi></mrow><mrow><mo>⁎</mo></mrow></msub></mrow></msup><mo>,</mo><mspace></mspace><mrow><mi>or</mi></mrow><mspace></mspace><mi>u</mi><
{"title":"A porous medium equation with spatially inhomogeneous absorption. Part II: Large time behavior","authors":"Razvan Gabriel Iagar , Diana-Rodica Munteanu","doi":"10.1016/j.jmaa.2026.130444","DOIUrl":"10.1016/j.jmaa.2026.130444","url":null,"abstract":"<div><div>We study the large time behavior of solutions to the Cauchy problem for the quasilinear absorption-diffusion equation<span><span><span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>=</mo><mi>Δ</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>σ</mi></mrow></msup><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>,</mo><mspace></mspace><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>×</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo><mo>,</mo></math></span></span></span> with exponents <span><math><mi>p</mi><mo>></mo><mi>m</mi><mo>></mo><mn>1</mn></math></span> and <span><math><mi>σ</mi><mo>></mo><mn>0</mn></math></span> and with initial conditions either satisfying<span><span><span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo><mo>∩</mo><mi>C</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo><mo>,</mo><mspace></mspace><munder><mi>lim</mi><mrow><mo>|</mo><mi>x</mi><mo>|</mo><mo>→</mo><mo>∞</mo></mrow></munder><mo></mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>θ</mi></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>A</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span></span></span> for some <span><math><mi>θ</mi><mo>≥</mo><mn>0</mn></math></span>. A number of different asymptotic profiles are identified, and uniform convergence on time-expanding sets towards them is established, according to the position of both <em>p</em> and <em>θ</em> with respect to the following critical exponents<span><span><span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>F</mi></mrow></msub><mo>(</mo><mi>σ</mi><mo>)</mo><mo>=</mo><mi>m</mi><mo>+</mo><mfrac><mrow><mi>σ</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>N</mi></mrow></mfrac><mo>,</mo><mspace></mspace><msub><mrow><mi>θ</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>=</mo><mfrac><mrow><mi>σ</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>p</mi><mo>−</mo><mi>m</mi></mrow></mfrac><mo>,</mo><mspace></mspace><msup><mrow><mi>θ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>=</mo><mi>N</mi><mo>.</mo></math></span></span></span> More precisely, solutions in radially symmetric self-similar form decaying as <span><math><mo>|</mo><mi>x</mi><mo>|</mo><mo>→</mo><mo>∞</mo></math></span> with the rates<span><span><span><math><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>∼</mo><mi>A</mi><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><msub><mrow><mi>θ</mi></mrow><mrow><mo>⁎</mo></mrow></msub></mrow></msup><mo>,</mo><mspace></mspace><mrow><mi>or</mi></mrow><mspace></mspace><mi>u</mi><","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 2","pages":"Article 130444"},"PeriodicalIF":1.2,"publicationDate":"2026-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023452","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 : 2026-01-16DOI: 10.1016/j.jmaa.2026.130442
Zhengchao Ji
In this paper, we establish some inequalities for the higher eigenvalues of the clamped plate problem on Riemannian manifolds with bounded sectional curvature. Our proofs are based on a Laplacian comparison and the Fourier transform. As an application of the Laplacian comparison, we obtain a generalized inequality of Cheng-Wei in . We also prove an improved lower bound for .
{"title":"Upper and lower bounds for the eigenvalues of the clamped plate problem on Riemannian manifolds","authors":"Zhengchao Ji","doi":"10.1016/j.jmaa.2026.130442","DOIUrl":"10.1016/j.jmaa.2026.130442","url":null,"abstract":"<div><div>In this paper, we establish some inequalities for the higher eigenvalues <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> of the clamped plate problem on Riemannian manifolds with bounded sectional curvature. Our proofs are based on a Laplacian comparison and the Fourier transform. As an application of the Laplacian comparison, we obtain a generalized inequality of Cheng-Wei in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. We also prove an improved lower bound for <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup><msub><mrow><mi>Λ</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"558 1","pages":"Article 130442"},"PeriodicalIF":1.2,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078040","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号