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Invariant measure of non-autonomous stochastic reaction-diffusion equations with infinite delay and additive white noise 具有无限延迟和加性白噪声的非自治随机反应扩散方程的不变测度
IF 1.2 3区 数学 Q1 MATHEMATICS Pub Date : 2026-01-22 DOI: 10.1016/j.jmaa.2026.130453
Wenqiang Zhao , Xia Liu
This article is concerned with the random dynamics of non-autonomous stochastic reaction-diffusion equations that incorporate additive white noise and infinite delay, with the delay term being globally Lipschitz continuous. We first establish the existence of (periodic) pullback random attractors for the corresponding non-autonomous dynamical system (NRDS). The asymptotical compactness of solutions is primarily achieved by applying the Arzelà-Ascoli theorem over a compact time interval, coupled with a limiting argument for the negative infinite part. Furthermore, we demonstrate that the solution to the underlying equations is jointly continuous in both the initial time and the initial data. This result allows us to construct a family of (periodic) invariant Borel probability measures that are supported within the pullback random attractors for the NRDS.
研究了包含加性白噪声和无限延迟的非自治随机反应扩散方程的随机动力学问题,该方程的延迟项为全局Lipschitz连续。我们首先建立了相应的非自治动力系统(NRDS)的(周期)回拉随机吸引子的存在性。解的渐近紧性主要是通过在紧时间区间上应用Arzelà-Ascoli定理,并结合负无穷部分的极限论证来实现的。进一步,我们证明了基础方程的解在初始时间和初始数据上是联合连续的。这个结果允许我们构造一组(周期)不变的Borel概率测度,这些测度在NRDS的回拉随机吸引子中得到支持。
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引用次数: 0
Simplicity of algebras and C⁎-algebras of self-similar groupoids 自相似群的代数和C -代数的简单性
IF 1.2 3区 数学 Q1 MATHEMATICS Pub Date : 2026-01-21 DOI: 10.1016/j.jmaa.2026.130446
Josiah Aakre
Many previously studied path algebras and self-similar group algebras may be viewed as Steinberg algebras of self-similar groupoids. By way of inverse semigroup algebras, we characterize when the Steinberg algebra of a self-similar groupoid is simple. We show that the simplicity of the reduced C-algebra of a contracting self-similar groupoid coincides with the simplicity of the Steinberg algebra. As an aside, we show that simplicity of the two algebras sometimes depends only on the skeleton of the self-similar groupoid acting on a strongly connected graph. Finally, we apply our methods to examples including a self-similar groupoid akin to multispinal self-similar groups and a self-similar groupoid built from the well-known Basilica group.
前人研究的许多路径代数和自相似群代数都可以看作是自相似群类群的Steinberg代数。利用逆半群代数,刻画了自相似群的Steinberg代数是否简单。我们证明了一个收缩自相似群的简化C -代数的简单性与Steinberg代数的简单性是一致的。作为题外话,我们证明了这两个代数的简单性有时只依赖于作用于强连通图的自相似群的骨架。最后,我们将我们的方法应用于实例,包括类似于多脊柱自相似群的自相似群和由著名的Basilica群构建的自相似群。
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引用次数: 0
Full stability characterization of shearing Cattaneo-Timoshenko systems 剪切Cattaneo-Timoshenko体系的全稳定性表征
IF 1.2 3区 数学 Q1 MATHEMATICS Pub Date : 2026-01-21 DOI: 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 χτ. Unlike previous results restricted to energy-based approaches or specific parameter configurations, we prove that χτ=0 is not only sufficient but also necessary for exponential stability. Otherwise, the system exhibits optimal algebraic decay rate t1/2 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 χτ and its effectiveness in predicting the complete system's asymptotic behavior.
我们提供了剪切Cattaneo-Timoshenko系统稳定行为的完整解析和数值表征,该系统由最近提出的稳定数控制,这里用χτ表示。与以往的结果局限于基于能量的方法或特定的参数配置不同,我们证明χτ=0不仅是充分的,而且是指数稳定性的必要条件。否则,系统表现出最佳的代数衰减率t−1/2的规则初始数据。理论结果与适当的数值分析相一致,其中有限差分格式证实了用χτ表示的稳定性特征及其在预测完整系统渐近行为方面的有效性。
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引用次数: 0
Ground state solutions and asymptotic behavior for a nonlocal Kirchhoff–Choquard equation with variable potential in R3 R3中具有变势的非局部Kirchhoff-Choquard方程的基态解和渐近行为
IF 1.2 3区 数学 Q1 MATHEMATICS Pub Date : 2026-01-20 DOI: 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</
研究了一类非线性非局部问题,该问题综合了两种复杂机制:kirchhoff型非局部扩散和涉及Hardy-Littlewood-Sobolev (HLS)临界指数的choquard型临界卷积非线性。具体地说,我们考虑R3中的下列方程:−(a+b(∫R3|∇u|2dx)θ)Δu+V(x)u=(Iα α| u|q)|u|q−2u,u∈H1(R3),u>0,其中a,b> 0,0 <θ<α 3,0 <α<3, Iα(x)= a α|x|3−α是Riesz势,V(x)是一个正的渐近常数势。该公式同时通过拉普拉斯函数的能量依赖系数捕获kirchhoff型非局域性,并通过具有临界指数q=3+α的卷积非线性捕获Choquard临界性。通过结合变分方法、Pohožaev-type恒等式和适应于这种双重非局部设置的全局紧性技术,我们证明了在亚临界和临界情况下正基态解的存在性。此外,我们分析了非局部参数逼近临界极限θ→1−和α→3−时的渐近行为,并证明了具有临界幂非线性的极限Kirchhoff方程的收敛性。我们的研究结果强调了Kirchhoff非局域性和Choquard临界性之间丰富的相互作用,揭示了它们在空间可变势的调制下的组合如何引起分析挑战和现象。
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Specifically, we consider the following equation in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;:&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;munder&gt;&lt;mo&gt;∫&lt;/mo&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; is the Riesz potential, and &lt;span&gt;&lt;math&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is a positive, asymptotically constant potential. This formulation simultaneously captures &lt;em&gt;Kirchhoff-type nonlocality&lt;/em&gt; through the energy-dependent coefficient of the Laplacian, and &lt;em&gt;Choquard criticality&lt;/em&gt; via a convolution nonlinearity with critical exponent &lt;span&gt;&lt;math&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;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, &lt;span&gt;&lt;math&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;α&lt;/","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}
引用次数: 0
Asymptotic behaviour of the heat equation in an exterior domain with general boundary conditions II. The case of bounded and of Lp data 具有一般边界条件的外域热方程的渐近行为[j]。有界和Lp数据的情况
IF 1.2 3区 数学 Q1 MATHEMATICS Pub Date : 2026-01-20 DOI: 10.1016/j.jmaa.2026.130448
Joaquín Domínguez-de-Tena , Aníbal Rodríguez-Bernal
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 RN. Different homogeneous boundary conditions are considered, including Dirichlet, Robin, and Neumann ones. In this second part of our work, for N3, 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 Lp with 1<p< where all solutions essentially decay to 0 and the convergence rate could be arbitrarily slow.
在这项工作中,我们研究了热方程的解在外部域的渐近行为,即域是RN中光滑紧集的补。考虑了不同的齐次边界条件,包括Dirichlet、Robin和Neumann边界条件。在第二部分的工作中,对于N≥3,我们考虑了有界初始数据的情况,并证明了经过一些修正项后,解在整个空间中变得接近解,并显示了复杂的行为是如何出现的。我们还分析了Lp中具有1<;p<;∞的初始数据的情况,其中所有解本质上都衰减到0,并且收敛速度可以任意慢。
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引用次数: 0
Volume preserving Willmore flow in a generalized Cahn-Hilliard flow 广义Cahn-Hilliard流中的体积保持Willmore流
IF 1.2 3区 数学 Q1 MATHEMATICS Pub Date : 2026-01-20 DOI: 10.1016/j.jmaa.2026.130451
Yuan Chen
We investigate the mass-preserving L2-gradient flow associated with a generalized Cahn–Hilliard equation. Our focus is on the sharp interface regime, where the interface width parameter ε>0 is small. For well-prepared initial data, we rigorously prove that, as ε0, solutions of the diffuse-interface model converge to the volume-preserving Willmore flow in arbitrary spatial dimensions N2. 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.
我们研究了与广义Cahn-Hilliard方程相关的保质量l2梯度流。我们的重点是尖锐界面区,其中界面宽度参数ε>;0很小。对于准备充分的初始数据,我们严格证明了当ε→0时,扩散界面模型的解收敛于任意空间维度N≥2的保体积Willmore流。该证明结合了匹配的渐近展开,频谱和能量估计,以建立远离界面的阶参数的收敛性,以及极限界面的精确运动律推导。这一结果将Fei和Liu[16]的分析从二维和三维环境扩展到一般的n维域,并适用于超越经典四次形式的广义对称双井势。
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引用次数: 0
Integral representation for a product of two Jacobi functions of the second kind 两个第二类雅可比函数积的积分表示
IF 1.2 3区 数学 Q1 MATHEMATICS Pub Date : 2026-01-20 DOI: 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 eixeix=1 to those functions.
从两个第二类雅可比函数乘积的Durand二重积分表示出发,导出了两个第二类雅可比函数乘积的核形式的积分表示。我们还导出了两个第二类雅可比函数的乘积的bateman型和。从这个积分表示导出了第一类雅可比函数在双曲和三角两种情况下的积分表示。在Jacobi函数的积分表示的基础上,我们还推导了一类和二类相关的Legendre函数、一类和二类相关的Ferrers函数以及一类和二类的Gegenbauer函数等极限函数积的积分表示。通过研究其中一个乘积在相关函数的奇异点附近的行为,我们也推导出单个函数的积分表示,包括第二类Jacobi函数的拉普拉斯型积分表示。最后,利用第二类函数的积公式导出了第一类和第二类Jacobi函数的平方和的nicholson型积分关系,并在合流极限下导出了第一类和第二类lagerre函数,将等式eixe−ix=1推广到这类函数。
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引用次数: 0
Kirchhoff-type equations involving the fractional (p,q)-Laplacian 涉及分数(p,q)-拉普拉斯式的kirchhoff型方程
IF 1.2 3区 数学 Q1 MATHEMATICS Pub Date : 2026-01-20 DOI: 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>
本文研究了下列kirchhofftype分数型(p-q)- laplace问题解的存在性和不存在性:{M([u]p,s1p)(−Δ)ps1u+M([u]q,s2q)(−Δ)qs2u=λ[a(x)|u|p−2u+b(x)|u|q−2u]+h(x),在Ω,u=0中,在RN∈Ω上,其中Ω∧RN (N≥1)是光滑边界0<;s1<s2<;1和s1p<;N的有界区域。我们假设1 & lt; q≤术;θ术;ps1⁎:= NpN型−s1p,和λ∈R。函数a(x)、b(x)、h(x)是非负的,且a,b∈L∞(Ω), h∈Lq(Ω)。利用变分方法,证明了至少两个弱解的存在性。第一个解是通过直接最小化相关能量泛函得到的,第二个解是通过应用山口定理得到的。我们还证明了参数λ>;0小值时的不存在性。
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We assume &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;θ&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;, and &lt;span&gt;&lt;math&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. The functions &lt;span&gt;&lt;math&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;","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}
引用次数: 0
Weak type (1,1) bounds for singular integrals with rough kernels on block spaces 块空间上粗糙核奇异积分的弱类型(1,1)界
IF 1.2 3区 数学 Q1 MATHEMATICS Pub Date : 2026-01-20 DOI: 10.1016/j.jmaa.2026.130447
Yanping Chen , Teng Wang , Huoxiong Wu
Let ΩL1(Sn1) be a homogeneous function of degree zero and have mean value zero. Consider the rough singular integralTΩf(x)=p.v.RnΩ(xy)|xy|nf(y)dy. We prove that TΩ is of weak type (1,1) if the rough kernel function Ω belongs to the block space Bq0,0(Sn1) for some q>1. This result substantially extends a classical theorem of Seeger (1996) [19].
设Ω∈L1(Sn−1)为0次齐次函数,均值为0。考虑粗糙奇异integralTΩf(x)=p.v.∫RnΩ(x−y)|x−y|nf(y)dy。如果粗糙核函数Ω对某些q>;1属于块空间Bq0,0(Sn−1),则证明TΩ是弱类型(1,1)。这一结果实质上扩展了Seeger(1996)的一个经典定理。
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引用次数: 0
Global positive bounded solutions for equations with regularly varying operator 正则变算子方程的整体正有界解
IF 1.2 3区 数学 Q1 MATHEMATICS Pub Date : 2026-01-19 DOI: 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.
考虑了在零处有规则变化的非齐次微分算子Φ非线性微分方程。算符Φ可以看作是p-拉普拉斯算符的扩展,在许多物理问题中出现,如几个例子所示。特别地,利用抽象不动点定理和相关半线性方程的某些性质,研究了具有Neumann型边界条件的半线上整体正有界解的存在性。结果不需要Φ逆算子的显式形式,并通过对这些解在无穷远处的渐近分析来完成。
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引用次数: 0
期刊
Journal of Mathematical Analysis and Applications
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