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Qualitative analysis for Hartree-Fock system with Hardy-Littlewood-Sobolev critical exponent 具有Hardy-Littlewood-Sobolev临界指数的Hartree-Fock系统的定性分析
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-06-20 DOI: 10.1016/j.jde.2025.113565
Meng Li , Haoyuan Xu , Meihua Yang , Maoding Zhen
{"title":"Qualitative analysis for Hartree-Fock system with Hardy-Littlewood-Sobolev critical exponent","authors":"Meng Li , Haoyuan Xu , Meihua Yang , Maoding Zhen","doi":"10.1016/j.jde.2025.113565","DOIUrl":"10.1016/j.jde.2025.113565","url":null,"abstract":"<div><div>In this paper, we develop an exhaustive analysis on standing waves with prescribed mass for the coupled Hartree-Fock system as following, which is introduced by Hartree in the 1920's and developed by Fock for describing large systems of identical fermions,<span><span><span><math><mrow><mrow><mo>{</mo><mtable><mtr><mtd></mtd><mtd><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>u</mi><mo>=</mo><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup><mo>⋆</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>u</mi><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>γ</mi></mrow></msup><mo>⋆</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>u</mi><mo>+</mo><mi>β</mi><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>γ</mi></mrow></msup><mo>⋆</mo><mo>|</mo><mi>v</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>u</mi></mtd></mtr><mtr><mtd></mtd><mtd><mo>−</mo><mi>Δ</mi><mi>v</mi><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>v</mi><mo>=</mo><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup><mo>⋆</mo><mo>|</mo><mi>v</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>v</mi><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>γ</mi></mrow></msup><mo>⋆</mo><mo>|</mo><mi>v</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>v</mi><mo>+</mo><mi>β</mi><mo>(</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>γ</mi></mrow></msup><mo>⋆</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mi>v</mi></mtd></mtr></mtable></mrow><mspace></mspace><mtext>in </mtext><msup><mrow><mi>R</mi></mrow><mrow><mn>5</mn></mrow></msup></mrow></math></span></span></span> under mass constraint conditions<span><span><span><math><mrow><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn>5</mn></mrow></msup></mrow></munder><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mn>5</mn></mrow></msup></mrow></munder><mo>|</mo><mi>v</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo></mrow></math></span></span></span> where <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><mn>0</mn><mo>(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo><mo>,</mo><mspace></ms","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"444 ","pages":"Article 113565"},"PeriodicalIF":2.4,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322436","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}
引用次数: 0
Small normalised solutions for a Schrödinger-Poisson system in expanding domains: Multiplicity and asymptotic behaviour 扩展域上Schrödinger-Poisson系统的小正则化解:多重性和渐近行为
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-06-20 DOI: 10.1016/j.jde.2025.113571
Edwin Gonzalo Murcia , Gaetano Siciliano
{"title":"Small normalised solutions for a Schrödinger-Poisson system in expanding domains: Multiplicity and asymptotic behaviour","authors":"Edwin Gonzalo Murcia ,&nbsp;Gaetano Siciliano","doi":"10.1016/j.jde.2025.113571","DOIUrl":"10.1016/j.jde.2025.113571","url":null,"abstract":"<div><div>Given a smooth bounded domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, we consider the following nonlinear Schrödinger-Poisson type system<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>ϕ</mi><mi>u</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>ω</mi><mi>u</mi><mspace></mspace></mtd><mtd><mspace></mspace><mtext>in </mtext><mi>λ</mi><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mo>−</mo><mi>Δ</mi><mi>ϕ</mi><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mspace></mspace></mtd><mtd><mspace></mspace><mtext>in </mtext><mi>λ</mi><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>&gt;</mo><mn>0</mn><mspace></mspace></mtd><mtd><mspace></mspace><mtext>in </mtext><mi>λ</mi><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mi>ϕ</mi><mo>=</mo><mn>0</mn><mspace></mspace></mtd><mtd><mspace></mspace><mtext>on </mtext><mo>∂</mo><mo>(</mo><mi>λ</mi><mi>Ω</mi><mo>)</mo><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mo>∫</mo></mrow><mrow><mi>λ</mi><mi>Ω</mi></mrow></msub><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup><mspace></mspace><mtext>d</mtext><mi>x</mi><mo>=</mo><msup><mrow><mi>ρ</mi></mrow><mrow><mn>2</mn></mrow></msup><mspace></mspace></mtd></mtr></mtable></mrow></math></span></span></span> in the expanding domain <span><math><mi>λ</mi><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo><mi>λ</mi><mo>&gt;</mo><mn>1</mn></math></span> and <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></span>, in the unknowns <span><math><mo>(</mo><mi>u</mi><mo>,</mo><mi>ϕ</mi><mo>,</mo><mi>ω</mi><mo>)</mo></math></span>. We show that, for arbitrary large values of the expanding parameter <em>λ</em> and arbitrary small values of the mass <span><math><mi>ρ</mi><mo>&gt;</mo><mn>0</mn></math></span>, the number of solutions is at least the Ljusternick-Schnirelmann category of <em>λ</em>Ω. Moreover we show that as <span><math><mi>λ</mi><mo>→</mo><mo>+</mo><mo>∞</mo></math></span> the solutions found converge to a ground state of the problem in the whole space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"444 ","pages":"Article 113571"},"PeriodicalIF":2.4,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322375","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}
引用次数: 0
Multiple normalized solutions for Schrödinger-Maxwell equation with Sobolev critical exponent and mixed nonlinearities 具有Sobolev临界指数和混合非线性的Schrödinger-Maxwell方程的多重归一化解
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-06-20 DOI: 10.1016/j.jde.2025.113564
Jin-Cai Kang , Yong-Yong Li , Chun-Lei Tang
{"title":"Multiple normalized solutions for Schrödinger-Maxwell equation with Sobolev critical exponent and mixed nonlinearities","authors":"Jin-Cai Kang ,&nbsp;Yong-Yong Li ,&nbsp;Chun-Lei Tang","doi":"10.1016/j.jde.2025.113564","DOIUrl":"10.1016/j.jde.2025.113564","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this paper, we study the Schrödinger-Maxwell equation with critical growth&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&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;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&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;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mi&gt;u&lt;/mi&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;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;μ&lt;/mi&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;mspace&gt;&lt;/mspace&gt;&lt;mtext&gt;in&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&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;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&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;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;mi&gt;d&lt;/mi&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/mrow&gt;&lt;/mrow&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;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; is prescribed, &lt;span&gt;&lt;math&gt;&lt;mi&gt;κ&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;mi&gt;μ&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, ⁎ denotes the convolution 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; appears as a Lagrange multiplier. Motivated by the works of Wei and Wu (2022) &lt;span&gt;&lt;span&gt;[41]&lt;/span&gt;&lt;/span&gt; for &lt;span&gt;&lt;math&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and Bellazzini and Siciliano (2011) &lt;span&gt;&lt;span&gt;[5]&lt;/span&gt;&lt;/span&gt; for homogeneous nonlinearity, we get two normalized solutions when &lt;span&gt;&lt;math&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, one of which is ground state normalized solution. It is worth emphasizing that the method of Schwarz spherical rearrangement is invalid for the case of &lt;span&gt;&lt;math&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, different from the case of &lt;span&gt;&lt;math&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, and it is hard to establish the strictly subadditive inequality of least energy to exclude the dichotomy of minimizing sequence standardly. To our knowledge, the existence of second solution for the above problem has not been addressed in the current literatures. Moreover, we will show a nonexistence result of positive normalized solution when &lt;span&gt;&lt;math&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, which can be regarded as a generalization and improvement of Jeanjean and Le (2021) &lt;span&gt;&lt;span&gt;[21]&lt;/span&gt;&lt;/span&gt; from the case of &lt;span&gt;&lt;math&gt;&lt;mi","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"443 ","pages":"Article 113564"},"PeriodicalIF":2.4,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321678","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}
引用次数: 0
The upper semi-continuity of random attractors to the stochastic evolution equations driven by rough path with Hurst index H∈(13,12] Hurst指数H∈(13,12)的粗糙路径驱动随机进化方程的随机吸引子的上半连续性
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-06-19 DOI: 10.1016/j.jde.2025.113552
Qiyong Cao, Hongjun Gao
{"title":"The upper semi-continuity of random attractors to the stochastic evolution equations driven by rough path with Hurst index H∈(13,12]","authors":"Qiyong Cao,&nbsp;Hongjun Gao","doi":"10.1016/j.jde.2025.113552","DOIUrl":"10.1016/j.jde.2025.113552","url":null,"abstract":"<div><div>In this paper, we consider the upper semi-continuity of the random attractors for a class of evolution equations driven by smooth stationary fractional noise with Hurst index <span><math><mi>H</mi><mo>∈</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>]</mo></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"444 ","pages":"Article 113552"},"PeriodicalIF":2.4,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313629","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}
引用次数: 0
Asymptotic stability of sharp fronts: Analysis and rigorous computation 尖锐锋面的渐近稳定性:分析与严格计算
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-06-19 DOI: 10.1016/j.jde.2025.113550
Blake Barker , Jared C. Bronski , Vera Mikyoung Hur , Zhao Yang
{"title":"Asymptotic stability of sharp fronts: Analysis and rigorous computation","authors":"Blake Barker ,&nbsp;Jared C. Bronski ,&nbsp;Vera Mikyoung Hur ,&nbsp;Zhao Yang","doi":"10.1016/j.jde.2025.113550","DOIUrl":"10.1016/j.jde.2025.113550","url":null,"abstract":"<div><div>We investigate the stability of traveling front solutions to nonlinear diffusive-dispersive equations of Burgers type, with a primary focus on the Korteweg-de Vries–Burgers (KdVB) equation, although our analytical findings extend more broadly. Manipulating the temporal modulation of the translation parameter of the front and employing the energy method, we establish asymptotic, nonlinear, and orbital stability, provided that an auxiliary Schrödinger equation possesses precisely one bound state. Notably, our result is independent of the monotonicity of the profile and does not necessitate the initial condition to be close to the front. We identify a sufficient condition for stability based on a functional that characterizes the ‘width’ of the traveling wave profile. Analytical verification for the KdVB equation confirms that this sufficient condition holds for the relative dispersion parameter within an open interval <span><math><mo>⊃</mo><mo>[</mo><mo>−</mo><mn>0.25</mn><mo>,</mo><mn>0.25</mn><mo>]</mo></math></span>, encompassing all monotone profiles. Utilizing validated numerics or rigorous computation, we present a computer-assisted proof demonstrating that the stability condition itself holds for parameter values within the interval <span><math><mo>[</mo><mn>0.2533</mn><mo>,</mo><mn>3.9</mn><mo>]</mo></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"444 ","pages":"Article 113550"},"PeriodicalIF":2.4,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144312767","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}
引用次数: 0
Symmetry breaking bifurcation and the stability of stationary solutions of a phase-field model 相场模型的对称破缺分岔与稳态解的稳定性
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-06-18 DOI: 10.1016/j.jde.2025.113500
Yasuhito Miyamoto , Tatsuki Mori , Sohei Tasaki , Tohru Tsujikawa , Shoji Yotsutani
{"title":"Symmetry breaking bifurcation and the stability of stationary solutions of a phase-field model","authors":"Yasuhito Miyamoto ,&nbsp;Tatsuki Mori ,&nbsp;Sohei Tasaki ,&nbsp;Tohru Tsujikawa ,&nbsp;Shoji Yotsutani","doi":"10.1016/j.jde.2025.113500","DOIUrl":"10.1016/j.jde.2025.113500","url":null,"abstract":"<div><div>We have been investigating the global bifurcation diagrams of stationary solutions for a phase field model proposed by Fix and Caginalp in a one-dimensional case. It has recently been shown that there exists a secondary bifurcation with a symmetry-breaking phenomenon from a branch consisting of symmetric solutions in the case where the total enthalpy equals zero. In this paper, we determine the stability/instability of all symmetric solutions and asymmetric solutions near the secondary bifurcation point. Moreover, we show representation formulas for all eigenvalues and eigenfunctions for the linearized eigenvalue problem around the symmetric solutions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"441 ","pages":"Article 113500"},"PeriodicalIF":2.4,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144306837","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}
引用次数: 0
Global existence and optimal decay rates of strong solutions to the equilibrium diffusion model arising in radiation hydrodynamics 辐射流体动力学中平衡扩散模型强解的整体存在性和最优衰减率
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-06-18 DOI: 10.1016/j.jde.2025.113557
Peng Jiang , Fucai Li , Jinkai Ni
{"title":"Global existence and optimal decay rates of strong solutions to the equilibrium diffusion model arising in radiation hydrodynamics","authors":"Peng Jiang ,&nbsp;Fucai Li ,&nbsp;Jinkai Ni","doi":"10.1016/j.jde.2025.113557","DOIUrl":"10.1016/j.jde.2025.113557","url":null,"abstract":"<div><div>In this paper, we investigate the global existence and optimal decay rates of strong solutions in the critical Sobolev space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span> to the equilibrium diffusion model arising in radiation hydrodynamics. This model is composed of the full compressible Navier-Stokes equations with radiation diffusion terms. Assuming that the initial data <span><math><mo>(</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> is sufficiently small in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm, we establish the global existence of strong solutions near the equilibrium state <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> by applying the refined energy method. Then, by performing Fourier analysis techniques and exploiting the frequency decomposition method, we get the optimal time-decay rates of strong solutions (including the highest order spatial derivatives) in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm. In particular, the lower bound of time-decay rates of strong solutions is also obtained by making use of Hodge decomposition, delicate spectral analysis and the theory of Besov space. In addition, we obtain the exponential decay of strong solutions in the periodic domain case.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"443 ","pages":"Article 113557"},"PeriodicalIF":2.4,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144307522","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}
引用次数: 0
The Gierer-Meinhardt system in the entire space with non-local proliferation rates 具有非局部扩散率的整个空间中的Gierer-Meinhardt系统
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-06-18 DOI: 10.1016/j.jde.2025.113559
Marius Ghergu , Nikos I. Kavallaris , Yasuhito Miyamoto
{"title":"The Gierer-Meinhardt system in the entire space with non-local proliferation rates","authors":"Marius Ghergu ,&nbsp;Nikos I. Kavallaris ,&nbsp;Yasuhito Miyamoto","doi":"10.1016/j.jde.2025.113559","DOIUrl":"10.1016/j.jde.2025.113559","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this work, we present a novel stationary Gierer-Meinhardt system incorporating non-local proliferation rates, defined as follows:&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mtext&gt; in &lt;/mtext&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mtext&gt; in &lt;/mtext&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; This system emerges in various contexts, such as biological morphogenesis, where two interacting chemicals, identified as an activator and an inhibitor, are described, and in ecological systems modeling the interaction between two species, classified as specialists and generalists. The non-local interspecies interactions are represented by the terms &lt;span&gt;&lt;math&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; where the ⁎-symbol denotes the convolution operation 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;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; with a kernel &lt;span&gt;&lt;math&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&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;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;∖&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. In the system, we assume that &lt;span&gt;&lt;math&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&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;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, while the parameters satisfy &lt;span&gt;&lt;math&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. Under various integrability conditions on the kernel &lt;em&gt;J&lt;/em&gt;, we establish the existence and non-existence of classical positive solutions in the function space &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;mi&gt;o&lt;/mi&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;δ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&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;mi&gt;N&lt;/mi&gt;&lt;","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"444 ","pages":"Article 113559"},"PeriodicalIF":2.4,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144308080","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}
引用次数: 0
Long-time behaviors of some stochastic differential equations driven by Lévy noise 由lsamvy噪声驱动的随机微分方程的长时间行为
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-06-16 DOI: 10.1016/j.jde.2025.113561
I. Orlovskyi , F. Proske , O. Tymoshenko
{"title":"Long-time behaviors of some stochastic differential equations driven by Lévy noise","authors":"I. Orlovskyi ,&nbsp;F. Proske ,&nbsp;O. Tymoshenko","doi":"10.1016/j.jde.2025.113561","DOIUrl":"10.1016/j.jde.2025.113561","url":null,"abstract":"<div><div>Using key tools such as Itô's formula for general semi-martingales, moment estimates for Lévy-type stochastic integrals, and properties of regularly varying functions we find conditions under which solutions of a stochastic differential equation with jumps are almost surely asymptotically equivalent to a nonrandom function as <span><math><mi>t</mi><mo>→</mo><mo>∞</mo></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"442 ","pages":"Article 113561"},"PeriodicalIF":2.4,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144290710","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}
引用次数: 0
Uniqueness of weak solutions to the primitive equations in some anisotropic spaces 各向异性空间中原始方程弱解的唯一性
IF 2.4 2区 数学
Journal of Differential Equations Pub Date : 2025-06-16 DOI: 10.1016/j.jde.2025.113554
Tim Binz , Yoshiki Iida
{"title":"Uniqueness of weak solutions to the primitive equations in some anisotropic spaces","authors":"Tim Binz ,&nbsp;Yoshiki Iida","doi":"10.1016/j.jde.2025.113554","DOIUrl":"10.1016/j.jde.2025.113554","url":null,"abstract":"<div><div>We consider the both 3D and 2D viscous primitive equations for ocean in the isothermal setting. While strong global well-posedness of the viscous primitive equations for large data in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> has already proved, the uniqueness of the weak solutions of Leray–Hope type for given initial data in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> remains an outstanding open problem. In this paper, we establish a new conditional uniqueness result for weak solutions to the primitive equations, that is, if a weak solution belongs some scaling invariant function spaces, and satisfies some additional assumptions, then the weak solution is unique. In particular, our result can be obtained as different one from <em>z</em>-weak solutions framework by adopting some anisotropic approaches with the homogeneous toroidal Besov spaces. As an application of the proof, we establish the energy equality for weak solutions in the uniqueness class given in the main theorem.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"444 ","pages":"Article 113554"},"PeriodicalIF":2.4,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144290763","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}
引用次数: 0
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