Journal of Differential Equations最新文献

{"title":"Multiplicity and profile of solutions for singularly perturbed Kirchhoff-type problems on closed manifolds","authors":"Xiaojin Bai,&nbsp;Hua Chen,&nbsp;Xiaochun Liu","doi":"10.1016/j.jde.2025.113727","DOIUrl":"10.1016/j.jde.2025.113727","url":null,"abstract":"<div><div>We investigate the existence of solutions for singularly perturbed Kirchhoff-type problems on a closed 3-dimensional Riemannian manifold, focusing on the relation between the number of solutions and the topological properties of the manifold. Our approach is based on the Lusternik–Schnirelmann category. We also provide a profile description of low energy solutions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"450 ","pages":"Article 113727"},"PeriodicalIF":2.3,"publicationDate":"2025-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144908229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Normalized solutions for a class of Sobolev critical Schrödinger systems","authors":"Houwang Li ,&nbsp;Tianhao Liu ,&nbsp;Wenming Zou","doi":"10.1016/j.jde.2025.113719","DOIUrl":"10.1016/j.jde.2025.113719","url":null,"abstract":"&lt;div&gt;&lt;div&gt;This paper focuses on the existence and multiplicity of normalized solutions for the following coupled Schrödinger system with Sobolev critical coupling term:&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;/mtd&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&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;u&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;p&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;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&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;α&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;β&lt;/mi&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;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;mtr&gt;&lt;mtd&gt;&lt;/mtd&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&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;v&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&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;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;β&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;v&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&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;mtr&gt;&lt;mtd&gt;&lt;/mtd&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;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&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;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;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;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&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;msup&gt;&lt;mrow&gt;&lt;mi&gt;b&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;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, &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;msub&gt;&lt;mrow&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;∖&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, and the exponents &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; satisfy&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mspace&gt;&lt;/mspa","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"450 ","pages":"Article 113719"},"PeriodicalIF":2.3,"publicationDate":"2025-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144903343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Abundance of periodic orbits for typical impulsive semiflows","authors":"Jaqueline Siqueira ,&nbsp;Maria Joana Torres ,&nbsp;Paulo Varandas","doi":"10.1016/j.jde.2025.113703","DOIUrl":"10.1016/j.jde.2025.113703","url":null,"abstract":"<div><div>Impulsive dynamical systems, modeled by a continuous semiflow and an impulse function, may be discontinuous and may have non-intuitive topological properties, as the non-invariance of the non-wandering set or the non-existence of invariant probability measures. In this paper we study dynamical features of impulsive flows parameterized by the space of impulses. We prove that impulsive semiflows determined by a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-Baire generic impulse are such that the set of hyperbolic periodic orbits is dense in the set of non-wandering points which meet the impulsive region. As a consequence, we provide sufficient conditions for the non-wandering set of a typical impulsive semiflow (except the discontinuity set) to be invariant. Several applications are given concerning impulsive semiflows obtained from billiard, Anosov and geometric Lorenz flows.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"448 ","pages":"Article 113703"},"PeriodicalIF":2.3,"publicationDate":"2025-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144902261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Global dynamics of a generalized van der Pol-Duffing system with arbitrary degree","authors":"Zhaoxia Wang ,&nbsp;Jueliang Zhou ,&nbsp;Lan Zou","doi":"10.1016/j.jde.2025.113722","DOIUrl":"10.1016/j.jde.2025.113722","url":null,"abstract":"<div><div>We study the global dynamics of a generalized van der Pol-Duffing system in this paper, which has four nonlinear terms with arbitrary degree. This generalized nonlinear system possesses complicated dynamics, including at most three limit cycles, a figure-eight loop, infinitely many heteroclinic bifurcations, Hopf bifurcation, double large limit cycle bifurcation, generalized pitchfork bifurcation and generalized Hopf bifurcation. In addition, these theoretical results are exhibited via numerical simulations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"450 ","pages":"Article 113722"},"PeriodicalIF":2.3,"publicationDate":"2025-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144903342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Local well-posedness in Gevrey function spaces for 3D Boussinesq boundary layer system","authors":"Qian Li ,&nbsp;Peixin Wang ,&nbsp;Xiaojing Xu","doi":"10.1016/j.jde.2025.113725","DOIUrl":"10.1016/j.jde.2025.113725","url":null,"abstract":"<div><div>In this paper, we consider the 3D Boussinesq boundary layer system in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, which is a coupling of the Prandtl type equations and a thermal layer equation due to the coupling of velocity and temperature in Boussinesq equations. We observe that there is also a cancellation mechanism in the temperature equation, which has been applied to the Prandtl equations in Li et al. (2022) <span><span>[14]</span></span>. Utilizing these cancellation mechanisms and constructing good unknowns, we overcome the loss of derivative arising in not only the velocity equations but also the temperature equation, then we show the local well-posedness of the Boussinesq boundary layer system in Gevrey function spaces. Furthermore, we obtain the optimal Gevrey index 2.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"450 ","pages":"Article 113725"},"PeriodicalIF":2.3,"publicationDate":"2025-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144895058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Nekhoroshev stability for random generalized Hamiltonian systems with different regularities","authors":"Bingqi Yu ,&nbsp;Yong Li","doi":"10.1016/j.jde.2025.113709","DOIUrl":"10.1016/j.jde.2025.113709","url":null,"abstract":"<div><div>In this article, we establish the Nekhoroshev stability of nearly integrable generalized Hamiltonian systems with bounded random perturbations possessing different regularity conditions. We generalize the original framework for proving the Nekhoroshev theorem. Using this unified framework, we can derive different normal form lemmas based on various regularity conditions, leading to results for stability times of different scales. Furthermore, this method allows perturbation functions with a certain degree of randomness and can be applied within the context of generalized Hamiltonian systems.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"448 ","pages":"Article 113709"},"PeriodicalIF":2.3,"publicationDate":"2025-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144893821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hölder regularity for nonlocal in time subdiffusion equations with general kernel","authors":"Adam Kubica ,&nbsp;Katarzyna Ryszewska ,&nbsp;Rico Zacher","doi":"10.1016/j.jde.2025.113716","DOIUrl":"10.1016/j.jde.2025.113716","url":null,"abstract":"<div><div>We study the regularity of weak solutions to nonlocal in time subdiffusion equations for a wide class of weakly singular kernels appearing in the generalised fractional derivative operator. We prove a weak Harnack inequality for nonnegative weak supersolutions and Hölder continuity of weak solutions to such problems. Our results substantially extend the results from our previous work <span><span>[11]</span></span> by leaving the framework of distributed order fractional time derivatives and considering a general <span><math><mi>PC</mi></math></span> kernel and by also allowing for an inhomogeneity in the PDE from a Lebesgue space of mixed type.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"450 ","pages":"Article 113716"},"PeriodicalIF":2.3,"publicationDate":"2025-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144893062","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Heteroclinic bifurcation near a loop tangent to an invariant line","authors":"Xianbo Sun ,&nbsp;Guilin Ji ,&nbsp;Qun Bin","doi":"10.1016/j.jde.2025.113713","DOIUrl":"10.1016/j.jde.2025.113713","url":null,"abstract":"<div><div>In this paper, we propose a method for examining the heteroclinic bifurcation near a loop tangent to an invariant line in near-Hamiltonian systems. Our objective is to derive the asymptotic expansion of a generalized Melnikov function, which encompasses not only the first-order Melnikov function but also higher-order Melnikov functions in a wider range of reversible Hamiltonian systems. We apply our findings to a cubic reversible Hamiltonian system with polynomial perturbations of degree <em>n</em>. Our contributions include:</div><div><strong>(i)</strong> Determining the precise number of limit cycles near the tangent loop by using the first-order Melnikov function for polynomial perturbations of arbitrary degree <em>n</em>.</div><div><strong>(ii)</strong> Deriving all-order Melnikov functions with simplified expressions and integrable conditions for the system under the cubic perturbation (<span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span>). Our analysis reveals that the first, second, third, and fourth-order Melnikov functions lead to the bifurcation of <span><math><mi>t</mi><mi>h</mi><mi>r</mi><mi>e</mi><mi>e</mi></math></span>, <span><math><mi>f</mi><mi>i</mi><mi>v</mi><mi>e</mi></math></span>, <em>six</em> limit cycles, and <em>one</em> limit cycle near the loop, respectively.</div><div><strong>(iii)</strong> Determining the exact upper bound on the maximum number of zeros of the first-order Melnikov function for the cubic perturbation by applying a modified Chebyshev criterion and an element-combination technique.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"449 ","pages":"Article 113713"},"PeriodicalIF":2.3,"publicationDate":"2025-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144893717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Almost sharp global wellposedness and scattering for the defocusing conformal wave equation on the hyperbolic space","authors":"Chutian Ma","doi":"10.1016/j.jde.2025.113714","DOIUrl":"10.1016/j.jde.2025.113714","url":null,"abstract":"<div><div>In this paper we prove a global well-posedness and scattering result for the defocusing conformal nonlinear wave equation in the hyperbolic space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>. We take advantage of the hyperbolic geometry which yields stronger Morawetz and Strichartz estimates. We show that the solution is globally wellposed and scatters if the initial data is radially symmetric and lies in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>ϵ</mi></mrow></msup><mo>(</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo><mo>×</mo><msup><mrow><mi>H</mi></mrow><mrow><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>ϵ</mi></mrow></msup><mo>(</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>, <span><math><mi>ϵ</mi><mo>&gt;</mo><mn>0</mn></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"448 ","pages":"Article 113714"},"PeriodicalIF":2.3,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144885364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On periodic nonlocal dispersal competition systems in heterogeneous shifting environments: Survival exchange and gap phenomena","authors":"Jia-Bing Wang ,&nbsp;Shao-Xia Qiao","doi":"10.1016/j.jde.2025.113708","DOIUrl":"10.1016/j.jde.2025.113708","url":null,"abstract":"<div><div>This is a continuation of our work <span><span>[26]</span></span> to investigate the joint influences of seasonal succession, climate change and long-distance free diffusion on the competitive dynamics, where we study the scenario that the growth rates of two competing species <strong>shift in the same trend</strong> with a periodically fluctuating speed. Since the effects of climate change on the habitats of the two competing species may be not synchronized, in this paper we consider the scenario where the two growth rates <strong>shift in opposite trends</strong> with a periodically fluctuating speed. Based on the monotone iterative technique, semigroup theory, sliding and squeezing skill as well as the upper- and lower-solution method, we establish the existence, uniqueness and exponential asymptotic stability of time-periodic and survival exchange type forced wave connecting the two semi-trivial periodic solutions associated to the corresponding limiting systems when the average shifting speed falls into a finite interval. On the contrary, when the average shifting speed is beyond this interval, we find that the gap phenomena will occur by using some comparison arguments.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"449 ","pages":"Article 113708"},"PeriodicalIF":2.3,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144890203","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}