Journal of Differential Equations最新文献

{"title":"An explicitly solvable NLS model with discontinuous standing waves","authors":"Riccardo Adami ,&nbsp;Filippo Boni ,&nbsp;Takaaki Nakamura ,&nbsp;Alice Ruighi","doi":"10.1016/j.jde.2025.113746","DOIUrl":"10.1016/j.jde.2025.113746","url":null,"abstract":"<div><div>We study the NLS Equation on the line with a point interaction given by the superposition of an attractive delta potential with a dipole interaction, in the cases of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-subcritical and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-critical nonlinearity.</div><div>For a subcritical nonlinearity we prove the existence and the uniqueness of Ground States at any mass. If the mass exceeds an explicit threshold, then there exists a positive excited state too.</div><div>For the critical nonlinearity we prove that Ground States exist only in a specific interval of masses, while in a different interval excited states exist. We provide the value of the optimal constant in the Gagliardo-Nirenberg estimate and describe in the dipole case the branches of the stationary states as the strength of the interaction varies.</div><div>Since all stationary states are explicitly computed, ours is a solvable model involving a non-standard interplay of a nonlinearity with a point interaction.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"451 ","pages":"Article 113746"},"PeriodicalIF":2.3,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145020202","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":"Morse index and non-degeneracy of double-tower solutions for prescribed scalar curvature problem","authors":"Yuxia Guo ,&nbsp;Yichen Hu ,&nbsp;Shaolong Peng","doi":"10.1016/j.jde.2025.113751","DOIUrl":"10.1016/j.jde.2025.113751","url":null,"abstract":"<div><div>We consider the following prescribed scalar curvature equations in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>:<span><span><span>(0.1)</span><span><math><mrow><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>=</mo><mi>K</mi><mo>(</mo><mo>|</mo><mi>y</mi><mo>|</mo><mo>)</mo><msup><mrow><mi>u</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mo>⁎</mo></mrow></msup><mo>−</mo><mn>1</mn></mrow></msup><mo>,</mo><mtext></mtext><mi>u</mi><mo>&gt;</mo><mn>0</mn><mtext> in </mtext><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>,</mo><mtext></mtext><mi>u</mi><mo>∈</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo><mo>,</mo></mrow></math></span></span></span> where <span><math><mi>K</mi><mo>(</mo><mi>r</mi><mo>)</mo></math></span> is a positive function, <span><math><mi>N</mi><mo>≥</mo><mn>5</mn></math></span> and <span><math><msup><mrow><mn>2</mn></mrow><mrow><mo>⁎</mo></mrow></msup><mo>=</mo><mfrac><mrow><mn>2</mn><mi>N</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></span>. We are concerned with the solutions which are invariant under some non-trivial sub-group of <span><math><mi>O</mi><mo>(</mo><mn>3</mn><mo>)</mo></math></span> to the above problem (we call them double-tower solutions). We first prove a non-degeneracy result for the positive double-tower solutions. As an application, we consider an eigenvalue problem related to prescribed scalar curvature equations and investigate the properties of the eigenvalues. And we compute the Morse index of the double-tower solutions. Our proof is based on the local Pohozaev identities, blow-up analysis, and the properties of the Green function.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"451 ","pages":"Article 113751"},"PeriodicalIF":2.3,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145020200","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":"Vanishing viscosity limit of compressible non-resistive magnetohydrodynamic equations with the no-slip boundary condition","authors":"Qiangchang Ju ,&nbsp;Jiawei Wang ,&nbsp;Feng Xie","doi":"10.1016/j.jde.2025.113749","DOIUrl":"10.1016/j.jde.2025.113749","url":null,"abstract":"<div><div>In this paper, we consider the vanishing viscosity limit of the three dimensional compressible non-resistive magnetohydrodynamic equations with the no-slip boundary condition in the half-space. Assuming that the initial normal magnetic field is non-degenerate, by identifying a new cancellation structure in the momentum equation, we can use the tangential derivatives of solutions to control the normal derivatives of the magnetic field and pressure. Furthermore, we establish uniform regularity estimates of solutions to the initial-boundary value problem of the compressible non-resistive magnetohydrodynamic equations in conormal Sobolev spaces. Then, based on these uniform regularity estimates and the compactness arguments, the vanishing viscosity limit of solutions to the compressible non-resistive magnetohydrodynamic equations is rigorously verified in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> sense.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"451 ","pages":"Article 113749"},"PeriodicalIF":2.3,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144997556","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":"Heat equation in a periodic domain with special initial data","authors":"Marcus Rosenberg,&nbsp;Jari Taskinen","doi":"10.1016/j.jde.2025.113754","DOIUrl":"10.1016/j.jde.2025.113754","url":null,"abstract":"<div><div>We consider the initial-boundary value problem with the Neumann boundary condition for the classical linear heat equation in unbounded domains <span><math><mi>Ω</mi><mo>⊊</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> which are periodic in all directions of the Cartesian coordinate system. Generalizing the results of a previous paper by the authors, we apply Floquet transform methods to obtain results on the large time decay rates of the solution in the sup-norm. We observe that for a general, integrable initial data, the solution decays at the same rate <span><math><msup><mrow><mi>t</mi></mrow><mrow><mo>−</mo><mi>d</mi><mo>/</mo><mn>2</mn></mrow></msup></math></span> as in the case of the Cauchy problem in the entire Euclidean space. We also consider special initial data with vanishing <em>x</em>-integral and obtain a faster decay rate. In the main results of the paper we pose for the initial data certain more detailed conditions, which are related to the lowest eigenvalue and eigenfunction of the model problem coming from the Floquet transform. Faster decay rates are obtained for such initial data.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"451 ","pages":"Article 113754"},"PeriodicalIF":2.3,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144997656","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-in-time maximal regularity for the Cauchy problem of the heat equation in BMO and applications","authors":"Xuan Thinh Duong ,&nbsp;Ji Li ,&nbsp;Liangchuan Wu ,&nbsp;Lixin Yan","doi":"10.1016/j.jde.2025.113748","DOIUrl":"10.1016/j.jde.2025.113748","url":null,"abstract":"<div><div>In this article, we establish global-in-time maximal regularity for the Cauchy problem of the classical heat equation <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> with <span><math><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></math></span> in a certain BMO setting, which improves the local-in-time result initially proposed by Ogawa and Shimizu in <span><span>[26]</span></span>, <span><span>[27]</span></span>. In further developing our method originally formulated for the heat equation, we obtain analogous global BMO-maximal regularity associated to the Schrödinger operator <span><math><mi>L</mi><mo>=</mo><mo>−</mo><mi>Δ</mi><mo>+</mo><mi>V</mi></math></span>, where the nonnegative potential <em>V</em> belongs to the reverse Hölder class <span><math><msub><mrow><mi>RH</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> for some <span><math><mi>q</mi><mo>&gt;</mo><mi>n</mi><mo>/</mo><mn>2</mn></math></span>. This extension includes several inhomogeneous estimates as ingredients, such as Carleson-type estimates for the external forces.</div><div>Our new methodology is to exploit elaborate heat kernel estimates, along with matched space-time decomposition on the involving integral-type structure of maximal operators, as well as some global techniques such as those from de Simon's work and Schur's lemma. One crucial trick is to utilize the mean oscillation therein to contribute a higher and necessary decay order for global-in-time estimates.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"451 ","pages":"Article 113748"},"PeriodicalIF":2.3,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144997555","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":"Foliations raised by quadratic-like polynomial submersions on the real plane and a sharp result on the real Jacobian conjecture","authors":"Francisco Braun ,&nbsp;Filipe Fernandes ,&nbsp;Ingrid S. Meza-Sarmiento","doi":"10.1016/j.jde.2025.113742","DOIUrl":"10.1016/j.jde.2025.113742","url":null,"abstract":"<div><div>We study the planar foliations whose leaves are the connected components of the fibers of polynomial submersion functions having degree 2 in one of the variables. As an application of this geometric study, we prove the following result on the so called real Jacobian conjecture in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>: it is true if one of the coordinate functions has degree 2 in one of the variables. This is sharp because of an existing counterexample with one of the coordinate functions having degree 3 in one of the variables.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"449 ","pages":"Article 113742"},"PeriodicalIF":2.3,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144996929","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":"Existence of axis-symmetric blow-up solution with multiple peak aggregations for the 2-D Keller-Segel systems coupled bipolar source and sink flow","authors":"Yukihiro Seki ,&nbsp;Kosuke Shibata ,&nbsp;Yoshie Sugiyama","doi":"10.1016/j.jde.2025.113745","DOIUrl":"10.1016/j.jde.2025.113745","url":null,"abstract":"<div><div>We consider the Keller-Segel systems in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, coupled with a bipolar source and sink flow. Focusing on the two-dimensional case (<span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>), we establish finite-time blow-up of solutions under an axis-symmetric setting, without requiring the solutions to be radial. In particular, we prove that multiple blow-up points appear in pairs (i.e., in even numbers) away from the origin, lying on the <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-axis and exhibiting axis-symmetry about the <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-axis. This result holds for initial data with total mass strictly greater than 16<em>π</em>, and stands in contrast to the classical radial setting, where blow-up is confined to the origin.</div><div>A crucial part of our analysis is a sharp <em>ε</em>-regularity theorem, originally developed for the classical Keller-Segel systems and first established by Luckhaus–Sugiyama–Velázquez <span><span>[12]</span></span>. This theorem states that if the local mass around <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is sufficiently small at some time <span><math><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, then the solution remains locally bounded in a suitable parabolic cylinder in space–time centered at <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span>. Compared to the classical <em>ε</em>-regularity theorem, it requires weaker assumptions and yields weaker conclusions, making it a form of partial regularity that is particularly essential for analyzing blow-up singularities.</div><div>Based on this sharp <em>ε</em>-regularity theorem, we further prove that only finitely many blow-up points appear as singular sets, and the asymptotic profile is characterized as the sum of a finite number of <em>δ</em>-functions and a regular part in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>. Moreover, our results reveal that multi-peak blow-up phenomena can occur with or without the presence of non-trivial flow, highlighting the intricate interplay between diffusion, chemotaxis, and persistent advection. By accounting for non-decaying flow and employing precise blow-up criteria, we establish that the blow-up time can be bounded above by any prescribed threshold. These findings are justified through the construction of a time-local existence and extension theory for strong solutions, which incorporates both advection and mass conservation.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"450 ","pages":"Article 113745"},"PeriodicalIF":2.3,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144989120","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":"Elliptic equations in divergence form with zero mass and critical exponential growth","authors":"J.C. de Albuquerque ,&nbsp;J. Carvalho ,&nbsp;E.D. Silva","doi":"10.1016/j.jde.2025.113743","DOIUrl":"10.1016/j.jde.2025.113743","url":null,"abstract":"<div><div>In this work, we consider a class of elliptic equations in divergence form with zero mass, involving weight functions that are not necessarily symmetric and nonlinearities satisfying critical exponential growth. For this purpose, we introduce a weighted Trudinger-Moser type inequality. We prove the existence of nonnegative least energy solutions and investigate their qualitative properties, including asymptotic behavior, growth estimates, regularity results, and the existence of strictly positive solutions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"451 ","pages":"Article 113743"},"PeriodicalIF":2.3,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144989805","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":"Sufficient conditions for the n-dimensional real Jacobian conjecture","authors":"Changjian Liu ,&nbsp;Yuzhou Tian","doi":"10.1016/j.jde.2025.113750","DOIUrl":"10.1016/j.jde.2025.113750","url":null,"abstract":"<div><div>The real Jacobian conjecture, proposed by Randall in 1983, asserts that a polynomial map <span><math><mi>F</mi><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> such that <span><math><mi>det</mi><mo>⁡</mo><mi>D</mi><mi>F</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≠</mo><mn>0</mn></math></span> for all <span><math><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is injective. However, this conjecture is disproven by Pinchuk's counterexample.</div><div>This investigation mainly consists of two parts. Firstly, we use the qualitative theory of dynamical systems to give an alternative proof of the polynomial version of the <em>n</em>-dimensional Hadamard's theorem. Secondly, we present some algebraic sufficient conditions for the <em>n</em>-dimensional real Jacobian conjecture. Our results not only extend the main result of [J. Differential Equations <strong>260</strong> (2016), 5250-5258] to quasi-homogeneous type, but also generalize it from <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> to <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. As a coproduct of our proof process, we solve an open problem formulated by Braun, Giné and Llibre in [J. Differential Equations <strong>260</strong> (2016), 5250-5258].</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"446 ","pages":"Article 113750"},"PeriodicalIF":2.3,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144988061","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":"The asymptotic of the Mullins-Sekerka and the area-preserving curvature flow in the planar flat torus","authors":"Vedansh Arya ,&nbsp;Daniele De Gennaro ,&nbsp;Anna Kubin","doi":"10.1016/j.jde.2025.113755","DOIUrl":"10.1016/j.jde.2025.113755","url":null,"abstract":"<div><div>We study the asymptotic behavior of flat flow solutions to the periodic and planar two-phase Mullins-Sekerka flow and area-preserving curvature flow. We show that flat flows converge to either a finite union of equally sized disjoint disks or to a finite union of disjoint strips or to the complement of these configurations exponentially fast. A key ingredient in our approach is the derivation of a sharp quantitative Alexandrov inequality for periodic smooth sets.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"451 ","pages":"Article 113755"},"PeriodicalIF":2.3,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144989806","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}