Journal of Differential Equations最新文献

{"title":"Geometric aspects of bifurcations for a classical predator-prey model","authors":"Wei Su ,&nbsp;Xiang Zhang","doi":"10.1016/j.jde.2025.113357","DOIUrl":"10.1016/j.jde.2025.113357","url":null,"abstract":"<div><div>The bifurcation and dynamics of the classical predator-prey model with the generalized Holling type III functional response have been studied from different aspects. When the denominator of the response function has at least one zero, its global dynamics has been classified. When the denominator does not vanish, its local bifurcation was classified in 2008 from analytic point of view. Here we first characterize the local bifurcation via geometry of the critical curve. Then utilizing these geometric aspects of the bifurcations, we further classify all global topological dynamics of this model in the slow-fast setting, where we can also exhibit not only the birth and disappearance but also the locations and shapes of the limit cycles.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"438 ","pages":"Article 113357"},"PeriodicalIF":2.4,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143907468","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 the regularity of axially-symmetric solutions to the incompressible Navier-Stokes equations in a cylinder","authors":"Wojciech S. Ożański ,&nbsp;Wojciech M. Zaja̧czkowski","doi":"10.1016/j.jde.2025.113373","DOIUrl":"10.1016/j.jde.2025.113373","url":null,"abstract":"<div><div>We consider the axisymmetric Navier-Stokes equations in a finite cylinder <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. We assume that <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span>, <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>φ</mi></mrow></msub></math></span>, <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>φ</mi></mrow></msub></math></span> vanish on the lateral boundary ∂Ω of the cylinder, and that <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>z</mi></mrow></msub></math></span>, <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>φ</mi></mrow></msub></math></span>, <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>z</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mi>φ</mi></mrow></msub></math></span> vanish on the top and bottom parts of the boundary ∂Ω, where we used standard cylindrical coordinates, and we denoted by <span><math><mi>ω</mi><mo>=</mo><mrow><mi>curl</mi></mrow><mspace></mspace><mi>v</mi></math></span> the vorticity field. We use weighted estimates and <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> Sobolev estimate on the modified stream function to derive three order-reduction estimates. These enable one to reduce the order of the nonlinear estimates of the equations, and help observe that the solutions to the equations are “almost regular”. We use the order-reduction estimates to show that the solution to the equations remains regular as long as, for any <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>6</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>, <span><math><msub><mrow><mo>‖</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo>‖</mo></mrow><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mo>∞</mo></mrow></msubsup><msubsup><mrow><mi>L</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msubsup></mrow></msub><mo>/</mo><msub><mrow><mo>‖</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>φ</mi></mrow></msub><mo>‖</mo></mrow><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mo>∞</mo></mrow></msubsup><msubsup><mrow><mi>L</mi></mrow><mrow><mi>x</mi></mrow><mrow><mo>∞</mo></mrow></msubsup></mrow></msub></math></span> remains bounded below by a positive number.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"438 ","pages":"Article 113373"},"PeriodicalIF":2.4,"publicationDate":"2025-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143907466","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 Cauchy problem for semi-linear Klein-Gordon equations in Friedmann-Lemaître-Robertson-Walker spacetimes","authors":"Makoto Nakamura,&nbsp;Takuma Yoshizumi","doi":"10.1016/j.jde.2025.113395","DOIUrl":"10.1016/j.jde.2025.113395","url":null,"abstract":"<div><div>The Cauchy problem for semi-linear Klein-Gordon equations is considered in Friedmann-Lemaître-Robertson-Walker spacetimes. The local and global well-posedness of the Cauchy problem is considered in Sobolev spaces. The non-existence of global solutions is also considered.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"438 ","pages":"Article 113395"},"PeriodicalIF":2.4,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143903582","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":"Ancient solutions to the parabolic Monge–Ampère equations with new asymptotic behavior at infinity","authors":"Jiguang Bao ,&nbsp;Zixiao Liu ,&nbsp;Ziwei Zhou","doi":"10.1016/j.jde.2025.113384","DOIUrl":"10.1016/j.jde.2025.113384","url":null,"abstract":"<div><div>Demonstrating new asymptotic behavior <span><math><mi>u</mi><mo>−</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> at infinity, we employ the Perron method to establish the existence of ancient solutions to exterior Dirichlet problems associated with the parabolic Monge–Ampère equation <span><math><mo>−</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>⋅</mo><mi>det</mi><mo>⁡</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>=</mo><mi>f</mi></math></span>, where <em>f</em> is asymptotic to 1 at infinity. The function <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> has unbounded and not strictly negative partial derivative with respect to time variable, and it represents a new family of smooth, parabolically convex, ancient entire solutions to the equation with <span><math><mi>f</mi><mo>≡</mo><mn>1</mn></math></span>. The family of solutions was recently discovered in [An–Bao–Liu, Nonlinear Anal., 2024], and their asymptotic behavior differs significantly from the scenario where <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> is bounded and strictly negative. In contrast to existing methodologies in the study of exterior Dirichlet problems, functions relying only on the value of <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> have been introduced to replace generalized symmetric functions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"438 ","pages":"Article 113384"},"PeriodicalIF":2.4,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143903583","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":"Dirichlet problem on perturbed conical domains via converging generalized power series","authors":"Martin Costabel ,&nbsp;Matteo Dalla Riva ,&nbsp;Monique Dauge ,&nbsp;Paolo Musolino","doi":"10.1016/j.jde.2025.113379","DOIUrl":"10.1016/j.jde.2025.113379","url":null,"abstract":"<div><div>We consider the Poisson equation with homogeneous Dirichlet conditions in a family of domains in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> indexed by a small parameter <em>ε</em>. The domains depend on <em>ε</em> only within a ball of radius proportional to <em>ε</em> and, as <em>ε</em> tends to zero, they converge in a self-similar way to a domain with a conical boundary singularity. We construct an expansion of the solution as a series of real positive powers of <em>ε</em>, and prove that it is not just an asymptotic expansion as <span><math><mi>ε</mi><mo>→</mo><mn>0</mn></math></span>, but that, for small values of <em>ε</em>, it converges normally in the Sobolev space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>. The phenomenon that solutions to boundary value problems on singularly perturbed domains may have <em>convergent</em> expansions is the subject of the Functional Analytic Approach by Lanza de Cristoforis and his collaborators. This approach was originally adopted to study small holes shrinking to interior points of a smooth domain and heavily relies on integral representations obtained through layer potentials. We choose a different technique that allows us to relax all regularity assumptions. We forgo boundary layer potentials and instead exploit expansions in terms of eigenfunctions of the Laplace-Beltrami operator on the intersection of the cone with the unit sphere. The basis for our analysis is a two-scale cross-cutoff ansatz for the solution that has similarities with the Maz'ya-Nazarov-Plamenevskij construction of a multiscale system for the asymptotic expansion of solutions of boundary value problems on domains singularly perturbed near singular points of the boundary. Specifically, we write the solution as a sum of a function in the slow variable multiplied by a cutoff function depending on the fast variable, plus a function in the fast variable multiplied by a cutoff function depending on the slow variable. While the cutoffs are considered fixed, the two unknown functions are solutions to a <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> system of partial differential equations that depend on <em>ε</em> in a way that can be analyzed in the framework of generalized power series when the right-hand side of the Poisson equation vanishes in a neighborhood of the perturbation. In this paper, we concentrate on this case. The treatment of more general right-hand sides requires a supplementary layer in the analysis and is postponed to a forthcoming paper.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"439 ","pages":"Article 113379"},"PeriodicalIF":2.4,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143899800","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 and nonexistence of minimizer for Thomas-Fermi-Dirac-von Weizsäcker model on lattice graph","authors":"Yong Liu ,&nbsp;Jun Wang ,&nbsp;Kun Wang ,&nbsp;Wen Yang ,&nbsp;Yanni Zhu","doi":"10.1016/j.jde.2025.113360","DOIUrl":"10.1016/j.jde.2025.113360","url":null,"abstract":"<div><div>The focus of our paper is to investigate the possibility of a minimizer for the Thomas-Fermi-Dirac-von Weizsäcker model on the lattice graph <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. The model is described by the following functional:<span><span><span><math><mi>E</mi><mo>(</mo><mi>φ</mi><mo>)</mo><mo>=</mo><munder><mo>∑</mo><mrow><mi>y</mi><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></munder><mrow><mo>(</mo><mo>|</mo><mi>∇</mi><mi>φ</mi><mo>(</mo><mi>y</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mo>(</mo><mi>φ</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>)</mo></mrow><mrow><mfrac><mrow><mn>10</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>−</mo><msup><mrow><mo>(</mo><mi>φ</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>)</mo></mrow><mrow><mfrac><mrow><mn>8</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup><mo>)</mo></mrow><mo>+</mo><munder><mo>∑</mo><mrow><mfrac><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow><mrow><mspace></mspace><mspace></mspace><mi>y</mi><mo>≠</mo><mi>x</mi></mrow></mfrac></mrow></munder><mfrac><mrow><msup><mrow><mi>φ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mi>φ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mrow><mo>|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo>|</mo></mrow></mfrac><mo>,</mo></math></span></span></span> with the additional constraint that <span><math><munder><mo>∑</mo><mrow><mi>y</mi><mo>∈</mo><msup><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></munder><msup><mrow><mi>φ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>m</mi></math></span>. We begin by establishing the existence of a minimizer for this model when <em>m</em> is sufficiently small. Conversely, we demonstrate that no minimizer exists when <em>m</em> exceeds a certain threshold. Additionally, we extend our analysis to a subset <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> and prove the nonexistence of a minimizer for the following functional:<span><span><span><math><mi>E</mi><mo>(</mo><mi>Ω</mi><mo>)</mo><mo>=</mo><mo>|</mo><mo>∂</mo><mi>Ω</mi><mo>|</mo><mo>+</mo><munder><mo>∑</mo><mrow><mfrac><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>Ω</mi></mrow><mrow><mspace></mspace><mi>y</mi><mo>≠</mo><mi>x</mi></mrow></mfrac></mrow></munder><mfrac><mrow><mn>1</mn></mrow><mrow><mo>|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo>|</mo></mrow></mfrac><mo>,</mo></math></span></span></span> under the constraint that <span><math><mo>|</mo><mi>Ω</mi><mo>|</mo><mo>=</mo><mi>V</mi></math></span> is sufficiently large.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"438 ","pages":"Article 113360"},"PeriodicalIF":2.4,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143899124","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 kernel estimates and their stabilities for symmetric jump processes with general mixed polynomial growths on metric measure spaces","authors":"Joohak Bae ,&nbsp;Jaehoon Kang ,&nbsp;Panki Kim ,&nbsp;Jaehun Lee","doi":"10.1016/j.jde.2025.113377","DOIUrl":"10.1016/j.jde.2025.113377","url":null,"abstract":"<div><div>In this paper, we consider a symmetric pure jump Markov process <em>X</em> on a metric measure space with volume doubling conditions. Our focus is on estimating the transition density <span><math><mi>p</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> of <em>X</em> and studying its stability when the jumping kernel exhibits general mixed polynomial growth.</div><div>Unlike previous work, in our setting, the rate function governing the jump growth may not be comparable to the scale function that determines whether <span><math><mi>p</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> has near-diagonal or off-diagonal estimates. Under the assumption that lower scaling index of scale function is greater than 1, we establish stabilities of heat kernel estimates. Additionally, if the metric measure space admits a conservative diffusion process with a transition density satisfying sub-Gaussian bounds, we generalize heat kernel estimates from <span><span>[3, Theorems 1.2 and 1.4]</span></span> using the rate function and the function <em>F</em> related to walk dimension of underlying space. As an application, we prove the equivalence between a finite moment condition based on <em>F</em> and a generalized Khintchine-type law of iterated logarithm at infinity for symmetric Markov processes.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"438 ","pages":"Article 113377"},"PeriodicalIF":2.4,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143899122","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 the fully nontrivial ground state solutions for the coupled nonlinear Brezis-Nirenberg Maxwell system","authors":"Cong Li ,&nbsp;Yong Liu ,&nbsp;Jun Wang ,&nbsp;Wen Yang","doi":"10.1016/j.jde.2025.113374","DOIUrl":"10.1016/j.jde.2025.113374","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this paper, we explore the existence of fully nontrivial solutions to the following nonlinear Brezis-Nirenberg Maxwell system&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;mi&gt;∇&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&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;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&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;κ&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;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&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;β&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&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;mspace&gt;&lt;/mspace&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mtext&gt;in&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&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;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;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;κ&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&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;β&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&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;mspace&gt;&lt;/mspace&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mtext&gt;in&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ν&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;E&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;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ν&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&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;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mtext&gt;on&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;mi&gt;Ω&lt;/mi&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;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;β&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;p&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;, and &lt;span&gt;&lt;math&gt;&lt;mi&gt;Ω&lt;/mi&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;/math&gt;&lt;/s","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"438 ","pages":"Article 113374"},"PeriodicalIF":2.4,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143888266","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":"Well-posedness of stochastic mSQG equations with Kraichnan noise and Lp data","authors":"Shuaijie Jiao ,&nbsp;Dejun Luo","doi":"10.1016/j.jde.2025.113362","DOIUrl":"10.1016/j.jde.2025.113362","url":null,"abstract":"<div><div>We consider stochastic mSQG (modified Surface Quasi-Geostrophic) equations with multiplicative transport noise of Kraichnan type, and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-initial conditions. Inspired by the recent work of Coghi and Maurelli <span><span>[11]</span></span>, we show weak existence and pathwise uniqueness of solutions to the equations for suitable choices of parameters in the nonlinearity, the noise and the integrability of initial data.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"438 ","pages":"Article 113362"},"PeriodicalIF":2.4,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143888265","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":"Multi-dimensional super-linear backward stochastic Volterra integral equations","authors":"Shengjun Fan ,&nbsp;Tianxiao Wang ,&nbsp;Jiongmin Yong","doi":"10.1016/j.jde.2025.113350","DOIUrl":"10.1016/j.jde.2025.113350","url":null,"abstract":"<div><div>In this paper, a systematic investigation is carried out for the general solvability of multi-dimensional backward stochastic Volterra integral equations (BSVIEs) with the generators being super-linear in the adjustment variable <em>Z</em>. Two major situations are discussed: (i) When the terminal term is bounded with the dependence of the generator on <em>Z</em> being of “diagonally strictly” quadratic growth and being sub-quadratically coupled with off-diagonal components; (ii) When the terminal term is unbounded having exponential moments of arbitrary order with the dependence of the generator on <em>Z</em> being diagonally no more than quadratic and being independent of off-diagonal components. Besides, for the case that the generator is super-quadratic in <em>Z</em>, some negative results are presented.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"437 ","pages":"Article 113350"},"PeriodicalIF":2.4,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143883004","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}