{"title":"Averaging principle for slow-fast SPDEs driven by mixed noises","authors":"Haoyuan Li, Hongjun Gao, Shiduo Qu","doi":"10.1016/j.jde.2025.02.080","DOIUrl":"10.1016/j.jde.2025.02.080","url":null,"abstract":"<div><div>This paper investigates a class of slow-fast stochastic partial differential equations driven by fractional Brownian motion and standard Brownian motion. Firstly, the well-posedness for such equations are established. Secondly, we provide the uniform <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-estimation for slow variable relying on the mild stochastic sewing Lemma. Finally, we obtain the approximate solution for slow variable via averaging principle.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"430 ","pages":"Article 113209"},"PeriodicalIF":2.4,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143550948","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":"Stability analysis for 1-D wave equation with delayed feedback control","authors":"Shijie Zhou , Hongyinping Feng , Zhiqiang Wang","doi":"10.1016/j.jde.2025.02.075","DOIUrl":"10.1016/j.jde.2025.02.075","url":null,"abstract":"<div><div>In this paper, we investigate the stability problem of 1-D wave equations with delayed feedback control on the boundary. By a delicate spectral analysis, the sufficient and necessary conditions for the feedback gain and the time delay are derived to guarantee the exponential stability of the closed-loop system. We discuss about all the situations for the time delay <span><math><mi>τ</mi><mo>></mo><mn>0</mn></math></span> including the case that <em>τ</em> is irrational. The stability region of the feedback gain exists if and only if the time delay <em>τ</em> is an even number. In this case, an explicit formula of the stability region is obtained accordingly and it characterizes the shrink of the stability region as <em>τ</em> tends to infinity. In addition, we find that the small perturbation of magnitude in the time delay can only trigger the excitation of high frequency modes. That completely proves the judgement in <span><span>[3, Page 5, Remark]</span></span> and gives a mathematical explanation why numerical experiments usually do not demonstrate the non-robustness when a small perturbation is added to the time delay.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"430 ","pages":"Article 113204"},"PeriodicalIF":2.4,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143550838","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 asymptotics for a class of Métivier groups","authors":"Hong-Quan Li , Ye Zhang","doi":"10.1016/j.jde.2025.02.070","DOIUrl":"10.1016/j.jde.2025.02.070","url":null,"abstract":"<div><div>We study the uniform asymptotic behaviour at infinity of the heat kernel, associated to the sub-Laplacian as well as the full Laplacian, on a large class of step-two Carnot groups. Moreover, sharp bounds for its derivatives will be obtained. As an application, precise estimates and small-time asymptotic behaviours for the heat kernel will be provided, as explicit as one can possibly hope for, in the setting of generalized Heisenberg-type groups.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"430 ","pages":"Article 113199"},"PeriodicalIF":2.4,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143534792","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":"Regularity of the free boundary for a semilinear vector-valued minimization problem","authors":"Lili Du , Yi Zhou","doi":"10.1016/j.jde.2025.02.068","DOIUrl":"10.1016/j.jde.2025.02.068","url":null,"abstract":"<div><div>In this paper, we consider the following vector-valued minimization problem<span><span><span><math><mi>min</mi><mo></mo><mrow><mo>{</mo><munder><mo>∫</mo><mrow><mi>D</mi></mrow></munder><mo>(</mo><mo>|</mo><mi>∇</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>F</mi><mo>(</mo><mo>|</mo><mi>u</mi><mo>|</mo><mo>)</mo><mo>)</mo><mi>d</mi><mi>x</mi><mo>:</mo><mspace></mspace><mspace></mspace><mi>u</mi><mo>∈</mo><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msup><mo>(</mo><mi>D</mi><mo>;</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo><mspace></mspace><mtext>and</mtext><mspace></mspace><mi>u</mi><mo>=</mo><mi>g</mi><mspace></mspace><mtext>on</mtext><mspace></mspace><mo>∂</mo><mi>D</mi><mo>}</mo></mrow></math></span></span></span> where <span><math><mi>u</mi><mo>:</mo><mi>D</mi><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> (<span><math><mi>m</mi><mo>≥</mo><mn>1</mn></math></span>) is a vector-valued function, <span><math><mi>D</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> (<span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>) is a bounded Lipschitz domain, <span><math><mi>g</mi><mo>∈</mo><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msup><mo>(</mo><mi>D</mi><mo>;</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></math></span> is a given vector-valued function and <span><math><mi>F</mi><mo>:</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo><mo>→</mo><mi>R</mi></math></span> is a given function. This minimization problem corresponds to the following semilinear elliptic system<span><span><span><math><mi>Δ</mi><mi>u</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msup><mrow><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mo>|</mo><mi>u</mi><mo>|</mo><mo>)</mo><mfrac><mrow><mi>u</mi></mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mfrac><msub><mrow><mi>χ</mi></mrow><mrow><mo>{</mo><mo>|</mo><mi>u</mi><mo>|</mo><mo>></mo><mn>0</mn><mo>}</mo></mrow></msub><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span> denotes the characteristic function of the set <em>A</em>. The linear case that <span><math><msup><mrow><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≡</mo><mn>2</mn></math></span> was studied in the previous elegant work by Andersson et al. (2015) <span><span>[3]</span></span>, in which an epiperimetric inequality played a crucial role to indicate an energy decay estimate and the uniqueness of blow-up limit. However, this epiperimetric inequality cannot be directly applied to our case due to the more general non-degenerate and non-homogeneous term <em>F</em> which leads to Weiss's energy functional does not have scaling properties. Motivated by the linear case, when <em>F</em> satisfies some ","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"430 ","pages":"Article 113197"},"PeriodicalIF":2.4,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143534811","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}
Diego Berti , Fabrice Planchon , Nikolay Tzvetkov , Nicola Visciglia
{"title":"New bounds on the high Sobolev norms of the 1D NLS solutions","authors":"Diego Berti , Fabrice Planchon , Nikolay Tzvetkov , Nicola Visciglia","doi":"10.1016/j.jde.2025.02.073","DOIUrl":"10.1016/j.jde.2025.02.073","url":null,"abstract":"<div><div>We introduce modified energies that are suitable to get upper bounds on the high Sobolev norms for solutions to the 1D periodic NLS. Our strategy is rather flexible and allows us to get a new and simpler proof of the bounds obtained by Bourgain in the case of the quintic nonlinearity, as well as its extension to the case of higher order nonlinearities. Our main ingredients are a combination of integration by parts and classical dispersive estimates.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"430 ","pages":"Article 113202"},"PeriodicalIF":2.4,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143534793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Liouville theorem for Lane-Emden equation of Baouendi-Grushin operators","authors":"Hua Chen, Xin Liao","doi":"10.1016/j.jde.2025.02.072","DOIUrl":"10.1016/j.jde.2025.02.072","url":null,"abstract":"<div><div>In this paper, we establish a Liouville theorem for solutions to the Lane-Emden equation involving Baouendi-Grushin operator:<span><span><span><math><mo>−</mo><mo>(</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>x</mi></mrow></msub><mi>u</mi><mo>+</mo><msup><mrow><mo>(</mo><mi>α</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn><mi>α</mi></mrow></msup><msub><mrow><mi>Δ</mi></mrow><mrow><mi>y</mi></mrow></msub><mi>u</mi><mo>)</mo><mo>=</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>u</mi><mo>,</mo></math></span></span></span> where <span><math><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>=</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with <span><math><mi>m</mi><mo>≥</mo><mn>1</mn></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>, and <span><math><mi>α</mi><mo>≥</mo><mn>0</mn></math></span>. We focus on solutions that are stable outside a compact set. Specifically, we prove that for <span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span>, when <em>p</em> is smaller than the Joseph–Lundgren exponent and differs from the Sobolev exponent, <span><math><mi>u</mi><mo>=</mo><mn>0</mn></math></span> is the unique <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> solution stable outside a compact set. This work extends the results obtained by Farina (J. Math. Pures Appl., <strong>87</strong> (5) (2007), 537–561).</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"430 ","pages":"Article 113201"},"PeriodicalIF":2.4,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143534810","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":"New global Carleman estimates and null controllability for a stochastic Cahn-Hilliard type equation","authors":"Sen Zhang , Hang Gao , Ganghua Yuan","doi":"10.1016/j.jde.2025.02.074","DOIUrl":"10.1016/j.jde.2025.02.074","url":null,"abstract":"<div><div>In this paper, we study the null controllability for a stochastic semilinear Cahn-Hilliard type equation, whose semilinear term contains first and second order derivatives of solutions. To start with, an improved global Carleman estimate for linear backward fourth order stochastic parabolic equations with <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-valued source terms is derived, which is based on a new fundamental identity for a fourth order stochastic parabolic operator. Based on it, we establish a new global Carleman estimate for linear backward fourth order stochastic parabolic equations with <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></math></span>-valued source terms, which, together with a fixed point argument, derive the desired null controllability for the stochastic Cahn-Hilliard type equation.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"430 ","pages":"Article 113203"},"PeriodicalIF":2.4,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143534714","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":"Pontryagin maximum principle for the deterministic mean field type optimal control problem via the Lagrangian approach","authors":"Yurii Averboukh, Dmitry Khlopin","doi":"10.1016/j.jde.2025.02.076","DOIUrl":"10.1016/j.jde.2025.02.076","url":null,"abstract":"<div><div>We study necessary optimality conditions for the deterministic mean field type free-endpoint optimal control problem. Our study relies on the Lagrangian approach that treats the mean field type control system as a crowd of infinitely many agents who are labeled by elements of some probability space. First, we derive the Pontryagin maximum principle in the Lagrangian form. Furthermore, we consider the Kantorovich and Eulerian formalizations which describe mean field type control systems via distributions on the set of trajectories and nonlocal continuity equation respectively. We prove that local minimizers in the Kantorovich or Eulerian formulations determine local minimizers within the Lagrangian approach. Using this, we deduce the Pontryagin maximum principle in the Kantorovich and Eulerian forms. To illustrate the general theory, we examine a model system of mean field type linear quadratic regulator. We show that the optimal strategy in this case is determined by a linear feedback.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"430 ","pages":"Article 113205"},"PeriodicalIF":2.4,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143534812","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":"Response tori in Hamiltonian systems of high order degeneracy - the super-critical case","authors":"Lu Xu , Wen Si , Yingfei Yi","doi":"10.1016/j.jde.2025.02.051","DOIUrl":"10.1016/j.jde.2025.02.051","url":null,"abstract":"<div><div>Consider the quasi-periodically forced, 2nd order differential equations<span><span><span><math><mover><mrow><mi>x</mi></mrow><mrow><mo>¨</mo></mrow></mover><mo>+</mo><mi>λ</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>l</mi></mrow></msup><mo>+</mo><mi>ε</mi><mi>f</mi><mo>(</mo><mi>ω</mi><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><mi>λ</mi><mo>≠</mo><mn>0</mn></math></span> is a constant, <span><math><mi>l</mi><mo>≥</mo><mn>2</mn></math></span> is an integer, <span><math><mi>ω</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> is a Diophantine frequency vector, <em>ε</em> is a small parameter, and <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> is real analytic. It is shown in <span><span>[18]</span></span> that if the leading order <em>p</em> of non-degeneracy of <span><math><mo>[</mo><mi>f</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mi>x</mi><mo>)</mo><mo>]</mo></math></span> satisfies <span><math><mn>0</mn><mo>≤</mo><mi>p</mi><mo><</mo><mi>l</mi><mo>/</mo><mn>2</mn></math></span>, then response solutions of the equation exist under some minor conditions. Indeed, <span><math><mi>l</mi><mo>/</mo><mn>2</mn></math></span> is the critical order of non-degeneracy of <span><math><mo>[</mo><mi>f</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mi>x</mi><mo>)</mo><mo>]</mo></math></span> such that relative equilibria of the equation can be solved from its averaged equation - a typical mechanism for the existence of response solutions in perturbed, quasi-periodically forced, 2nd order nonlinear equations. In this paper, we consider the existence of response solutions of the equation for the super-critical case, i.e., <span><math><mo>[</mo><mi>f</mi><mo>(</mo><mo>⋅</mo><mo>,</mo><mi>x</mi><mo>)</mo><mo>]</mo></math></span> is degenerate at least up to an order <span><math><mi>p</mi><mo>≥</mo><mi>l</mi><mo>/</mo><mn>2</mn></math></span>. We will show in this case that response solutions can still exist around perturbed relative equilibria of the normalized equation by considering non-degeneracy of the new perturbation after the normalization that is of at least <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> order. This reveals a mechanism for the existence of response solutions of the equation in the super-critical case.</div><div>For the sake of generality, we will actually consider a general Hamiltonian normal form containing the normalized equation as a particular case. We will prove a general theorem concerning the existence of response tori of the normal form through averaging, finding rela","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 612-645"},"PeriodicalIF":2.4,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143487184","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}
Anne Boutet de Monvel , Jonatan Lenells , Dmitry Shepelsky
{"title":"The focusing NLS equation with step-like oscillating background: Asymptotics in a transition zone","authors":"Anne Boutet de Monvel , Jonatan Lenells , Dmitry Shepelsky","doi":"10.1016/j.jde.2025.02.016","DOIUrl":"10.1016/j.jde.2025.02.016","url":null,"abstract":"<div><div>In a recent paper, we presented scenarios of long-time asymptotics for the solution of the focusing nonlinear Schrödinger equation with initial data approaching plane waves of the form <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mn>2</mn><mi>i</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>x</mi></mrow></msup></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>ϕ</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mn>2</mn><mi>i</mi><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>x</mi></mrow></msup></math></span> at minus and plus infinity, respectively. In the shock case <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> some scenarios include sectors of genus 3, that is, sectors <span><math><msub><mrow><mi>ξ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><mi>ξ</mi><mo><</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, <span><math><mi>ξ</mi><mo>≔</mo><mi>x</mi><mo>/</mo><mi>t</mi></math></span>, where the leading term of the asymptotics is expressed in terms of hyperelliptic functions attached to a Riemann surface of genus 3. The present paper deals with the asymptotic analysis in a transition zone between two genus 3 sectors. The leading term is expressed in terms of elliptic functions attached to a Riemann surface of genus 1. A central step in the derivation is the construction of a local parametrix in a neighborhood of two merging branch points.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"429 ","pages":"Pages 747-801"},"PeriodicalIF":2.4,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143508302","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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