Journal of Differential Equations最新文献

{"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}

{"title":"Nonintegrability of time-periodic perturbations of analytically integrable systems near homo- and heteroclinic orbits","authors":"Kazuyuki Yagasaki","doi":"10.1016/j.jde.2025.113693","DOIUrl":"10.1016/j.jde.2025.113693","url":null,"abstract":"<div><div>We consider time-periodic perturbations of analytically integrable systems in the sense of Bogoyavlenskij and study their <em>real-meromorphic</em> nonintegrability, using a generalized version due to Ayoul and Zung of the Morales-Ramis theory. The perturbation terms are assumed to have finite Fourier series in time, and the perturbed systems are rewritten as higher-dimensional autonomous systems having the small parameter as a state variable. We show that if the Melnikov functions are not constant, then the autonomous systems are not <em>real-meromorphically</em> integrable near homo- and heteroclinic orbits. We illustrate the theory for rotational motions of a periodically forced rigid body which provides a mathematical model of a quadrotor helicopter.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"446 ","pages":"Article 113693"},"PeriodicalIF":2.3,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144879592","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 model example of wave equation with oscillating scale-invariant damping","authors":"Marina Ghisi,&nbsp;Massimo Gobbino","doi":"10.1016/j.jde.2025.113695","DOIUrl":"10.1016/j.jde.2025.113695","url":null,"abstract":"<div><div>We analyze a simple example of wave equation with a time-dependent damping term, whose coefficient decays at infinity at the scale-invariant rate and includes an oscillatory component that is integrable but not absolutely integrable.</div><div>We show that the oscillations in the damping coefficient induce a resonance effect with a fundamental solution of the elastic term, altering the energy decay rate of solutions. In particular, some solutions exhibit slower decay compared to the case without the oscillatory component.</div><div>Our proof relies on Fourier analysis and a representation of solutions in polar coordinates, reducing the problem to a detailed study of the asymptotic behavior of solutions to a family of ordinary differential equations and suitable oscillatory integrals.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"448 ","pages":"Article 113695"},"PeriodicalIF":2.3,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144866041","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":"Nonlinear stability results for stationary solutions of reaction-diffusion-ODE systems","authors":"Chris Kowall ,&nbsp;Anna Marciniak-Czochra ,&nbsp;Finn Münnich","doi":"10.1016/j.jde.2025.113704","DOIUrl":"10.1016/j.jde.2025.113704","url":null,"abstract":"<div><div>Reaction-diffusion-ODE systems are emerging in modeling of biological pattern formation based on the coupling of diffusive and non-diffusive spatially heterogeneous processes. They may exhibit patterns with singularities such as jump-discontinuities. This work provides nonlinear stability and instability conditions for bounded stationary solutions of reaction-diffusion-ODE systems consisting of <em>m</em> ODEs coupled with <em>k</em> reaction-diffusion equations. We characterize the spectrum of the linearized operator and relate its spectral properties to the corresponding semigroup properties. Considering the function spaces <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mrow><mi>m</mi><mo>+</mo><mi>k</mi></mrow></msup><mo>,</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mrow><mi>m</mi></mrow></msup><mo>×</mo><mi>C</mi><msup><mrow><mo>(</mo><mover><mrow><mi>Ω</mi></mrow><mo>‾</mo></mover><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msup></math></span> and <span><math><mi>C</mi><msup><mrow><mo>(</mo><mover><mrow><mi>Ω</mi></mrow><mo>‾</mo></mover><mo>)</mo></mrow><mrow><mi>m</mi><mo>+</mo><mi>k</mi></mrow></msup></math></span>, we establish a sign condition on the spectral bound of the linearized operator, which implies nonlinear stability or instability of the stationary pattern.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"448 ","pages":"Article 113704"},"PeriodicalIF":2.3,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144866040","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":"Two-dimensional signal-dependent parabolic-elliptic Keller-Segel system and its mean-field derivation","authors":"Lukas Bol ,&nbsp;Li Chen ,&nbsp;Yue Li","doi":"10.1016/j.jde.2025.113712","DOIUrl":"10.1016/j.jde.2025.113712","url":null,"abstract":"<div><div>In this paper, the well-posedness of two-dimensional signal-dependent Keller-Segel system and its mean-field derivation from a interacting particle system on the whole space are investigated. The signal dependence effect is reflected by the fact that the diffusion coefficient in the particle system depends nonlinearly on the interactions between the individuals. Therefore, the mathematical challenge in studying the well-posedness of this system lies in the possible degeneracy and the aggregation effect when the concentration of signal becomes unbounded. The well-established method on bounded domains, to obtain the appropriate estimates for the signal concentration, is invalid for the whole space case. Motivated by the entropy minimization method and Onofri's inequality, which has been successfully applied for the parabolic-parabolic Keller-Segel system, we establish a complete entropy estimate benefited from the linear diffusion term, which plays an important role in obtaining the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> estimates for the solution. Furthermore, the upper bound for the concentration of signal is obtained. Based on the estimates we obtained for the density of bacteria, the rigorous mean-field derivation is proved by introducing an intermediate particle system with a mollified interaction potential with logarithmic scaling. By using this mollification, we obtain the convergence of the particle trajectories in expectation, which implies the weak propagation of chaos. Additionally, under a regularity assumption of the initial data, we infer higher regularity for the solutions, which allows us to use the relative entropy method to derive the strong <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> convergence for the propagation of chaos.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"450 ","pages":"Article 113712"},"PeriodicalIF":2.3,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144866107","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":"Symplectic normal form and growth of Sobolev norm","authors":"Zhenguo Liang ,&nbsp;Jiawen Luo ,&nbsp;Zhiyan Zhao","doi":"10.1016/j.jde.2025.113702","DOIUrl":"10.1016/j.jde.2025.113702","url":null,"abstract":"<div><div>For a class of reducible Hamiltonian partial differential equations (PDEs) with arbitrary spatial dimension, quantified by a quadratic polynomial with time-dependent coefficients, we present a comprehensive classification of long-term solution behaviors within Sobolev space. This classification is achieved through the utilization of Metaplectic and Schrödinger representations. Each pattern of Sobolev norm behavior corresponds to a specific <em>n</em>−dimensional symplectic normal form, as detailed in Theorems 1.1 and 1.2.</div><div>When applied to periodically or quasi-periodically forced <em>n</em>−dimensional quantum harmonic oscillators, we identify novel growth rates for the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>−</mo></math></span>norm as <em>t</em> tends to infinity, such as <span><math><msup><mrow><mi>t</mi></mrow><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>s</mi></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mi>λ</mi><mi>s</mi><mi>t</mi></mrow></msup></math></span> (with <span><math><mi>λ</mi><mo>&gt;</mo><mn>0</mn></math></span>) and <span><math><msup><mrow><mi>t</mi></mrow><mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>s</mi></mrow></msup><mo>+</mo><mi>ι</mi><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn><mi>n</mi><mi>s</mi></mrow></msup></math></span> (with <span><math><mi>ι</mi><mo>≥</mo><mn>0</mn></math></span>). Notably, we demonstrate that stability in Sobolev space, defined as the boundedness of the Sobolev norm, is essentially a unique characteristic of one-dimensional scenarios, as outlined in Theorem 1.3.</div><div>As a byproduct, we discover that the growth rate of the Sobolev norm for the quantum Hamiltonian can be directly described by that of the solution to the classical Hamiltonian which exhibits the optimal growth, as articulated in Theorem 1.4.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"449 ","pages":"Article 113702"},"PeriodicalIF":2.3,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144863657","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":"Veech's theorem of higher order","authors":"Jiahao Qiu,&nbsp;Xiangdong Ye","doi":"10.1016/j.jde.2025.113707","DOIUrl":"10.1016/j.jde.2025.113707","url":null,"abstract":"&lt;div&gt;&lt;div&gt;For an abelian group &lt;em&gt;G&lt;/em&gt;, &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&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;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, let &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∏&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, it is shown that for a minimal system &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; with &lt;em&gt;G&lt;/em&gt; being abelian, &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;RP&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; if and only if there exists a sequence &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; and points &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; such that for every &lt;span&gt;&lt;math&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;﹨&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;,&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;munder&gt;&lt;mi&gt;lim&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mi&gt;and&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;munder&gt;&lt;mi&gt;lim&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"449 ","pages":"Article 113707"},"PeriodicalIF":2.3,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144863658","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 A-function related to the Titchmarsh-Weyl m-function of Sturm-Liouville operators","authors":"Xuewen Wu ,&nbsp;Guangsheng Wei","doi":"10.1016/j.jde.2025.113691","DOIUrl":"10.1016/j.jde.2025.113691","url":null,"abstract":"<div><div>We provide a new method to establish Simon's <em>A</em>-functions associated with the Titchmarsh-Weyl <em>m</em>-functions of Sturm-Liouville differential operators. This method is based on a connection with the kernel of transformation operators. We also obtain some properties of the <em>A</em>-functions on finite intervals, which yield a high-energy asymptotic representation of the Titchmarsh-Weyl <em>m</em>-function.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"448 ","pages":"Article 113691"},"PeriodicalIF":2.3,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865609","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 an equilibrium with limited stock market participation and power utilities","authors":"Paolo Guasoni ,&nbsp;Kasper Larsen ,&nbsp;Giovanni Leoni","doi":"10.1016/j.jde.2025.113694","DOIUrl":"10.1016/j.jde.2025.113694","url":null,"abstract":"<div><div>For constants <span><math><mi>γ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><mi>A</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>, we prove existence and uniqueness of a solution to the singular and path-dependent Riccati-type ODE<span><span><span><math><mrow><mrow><mo>{</mo><mtable><mtr><mtd><msup><mrow><mi>h</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>y</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>γ</mi></mrow><mrow><mi>y</mi></mrow></mfrac><mo>(</mo><mi>γ</mi><mo>−</mo><mi>h</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>)</mo><mo>+</mo><mi>h</mi><mo>(</mo><mi>y</mi><mo>)</mo><mfrac><mrow><mi>γ</mi><mo>+</mo><mo>(</mo><mo>(</mo><mi>A</mi><mo>−</mo><mi>γ</mi><mo>)</mo><mi>exp</mi><mo>⁡</mo><mo>{</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msubsup><mfrac><mrow><mn>1</mn><mo>−</mo><mi>h</mi><mo>(</mo><mi>q</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>−</mo><mi>q</mi></mrow></mfrac><mi>d</mi><mi>q</mi><mo>}</mo><mo>−</mo><mi>A</mi><mo>)</mo><mi>h</mi><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>−</mo><mi>y</mi></mrow></mfrac><mo>,</mo><mspace></mspace><mi>y</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo></mtd></mtr><mtr><mtd><mi>h</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mi>γ</mi><mo>,</mo><mspace></mspace><mi>h</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>1</mn><mo>.</mo></mtd></mtr></mtable></mrow></mrow></math></span></span></span> As an application, we use the ODE solution to prove existence of a Radner equilibrium with homogeneous power-utility investors in the limited participation model from Basak and Cuoco (1998).</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"448 ","pages":"Article 113694"},"PeriodicalIF":2.3,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144866039","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":"Constructive approaches to QP-time-dependent KAM theory for Lagrangian tori in Hamiltonian systems","authors":"Renato C. Calleja ,&nbsp;Alex Haro ,&nbsp;Pedro Porras","doi":"10.1016/j.jde.2025.113681","DOIUrl":"10.1016/j.jde.2025.113681","url":null,"abstract":"<div><div>In this paper, we prove a KAM theorem in a-posteriori format, using the parameterization method to look invariant tori in non-autonomous Hamiltonian systems with <em>n</em> degrees of freedom that depend periodically or quasi-periodically (QP) on time, with <em>ℓ</em> external frequencies. Such a system is described by a Hamiltonian function in the 2<em>n</em>-dimensional phase space, <span><math><mi>M</mi></math></span>, that depends also on <em>ℓ</em> angles, <span><math><mi>φ</mi><mo>∈</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>ℓ</mi></mrow></msup></math></span>. We take advantage of the fibered structure of the extended phase space <span><math><mi>M</mi><mo>×</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>ℓ</mi></mrow></msup></math></span>. As a result of our approach, the parameterization of tori requires the last <em>ℓ</em> variables, to be precise <em>φ</em>, while the first 2<em>n</em> components are determined by an invariance equation. This reduction decreases the dimension of the problem where the unknown is a parameterization from <span><math><mn>2</mn><mo>(</mo><mi>n</mi><mo>+</mo><mi>ℓ</mi><mo>)</mo></math></span> to 2<em>n</em>.</div><div>We employ a quasi-Newton method, in order to prove the KAM theorem. This iterative method begins with an initial parameterization of an approximately invariant torus, meaning it approximately satisfies the invariance equation. The approximation is refined by applying corrections that reduce quadratically the invariance equation error. This process converges to a torus in a complex strip of size <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>, provided suitable Diophantine <span><math><mo>(</mo><mi>γ</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math></span> conditions and a non-degeneracy condition on the torsion are met. Given the nature of the proof, this provides a numerical method that can be effectively implemented on a computer, the details are given in the companion paper <span><span>[9]</span></span>. This approach leverages precision and efficiency to compute invariant tori.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"449 ","pages":"Article 113681"},"PeriodicalIF":2.3,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144863656","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}