Journal of Differential Equations最新文献

{"title":"Spectrum for a weighted one-dimensional fractional Laplace operator","authors":"Bianxia Yang , Zhijiang Zhang , Ruyun Ma","doi":"10.1016/j.jde.2025.113501","DOIUrl":"10.1016/j.jde.2025.113501","url":null,"abstract":"<div><div>In this paper, we study the spectrum of the one-dimensional fractional Laplace operator with a definite weight<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msup><mrow><mo>(</mo><mo>−</mo><mfrac><mrow><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>d</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>)</mo></mrow><mrow><mi>α</mi></mrow></msup><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>λ</mi><mi>a</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mi>x</mi><mo>∈</mo><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mi>x</mi><mspace></mspace><mspace></mspace><mtext>in</mtext><mspace></mspace><mi>R</mi><mo>∖</mo><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>,</mo><mi>a</mi><mo>∈</mo><mi>C</mi><mo>(</mo><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>,</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>+</mo><mo>∞</mo><mo>)</mo><mo>)</mo></math></span> and <span><math><msup><mrow><mo>(</mo><mo>−</mo><mfrac><mrow><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>d</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>)</mo></mrow><mrow><mi>α</mi></mrow></msup></math></span> is the one-dimensional fractional Laplace nonlocal operator. By virtue of Γ-convergence arguments, we investigate, in the singular limit, that the eigenvalue and eigenfunction of the nonlocal operator converge to those of the corresponding classical second-order two-point boundary value problem in the first place, and then, building upon the continuity of the eigenvalues and eigenfunctions as a function of fractional index, we derive the simplicity of the eigenvalues of the fractional Laplace nonlocal operator for all fractional index <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> by adopting a stet-by-step iterative approach. Furthermore, using the <em>α</em>-harmonic extension, we receive that the corresponding eigenfunction <span><math><msubsup><mrow><mi>φ</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>α</mi></mrow></msubsup></math></span> has at most <span><math><mn>2</mn><mi>k</mi><mo>−</mo><mn>2</mn></math></span> zeros in <span><math><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. At last, from an experiment point of view, we give the numerical eigenvalues and eigenfunctions of the weighted fractional Laplace problem by means of the finite element method in some special cases, which enriches","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"443 ","pages":"Article 113501"},"PeriodicalIF":2.4,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144253679","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":"Population dynamics in closed polluted aquatic ecosystems with time-periodic input of toxicants","authors":"Zhenzhen Li,&nbsp;Zhi-An Wang","doi":"10.1016/j.jde.2025.113502","DOIUrl":"10.1016/j.jde.2025.113502","url":null,"abstract":"<div><div>This paper is concerned with a diffusive population-toxicant system in a polluted aquatic environment with temporally periodic and spatially heterogeneous input of toxicants. By a variety of mathematical tools, such as the principal eigenvalue theory, method of upper-lower solutions, theory of monotone semi-flow, implicit function theorem, etc., we derive sufficient conditions on the existence and global stability of periodic solutions with fixed diffusion rates and explore the asymptotic profiles of positive periodic solutions for large and small diffusion rates. Our results show that if the toxicity of toxicants is low (resp. high), then the aquatic population persists (resp. becomes extinct), while both persistence and extinction may be locally stable (i.e. bi-stability) for moderate toxicity of toxicants. We also find that the spatial distribution of positive periodic solutions with small diffusion rates is quite different from that with large diffusion rates.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"442 ","pages":"Article 113502"},"PeriodicalIF":2.4,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144254905","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":"Topological classification of global dynamics of planar polynomial Hamiltonian systems with separable variables","authors":"Xuemeng Sun,&nbsp;Dongmei Xiao","doi":"10.1016/j.jde.2025.113496","DOIUrl":"10.1016/j.jde.2025.113496","url":null,"abstract":"<div><div>In the paper, we completely characterize the local dynamics of polynomial Hamiltonian systems with separable variables and provide a method to determine its global dynamics on Poincaré disk. It is shown that there are three (four) topological classifications for finite (infinite, resp.) singular points of the Hamiltonian system with any degree <em>n</em>, and its global dynamics can be determined by the number of singular points and their separatrix skeleton. This provides an approach to characterize the topological classification of real algebraic curves <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>y</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, where <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>y</mi><mo>)</mo></math></span> are real polynomials of degrees <em>m</em> and <em>n</em>, respectively.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"443 ","pages":"Article 113496"},"PeriodicalIF":2.4,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144241196","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 boundedness and blow-up in a repulsive chemotaxis-consumption system in higher dimensions","authors":"Jaewook Ahn ,&nbsp;Kyungkeun Kang ,&nbsp;Dongkwang Kim","doi":"10.1016/j.jde.2025.113503","DOIUrl":"10.1016/j.jde.2025.113503","url":null,"abstract":"<div><div>This paper investigates the repulsive chemotaxis-consumption model<span><span><span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>=</mo><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>D</mi><mo>(</mo><mi>u</mi><mo>)</mo><mi>∇</mi><mi>u</mi><mo>)</mo><mo>+</mo><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>u</mi><mi>∇</mi><mi>v</mi><mo>)</mo><mo>,</mo><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>u</mi><mi>v</mi><mo>,</mo></math></span></span></span> in an <em>n</em>-dimensional ball, <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>, where the diffusion coefficient <em>D</em> is an appropriate extension of the function <span><math><mn>0</mn><mo>≤</mo><mi>ξ</mi><mo>↦</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>ξ</mi><mo>)</mo></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> for <span><math><mi>m</mi><mo>&gt;</mo><mn>0</mn></math></span>. Under the boundary conditions<span><span><span><math><mi>ν</mi><mo>⋅</mo><mo>(</mo><mi>D</mi><mo>(</mo><mi>u</mi><mo>)</mo><mi>∇</mi><mi>u</mi><mo>+</mo><mi>u</mi><mi>∇</mi><mi>v</mi><mo>)</mo><mo>=</mo><mn>0</mn><mspace></mspace><mtext> and </mtext><mspace></mspace><mi>v</mi><mo>=</mo><mi>M</mi><mo>&gt;</mo><mn>0</mn><mo>,</mo></math></span></span></span> we demonstrate that for <span><math><mi>m</mi><mo>&gt;</mo><mn>1</mn></math></span>, or <span><math><mi>m</mi><mo>=</mo><mn>1</mn></math></span> and <span><math><mn>0</mn><mo>&lt;</mo><mi>M</mi><mo>&lt;</mo><mn>2</mn><mo>/</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></math></span>, the system admits globally bounded classical solutions for any choice of sufficiently smooth radial initial data. This result is further extended to the case <span><math><mn>0</mn><mo>&lt;</mo><mi>m</mi><mo>&lt;</mo><mn>1</mn></math></span> when <em>M</em> is chosen to be sufficiently small, depending on the initial conditions. In contrast, it is shown that for <span><math><mn>0</mn><mo>&lt;</mo><mi>m</mi><mo>&lt;</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></mfrac></math></span>, the system exhibits blow-up behavior for sufficiently large <em>M</em>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"443 ","pages":"Article 113503"},"PeriodicalIF":2.4,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144241197","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":"Decay rates for star-shaped degenerate heat-wave coupled networks","authors":"Jia-Xian Guang,&nbsp;Zhong-Jie Han","doi":"10.1016/j.jde.2025.113505","DOIUrl":"10.1016/j.jde.2025.113505","url":null,"abstract":"<div><div>This work investigates the long-time dynamics of a star-shaped network composed of degenerate heat and wave equations. The well-posedness of the system is proved by standard semigroup theories and a comprehensive criterion for strong stability in such degenerate partial differential equations (PDE) networks is established. Through frequency domain analysis, the polynomial decay rate is explored in two scenarios: networks with a single wave equation, where the explicit decay rate depends solely on the degree of degeneration in those diffusion coefficients of the heat parts, and networks with multiple wave equations, where the explicit decay rates are derived under specific irrationality conditions on the spatial lengths of the wave equations involved in the network using Diophantine approximation arguments. Finally, a generalized slow decay rate is derived, providing a broader understanding of the long-time behavior of this complex degenerate heat-wave networks.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"443 ","pages":"Article 113505"},"PeriodicalIF":2.4,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144241195","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":"Asymptotic behavior of degenerate linear kinetic equations with non-isothermal boundary conditions","authors":"Armand Bernou","doi":"10.1016/j.jde.2025.113470","DOIUrl":"10.1016/j.jde.2025.113470","url":null,"abstract":"<div><div>We study the degenerate linear Boltzmann equation inside a bounded domain with a generalized diffuse reflection at the boundary and variable temperature, including the Maxwell boundary conditions with the wall Maxwellian or heavy-tailed reflection kernel and the Cercignani-Lampis boundary condition. Our abstract collisional setting applies to the linear BGK model, the relaxation towards a space-dependent steady state, and collision kernels with fat tails. We prove for the first time the existence of a steady state and a rate of convergence towards it without assumptions on the temperature variations. Our results for the Cercignani-Lampis boundary condition make also no hypotheses on the accommodation coefficients. The proven rate is exponential when a control condition on the degeneracy of the collision operator is satisfied, and only polynomial when this assumption is not met, in line with our previous results regarding the free-transport equation. We also provide a precise description of the different convergence rates, including lower bounds, when the steady state is bounded. Our method yields constructive constants.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"442 ","pages":"Article 113470"},"PeriodicalIF":2.4,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144239538","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 structure of the geometric tangent cone to the Wasserstein space","authors":"Averil Aussedat","doi":"10.1016/j.jde.2025.113520","DOIUrl":"10.1016/j.jde.2025.113520","url":null,"abstract":"<div><div>This article focuses on the metric orthogonal of the geometric tangent cone to the Wasserstein space. Some algebraic and topological properties are given, as well as a complete characterization and weak closedness property in dimension 1. It is shown that in general, the directional derivative of the Wasserstein distance is not sufficient to differentiate between the tangent cone and its orthogonal. To conclude, a general Helmholtz-Hodge decomposition is proved for measure fields.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"442 ","pages":"Article 113520"},"PeriodicalIF":2.4,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144239795","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 length-preserving and area-preserving inverse curvature flow of planar curves with singularities","authors":"Yunlong Yang ,&nbsp;Yanwen Zhao ,&nbsp;Jianbo Fang ,&nbsp;Yanlong Zhang","doi":"10.1016/j.jde.2025.113517","DOIUrl":"10.1016/j.jde.2025.113517","url":null,"abstract":"<div><div>This paper aims to investigate the evolution problem for planar curves with singularities. Motivated by the inverse curvature flow introduced by Li and Wang (2023) <span><span>[33]</span></span>, we intend to consider the area-preserving and length-preserving inverse curvature flow with nonlocal term for <em>ℓ</em>-convex Legendre curves. For the area-preserving flow, an <em>ℓ</em>-convex Legendre curve with initial algebraic area <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>&gt;</mo><mn>0</mn></math></span> evolves to a circle of radius <span><math><msqrt><mrow><mfrac><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow><mrow><mi>π</mi></mrow></mfrac></mrow></msqrt></math></span>. For the length-preserving flow, an <em>ℓ</em>-convex Legendre curve with initial algebraic length <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> evolves to a circle of radius <span><math><mfrac><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow><mrow><mn>2</mn><mi>π</mi></mrow></mfrac></math></span>. As the by-product, we obtain some geometric inequalities for <em>ℓ</em>-convex Legendre curves through the length-preserving flow.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"443 ","pages":"Article 113517"},"PeriodicalIF":2.4,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144241005","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":"KAM theorem for degenerate generalized Hamiltonian systems with continuous parameters","authors":"Jiayin Du ,&nbsp;Shuguan Ji ,&nbsp;Yong Li","doi":"10.1016/j.jde.2025.113504","DOIUrl":"10.1016/j.jde.2025.113504","url":null,"abstract":"<div><div>In this paper, we prove the persistence of invariant tori for degenerate generalized Hamiltonian systems with continuous parameters. We demonstrate that the persistent invariant tori retain the same frequency as the unperturbed tori under a certain transversality condition and a weak convexity condition for the frequency mapping. Generally speaking, at least Lipschitz continuity with respect to the parameter is needed in KAM-type results, while in this paper, we only require it to be continuous. Additionally, the system we consider is degenerate. Therefore, this paper can also be seen as an extension of KAM-type theory from non-degenerate generalized Hamiltonian systems to degenerate generalized Hamiltonian systems.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"441 ","pages":"Article 113504"},"PeriodicalIF":2.4,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144254784","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":"Accelerating or not in the spatial propagation of nonlocal dispersal cooperative reducible systems","authors":"Teng-Long Cui ,&nbsp;Wan-Tong Li ,&nbsp;Wen-Bing Xu","doi":"10.1016/j.jde.2025.113519","DOIUrl":"10.1016/j.jde.2025.113519","url":null,"abstract":"<div><div>This paper investigates the spatial propagation problem in cooperative recursive systems in the absence of irreducibility, which is a critical assumption to guarantee uniform spatial propagation across all components. When the linearization at zero vector of the reducible system is in Frobenius form, we demonstrate that the <em>i</em>-th component exhibiting accelerated propagation could accelerate the spatial propagation of all other components and the spreading speeds of all components are infinite, provided that the <span><math><mo>(</mo><mi>i</mi><mo>,</mo><mi>i</mi><mo>)</mo></math></span>-entry in the Frobenius matrix belongs to the first diagonal block. This result reveals that uniform propagation of all components can occur even when the irreducibility condition is not satisfied. However, when the <span><math><mo>(</mo><mi>i</mi><mo>,</mo><mi>i</mi><mo>)</mo></math></span>-entry is not in the first diagonal block, some components have finite spreading speeds while others have infinite ones, which implies that the propagation of the system is non-uniform. Moreover, we extend our analysis to nonlocal dispersal cooperative systems and explore a special case where the dispersal kernel of a component has an algebraically decaying tail.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"443 ","pages":"Article 113519"},"PeriodicalIF":2.4,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144241194","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}