Pub Date : 2026-01-13DOI: 10.1016/j.jde.2025.114092
Wen-Xin Qin , Tong Zhou
The depinning force for the Frenkel-Kontorova chain is a critical value of the driving force F up to which there continue to be Birkhoff equilibria of rotation symbol ω and above which there are none. In this paper we investigate the modulus of continuity for the depinning force at rational rotation symbols and and obtain the estimate where C is a constant and denotes the underlying number associated to the rotation symbol ω. A similar conclusion for also holds true.
As an application, we give an open and dense result for , a threshold of driving force such that there exist stationary fronts for and traveling fronts for .
{"title":"Modulus of continuity for depinning force at rational rotation symbols and application","authors":"Wen-Xin Qin , Tong Zhou","doi":"10.1016/j.jde.2025.114092","DOIUrl":"10.1016/j.jde.2025.114092","url":null,"abstract":"<div><div>The depinning force for the Frenkel-Kontorova chain is a critical value <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>ω</mi><mo>)</mo></math></span> of the driving force <em>F</em> up to which there continue to be Birkhoff equilibria of rotation symbol <em>ω</em> and above which there are none. In this paper we investigate the modulus of continuity for the depinning force at rational rotation symbols <span><math><mi>p</mi><mo>/</mo><mi>q</mi><mo>+</mo></math></span> and <span><math><mi>p</mi><mo>/</mo><mi>q</mi><mo>−</mo></math></span> and obtain the estimate<span><span><span><math><mo>|</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>p</mi><mo>/</mo><mi>q</mi><mo>+</mo><mo>)</mo><mo>−</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>ω</mi><mo>)</mo><mo>|</mo><mo>≤</mo><mi>C</mi><mo>|</mo><mi>q</mi><msup><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>−</mo><mi>p</mi><mo>|</mo><mo>,</mo><mspace></mspace><mtext> for </mtext><mspace></mspace><mi>ω</mi><mo>></mo><mi>p</mi><mo>/</mo><mi>q</mi><mo>+</mo><mo>,</mo></math></span></span></span> where <em>C</em> is a constant and <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> denotes the underlying number associated to the rotation symbol <em>ω</em>. A similar conclusion for <span><math><mi>p</mi><mo>/</mo><mi>q</mi><mo>−</mo></math></span> also holds true.</div><div>As an application, we give an open and dense result for <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mn>0</mn><mo>/</mo><mn>1</mn><mo>+</mo><mo>)</mo><mo>></mo><mn>0</mn></math></span>, a threshold of driving force such that there exist stationary fronts for <span><math><mi>F</mi><mo>≤</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mn>0</mn><mo>/</mo><mn>1</mn><mo>+</mo><mo>)</mo></math></span> and traveling fronts for <span><math><mi>F</mi><mo>></mo><msub><mrow><mi>F</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mn>0</mn><mo>/</mo><mn>1</mn><mo>+</mo><mo>)</mo></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114092"},"PeriodicalIF":2.3,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977972","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}
Pub Date : 2026-01-13DOI: 10.1016/j.jde.2026.114095
Tuowei Chen , Qiangchang Ju
This paper is concerned with the two-dimensional full compressible Navier-Stokes equations between two infinite parallel isothermal walls, where the upper wall is moving with a horizontal velocity, while the lower wall is stationary, and there allows a temperature difference between the two walls. It is shown that if the initial state is close to the Couette flow with a temperature gradient, then the global strong solutions exist, provided that the Reynolds and Mach numbers are low and the temperature difference between the two walls is small. The low Mach number limit of the global strong solutions is also shown for the case that both walls maintain the same temperature.
{"title":"Global existence for full compressible Navier-Stokes equations around the Couette flow with a temperature gradient in an infinite channel","authors":"Tuowei Chen , Qiangchang Ju","doi":"10.1016/j.jde.2026.114095","DOIUrl":"10.1016/j.jde.2026.114095","url":null,"abstract":"<div><div>This paper is concerned with the two-dimensional full compressible Navier-Stokes equations between two infinite parallel isothermal walls, where the upper wall is moving with a horizontal velocity, while the lower wall is stationary, and there allows a temperature difference between the two walls. It is shown that if the initial state is close to the Couette flow with a temperature gradient, then the global strong solutions exist, provided that the Reynolds and Mach numbers are low and the temperature difference between the two walls is small. The low Mach number limit of the global strong solutions is also shown for the case that both walls maintain the same temperature.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114095"},"PeriodicalIF":2.3,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977973","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}
Pub Date : 2026-01-12DOI: 10.1016/j.jde.2026.114102
Xiang Lv
We prove a concise and easily verifiable criterion on the existence and global stability of stationary solutions for random dynamical systems (RDSs), which is very useful in a wide range of applications. As a consequence, we can show that the ω-limit sets of all pullback trajectories of semilinear/nonlinear stochastic differential equations (SDEs) with additive/multiplicative white noise are composed of nontrivial random equilibria. Furthermore, in the applications of stability analysis for SDEs, our conditions are not only sufficient but indeed sharp.
{"title":"An abstract criterion on the existence and global stability of stationary solutions for random dynamical systems and its applications","authors":"Xiang Lv","doi":"10.1016/j.jde.2026.114102","DOIUrl":"10.1016/j.jde.2026.114102","url":null,"abstract":"<div><div>We prove a concise and easily verifiable criterion on the existence and global stability of stationary solutions for random dynamical systems (RDSs), which is very useful in a wide range of applications. As a consequence, we can show that the <em>ω</em>-limit sets of all pullback trajectories of semilinear/nonlinear stochastic differential equations (SDEs) with additive/multiplicative white noise are composed of nontrivial random equilibria. Furthermore, in the applications of stability analysis for SDEs, our conditions are not only sufficient but indeed sharp.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"458 ","pages":"Article 114102"},"PeriodicalIF":2.3,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977524","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}
Pub Date : 2026-01-12DOI: 10.1016/j.jde.2026.114097
Bin Qian , Min Wang , Ran Wang , Yimin Xiao
Consider the nonlinear stochastic heat equation where is a Gaussian noise which is white in time and fractional in space with Hurst parameter . The existence and uniqueness of the solutions to this equation were proved by Balan et al. [1] when is an affine function, and by Hu et al. [19] when σ is differentiable with Lipschitz derivative and . In both settings, the Hölder continuity of the solution has been proved by Balan et al. [2] and Hu et al. [19], respectively.
In this paper, we study the asymptotic behavior of the temporal increment for fixed and as , within the framework of [19]. As applications, we derive Khinchin's law of the iterated logarithm, Chung's law of the iterated logarithm, and the quadratic variation of the temporal process , where is fixed.
考虑非线性随机热方程∂u(t,x)∂t=∂2u(t,x)∂x2+σ(u(t,x))W˙（t,x），t>0,x∈R，其中W˙是高斯噪声，在时间上是白的，在空间上是分数的，Hurst参数H∈(14,12)。当σ(u)=au+b是仿射函数时，Balan et al.[1]证明了该方程解的存在唯一性；当σ(0)=0时，σ可与Lipschitz导数微分时，Hu et al.[19]证明了该方程解的存在唯一性。在这两种情况下，分别由Balan et al.[2]和Hu et al.[19]证明了解的Hölder连续性。本文在[19]的框架下，研究了固定t≥0且x∈R为ε↓0时，时间增量u(t+ε,x)−u（t,x）的渐近性。作为应用，我们导出了迭代对数的Khinchin定律，迭代对数的Chung定律，以及时间过程{u(t,x)}t≥0的二次变分，其中x∈R是固定的。
{"title":"Temporal regularity for the nonlinear stochastic heat equation with spatially rough noise","authors":"Bin Qian , Min Wang , Ran Wang , Yimin Xiao","doi":"10.1016/j.jde.2026.114097","DOIUrl":"10.1016/j.jde.2026.114097","url":null,"abstract":"<div><div>Consider the nonlinear stochastic heat equation<span><span><span><math><mfrac><mrow><mo>∂</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>∂</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mi>σ</mi><mo>(</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>)</mo><mover><mrow><mi>W</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace></mspace><mi>x</mi><mo>∈</mo><mi>R</mi><mo>,</mo></math></span></span></span> where <span><math><mover><mrow><mi>W</mi></mrow><mrow><mo>˙</mo></mrow></mover></math></span> is a Gaussian noise which is white in time and fractional in space with Hurst parameter <span><math><mi>H</mi><mo>∈</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></math></span>. The existence and uniqueness of the solutions to this equation were proved by Balan et al. <span><span>[1]</span></span> when <span><math><mi>σ</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><mi>a</mi><mi>u</mi><mo>+</mo><mi>b</mi></math></span> is an affine function, and by Hu et al. <span><span>[19]</span></span> when <em>σ</em> is differentiable with Lipschitz derivative and <span><math><mi>σ</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></math></span>. In both settings, the Hölder continuity of the solution has been proved by Balan et al. <span><span>[2]</span></span> and Hu et al. <span><span>[19]</span></span>, respectively.</div><div>In this paper, we study the asymptotic behavior of the temporal increment <span><math><mi>u</mi><mo>(</mo><mi>t</mi><mo>+</mo><mi>ε</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span> for fixed <span><math><mi>t</mi><mo>≥</mo><mn>0</mn></math></span> and <span><math><mi>x</mi><mo>∈</mo><mi>R</mi></math></span> as <span><math><mi>ε</mi><mo>↓</mo><mn>0</mn></math></span>, within the framework of <span><span>[19]</span></span>. As applications, we derive Khinchin's law of the iterated logarithm, Chung's law of the iterated logarithm, and the quadratic variation of the temporal process <span><math><msub><mrow><mo>{</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>}</mo></mrow><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>, where <span><math><mi>x</mi><mo>∈</mo><mi>R</mi></math></span> is fixed.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114097"},"PeriodicalIF":2.3,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980540","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}
Pub Date : 2026-01-09DOI: 10.1016/j.jde.2026.114096
Peng Shi , Yan-Xia Feng , Wan-Tong Li , Fei-Ying Yang
Recent studies indicate that in many epidemics, the strains (bacterial or viral) of disease-causing pathogens exhibit significant diversity, and human mobility patterns follow scale-free, nonlocal dynamics characterized by heavy-tailed distributions such as Lévy flights. To investigate the long-range geographical spread of multi-strain epidemics, this article proposes a multi-strain susceptible-infected-susceptible (SIS) model incorporating fractional diffusion. The central questions addressed in our study include the competitive exclusion and coexistence of multiple strains, as well as the influence of fractional powers and dispersal rates on the asymptotic behavior of equilibrium solutions. Our analysis demonstrates that: (i) the basic reproduction number acts as a threshold for disease extinction; (ii) the invasion number serves as a threshold for both the existence and stability of the coexistence equilibrium and the stability of single-strain endemic equilibria. Additionally, we examine the effect of home and hospital isolation measures on disease transmission.
{"title":"Spatiotemporal dynamics in a multi-strain epidemic model with fractional diffusion","authors":"Peng Shi , Yan-Xia Feng , Wan-Tong Li , Fei-Ying Yang","doi":"10.1016/j.jde.2026.114096","DOIUrl":"10.1016/j.jde.2026.114096","url":null,"abstract":"<div><div>Recent studies indicate that in many epidemics, the strains (bacterial or viral) of disease-causing pathogens exhibit significant diversity, and human mobility patterns follow scale-free, nonlocal dynamics characterized by heavy-tailed distributions such as Lévy flights. To investigate the long-range geographical spread of multi-strain epidemics, this article proposes a multi-strain susceptible-infected-susceptible (SIS) model incorporating fractional diffusion. The central questions addressed in our study include the competitive exclusion and coexistence of multiple strains, as well as the influence of fractional powers and dispersal rates on the asymptotic behavior of equilibrium solutions. Our analysis demonstrates that: (i) the basic reproduction number acts as a threshold for disease extinction; (ii) the invasion number serves as a threshold for both the existence and stability of the coexistence equilibrium and the stability of single-strain endemic equilibria. Additionally, we examine the effect of home and hospital isolation measures on disease transmission.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114096"},"PeriodicalIF":2.3,"publicationDate":"2026-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145924239","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}
Pub Date : 2026-01-08DOI: 10.1016/j.jde.2025.114083
Nguyen Huy Tuan , Nguyen Anh Tuan
This study analyzes a nonlocal-in-time Keller-Segel (KS) chemotaxis system describing organism movement with memory effects. Two distinct regimes are tackled. Firstly, for the time-fractional KS equation augmented by a logistic source, we show that sufficiently dominant damping guarantees existence of a unique global mild solution that remains uniformly bounded for all time. The proof blends a priori estimates in uniformly local Lebesgue spaces with new semigroup bounds for solution operators involving Mittag-Leffler kernels. Secondly, removing the logistic term, we investigate singular behavior. Via Fourier analysis and Besov-Triebel-Lizorkin embeddings we construct initial data leading to finite-time blowup. Additionally, Littlewood-Paley decompositions reveal norm inflation: arbitrarily small data in rough topologies can produce nonzero solution norms instantaneously, signaling ill-posedness. Together, these results shed light on open issues regarding the global boundedness and singular solutions for memory-driven chemotaxis system.
{"title":"Global dynamics of the nonlocal Keller-Segel system: Uniform boundedness and singular behavior","authors":"Nguyen Huy Tuan , Nguyen Anh Tuan","doi":"10.1016/j.jde.2025.114083","DOIUrl":"10.1016/j.jde.2025.114083","url":null,"abstract":"<div><div>This study analyzes a nonlocal-in-time Keller-Segel (KS) chemotaxis system describing organism movement with memory effects. Two distinct regimes are tackled. Firstly, for the time-fractional KS equation augmented by a logistic source, we show that sufficiently dominant damping guarantees existence of a unique global mild solution that remains uniformly bounded for all time. The proof blends a priori estimates in uniformly local Lebesgue spaces with new semigroup bounds for solution operators involving Mittag-Leffler kernels. Secondly, removing the logistic term, we investigate singular behavior. Via Fourier analysis and Besov-Triebel-Lizorkin embeddings we construct initial data leading to finite-time blowup. Additionally, Littlewood-Paley decompositions reveal norm inflation: arbitrarily small data in rough topologies can produce nonzero solution norms instantaneously, signaling ill-posedness. Together, these results shed light on open issues regarding the global boundedness and singular solutions for memory-driven chemotaxis system.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114083"},"PeriodicalIF":2.3,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145921895","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}
Pub Date : 2026-01-08DOI: 10.1016/j.jde.2025.114093
Haiyun Deng , Hairong Liu , Xiaoping Yang
In this paper, one of our aims is to investigate the instability of the distribution of the critical point set of a solution u to a semilinear equation with Dirichlet boundary condition in the planar annular domains. Precisely, we prove that in an eccentric circle annular domain, or a petal-like domain, or an annular domain where the interior and exterior boundaries are equally scaled ellipses contains only finitely many points rather than a Jordan curve. This result indicates that the critical point set is unstable when any boundary of planar concentric circle annular domain Ω has some small deformation or minor perturbation. Based on studying the distribution of the nodal sets and , we prove that the solution u on each symmetric axis has exactly two critical points under some conditions. Meanwhile, we further obtain that only has two critical points in an eccentric circle annular domain, has four critical points in an exterior petal-like domain with the exterior boundary is an ellipse, and the maximum points are distributed on the long symmetric semi-axis and the saddle points on the short symmetric semi-axis. Moreover, we describe the geometric location of critical points of the solution u by the moving plane method.
{"title":"On the number and geometric location of critical points of solutions to a semilinear elliptic equation in annular domains","authors":"Haiyun Deng , Hairong Liu , Xiaoping Yang","doi":"10.1016/j.jde.2025.114093","DOIUrl":"10.1016/j.jde.2025.114093","url":null,"abstract":"<div><div>In this paper, one of our aims is to investigate the instability of the distribution of the critical point set <span><math><mi>C</mi><mo>(</mo><mi>u</mi><mo>)</mo></math></span> of a solution <em>u</em> to a semilinear equation with Dirichlet boundary condition in the planar annular domains. Precisely, we prove that <span><math><mi>C</mi><mo>(</mo><mi>u</mi><mo>)</mo></math></span> in an eccentric circle annular domain, or a petal-like domain, or an annular domain where the interior and exterior boundaries are equally scaled ellipses contains only finitely many points rather than a Jordan curve. This result indicates that the critical point set <span><math><mi>C</mi><mo>(</mo><mi>u</mi><mo>)</mo></math></span> is unstable when any boundary of planar concentric circle annular domain Ω has some small deformation or minor perturbation. Based on studying the distribution of the nodal sets <span><math><msubsup><mrow><mi>u</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>(</mo><mn>0</mn><mo>)</mo><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>=</mo><mi>∇</mi><mi>u</mi><mo>⋅</mo><mi>θ</mi><mo>)</mo></math></span> and <span><math><msup><mrow><mi>u</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mn>0</mn><mo>)</mo></math></span>, we prove that the solution <em>u</em> on each symmetric axis has exactly two critical points under some conditions. Meanwhile, we further obtain that <span><math><mi>C</mi><mo>(</mo><mi>u</mi><mo>)</mo></math></span> only has two critical points in an eccentric circle annular domain, has four critical points in an exterior petal-like domain with the exterior boundary <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>E</mi></mrow></msub></math></span> is an ellipse, and the maximum points are distributed on the long symmetric semi-axis and the saddle points on the short symmetric semi-axis. Moreover, we describe the geometric location of critical points of the solution <em>u</em> by the moving plane method.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"458 ","pages":"Article 114093"},"PeriodicalIF":2.3,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145921471","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}
We investigate dispersive and Strichartz estimates for the Schrödinger equation involving the fractional Laplacian in real hyperbolic spaces and their discrete analogues, homogeneous trees. Due to the Knapp phenomenon, the Strichartz estimates on Euclidean spaces for the fractional Laplacian exhibit some loss of derivatives. A similar phenomenon appears on real hyperbolic spaces. However, such a loss disappears on homogeneous trees, due to the triviality of the estimates for small times.
{"title":"The Schrödinger equation with fractional Laplacian on hyperbolic spaces and homogeneous trees","authors":"Jean-Philippe Anker , Guendalina Palmirotta , Yannick Sire","doi":"10.1016/j.jde.2025.114065","DOIUrl":"10.1016/j.jde.2025.114065","url":null,"abstract":"<div><div>We investigate dispersive and Strichartz estimates for the Schrödinger equation involving the fractional Laplacian in real hyperbolic spaces and their discrete analogues, homogeneous trees. Due to the Knapp phenomenon, the Strichartz estimates on Euclidean spaces for the fractional Laplacian exhibit some loss of derivatives. A similar phenomenon appears on real hyperbolic spaces. However, such a loss disappears on homogeneous trees, due to the triviality of the estimates for small times.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114065"},"PeriodicalIF":2.3,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940024","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}
Pub Date : 2026-01-06DOI: 10.1016/j.jde.2025.114084
Qian Guo , Taishan Yi , Yurong Zhang , Xingfu Zou
In this paper, we study the threshold dynamics of a class of reaction-diffusion systems in a cylindrical domain with shifting effect. We first transform the reaction-diffusion system into a spatially inhomogeneous autonomous system using moving coordinates and analyze the fundamental properties of the solution to this new system. Then, we establish uniform asymptotic annihilation of the autonomous system by constructing an upper system sequence. Finally, employing the theory of asymptotic spectral radius, we investigate the threshold dynamics of the system, including existence/nonexistence and uniqueness of forced wave, as well as its global stability. Particularly, we establish a logarithmic relation between the asymptotic spectral radius and the standard generalized principal eigenvalue, thereby characterizing the influence of the climate shifting speed c on the asymptotic spectral radius.
{"title":"Threshold dynamics of a reaction-diffusion system in a cylinder with shifting effect","authors":"Qian Guo , Taishan Yi , Yurong Zhang , Xingfu Zou","doi":"10.1016/j.jde.2025.114084","DOIUrl":"10.1016/j.jde.2025.114084","url":null,"abstract":"<div><div>In this paper, we study the threshold dynamics of a class of reaction-diffusion systems in a cylindrical domain with shifting effect. We first transform the reaction-diffusion system into a spatially inhomogeneous autonomous system using moving coordinates and analyze the fundamental properties of the solution to this new system. Then, we establish uniform asymptotic annihilation of the autonomous system by constructing an upper system sequence. Finally, employing the theory of asymptotic spectral radius, we investigate the threshold dynamics of the system, including existence/nonexistence and uniqueness of forced wave, as well as its global stability. Particularly, we establish a logarithmic relation between the asymptotic spectral radius and the standard generalized principal eigenvalue, thereby characterizing the influence of the climate shifting speed <em>c</em> on the asymptotic spectral radius.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"458 ","pages":"Article 114084"},"PeriodicalIF":2.3,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145921472","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}
Pub Date : 2026-01-05DOI: 10.1016/j.jde.2025.114061
Walter Craig , Carlos García-Azpeitia
We investigate the formation of steady waves in two-dimensional fluids flowing with a mean velocity c over a periodic bottom. Adopting a Dirichlet–Neumann operator formulation, we prove that—apart from a discrete sequence of critical speeds at which classical Stokes waves bifurcate— the flat-surface solution continues uniquely to a nontrivial steady state under a small bathymetric variation. Furthermore, our main theorem proves that each nondegenerate –orbit of steady waves on the flat bottom (including Stokes waves) gives rise to at least two distinct steady solutions when a small bathymetric variation is introduced.
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