Journal of Differential Equations最新文献

英文中文

On the number and geometric location of critical points of solutions to a semilinear elliptic equation in annular domains 环形区域上半线性椭圆方程解的临界点的数目和几何位置

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-01-08 DOI: 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

C (u)

of a solution u to a semilinear equation with Dirichlet boundary condition in the planar annular domains. Precisely, we prove that

C (u)

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

C (u)

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

u_{θ}^{- 1} (0) (u_{θ} = \nabla u \cdot θ)

and

u^{- 1} (0)

, we prove that the solution u on each symmetric axis has exactly two critical points under some conditions. Meanwhile, we further obtain that

C (u)

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

γ_{E}

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.

本文的目的之一是研究平面环形区域上具有Dirichlet边界条件的半线性方程解的临界点集C(u)的分布的不稳定性。准确地说，我们证明了C(u)在偏心圆环形区域，或花瓣状区域，或内外边界为等比例椭圆的环形区域中只包含有限多个点，而不是约旦曲线。这一结果表明，当平面同心圆环形畴Ω的任一边界有较小的变形或微扰时，临界点集C(u)是不稳定的。通过研究节点集uθ−1(0)（uθ=∇u⋅θ）和u−1(0)的分布，证明了在某些条件下，每个对称轴上的解u恰好有两个临界点。同时，我们进一步得到C(u)在偏心圆环形域中只有2个临界点，在外边界γE为椭圆的外花瓣状域中有4个临界点，最大值点分布在长对称半轴上，鞍点分布在短对称半轴上。此外，我们用移动平面法描述了解u的关键点的几何位置。

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引用次数: 0

The Schrödinger equation with fractional Laplacian on hyperbolic spaces and homogeneous trees 双曲空间和齐次树上的分数阶拉普拉斯方程Schrödinger

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-01-06 DOI: 10.1016/j.jde.2025.114065

Jean-Philippe Anker , Guendalina Palmirotta , Yannick Sire

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.

我们研究了真实双曲空间及其离散类似物齐次树中涉及分数阶拉普拉斯方程Schrödinger的色散估计和Strichartz估计。由于Knapp现象的存在，分数阶拉普拉斯算子在欧几里得空间上的Strichartz估计具有一定的导数损失。在实双曲空间中也出现了类似的现象。然而，由于小时间估计的琐碎性，这种损失在齐次树上消失。

引用次数: 0

Threshold dynamics of a reaction-diffusion system in a cylinder with shifting effect 具有位移效应的汽缸反应扩散系统的阈值动力学

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-01-06 DOI: 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.

本文研究了一类具有位移效应的反应扩散系统在圆柱形域上的阈值动力学问题。我们首先利用移动坐标将反应扩散系统转化为空间非齐次自治系统，并分析了该系统解的基本性质。然后，通过构造上系统序列，建立了自治系统的一致渐近湮灭。最后，利用渐近谱半径理论，研究了系统的阈值动力学，包括强迫波的存在/不存在性和唯一性，以及系统的全局稳定性。特别地，我们建立了渐近谱半径与标准广义主特征值之间的对数关系，从而表征了气候变化速度c对渐近谱半径的影响。

引用次数: 0

Steady waves in flows over periodic bottoms 在周期性底部流动的稳定波浪

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-01-05 DOI: 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

c_{k}

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

S^{1}

–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.

我们研究了以平均速度c流过周期底的二维流体中稳定波的形成。采用Dirichlet-Neumann算子公式，我们证明了，除了经典Stokes波分叉的临界速度ck的离散序列外，平面解在一个小的水深变化下唯一地持续到一个非平凡的稳态。此外，我们的主要定理证明了当引入一个小的水深变化时，在平底上的稳定波（包括Stokes波）的每个非简并s1轨道至少产生两个不同的稳定解。

引用次数: 0

Canard cycles of non-linearly regularized piecewise smooth vector fields 非线性正则化分段光滑向量场的鸭纳德循环

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-01-05 DOI: 10.1016/j.jde.2025.114079

Peter De Maesschalck , Renato Huzak , Otavio Henrique Perez

The main purpose of this paper is to study limit cycles of non-linear regularizations of planar piecewise smooth systems. We deal with a slow-fast Hopf point after non-linear regularization and blow-up. We give a simple criterion for the existence of limit cycles of canard type blue for a class of (non-linearly) regularized piecewise smooth systems, expressed in terms of zeros of the slow divergence integral. Using the criterion we can construct a quadratic regularization of a piecewise linear center such that for any integer

k > 0

it has at least

k + 1

limit cycles, for a suitably chosen monotonic transition function

φ_{k} : R \to R

. We prove a similar result for regularized codimension 1 invisible-invisible fold-fold singularities of type

I I_{2}

. Canard cycles of dodging layer are also considered, and we prove that there can be at most 2 limit cycles (born in a saddle-node bifurcation).

本文的主要目的是研究平面分段光滑系统非线性正则化的极限环。我们处理了一个经过非线性正则化和爆破后的慢速Hopf点。对于一类用慢散度积分的零点表示的（非线性）正则分段光滑系统，给出了蓝鸭型极限环存在的一个简单判据。利用该准则，我们可以构造分段线性中心的二次正则化，使得对于任意整数k>；0，对于适当选择的单调过渡函数φk:R→R，它至少有k+1个极限环。我们证明了正则余维1型II2的不可见-不可见折叠奇异性的类似结果。同时考虑了闪避层的Canard环，并证明了存在最多2个极限环（生于鞍节点分岔）。

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引用次数: 0

Resonant Turing-Turing bifurcation and hexagonal patterns of nonlocal reaction-diffusion delayed equations on 2D rectangular domain 二维矩形域非局部反应扩散延迟方程的共振图灵-图灵分岔和六边形图

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-01-05 DOI: 10.1016/j.jde.2025.114082

Xun Cao, Weihua Jiang

Hexagonal patterns are very common in the real world. This paper focuses on the structure, stability, and coexistence of spatial hexagonal patterns arising from resonant Turing-Turing bifurcation on 2D rectangular domain. First, we establish the third-order normal form of resonant/nonresonant Turing-Turing bifurcation for the nonlocal partial functional differential equations (PFDEs) on 2D rectangular domain, and show 14 simpler forms describing distinct spatial phenomena. Using the original system parameters, we explicitly provide formulae for computing the coefficients of these 14 normal forms. Finally, we investigate spatial hexagonal patterns of a nonlocal Holling-Tanner model near resonant Turing-Turing singularity, by means of the established normal forms. Specifically, on 2D rectangular domain, resonant Turing-Turing bifurcation can occur, and there emerge bistable spatial hexagonal patterns, their coexistence with uniform pattern (i.e., the coexistence equilibrium) or a pair of/one stripe pattern, as well as transient flipped hexagonal patterns. Numerical simulations further validate the theoretical analysis.

六边形图案在现实世界中非常常见。本文主要研究二维矩形域上由共振图灵-图灵分岔引起的空间六边形图形的结构、稳定性和共存性。首先，我们建立了二维矩形域上非局部偏泛函微分方程的共振/非共振图灵-图灵分岔的三阶范式，并给出了描述不同空间现象的14种简化形式。利用原有的系统参数，明确地给出了这14种范式系数的计算公式。最后，我们利用已建立的范式研究了谐振图灵-图灵奇点附近非局部Holling-Tanner模型的空间六角形。具体而言，在二维矩形域上，可以发生共振图灵-图灵分岔，出现双稳态空间六边形图形，它们与均匀图形共存（即共存平衡）或一对/一条条纹图形共存，以及瞬态翻转六边形图形。数值模拟进一步验证了理论分析的正确性。

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引用次数: 0

Slowly oscillating periodic solutions in a nonlinear Volterra equation with non-symmetric feedback 非对称反馈非线性Volterra方程的慢振荡周期解

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-01-05 DOI: 10.1016/j.jde.2025.114071

Quentin Griette , Franco Herrera

In this work we study a nonlinear Volterra equation with non-symmetric feedback that arises as a particular case of the Gurtin-MacCamy model in population dynamics. We are particularly interested in the existence of slowly oscillating periodic solutions when the trivial stationary state is unstable. Here the absence of symmetry of the nonlinearity prevents the use of many traditional strategies to obtain a priori estimates on the solution. Without a precise knowledge of the period of the solution, we manage to prove the forward invariance of a carefully constructed set of initial data whose properties imply the slowly oscillating character of all continuations. We prove the existence of periodic solutions by constructing a homeomorphism between our set and a convex subset of a different Banach space, thereby showing that it possesses the fixed-point property. Finally, in a singular limit of a parameter, we show that this periodic solution converges to the solution of a well-known discrete difference equation. We conclude the paper with some numerical simulations to illustrate the existence of the periodic orbit as well as the singular limit behavior.

在这项工作中，我们研究了具有非对称反馈的非线性Volterra方程，它是种群动力学中Gurtin-MacCamy模型的一个特殊情况。当平凡的定态不稳定时，我们对慢振荡周期解的存在性特别感兴趣。在这里，非线性的不对称性阻碍了使用许多传统策略来获得对解的先验估计。在没有解周期的精确知识的情况下，我们设法证明了一组精心构造的初始数据的前向不变性，这些初始数据的性质暗示了所有延拓的缓慢振荡特征。通过构造该集合与另一个Banach空间的凸子集之间的同胚，证明了周期解的存在性，从而证明了它具有不动点的性质。最后，在一个参数的奇异极限下，我们证明了这个周期解收敛于一个众所周知的离散差分方程的解。最后用一些数值模拟来说明周期轨道的存在性和奇异极限行为。

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引用次数: 0

Traveling waves for highly degenerate and singular reaction-diffusion-advection equations with discontinuous coefficients 具有不连续系数的高度简并奇异反应扩散平流方程的行波

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-01-02 DOI: 10.1016/j.jde.2025.114075

Umberto Guarnotta, Cristina Marcelli

Sufficient conditions for either existence or non-existence of traveling wave solutions for a general quasi-linear reaction-diffusion-convection equation, possibly highly degenerate or singular, with discontinuous coefficients are furnished. Under an additional hypothesis on the convection term, the set of admissible wave speeds is characterized in terms of the minimum wave speed, which is estimated through a double-sided bound.

给出了具有不连续系数的一般准线性反应-扩散-对流方程行波解存在或不存在的充分条件，该方程可能高度简并或奇异。在对流项的附加假设下，允许波速集合用最小波速来表示，最小波速通过一个双面边界来估计。

引用次数: 0

Soliton resolution and long-time asymptotics for the Camassa-Holm positive flow with the presence of discrete spectra 离散谱存在下Camassa-Holm正流的孤子分辨率和长时间渐近性

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-01-02 DOI: 10.1016/j.jde.2025.114078

Jia Wang , Xianguo Geng , Kedong Wang , Huan Liu

Based on the Riemann-Hilbert approach and the

\overline{\partial}

steepest descent method, the Cauchy problem of the Camassa-Holm positive flow with the existence of discrete spectra is studied. Starting from the Lax pair of the Camassa-Holm positive flow, we perform spectral analysis and eliminate the singularities in the spectral problem. According to the distribution of the stationary points on the initial jump contour, the half-plane is partitioned into three regions. Using the

\overline{\partial}

steepest descent method, we derive the soliton resolution or the long-time asymptotics in different regions: (1) In the soliton region, the soliton resolution is expressed as soliton solutions with residual error; (2) In the similarity region, the long-time asymptotics of the solution are characterized by the dispersion term with residual error; (3) In the transition region, the Painlevé asymptotics of the solution are characterized by solutions to the Painlevé II equation and the residual error.

基于Riemann-Hilbert方法和∂∂最陡下降法，研究了离散谱存在的Camassa-Holm正流的Cauchy问题。从Camassa-Holm正流的Lax对出发，进行了谱分析，消除了谱问题中的奇异性。根据初始跳跃轮廓上驻点的分布，将半平面划分为三个区域。利用∂∂最陡下降法，导出了不同区域内的孤子解析或长时间渐近性：(1)在孤子区域内，将孤子解析表示为残差孤子解；(2)在相似区域，解的长时间渐近性用带残差的离散项来表征；(3)在过渡区域，解的painlev渐近性由painlev方程的解和残差来表征。

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引用次数: 0

Smooth solution of the compressible Navier-Stokes-Korteweg system with large initial data 具有大初始数据的可压缩Navier-Stokes-Korteweg系统的光滑解

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-01-02 DOI: 10.1016/j.jde.2025.114076

Wenchao Dong

In this paper, we investigate the Cauchy problem for the one-dimensional compressible Navier-Stokes-Korteweg system with degenerate transport coefficients, considering large initial data. For the isothermal case, we establish the global well-posedness for transport coefficients dependent on density, without requiring the pressure to be strictly monotone decreasing. In the non-isothermal case, where transport coefficients are dependent on both density and temperature, we obtain a novel Nishida-Smoller type stability result. It is worth highlighting that the strong solution, achievable with initial data in the

H^{2} \times H^{1} \times H^{1}

space, is indeed smooth. This is attributed to the capillary term's smoothing influence, a characteristic that markedly distinguishes it from the Navier-Stokes system. These findings significantly advance the understanding established in Germain and LeFloch (2016) [25].

本文研究了具有退化输运系数的一维可压缩Navier-Stokes-Korteweg系统在大初始数据条件下的Cauchy问题。对于等温情况，我们建立了输运系数随密度的全局适定性，而不要求压力是严格单调减小的。在非等温情况下，输运系数同时依赖于密度和温度，我们得到了一个新的Nishida-Smoller型稳定性结果。值得强调的是，使用H2×H1×H1空间中的初始数据可以实现的强解确实是平滑的。这归因于毛细项的平滑影响，这是它与Navier-Stokes系统明显不同的特征。这些发现显著推进了Germain和LeFloch(2016)[25]所建立的理解。

引用次数: 0

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