Journal of Differential Equations最新文献

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Well–posedness and wave breaking of solutions for the Degasperis–Procesi equation including Coriolis effects 含科里奥利效应的Degasperis-Procesi方程解的适定性和破波性

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-31 DOI: 10.1016/j.jde.2025.114077

Joachim Escher , Baihong Li , Yuanhong Wei

This paper is devoted to the mathematical study of shallow water equations with Coriolis effects due to the Earth's rotation. Considering shallow water waves flowing along the zonal direction near the equator and performing a double asymptotic expansion in the amplitude and shallowness parameter in the full three–dimensional Euler equations, a Degasperis–Procesi type equation with Coriolis correction is obtained. In a first step a precise well–posedness result is provided, based on Kato's theory of quasi-linear evolution equations. Relying on this result, a blow–up criterion of the corresponding solutions is established. For bounded solutions, that is, solutions which are a priori bounded in

L^{\infty}

, a sufficient condition for the blow–up behaviour in the form of a wave breaking phenomenon is provided. In addition, the blow–up rate is obtained as well.

本文对地球自转引起的含科里奥利效应的浅水方程进行了数学研究。考虑赤道附近沿纬向流动的浅水波浪，对全三维欧拉方程的振幅和浅度参数进行双渐近展开，得到了具有科里奥利校正的Degasperis-Procesi型方程。第一步，基于Kato的准线性演化方程理论，给出了一个精确的适定性结果。在此基础上，建立了相应解的爆破判据。对于有界解，即在L∞上先验有界的解，给出了破波现象形式的爆破行为的充分条件。此外，还得到了爆破率。

引用次数: 0

Advances in solving nonlinear Schrödinger equations with general potentials 广义势非线性Schrödinger方程的求解进展

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-30 DOI: 10.1016/j.jde.2025.114066

Romildo Lima , Liliane Maia , Mayra Soares

We establish the existence of a positive solution to nonlinear Schrödinger equations(P_V)

- Δ u + V (x) u = f (u) in R^{N},

for very general potentials V, with positive or zero limit at infinity (allowing convergence from above, below, or oscillations), but imposing no decay rate assumption. Also, the nonlinearities f may satisfy mild hypotheses, including superlinear or asymptotically linear growth at infinity. This is possible due to the application of the classical Monotonicity Trick approach.

我们建立了非线性Schrödinger方程(PV) - Δu+V(x)u=f(u)在RN中的正解的存在性，对于非常一般的电位V，在无穷远处具有正极限或零极限（允许从上、下或振荡收敛），但不施加衰减率假设。此外，非线性f可以满足温和的假设，包括在无穷远处的超线性或渐近线性增长。由于应用了经典的单调性技巧方法，这是可能的。

引用次数: 0

A bilinear pointwise tracking optimal control problem for a semilinear elliptic PDE 半线性椭圆型PDE的双线性点跟踪最优控制问题

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-30 DOI: 10.1016/j.jde.2025.114073

Enrique Otárola, Daniel Quero, Matías Sasso

We consider a bilinear optimal control problem with pointwise tracking for a semilinear elliptic PDE in two and three dimensions. The control variable enters the PDE as a (reaction) coefficient and the cost functional contains point evaluations of the state variable. These point evaluations lead to an adjoint problem with a linear combination of Dirac measures as a forcing term. In Lipschitz domains, we derive the existence of optimal solutions and analyze first and necessary and sufficient second order optimality conditions. We also prove that every locally optimal control

\bar{u}

belongs to

H^{1} (Ω)

. Finally, assuming that the domain

Ω \subset R^{2}

is a convex polygon, we prove that

\bar{u} \in C^{0, 1} (\bar{Ω})

研究二维和三维半线性椭圆型偏微分方程的双线性点跟踪最优控制问题。控制变量作为（反应）系数进入PDE，代价函数包含状态变量的点评估。这些点的评估导致了狄拉克测度的线性组合作为强迫项的伴随问题。在Lipschitz域上，我们得到了最优解的存在性，并分析了二阶最优性的一、充分必要条件。我们还证明了每一个局部最优控制u¯都属于H1（Ω）。最后，假设域Ω∧R2是一个凸多边形，我们证明u¯∈C0,1（Ω¯）。

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引用次数: 0

Segregated solutions for a class of systems with Lotka-Volterra interaction 一类具有Lotka-Volterra相互作用的系统的分离解

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-30 DOI: 10.1016/j.jde.2025.114074

Qing Guo , Angela Pistoia , Shixin Wen

This paper deals with the existence of positive solutions to the system

{\begin{matrix} - Δ w_{1} - ε w_{1} = μ_{1} w_{1}^{p} + β w_{1} w_{2}, w_{1} > 0, & in Ω, \\ - Δ w_{2} - ε w_{2} = μ_{2} w_{2}^{p} + β w_{1} w_{2}, w_{2} > 0, & in Ω, \\ w_{1} = w_{2} = 0, & on \partial Ω, \end{matrix}

where

Ω \subseteq R^{N}

N \geq 4

p = 2^{⁎} - 1

, and

ε \in (0, Λ_{1} (Ω))

is sufficiently small. The interaction coefficient

β = β (ε) \to 0

ε \to 0

We construct a family of segregated solutions to this system, where each component blows-up at a different critical point of the Robin function as

ε \to 0

. The system lacks a variational formulation due to its specific coupling form, which leads to essentially different behaviors in the subcritical, critical, and supercritical regimes and requires an appropriate functional settings to carry out the construction.

本文讨论了∂Ω上{−Δw1−εw1=μ1w1p+βw1w2,w1>0，在Ω，−Δw2−εw2=μ2w2p+βw1w2,w2>0，在Ω，w1=w2=0，其中Ω RN， N≥4,p=2 - 1, ε∈（0，Λ1（Ω））足够小的正解的存在性。相互作用系数β=β(ε)→0为ε→0。我们构造了该系统的一组分离解，其中每个分量在Robin函数ε→0的不同临界点处爆炸。由于其特定的耦合形式，该系统缺乏变分公式，这导致在亚临界、临界和超临界状态下本质上不同的行为，需要适当的功能设置来进行构建。

引用次数: 0

Bifurcation from periodic solutions in delay differential equations 时滞微分方程周期解的分岔

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-30 DOI: 10.1016/j.jde.2025.114072

Shangjiang Guo

In this paper, without establishing the Poincaré map, we employ Lyapunov-Schmidt procedure to investigate the one-codimensional bifurcations from the periodic orbits in delay differential equations, and obtain some important formulas giving the relevant coefficients for the determinations of bifurcation direction and stability of the bifurcating periodic solutions.

本文在不建立poincar映射的情况下，利用Lyapunov-Schmidt过程研究了时滞微分方程周期轨道的一协维分岔问题，得到了确定分岔方向和分岔周期解稳定性的相关系数的一些重要公式。

引用次数: 0

Differentiability and kernel estimates for Robin operators Robin算子的可微性和核估计

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-30 DOI: 10.1016/j.jde.2025.114055

A.F.M. ter Elst, M.F. Wong

Consider the elliptic operator

A = - \sum_{k, l = 1}^{d} \partial_{k} c_{k l} \partial_{l} - \sum_{k = 1}^{d} \partial_{k} b_{k} + \sum_{k = 1}^{d} c_{k} \partial_{k} + c_{0}

on a bounded connected open set

Ω \subset R^{d}

of class

C^{1 + κ}

, where the

c_{k l}, b_{k}, c_{k} \in C^{κ} (Ω, C)

are Hölder continuous of order κ and

c_{0} \in L_{\infty} (Ω, C)

, subject to Robin boundary conditions

\partial_{ν} u + β Tr u = 0

, with

β \in C^{κ} (\partial Ω, C)

is complex valued and

κ \in (0, 1)

. We show that the kernel of the semigroup generated by −A is differentiable in each variable and that the derivatives are Hölder continuous of order κ. Moreover, we prove Gaussian kernel bounds and Hölder Gaussian bounds for the derivatives of the kernel.

考虑有界连通开集Ω +κ类的Rd上的椭圆算子a =−∑k,l=1d∂kckl∂l−∑k=1d∂kbk+∑k=1dck∂k+c0，其中ckl，bk,ck∈Cκ（Ω，C）是κ阶连续的Hölder，且c0∈l∞（Ω，C），服从Robin边界条件∂νu+β tru =0，其中β∈Cκ（∂Ω，C）是复值，κ∈(0,1)。我们证明了−A生成的半群的核在每个变量上都是可微的，并且其导数是κ阶Hölder连续的。此外，我们还证明了高斯核界和Hölder核导数的高斯界。

引用次数: 0

The space of Hardy weights for quasilinear operators on discrete graphs 离散图上拟线性算子的Hardy权空间

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-29 DOI: 10.1016/j.jde.2025.114057

Ujjal Das , Matthias Keller , Yehuda Pinchover

We study Hardy inequalities for p-Schrödinger operators on general weighted graphs. Specifically, we prove a Maz'ya-type result, where we characterize the space of Hardy weights for p-Schrödinger operators via a generalized capacity. The novel ingredient in the proof is the demonstration that the simplified energy of the p-Schrödinger energy functional is compatible with certain normal contractions. As a consequence, we obtain a necessary integrability criterion for Hardy weights. Finally, using some tools of criticality theory, we investigate the existence of minimizers in the Hardy inequalities and discuss relations to Cheeger-type estimates.

研究了广义加权图上p-Schrödinger算子的Hardy不等式。具体来说，我们证明了一个Maz'ya型结果，其中我们通过广义容量刻画了p-Schrödinger算子的Hardy权空间。该证明的新颖之处是证明了p-Schrödinger能量泛函的简化能量与某些正常收缩相容。由此，我们得到了Hardy权值的一个必要的可积准则。最后，利用临界理论的一些工具，研究了Hardy不等式中极小值的存在性，并讨论了其与cheeger型估计的关系。

引用次数: 0

Mean-field limits for stochastic interacting particles via digraph measures 用有向图测量随机相互作用粒子的平均场极限

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-29 DOI: 10.1016/j.jde.2025.114054

Christian Kuehn , Carlos Pulido

Many natural phenomena are effectively described by interacting particle systems, which can be modeled using either deterministic or stochastic differential equations (SDEs). In this study, we specifically investigate particle systems modeled by SDEs, wherein the mean field limit converges to a Vlasov-Fokker-Planck-type equation. Departing from conventional approaches in stochastic analysis, we explore the network connectivity between particles using digraph measures (DGMs). DGMs are one possible tool to capture sparse, intermediate and dense network/graph interactions in the mean-field thereby going beyond more classical approaches such as graphons. Since the main goal is to capture large classes of mean-field limits, we set up our approach using measure-theoretic arguments and combine them with suitable moment estimates to ensure approximation results for the mean-field.

相互作用的粒子系统可以有效地描述许多自然现象，它们可以用确定性或随机微分方程（SDEs）来建模。在本研究中，我们专门研究了由SDEs建模的粒子系统，其中平均场极限收敛于vlasov - fokker - planck型方程。从传统的随机分析方法出发，我们使用有向图度量（DGMs）来探索粒子之间的网络连通性。dgm是一种可能的工具，可以在平均场中捕获稀疏、中间和密集的网络/图交互，从而超越更经典的方法，如图元。由于主要目标是捕获大类别的平均场极限，我们使用测量理论参数建立我们的方法，并将它们与合适的矩估计相结合，以确保平均场的近似结果。

引用次数: 0

Corrigendum to: “Sampling and equidistribution theorems for elliptic second order operators, lifting of eigenvalues, and applications” [J. Differ. Equ. 268 (12) (2020) 7669–7714] “椭圆二阶算子的抽样和等分布定理，特征值的提升及其应用”[J]。是不同的。法典第268(12)(2020)7669-7714条]

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-29 DOI: 10.1016/j.jde.2025.114060

Martin Tautenhahn , Ivan Veselić

We consider elliptic second order partial differential operators with Lipschitz continuous leading order coefficients on finite cubes and the whole Euclidean space. We prove quantitative sampling and equidistribution theorems for eigenfunctions. The estimates are scale-free, in the sense that for a sequence of growing cubes we obtain uniform estimates. These results are applied to prove lifting of eigenvalues as well as the infimum of the essential spectrum, and an uncertainty relation (aka spectral inequality) for short energy interval spectral projectors. Several applications including random operators are discussed. In the proof we have to overcome several challenges posed by the variable coefficients of the leading term.

研究有限立方空间和整个欧几里德空间上具有Lipschitz连续序次系数的椭圆型二阶偏微分算子。证明了特征函数的定量抽样定理和等分布定理。估计是无标度的，从某种意义上说，对于一个不断增长的立方体序列，我们得到了一致的估计。这些结果被应用于证明特征值的提升和本质谱的最小值，以及短能量间隔谱投影的不确定关系（又称谱不等式）。讨论了包括随机算子在内的几种应用。在证明中，我们必须克服几个由前项的可变系数带来的挑战。

引用次数: 0

Boundary reconstruction for the anisotropic fractional Calderón problem 各向异性分数阶Calderón问题的边界重建

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2025-12-29 DOI: 10.1016/j.jde.2025.114052

Xiaopeng Cheng , Angkana Rüland

In this article, we provide a boundary reconstruction result for the anisotropic fractional Calderón problem and its associated degenerate elliptic extension into the upper half plane. More precisely, considering the setting from Feizmohammadi et al. [23], we show that the metric on the measurement set can be reconstructed from the source-to-solution data. To this end, we rely on the approach by Brown [6] in the framework developed by Nakamura and Tanuma [44] (see also Kang and Yun [32]) after localizing the problem by considering it through an extension perspective.

本文给出了各向异性分数阶Calderón问题及其在上半平面上的退化椭圆扩展的边界重建结果。更准确地说，考虑Feizmohammadi等人[23]的设置，我们证明了测量集上的度量可以从源数据到解数据进行重构。为此，我们在通过扩展的角度来考虑问题的局部化之后，在Nakamura和Tanuma[44]（另见Kang和Yun[32]）开发的框架中依赖Brown[6]的方法。

引用次数: 0

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