Journal of Differential Equations最新文献

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Relative Morse index of the discrete nonlinear Schrödinger equations with strongly indefinite potentials and applications 具有强不定势的离散非线性Schrödinger方程的相对莫尔斯指数及其应用

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-02-11 DOI: 10.1016/j.jde.2026.114202

Ben-Xing Zhou , Qinglong Zhou

In this paper, we study the relative Morse index theory of discrete nonlinear Schrödinger equations

- Δ u_{n} + v_{n} u_{n} - ω u_{n} = f_{n} (u_{n})

with strongly indefinite potential functions

V = {v_{n} : n \in Z}

satisfying

\lim_{| n | \to \infty} | v_{n} | = + \infty

. As applications, we study the existence and multiplicity of homoclinic solutions for discrete asymptotically linear Schrödinger equations with saturable nonlinearity

{f_{n} : n \in Z}

. In previous works, the prevalent assumption was confined to coercive potential functions (satisfying

\lim_{| n | \to \infty} v_{n} = + \infty

), in contrast to the strongly indefinite potential functions considered herein (with

\lim_{| n | \to \infty} | v_{n} | = + \infty

本文研究了具有强不定势函数V={vn:n∈Z}满足lim|n|→∞(|)vn|=+∞的离散非线性Schrödinger方程- Δun+vnun - ωun=fn（un）的相对莫尔斯指数理论。作为应用，我们研究了具有可饱和非线性{fn:n∈Z}的离散渐近线性Schrödinger方程同宿解的存在性和多重性。在以往的工作中，普遍的假设局限于强制势函数（满足lim|n|→∞(|→∞)），而本文考虑的是强不定势函数（满足lim|n|→∞(|)vn|=+∞）。

引用次数: 0

Self-similar solutions of semilinear heat equations with positive speed 具有正速度的半线性热方程的自相似解

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-02-11 DOI: 10.1016/j.jde.2026.114201

Kyeongsu Choi, Jiuzhou Huang

We classify the smooth self-similar solutions of the semilinear heat equation

u_{t} = Δ u + | u |^{p - 1} u

R^{n} \times (0, T)

satisfying an integral condition for all

p > 1

with positive speed. As a corollary, we prove that finite time blowing up solutions of this equation on a bounded convex domain with

u (\cdot, 0) \geq 0

and

u_{t} (\cdot, 0) \geq 0

converges to a positive constant after rescaling at the blow-up point for all

p > 1

在rnx （0，T）中，我们对半线性热方程ut=Δu+|u|p−1u的光滑自相似解进行了分类，这些解对所有速度为正的p>；1满足积分条件。作为一个推论，我们证明了该方程在u(⋅,0)≥0和ut(⋅,0)≥0的有界凸域上的有限时间爆破解在爆破点对所有p>；1重新缩放后收敛于一个正常数。

引用次数: 0

Finite time blow-up analysis for the generalized Proudman-Johnson model 广义Proudman-Johnson模型的有限时间爆破分析

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-02-06 DOI: 10.1016/j.jde.2026.114187

Jie Guo, Quansen Jiu

In this paper, we study the generalized Proudman-Johnson equation posed on the torus. In the critical regime where the parameter a is close to and slightly greater than 1, we establish finite time blow-up of smooth solutions to the inviscid case. Moreover, we show that the blow-up is asymptotically self-similar for a class of smooth initial data. In contrast, when the parameter a lies slightly below 1, we prove the global in time existence for the same initial data. In addition, we demonstrate that inviscid Proudman-Johnson equation with Hölder continuous data also develops a self-similar blow-up. Finally, for the viscous case with

a > 1

, we prove that smooth initial data can still lead to finite time blow-up.

本文研究了环面上的广义Proudman-Johnson方程。在参数a接近且略大于1的临界区域，我们建立了无粘情况下光滑解的有限时间爆破。此外，我们还证明了对一类光滑初始数据的爆破是渐近自相似的。相反，当参数a略小于1时，对于相同的初始数据，我们证明了全局的时间存在性。此外，我们还证明了具有Hölder连续数据的无粘Proudman-Johnson方程也具有自相似爆破。最后，对于a>；1的粘性情况，我们证明了光滑初始数据仍然可以导致有限时间爆炸。

引用次数: 0

Riesz potential estimates for double obstacle problems with Orlicz growth Riesz对Orlicz增长的双重障碍问题的潜在估计

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-02-06 DOI: 10.1016/j.jde.2026.114192

Qi Xiong , Zhenqiu Zhang , Lingwei Ma

In this paper, we consider the solutions to the non-homogeneous double obstacle problems with Orlicz growth involving measure data. After establishing the existence of the solutions to this problem in the Orlicz-Sobolev space, we derive a pointwise gradient estimate for these solutions by Riesz potential, which leads to the result on the

C^{1}

regularity criterion.

本文研究了包含测量数据的具有Orlicz增长的非齐次双障碍问题的解。在建立了该问题在Orlicz-Sobolev空间中解的存在性之后，利用Riesz势导出了这些解的点向梯度估计，从而得到了C1正则性准则的结果。

引用次数: 0

Hydrodynamic limit to the rarefaction wave for the Boltzmann equation 玻尔兹曼方程中稀疏波的水动力极限

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-01-30 DOI: 10.1016/j.jde.2026.114161

Guanghui Wang , Lingda Xu , Mingying Zhong

In this paper, we study the hydrodynamic limit for rarefaction wave from the Boltzmann equation to Euler equations. We obtain the convergence rate of ϵ in

L^{\infty}

norm on finite time interval

[0, T]

, where

ϵ > 0

is the Knudsen number and

T > 0

is any fixed constant. This convergence rate coincides with Caflisch 1980, cf. [1], which studied the hydrodynamic limit for smooth Euler solutions. This rate improves the result of Xin-Zeng 2010, where the convergence rate is

ϵ^{\frac{1}{2}}

L^{\infty}

norm, cf. [25]. The result is obtained by a refined energy estimate and the better rates are obtained for the higher-order derivatives.

本文从玻尔兹曼方程到欧拉方程研究了稀疏波的水动力极限。我们得到了L∞范数在有限时间区间[0，T]上的收敛速率，其中ϵ>；0为Knudsen数，T >；0为任意固定常数。这一收敛速度与Caflisch 1980， cf.[1]研究光滑欧拉解的水动力极限相一致。该速率改进了Xin-Zeng 2010的结果，在L∞范数下收敛速率为ϵ12，参见[25]。通过改进的能量估计得到了结果，并对高阶导数得到了较好的速率。

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引用次数: 0

Two positive normalized solutions on star-shaped bounded domains to the Brézis-Nirenberg problem brsamzis - nirenberg问题在星形有界区域上的两个正规范化解

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-01-30 DOI: 10.1016/j.jde.2026.114169

Linjie Song, Wenming Zou

We establish the existence of two positive solutions with prescribed mass for NLS on star-shaped bounded domains: one is the normalized ground state and another is at a mountain pass level. We merely address the Sobolev critical case since the Sobolev subcritical one can be addressed by following similar arguments and is easier.

我们建立了星形有界域上NLS的两个规定质量正解的存在性：一个是归一化基态，另一个是在山口水平。我们只讨论Sobolev临界情况，因为Sobolev次临界情况可以通过类似的论证来解决，而且更容易。

引用次数: 0

The strongly nonlocal Allen–Cahn problem 强非局部Allen-Cahn问题

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-01-30 DOI: 10.1016/j.jde.2026.114168

Erisa Hasani, Stefania Patrizi

We study the sharp interface limit of the fractional Allen–Cahn equation

ε \partial_{t} u^{ε} = I_{n}^{s} [u^{ε}] - \frac{1}{ε^{2 s}} W^{'} (u^{ε}) in (0, \infty) \times R^{n}, n \geq 2,

where

ε > 0

I_{n}^{s} = - c_{n, s} {(- Δ)}^{s}

is the fractional Laplacian of order

2 s \in (0, 1)

R^{n}

, and W is a smooth double-well potential with minima at 0 and 1. We focus on the singular regime

s \in (0, \frac{1}{2})

, corresponding to strongly nonlocal diffusion. For suitably prepared initial data, we prove that the solution

u^{ε}

converges, as

ε \to 0

, to the minima of W with the interface evolving by fractional mean curvature flow. This establishes the first rigorous convergence result in this regime, complementing and completing previous work for

s \geq \frac{1}{2}

我们研究了分数阶Allen-Cahn方程ε∂tuε=Ins[uε]−1ε2sW ' (uε)in（0,∞）×Rn,n≥2，其中ε>；0, Ins=−cn,s（−Δ）s是Rn中2s阶的分数阶拉普拉斯算子∈(0,1)，W是一个在0和1处有极小值的光滑双阱势。我们关注奇异区域s∈(0,12)，对应于强非局部扩散。对于适当准备的初始数据，我们证明了当ε→0时，随着分数阶平均曲率流的界面演化，解收敛到W的最小值。这建立了该区域的第一个严格的收敛结果，补充并完成了先前关于s≥12的工作。

引用次数: 0

Dynamical versions of Morgan's uncertainty principle and electromagnetic Schrödinger evolutions 摩根测不准原理的动力学版本和电磁Schrödinger演化

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-01-30 DOI: 10.1016/j.jde.2026.114159

Shanlin Huang , Zhenqiang Wang

This paper investigates the unique continuation properties of solutions of the electromagnetic Schrödinger equation

i \partial_{t} u (x, t) + {(\nabla - i A)}^{2} u (x, t) = V (x, t) u (x, t) in R^{d} \times [0, 1],

where A represents a time-independent magnetic vector potential and V is a bounded, complex valued time-dependent potential. Given

1 < p < 2

and

1 / p + 1 / q = 1

, we prove that there exists

N_{p} > 0

such that if

\int_{R^{d}} | u (x, 0) |^{2} e^{2 α^{p} | x |^{p} / p} d x + \int_{R^{d}} | u (x, 1) |^{2} e^{2 β^{q} | x |^{q} / q} d x < \infty

for some

α, β > 0

, and if

α β > N_{p}

, then

u \equiv 0

. These results can be interpreted as dynamical versions of the uncertainty principle of Morgan's type. Furthermore, as an application, our results extend to a large class of semi-linear Schrödinger equations.

本文研究了电磁Schrödinger方程i∂tu(x,t)+(∇−iA)2u(x,t)=V(x,t)u(x,t) inrdx[0,1]解的唯一连续性质，其中A表示时无关的磁矢量势，V是有界的复值时相关势。给定1<；p<；2和1/p+1/q=1，我们证明了Np>；0的存在，使得∫Rd|u(x,0) |e2α - p|x|p/pdx+∫Rd|u(x,1)|2e2βq|x|q/qdx<；∞对于某些α，β>0，且αβ>；Np，则u≡0。这些结果可以解释为摩根不确定性原理的动态版本。此外，作为一个应用，我们的结果推广到一类半线性Schrödinger方程。

引用次数: 0

Large time behavior of solutions to unipolar Euler-Poisson equations with space-dependent damping 具有空间相关阻尼的单极欧拉-泊松方程解的大时间行为

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-01-30 DOI: 10.1016/j.jde.2026.114172

Chunpeng Wang, Jianing Xu

This paper is concerned with the Cauchy problem to Euler-Poisson equations for one-dimensional unipolar hydrodynamic model of semiconductors with damping of space-dependent coefficient. Under some smallness assumptions on the initial data, we establish the global existence of smooth solutions to the Cauchy problem by applying the energy methods. It is shown that the solutions to unipolar Euler-Poisson equations with space-dependent damping time-exponentially converge to the stationary solutions. No smallness assumption is imposed on the space-dependent coefficient of damping.

本文研究了具有空间依赖系数阻尼的一维单极半导体流体力学模型的欧拉-泊松方程的Cauchy问题。在初始数据的一些较小的假设条件下，利用能量方法建立了柯西问题光滑解的全局存在性。证明了具有空间相关阻尼的单极欧拉-泊松方程的解在时间指数上收敛于平稳解。对阻尼的空间相关系数不作小的假设。

引用次数: 0

Quantitative blow-up via renormalized Kato theory: Resolving Nakao-type systems 通过重整加藤理论的定量爆炸：解决中尾型系统

IF 2.3 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2026-01-29 DOI: 10.1016/j.jde.2026.114165

Mengyun Liu

We address the fundamental obstruction identified in [10, Remark 3] for system (1) where sign-changing kernels when

b^{2} < 4 k

preclude blow-up arguments via nonnegative functionals—by partially resolving it in the regime

b^{2} < k

Building upon Kato-type techniques, we develop a renormalized iteration scheme establishing the quantitative upper bounds for blow-up times. This framework resolves the critical case

b^{2} < k

for

b, k > 0

under

θ (p, q, n) : = \frac{1}{p q - 1} - \frac{n - 1}{2} \geq 0

. When combined with [10]'s results for

b^{2} \geq 4 k

, it completes the blow-up theory for the subregime

b^{2} < k

. For

b, k < 0

, we prove blow-up in the extended critical region

Γ_{_{G G}} (p, q, n) : = \max {\frac{p + 1}{p q - 1}, \frac{q + 1}{p q - 1}} - \frac{n - 1}{2} \geq 0,

strictly containing the classical critical set.

我们解决了在系统(1)中[10，Remark 3]中发现的基本障碍，其中当b2<；4k时的符号变化核通过非负泛函排除了爆炸参数-通过在b2<；k中部分解决了它。在加藤型技术的基础上，我们开发了一种重归一化的迭代方案，建立了爆炸时间的定量上界。该框架解决了在θ(p,q,n) =1pq−1−n−12≥0的情况下，b,k, b2<；k的临界情况。结合[10]在b2≥4k时的结果，完成了子区b2<；k的爆破理论。对于b，k<0，证明了扩展临界regionΓGG（p,q,n）的爆破性：=max (p +1pq−1,q+1pq−1}−n−12≥0，严格包含经典临界集。

引用次数: 0

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