Journal of Differential Equations最新文献

英文中文

Sobolev instability in the cubic NLS equation with convolution potentials on irrational tori 无理环上具有卷积势的立方 NLS 方程中的索波列夫不稳定性

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-27 DOI: 10.1016/j.jde.2024.09.044

In this paper we prove the existence of solutions to the cubic NLS equation with convolution potentials on two dimensional irrational tori undergoing an arbitrarily large growth of Sobolev norms as time evolves. Our results apply also to the case of square (and rational) tori. We weaken the regularity assumptions on the convolution potentials, required in a previous work by Guardia (2014) [11] for the square case, to obtain the

H^{s}

-instability (

s > 1

) of the elliptic equilibrium

u = 0

. We also provide the existence of solutions

u (t)

with arbitrarily small

L^{2}

norm which achieve a prescribed growth, say

{‖ u (T) ‖}_{H^{s}} \geq K {‖ u (0) ‖}_{H^{s}}

K ≫ 1

, within a time T satisfying polynomial estimates, namely

0 < T < K^{c}

for some

c > 0

在本文中，我们证明了二维无理环上具有卷积势的立方 NLS 方程的解的存在性，随着时间的推移，这些解的索波列夫规范会发生任意大的增长。我们的结果也适用于平方（和有理）环的情况。我们弱化了 Guardia（2014）[11] 之前针对正方形情形的工作中所要求的卷积势的正则性假设，从而得到了椭圆均衡 u=0 的 Hs-不稳定性 (s>1)。我们还提供了具有任意小 L2 准则的解 u(t)的存在性，这些解在满足多项式估计（即 0<T<Kc for some c>0）的时间 T 内实现了规定增长，即‖u(T)‖Hs≥K‖u(0)‖Hs, K≫1。

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引用次数: 0

The central limit theorems for integrable Hamiltonian systems perturbed by white noise 受白噪声扰动的可积分哈密顿系统的中心极限定理

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-27 DOI: 10.1016/j.jde.2024.09.047

In this paper, we consider the dynamics of integrable stochastic Hamiltonian systems. Utilizing the Nagaev-Guivarc'h method, we obtain several generalized results of the central limit theorem. Making use of this technique and the Birkhoff ergodic theorem, we prove that the invariant tori persist under stochastic perturbations. Moreover, they asymptotically follow a Gaussian distribution, which gives a positive answer to the stability of integrable stochastic Hamiltonian systems over time. Our results hold true for both Gaussian and non-Gaussian noises, and their intensities can be not small.

在本文中，我们考虑了可积分随机哈密尔顿系统的动力学问题。利用 Nagaev-Guivarc'h 方法，我们得到了中心极限定理的几个广义结果。利用这一技术和伯克霍夫遍历定理，我们证明了不变环在随机扰动下持续存在。此外，它们渐近地服从高斯分布，这给出了可积分随机哈密顿系统随时间变化的稳定性的正面答案。我们的结果对高斯和非高斯噪声都适用，而且它们的强度可以不小。

引用次数: 0

On the Borel summability of formal solutions of certain higher-order linear ordinary differential equations 论某些高阶线性常微分方程形式解的伯累尔求和性

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-26 DOI: 10.1016/j.jde.2024.09.041

We consider a class of

n^{th}

-order linear ordinary differential equations with a large parameter u. Analytic solutions of these equations can be described by (divergent) formal series in descending powers of u. We demonstrate that, given mild conditions on the potential functions of the equation, the formal solutions are Borel summable with respect to the parameter u in large, unbounded domains of the independent variable. We establish that the formal series expansions serve as asymptotic expansions, uniform with respect to the independent variable, for the Borel re-summed exact solutions. Additionally, we show that the exact solutions can be expressed using factorial series in the parameter, and these expansions converge in half-planes, uniformly with respect to the independent variable. To illustrate our theory, we apply it to an

n^{th}

-order Airy-type equation.

我们考虑了一类具有大参数 u 的 n 次阶线性常微分方程。这些方程的解析解可以用 u 的降幂（发散）形式数列来描述。我们证明，给定方程势函数的温和条件，形式解在自变量的大无界域中关于参数 u 是伯尔可求和的。我们确定，形式级数展开可作为关于自变量的渐近展开，与 Borel 重求和精确解一致。此外，我们还证明了精确解可以用参数中的阶乘级数来表示，并且这些展开在半平面上收敛，与自变量保持一致。为了说明我们的理论，我们将其应用于 n 次阶 Airy 型方程。

引用次数: 0

Boundedness for the chemotaxis system with logistic growth 具有逻辑增长的趋化系统的有界性

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-25 DOI: 10.1016/j.jde.2024.09.040

In this paper, we consider a mathematical model motivated by the studies of coral broadcast spawning

{\begin{matrix} \partial_{t} n + u \cdot \nabla n - Δ n & = - χ \nabla \cdot (n \nabla c) + n - ϵ n^{q} \\ \partial_{t} c + u \cdot \nabla c - Δ c & = - c + n \end{matrix} in R^{d} \times R^{+},

where

d = 2, 3

ϵ > 0

, and

q \geq 2

. We establish global-in-time well-posedness and boundedness of the solution to the Cauchy problem of this system by developing local-in-space estimates. The crux point of our proof depends intensely on localization in the space of solutions induced by “local effect” of the

L^{\infty} (R^{d})

-norm.

在本文中，我们考虑了一个由珊瑚广播产卵研究激发的数学模型{∂tn+u⋅∇n-Δn=-χ∇⋅(n∇c)+n-ϵnq∂tc+u⋅∇c-Δc=-c+n in Rd×R+，其中 d=2,3，ϵ>0，q≥2。我们通过建立局部空间估计，建立了该系统的考希问题解的全局时间拟合性和有界性。我们证明的关键点主要取决于 L∞(Rd)-norm 的 "局部效应 "所引起的解空间的局部性。

引用次数: 0

Spectral instability of peakons for the b-family of Novikov equations 诺维科夫方程 b 族的峰子谱不稳定性

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-25 DOI: 10.1016/j.jde.2024.09.031

In this paper, we are concerned with a one-parameter family of peakon equations with cubic nonlinearity parametrized by a parameter usually denoted by the letter b. This family is called the “b-Novikov” since it reduces to the integrable Novikov equation in the case

b = 3

. By extending the corresponding linearized operator defined on functions in

H^{1} (R)

to one defined on weaker functions on

L^{2} (R)

, we prove spectral and linear instability on

L^{2} (R)

of peakons in the b-Novikov equations for any b. We also consider the stability on

H^{1} (R)

and show that the peakons are spectrally or linearly stable only in the case

b = 3

在本文中，我们关注的是具有立方非线性参数的峰值子方程的一参数族，其参数通常用字母 b 表示。这个族被称为 "b-Novikov"，因为它在 b=3 的情况下简化为可积分的 Novikov 方程。通过将定义在 H1(R) 中函数上的相应线性化算子扩展到定义在 L2(R) 上较弱函数上的算子，我们证明了 b-Novikov 方程中任何 b 的峰值子在 L2(R) 上的谱和线性不稳定性。

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引用次数: 0

An Lq(Lp)-regularity theory for parabolic equations with integro-differential operators having low intensity kernels 具有低强度核的积分微分算子抛物方程的 Lq(Lp)- 规则性理论

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-24 DOI: 10.1016/j.jde.2024.09.033

<div><div>In this article, we present the existence, uniqueness, and regularity of solutions to parabolic equations with non-local operators<math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>a</mi></mrow></msup><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn></math> in <math><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math> spaces. Our spatial operator <math><msup><mrow><mi>L</mi></mrow><mrow><mi>a</mi></mrow></msup></math> is an integro-differential operator of the form<math><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></munder><mrow><mo>(</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo><mo>−</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>∇</mi><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>⋅</mo><mi>y</mi><msub><mrow><mn>1</mn></mrow><mrow><mo>|</mo><mi>y</mi><mo>|</mo><mo>≤</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow><mi>a</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>y</mi><mo>)</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mo>|</mo><mi>y</mi><mo>|</mo><mo>)</mo><mi>d</mi><mi>y</mi><mo>.</mo></math> Here, <math><mi>a</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>y</mi><mo>)</mo></math> is a merely bounded measurable coefficient, and we employed the theory of additive process to handle it. We investigate conditions on <math><msub><mrow><mi>j</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>)</mo></math> which yield <math><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math>-regularity of solutions. Our assumptions on <math><msub><mrow><mi>j</mi></mrow><mrow><mi>d</mi></mrow></msub></math> are general so that <math><msub><mrow><mi>j</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>)</mo></math> may be comparable to <math><msup><mrow><mi>r</mi></mrow><mrow><mo>−</mo><mi>d</mi></mrow></msup><mi>ℓ</mi><mo>(</mo><msup><mrow><mi>r</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math> for a function ℓ which is slowly varying at infinity. For example, we can take <math><mi>ℓ</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>=</mo><mi>log</mi><mo>⁡</mo><mo>(</mo><mn>1</mn><mo>+</mo><msup><mrow><mi>r</mi></mrow><mrow><mi>α</mi></mrow></msup><mo>)</mo></math> or <math><mi>ℓ</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>=</mo><mi>min</mi><mo>⁡</mo><mo>{</mo><msup><mrow><mi>r</mi></mrow><mrow><mi>α<

本文提出了 Lq(Lp) 空间中带有非局部算子∂tu(t,x)=Lau(t,x)+f(t,x),t>0 的抛物方程解的存在性、唯一性和正则性。我们的空间算子 La 是一个形式为∫Rd(u(x+y)-u(x)-∇u(x)⋅y1|y|≤1)a(t,y)jd(|y|)dy 的积分微分算子。这里，a(t,y) 只是一个有界的可测系数，我们用加法过程理论来处理它。我们研究了 jd(r) 的条件，这些条件产生了 Lq(Lp) 规则性解。我们对 jd 的假设是一般性的，因此对于在无穷远处缓慢变化的函数 ℓ 而言，jd(r) 可能与 r-dℓ(r-1)相当。例如，我们可以取 ℓ(r)=log(1+rα) 或 ℓ(r)=min{rα,1} (α∈(0,2)) 。事实上，我们的结果涵盖了傅里叶乘数ψ(ξ)对|ξ|≥1不存在任何缩放条件的算子。此外，我们还给出了一些算子的例子，这些算子无法被之前考虑了ψ的平滑性或缩放条件的结果所涵盖。

引用次数: 0

Limiting behavior of invariant foliations for SPDEs in singularly perturbed spaces 奇异扰动空间中 SPDEs 不变叶形的极限行为

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-24 DOI: 10.1016/j.jde.2024.09.039

In this paper, we investigate a class of stochastic semilinear parabolic equations subjected to multiplicative noise within singularly perturbed phase spaces. We first establish the existence and smoothness of stable foliations. Then we prove that the long-term behavior of each solution is determined by a solution residing on the pseudo-unstable manifold via a leaf of the stable foliation. Finally, we present the convergence of

C^{1}

invariant foliations as the high dimensional region collapse to low dimensional region. In contrast to the convergence of pseudo-unstable manifolds, we introduce a novel technique to address challenges arising from the singularity of the stable term of hyperbolicity in the proof of convergence of stable manifolds and stable foliations as the space collapses.

在本文中，我们研究了一类在奇异扰动相空间内受乘法噪声影响的随机半线性抛物方程。我们首先确定了稳定对折的存在性和平滑性。然后，我们证明每个解的长期行为都是由驻留在伪不稳定流形上的解通过稳定叶子决定的。最后，我们提出了 C1 不变叶形在高维区域向低维区域坍缩时的收敛性。与伪不稳定流形的收敛性不同，我们引入了一种新技术，以解决在证明稳定流形和稳定叶状体在空间塌缩时的收敛性时，双曲线稳定项的奇异性所带来的挑战。

引用次数: 0

Exponential stability of a diffuse interface model of incompressible two-phase flow with phase variable dependent viscosity and vacuum 具有相变粘度和真空的不可压缩两相流扩散界面模型的指数稳定性

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-24 DOI: 10.1016/j.jde.2024.09.036

This paper is concerned with a simplified model for two-phase fluids with diffuse interface. The model couples the nonhomogeneous incompressible Navier-Stokes equations with the Allen-Cahn equation. The viscosity coefficient is allowed to depend both on the phase variable and on the density. Under some smallness assumptions on initial data, the global existence of unique strong solutions to the 3D Cauchy problem and the initial boundary value problem is established. Meanwhile, we obtain the exponential decay-in-time properties of the solutions. Here, the initial vacuum is allowed and no compatibility conditions are required for the initial data via time weighted techniques.

本文涉及一种具有扩散界面的两相流体简化模型。该模型将非均质不可压缩纳维-斯托克斯方程与艾伦-卡恩方程耦合在一起。允许粘度系数同时取决于相变量和密度。在初始数据很小的假设条件下，建立了三维 Cauchy 问题和初始边界值问题的唯一强解的全局存在性。同时，我们还得到了解的时间指数衰减特性。在这里，通过时间加权技术，允许初始真空，且不要求初始数据的相容性条件。

引用次数: 0

On the periodic solutions of switching scalar dynamical systems 论开关标量动力系统的周期解

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-24 DOI: 10.1016/j.jde.2024.09.032

In this paper, we investigate the existence and stability of periodic solutions of switching dynamical systems consisting of two sub-equations. We first establish a general criterion to determine the stability of periodic solutions; namely, we derive the conditions under which the periodic solution is locally asymptotically stable, globally asymptotically stable, or unstable. Next, we develop general theorems to count the number of periodic solutions and find the basins of attractions for the periodic solutions and the trivial solution, respectively. As applications, we analyze two biological models in recent literature. Our general theorems not only reproduce the existing results in a unified and simpler manner but also lead to new and complete dynamical results including bistability of the periodic solution and the trivial solution. Numerical examples are also given to illustrate our theoretical results.

在本文中，我们研究了由两个子方程组成的切换动力系统的周期解的存在性和稳定性。我们首先建立了确定周期解稳定性的一般标准，即推导出周期解局部渐近稳定、全局渐近稳定或不稳定的条件。接着，我们提出了计算周期解数量的一般定理，并分别找到了周期解和三解的吸引盆地。作为应用，我们分析了近期文献中的两个生物模型。我们的一般定理不仅以统一和更简单的方式重现了现有结果，而且还带来了新的和完整的动力学结果，包括周期解和三元解的双稳态性。我们还给出了数值例子来说明我们的理论结果。

引用次数: 0

Maslov-type (L,P)-index and subharmonic P-symmetric brake solutions for Hamiltonian systems 哈密尔顿系统的马斯洛夫型（L，P）指数和亚谐波 P 对称制动解

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-09-24 DOI: 10.1016/j.jde.2024.09.037

This paper introduces a novel iteration inequality for the Maslov-type

(L, P)

-index of iterated symplectic paths. Here, P is a fixed 2n-dimensional symplectic and orthogonal matrix satisfying

P^{m} = I

. These advancements in index theory are then applied to investigate the multiplicity of subharmonic solutions in Hamiltonian systems exhibiting dihedral equivariance with period mτ. Notably, a criterion of geometric distinction is established for two subharmonic P-symmetric brake orbits with periods kmτ and lmτ within the set

{k m τ | k \equiv 1 (mod m)}

. This criterion is based on a lower bound estimate for the ratio

l / k

. Specifically, for odd k, the lower bound must be not less than

(\frac{1}{2} \dim \ker (P - I) + 2) m + 1

, while for even k, it must be not less than

(\frac{1}{2} \dim \ker (P - I) + n + 2) m + 1

本文针对迭代交映路径的马斯洛夫型（L,P）指数提出了一种新的迭代不等式。这里，P 是一个固定的 2n 维交映和正交矩阵，满足 Pm=I。指数理论的这些进展随后被应用于研究哈密顿系统中的次谐波解的多重性，该系统表现出周期为 mτ 的二面等差性。值得注意的是，在{kmτ|k≡1 (mod m)}集合内，建立了周期分别为 kmτ 和 lmτ 的两个次谐波 P 对称制动轨道的几何区分标准。这一标准基于 l/k 比率的下限估计值。具体来说，对于奇数 k，下限必须不小于 (12dimker(P-I)+2)m+1，而对于偶数 k，下限必须不小于 (12dimker(P-I)+n+2)m+1。

引用次数: 0

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全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

期刊

Journal of Differential Equations

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