带非线性平流项的非局部逻辑斯蒂方程

非局部种群动力学方程 $L^2$ 分岔曲线的渐近公式

Asymptotic formulas for $L^2$ bifurcation curves of nonlocal logistic equation of population dynamics

本文研究了非局部种群动力学方程中 $L^2$ 分岔曲线的渐近行为，推导了当 $α$ 趋近于无穷大时的精确渐近公式。

A logistic equation with nonlocal interactions

We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms. More precisely, for populations that propagate according to a Lévy process and can reach resources in a neighborhood of their position, we compare (and find explicit threshold for survival) the local and nonlocal case. As ambient space, we can consider: bounded domains, periodic environments, and transition problems, where the environment consists of a block of infinitesimal diffusion and an adjacent nonlocal one. In each of these cases, we analyze the existence/nonexis…

A study of nonlocal spatially heterogeneous logistic equation with harvesting

We study a class of nonlocal reaction-diffusion equations with a harvesting term where the nonlocal operator is given by a Bernstein function of the Laplacian. In particular, it includes the fractional Laplacian, fractional relativistic operators, sum of fractional Laplacians of different order etc. We study existence, uniqueness and multiplicity results of the solutions to the steady state equation. We also consider the parabolic counterpart and establish the long time asymptotic of the solutions. Our proof techniques rely on both analytic and probabilistic arguments.

一个异质非局部对流-扩散系统

A heterogeneous nonlocal advection--diffusion system

本文研究了非局部对流-扩散方程组在异质种群上的局部和全局适定性，通过半群理论和压缩映射原理获得弱解的局部适定性及正则性，利用Nash不等式和初始数据小性条件分别建立正则核和非正则核的全局界。

关于带非线性平流项的非局部逻辑斯谛系统正解的存在性

On the existence of a positive solution to a nonlocal logistic system with nonlinear advection terms

本文研究带非线性平流项的非局部逻辑斯谛系统，通过分岔理论证明了该问题正解的存在性，并指出积分非局部项使问题更具现实代表性。

非局部平流-反应-扩散方程中的图灵分岔、临界性与多稳态性

Turing bifurcation, criticality, and multistability in a nonlocal advection-reaction-diffusion equation

本文研究带逻辑增长的非局部平流-反应-扩散方程，分析相互作用核对图灵分岔临界性与多稳态性的影响，推导分岔判据与亚临界条件，经数值分析发现稳定齐次解可与稳定非齐次解或时间周期解共存。

Well-posedness of fully nonlinear and nonlocal critical parabolic equations

In this paper we prove the existence of smooth solutions to fully nonlinear and nonlocal parabolic equations with critical index. The proof relies on the apriori Hölder estimate for advection fractional-diffusion equation established by Silvestre \cite{Si2}.

源自空间记忆与非局部平流的生物聚集

Biological Aggregations from Spatial Memory and Nonlocal Advection

该研究构建融合空间记忆与非局部探测的单物种非局部反应-扩散-平流耦合PDE-ODE模型，证明一维赫尔德连续弱解的存在唯一性，完成谱与分岔分析，并用伪谱法数值验证补充解析结论。

由非局部平流形成的生物模式的系统发育

A phylogeny of biological patterns formed by nonlocal advection

本文详细研究了带非局部平流的偏微分方程在2D域上形成的生物模式，通过系统改变参数分类了涌现模式与运动机制的联系，并展示了该框架在揭示生命科学各领域自组织机制方面的应用价值。

一类具有logistic项的局部与非局部非线性吸引-排斥趋化模型的一致时间有界性

Uniform-in-time boundedness in a class of local and nonlocal nonlinear attraction-repulsion chemotaxis models with logistics

本文研究一类带logistic项的局部与非局部非线性吸引-排斥趋化模型，通过参数条件确保解的全局有界性，并扩展了已有文献结果。

Optimisation of total population in logistic model with nonlocal dispersals and heterogeneous environments

In this paper, we investigate the issue of maximizing the total equilibrium population with respect to resources distribution m(x) and diffusion rates d under the prescribed total amount of resources in a logistic model with nonlocal dispersals. Among other things, we show that for $d\ge1$, there exist $C_0, C_1>0$, depending on the $\|m\|_{L^1}$ only, such that $$C_0\sqrt{d}<\mbox{supremum~ of~ total~ population}<C_1\sqrt{d}.$$ However, when replaced by random diffusion, a conjecture, proposed by Ni and justified in [3], indicates that in the one-dimensional case, supremum of total population…

平流环境中的非局部扩散单种群模型

A nonlocal diffusion single population model in advective environment

本文建立非局部反应-扩散-平流单种群模型，通过适定性、主特征值及稳态解分析，给出物种持续生存或灭绝的精确判据，并证明大平流速率终将导致灭绝。

带平流项的非局部算子的主特征值：精确标度极限与谱相变

Principal eigenvalues of nonlocal operators with advection: sharp scaling limits and spectral phase transitions

该研究针对带固定符号漂移的一维非局部扩散算子，通过直接方法和多种分析技术确定其主特征值的精确标度极限与谱相变，填补了非自伴情形下变分刻画缺失的研究空白。

Boundedness of large-time solutions to a chemotaxis model with nonlocal and semilinear flux

A semilinear version of parabolic-elliptic Keller-Segel system with the \emph{critical} nonlocal diffusion is considered in one space dimension. We show boundedness of weak solutions under very general conditions on our semilinearity. It can degenerate, but has to provide a stronger dissipation for large values of a solution than in the critical linear case or we need to assume certain (explicit) data smallness. Moreover, when one considers a logistic term with a parameter $r$, we obtain our results even for diffusions slightly weaker than the critical linear one and for arbitrarily large init…

具有退化各向异性扩散的非局部方程解的全局存在性

Global existence of solutions to a nonlocal equation with degenerate anisotropic diffusion

在高维无通量边界条件下，建立了具退化近视扩散与非局部粘附的非局部扩散-对流-反应方程极弱解的全局存在性，并以上盒分形维数刻画扩散张量的容许退化性。

Analysis of Anisotropic Nonlocal Diffusion Models: Well-posedness of Fractional Problems for Anomalous Transport

We analyze the well-posedness of an anisotropic, nonlocal diffusion equation. Establishing an equivalence between weighted and unweighted anisotropic nonlocal diffusion operators in the vein of unified nonlocal vector calculus, we apply our analysis to a class of fractional-order operators and present rigorous estimates for the solution of the corresponding anisotropic anomalous diffusion equation. Furthermore, we extend our analysis to the anisotropic diffusion-advection equation and prove well-posedness for fractional orders s in [0.5,1). We also present an application of the advection-diffu…

一个双物种非局部对流扩散系统的弱非线性分析

Weakly nonlinear analysis of a two-species non-local advection-diffusion system

本文对包含非局部对流项的双物种对流扩散模型进行弱非线性分岔分析，推导出耦合零模态的 Ginzburg-Landau 振幅方程，揭示了不同参数下亚临界与超临界分岔的稳定性，并发现小振幅模式不稳定时会出现大振幅模式与滞后现象。

Regularity theory for time-fractional advection-diffusion-reaction equations

We investigate the behavior of the time derivatives of the solution to a linear time-fractional, advection-diffusion-reaction equation, allowing space- and time-dependent coefficients as well as initial data that may have low regularity. Our focus is on proving estimates that are needed for the error analysis of numerical methods. The nonlocal nature of the fractional derivative creates substantial difficulties compared with the case of a classical parabolic PDE. In our analysis, we rely on novel energy methods in combination with a fractional Gronwall inequality and certain properties of frac…

On the logistic equation for the fractional p-Laplacian

We consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, with a logistic type reaction depending on a positive parameter. In the subdiffusive and equidiffusive cases, we prove existence and uniqueness of the positive solution when the parameter lies in convenient intervals. In the superdiffusive case, we establish a bifurcation result. A new strong comparison result, of independent interest, plays a crucial role in the proof of such bifurcation result.

具有保护区和非局部扩散的完全非线性Logistic方程

Fully Nonlinear Logistic Equations with Sanctuary Regions and Nonlocal Diffusion

研究带保护区和完全非线性非局部扩散的Logistic方程正解，刻画存在区间并证明唯一性，揭示下端点消失与上端点双尺度爆破现象。

长期动力学与阈值现象：具有不对称内核的自由边界SIS模型在输运周期性环境中

Long-time dynamics and threshold Phenomena for a free-boundary SIS Model with asymmetric kernels in advective periodic environments

本文研究了具有自由边界、输运和空间异质性的非局部SIS流行病模型，探讨了疾病传播与消失之间的临界阈值及长期行为。

带局部反应项的非线性非局部反应扩散问题

Nonlinear nonlocal reaction-diffusion problem with local reaction

本文研究一般度量测度空间中带局部反应项的非局部扩散问题，证明了解的全局存在性、比较原理等性质，分析了平衡解与渐近行为，并对比了其与局部扩散问题的差异。

分数 $p$-拉普拉斯的对数椭圆与抛物问题

Logistic elliptic and parabolic problem for the fractional $p$-Laplacian

本文研究了分数 $p$-拉普拉斯的对数椭圆与抛物问题，证明了弱解的存在性、唯一性以及相关行为，包括稳定、有限时间消失和爆炸。

(Non)local logistic equations with Neumann conditions

We consider here a problem of population dynamics modeled on a logistic equation with both classical and nonlocal diffusion, possibly in combination with a pollination term. The environment considered is a niche with zero-flux, according to a new type of Neumann condition. We discuss the situations that are more favorable for the survival of the species, in terms of the first positive eigenvalue. Quite surprisingly, the eigenvalue analysis for the one dimensional case is structurally different than the higher dimensional setting, and it sensibly depends on the nonlocal character of the dispers…

非局部物流与非线性生产在吸引-排斥趋化模型中的分析：全局well-posedness的分析

Nonlocal logistics and nonlinear productions in an attraction-repulsion chemotaxis model: analysis of the global well-posedness

本文研究了包含吸引-排斥效应和非局部生产项的趋化模型，分析了经典解的全局存在性和有界性，通过非局部阻尼防止时间奇异性的形成。

Detecting minimum energy states and multi-stability in nonlocal advection-diffusion models for interacting species

Deriving emergent patterns from models of biological processes is a core concern of mathematical biology. In the context of partial differential equations (PDEs), these emergent patterns sometimes appear as local minimisers of a corresponding energy functional. Here we give methods for determining the qualitative structure of local minimum energy states of a broad class of multi-species nonlocal advection-diffusion models, recently proposed for modelling the spatial structure of ecosystems. We show that when each pair of species respond to one another in a symmetric fashion (i.e. via mutual av…

关于平流非局部算子：主本征对的多重性

On advective nonlocal operators: multiplicity of principal eigenpairs

本文研究带平流的周期异质非局部扩散模型的主本征对多重性，因算子非预解紧无法用经典Krein-Rutman理论，分平流系数常号与非常号情形分析，还讨论其在非线性KPP型方程中的应用。

Existence and stability of steady state solutions of reaction-diffusion equations with nonlocal delay effect

A general reaction-diffusion equation with spatiotemporal delay and homogeneous Dirichlet boundary condition is considered. The existence and stability of positive steady state solutions are proved via studying an equivalent reaction-diffusion system without nonlocal and delay structure and applying local and global bifurcation theory. The global structure of the set of steady states is characterized according to type of nonlinearities and diffusion coefficient. Our general results are applied to diffusive logistic growth models and Nicholson's blowflies type models.

Fractional diffusion with Neumann boundary conditions: the logistic equation

Motivated by experimental studies on the anomalous diffusion of biological populations, we introduce a nonlocal differential operator which can be interpreted as the spectral square root of the Laplacian in bounded domains with Neumann homogeneous boundary conditions. Moreover, we study related linear and nonlinear problems exploiting a local realization of such operator as performed in [X. Cabre' and J. Tan. Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 2010] for Dirichlet homogeneous data. In particular we tackle a class of nonautonomous nonl…

具有非局部Logistic源和次线性生产的吸引-排斥趋化系统的全局存在性与有界性

Global existence and boundedness in an attraction-repulsion chemotaxis system with nonlocal logistic source and sublinear productions

本文研究带非局部Logistic源和次线性生产的吸引-排斥趋化系统，在参数满足适当条件时，证明其存在唯一的全局有界经典解。

Long-time behaviour of an advection-selection equation

We study the long-time behaviour of the advection-selection equation $$\partial_tn(t,x)+\nabla \cdot \left(f(x)n(t,x)\right)=\left(r(x)-ρ(t)\right)n(t,x),\quad ρ(t)=\int_{\mathbb{R}^d}{n(t,x)dx}\quad t\geq 0, \; x\in \mathbb{R}^d,$$ with an initial condition $n(0, \cdot)=n^0$. In the field of adaptive dynamics, this equation typically describes the evolution of a phenotype-structured population over time. In this case, $x\mapsto n(t,x)$ represents the density of the population characterised by a phenotypic trait $x$, the advection term `$\nabla \cdot \left(f(x)n(t,x)\right)$' a cell differenti…

一维有界域中非局部 Fokker-Planck 方程的 Keller-Segel 型逼近

Keller-Segel type approximation for nonlocal Fokker-Planck equations in one-dimensional bounded domain

本文研究一维有界域中非局部Fokker-Planck方程的Keller-Segel型逼近问题。通过奇异极限分析，证明该逼近对任意势函数均可行并确定了收敛速度，同时利用Chebyshev节点Lagrange插值多项式显式确定逼近所需的Keller-Segel系统参数。

对数扩散方程由混合分数阶算子的叠加所支配

Logistic diffusion equations governed by the superposition of operators of mixed fractional order

本文研究了由混合分数阶算子叠加驱动的对数扩散方程，在敌对环境中探讨种群生存与灭绝的条件及机制。

非局部种群模型中认知地图的持续现象与图案形成

The lingering phenomenon and pattern formation in a nonlocal population model with cognitive map

本文提出非局部种群模型，研究认知地图对种群空间分布和持续现象的影响，揭示了学习与遗忘速率对种群动态的调控作用。

Mathematical models for cell migration: a nonlocal perspective

We provide a review of recent advancements in nonlocal continuous models for migration, mainly from the perspective of its involvement in embryonal development and cancer invasion. Particular emphasis is placed on spatial nonlocality occurring in advection terms, used to characterise a cell's motility bias according to its interactions with other cellular and acellular components in its vicinity (e.g., cell-cell and cell-tissue adhesions, nonlocal chemotaxis), but we also shortly address spatially nonlocal source terms. Following a brief introduction and motivation, we give a systematic classi…

涉及对数拉普拉斯算子的非局部椭圆方程：存在性、非存在性与唯一性结果

Nonlocal elliptic equations involving logarithmic Laplacian: Existence, non-existence and uniqueness results

本文研究涉及对数拉普拉斯算子的非局部椭圆方程的存在性、非存在性与唯一性，证明Pohozaev恒等式和Díaz-Saa型不等式，并揭示加权非局部问题的极小能量解在超线性和次线性增长下分别收敛于Brézis-Nirenberg型和逻辑型极限问题的解。

具有退化迁移率的非局部Cahn-Hilliard方程：不可压缩极限与稳态收敛性

Nonlocal Cahn-Hilliard equation with degenerate mobility: Incompressible limit and convergence to stationary states

本文首次在组织生长模型中引入非局部表面张力项，研究退化非局部Cahn-Hilliard方程的不可压缩极限，证明弱解长期收敛到常数稳态，且表面张力不阻碍肿瘤完全侵袭。

随机停止下被动示踪剂弥散的非局部动力学

Nonlocal Dynamics of Passive Tracer Dispersion with Random Stopping

研究流体中随机停止下被动示踪剂弥散的非局部动力学，用带记忆项的积分偏微分方程建模，并在适当条件下证明浓度指数衰减。

具有非局部扩散的非自治 Fisher-KPP 方程的传播性质

Spreading properties for non-autonomous Fisher-KPP equations with nonlocal diffusion

本文研究了由细尾核驱动的具有非局部扩散的非自治 Fisher-KPP 方程的传播速度，针对紧支撑和指数衰减初始数据，在一般时间异质性下给出了用最小均值表示的上下界估计，并在更强假设下证明了确定传播速度，通过精细分析克服了非局部扩散解缺乏正则性的困难。

关于logistic Choquard方程的若干存在性与唯一性结果

Some existence and uniqueness results for logistic Choquard equation

该文研究分数阶$p$-Laplacian驱动的双重非局部logistic Choquard方程，在一般非线性假设下分析弱解的存在性、唯一性等性质，应用于次扩散、超扩散问题并得到特定非线性项下能量变号解的存在性。

Global stability in a nonlocal reaction-diffusion equation

We study stability of stationary solutions for a class of non-local semilinear parabolic equations. To this end, we prove the Feynman--Kac formula for a Lévy processes with time-dependent potentials and arbitrary initial condition. We propose sufficient conditions for asymptotic stability of the zero solution, and use them to the study of the spatial logistic equation arising in population ecology. For this equation, we find conditions which imply that its positive stationary solution is asymptotically stable. We consider also the case when the initial condition is given by a random field.

On the Effects of Advection and Vortex Stretching

We prove finite-time singularity formation for De Gregorio's model of the three-dimensional vorticity equation in the class of $L^p\cap C^α(\mathbb{R})$ vorticities for some $α>0$ and $p<\infty$. We also prove finite-time singularity formation from smooth initial data for the Okamoto-Sakajo-Wunsch models in a new range of parameter values. As a consequence, we have finite-time singularity for certain infinite-energy solutions of the surface quasi-geostrophic equation which are $C^α$-regular. One of the difficulties in the models we consider is that there are competing \emph{nonlocal} stabilizi…

适定主动标量方程的算子分裂

Operator splitting for well-posed active scalar equations

针对非线性平流与线性扩散/色散耦合的主动标量方程，证明了Godunov和Strang分裂方法在初始数据足够正则时具有预期收敛阶。

整数格上分数阶Fisher-KPP方程的尖锐传播与局域到非局域转变

Sharp Propagation and the Local-to-Nonlocal Transition for Fractional Fisher-KPP Equations on Integer Lattices

该研究针对整数格上的分数阶Fisher-KPP方程，分析了紧支集与类前沿数据的传播速率、局域到非局域的转变行为，通过次核界等方法完成相关分析并排除有限速度平面行波前沿。

Well-Posedness and Equilibrium Behaviour of Overdamped Dynamic Density Functional Theory

We establish the global well-posedness of overdamped dynamic density functional theory (DDFT): a nonlinear, nonlocal integro-partial differential equation used in statistical mechanical models of colloidal fluids, and other applications including nonlinear reaction-diffusion systems and opinion dynamics. With nonlinear no-flux boundary conditions, we determine the existence and uniqueness of the weak density and flux, subject to two-body hydrodynamic interactions (HI). We also show that the density is Lyapunov stable with respect to the usual (Helmholtz) free energy functional. Principally, th…

一种预条件边界值方法用于具有半拉普拉斯算子的对流-扩散方程

A preconditioned boundary value method for advection-diffusion equations with half Laplacian via spectrum doubling

本文提出一种预条件边界值方法，用于求解包含半拉普拉斯算子的对流-扩散方程，通过谱倍增方法将半扩散方程转化为一阶倍增系统，并在合适稳定不变子空间中恢复原始半扩散动力学。

基于Westervelt模型的超声增强药物递送建模

Westervelt-based modeling of ultrasound-enhanced drug delivery

本研究建立了超声增强药物递送的非线性多物理场模型，通过不动点论证和能量估计证明其适定性，并通过数值实验验证了考虑非线性效应的重要性。