Special Session 55: Nonlocal effects in diffusion equations

Some Progress on Single Species Models with Nonlocal Dispersal Strategies in Heterogeneous Environments

Xueli Bai Northwestern Polytechnical University Peoples Rep of China Co-Author(s):    Fang Li, Jiale Shi

Abstract: In this talk, we consider a single species model with nonlocal dispersal strategy and discuss how the dispersal rate and the distribution of resources affect the total population and survival chances by summarizing some previous results and demonstrate some relevant progress. The first topic is about the monotonicity of total population upon dispersal rate. For the nonlocal model, we prove a new result, which reveals essential difference between local and nonlocal models for certain distribution of resources. Secondly, we discuss optimal spatial arrangement for survival chances and total populations. The results for both local and nonlocal models demonstrate that the concentration of resources is beneficial for species.

Nonlocal Effects in Deterministic and Stochastic Pseudo-Parabolic Equations

Yang Cao Dalian University of Technology Peoples Rep of China Co-Author(s):    Haomeng Chen, Chunhua Jin, Jingxue Yin

Abstract: This report examines nonlocal effects in pseudo-parabolic equations across deterministic and stochastic settings. In the deterministic case, When $\tau/D\leq1$, the pseudo-parabolic term acts as a structure-preserving regularization, and the traveling waves retain the monotonicity of classical Fisher-KPP fronts; when $\tau/D>1$, it actively induces oscillations in the wave profile near the equilibrium $u=1$. This predicts the saturation overshoot observed in porous media but absent in classical diffusion. In the stochastic counterpart, the nonlocal effect results in a bounded operator spectrum, which restricts the dissipation rate of high-frequency modes to a finite level and forces trajectory estimators to depend on long-time observations, while sustaining noise responses that make MLE asymptotic variance dimension-independent. Both manifestations reveal how nonlocal structure fundamentally reshapes wave dynamics and statistical estimability.

A study on the regularity of mixed local-nonlocal problems with measure data

Mengyao Ding Harbin institute of Technology Peoples Rep of China Co-Author(s):    YingLi, ChaoZhang

Abstract: We investigate the existence, uniqueness, and regularity of solutions in the framework of fractional Sobolev spaces for mixed local-nonlocal problems with measure data. By combining techniques from the theory of linear elliptic equations with nonlocal operators and classical potential theory, we derive some regularity estimates for the equation.

Self-similar Singular Solutions for the Critical Keller-Segel Model with $p$-Laplacian Diffusion

Chunhua Jin South China Normal University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we focus on the critical dynamics of the Keller-Segel model with $p$-Laplacian diffusion. We first establish a sharp threshold that distinguishes globally bounded solutions from finite-time blow-up. In the critical case, we rigorously construct backward self-similar blow-up solutions with compact support and radial monotonicity. These solutions exhibit concentration into a Dirac $\delta$-singularity at the blow-up time as their support shrinks to the origin. Additionally, we explore forward self-similar singular solutions, further characterizing their initial singularity.

Existence and nonexistence of stable patterns in semilinear nonlocal diffusion equations

Fang Li Sun Yat-sen University Peoples Rep of China Co-Author(s):    Xueli Bai, Xuefeng Wang

Abstract: In this talk, we consider the dynamics of semilinear nonlocal diffusion equations on bounded domains with no-flux boundary conditions, specifically focusing on the existence and stability of non-constant steady states, referred to as patterns. According to the results of Casten, Holland, and Matano regarding semilinear local diffusion equations, we know that stable patterns do not exist in convex domains, while they do emerge in dumbbell-shaped geometries, particularly when the kinetic term is bistable. We extend these findings to nonlocal diffusion analogs, demonstrating the absence of stable smooth patterns in both one-dimensional intervals and multi-dimensional balls. In addition, we construct discontinuous, asymptotically stable patterns when the kinetic term is bistable. Our results reveal a significant principle: large nonlocal diffusion tends to destabilize patterns, whereas weak nonlocal diffusion stabilizes them, especially in cases with bistable kinetic terms. Importantly, the geometry of the domain appears to play a less critical role in this process of stabilization.

Cauchy problem of the parabolic-elliptic density suppressed motility model with logistic source

Jing Li Minzu University of China Peoples Rep of China Co-Author(s):    Wang Zhi-an, Xu Wenbing

Abstract: To understand the ``self-trapping`` mechanism inducing spatio-temporal pattern formations observed in the experiment of for bacterial motion, the density-suppressed motility model was proposed. The purpose of this paper is to consider the Cauchy problem with the motility function $\gamma(v)=\frac{1}{(1+v)^m}$, we obtain that the condition $0

On length-preserving and area-preserving anisotropic curvature flow of convex closed plane curves

Lin Liu Southeast University Peoples Rep of China Co-Author(s):    Lin Liu, Dong-Ho Tsai, Xiao-Liu Wang

Abstract: In this talk, we introduce the evolution problem of area-preserving and length-preserving flows with the positive power and negative power of anisotropic curvature. For the case of the positive power anisotropic curvature flows, by utilizing Tso$^{\prime}$s method, we establish time-independent upper bound estimate for curvature and then derive estimates for the derivatives of curvature. Both flows are shown to exist globally and converge to the boundary of the homothety of Wulff shapes. For the case of negative power, we apply Tso$^{\prime}$s method to establish the time-independent lower bound estimate of curvature for the area-preserving curvature flow. And, we apply Bernstein$^{\prime}$s estimate or Moser iteration method to establish the lower bound of curvature for the length-preserving curvature flow. Under the assumption of global existence, it is shown that the flows converge to the boundary of the homothety of Wulff shapes.

On principal eigenvalues for elliptic operators with divergence-free flow

Shuang Liu Beijing Institute of Technology Peoples Rep of China Co-Author(s):    Yuan Lou

Abstract: In this talk, we will discuss some progress on principal eigenvalues of second order elliptic operators with the divergence-free flow. Some monotonicity and asymptotic behaviors of principal eigenvalues, with respect to diffusion rate and flow amplitude, are established. These local asymptotic analysis can help us find some global information on the principal eigenvalue, which enables us to better understand qualitative properties of the principal eigenvalues. This is a joint work with Professor Yuan Lou.

Precise propagation dynamics of the nonlocal KPP free boundary problem with non-symmetric kernels

wenjie ni University of New England Australia Co-Author(s):

Abstract: This talk presents our recent joint work with Professor Yihong Du and Professor Xiangdong Fang on the asymptotic limit of the principal eigenvalue of asymmetric nonlocal diffusion operators and its implications for propagation dynamics in Fisher KPP type equations. We first establish an explicit formula for the asymptotic limit of the principal eigenvalue of a nonlocal diffusion operator with drift, revealing how the asymmetry of the dispersal kernel affects the spectral structure. Building on this result, we analyze the long-term behavior of nonlocal KPP equations using a new eigenvalue-based approach that avoids the reliance on traveling waves.

Multiscale Nonlocal Modeling of Phenotypic Plasticity and Drug Resistance in Glioblastoma

Xiaoqiang Sun Sun Yat-sen University Peoples Rep of China Co-Author(s):    Ming Yan, Xiaoqiang Sun

Abstract: Glioblastoma (GBM) progression is driven by the dynamic interplay between tumor cells, which transition between proneural (PN) and mesenchymal (MES) states, and tumor-associated macrophages (TAMs), which shift between M1 and M2 phenotypes. In this study, we developed a multiscale spatiotemporal model based on nonlocal reaction-diffusion equations to investigate the intrinsic mechanisms of GBM phenotypic transitions during both natural progression and drug treatment. The model integrates continuous tumor and TAM phenotypic shifts and their interplay via microenvironment-mediated signaling feedback loops. Through rigorous mathematical analysis, we established the well-posedness of the model and constructed an efficient numerical scheme for its solution. The model demonstrates excellent agreement with experimental observations across multiple validation metrics, confirming its biological plausibility and predictive capability. Using global sensitivity analysis and parameter stability analysis, we systematically identified and quantitatively characterized key parameters regulating tumor growth dynamics and treatment response. Finally, we evaluated the efficacy of combination therapy regimens and proposed strategic approaches for optimizing GBM treatment. Our study establishes a novel nonlocal modeling framework for investigating phenotypic plasticity in tumor and immune cells and their crosstalk, providing valuable insights for optimizing combination therapies in cancer treatment.

Principal Eigenvalue of Second Order Elliptic Operators

Xin Xu South China Normal University Peoples Rep of China Co-Author(s):    Xueli Bai, Rui Peng, Zhi-an Wang, Kexin Zhang, Maolin Zhou

Abstract: In this talk, we would like to report our recent work on the the convergence of the principal eigenvalue of the elliptic operator \begin{equation*} -d\Delta \varphi(x)-2 s v(x)\cdot \nabla \varphi(x)+c(x)\varphi(x)=\lambda(d,s)\varphi(x) \end{equation*} Firstly, we constructed an infinitely oscillating gradient advection term such that the principal eigenvalue does not converge as $s\to+\infty$. Secondly, we considered the principal eigenvalue of the elliptic operator with a constant advection term and the boundary conditions $\varphi'(0)=s(1+b_0)\varphi(0)$ and $\varphi'(1)=s(1-b_1)\varphi(1)$ in one dimension. Then we determined the limits of these principal eigenvalues as $s$ tends to infinity for all parameters $b_0,b_1\in(-\infty,+\infty)$. Recently, we have further investigated the asymptotic eigenvalue problem on manifolds and obtained some results.

A Liouville-Type Theorem for the Weighted Higher-Order Elliptic System with Navier Boundary Conditions

Weiwei Zhao Hainan University Germany Co-Author(s):    Xiaoling Shao, Changhui Hu, Zhiyu Cheng

Abstract: In this talk, we present a Liouville-type theorem for a weighted higher-order elliptic system, which holds in a wider range of exponents characterized by a critical curve. Our approach consists of two main steps. First, we establish a corresponding Liouville-type result for the associated integral system. Then, using superharmonic properties of the differential system, we show the equivalence between the elliptic system and the integral system. This extends and improves existing results in the literature.

Spreading speeds of nonlocal Fisher-KPP equations in heterogeneous media

Tao Zhou Anhui University Peoples Rep of China Co-Author(s):

Abstract: Spreading speeds of nonlocal Fisher-KPP equations in heterogeneous media will be presented, including both temporally and spatially heterogeneous cases. The characterization of the speeds relies on the generalized principal eigenvalue and time averaging, respectively. In addition, some interesting examples will be introduced.