Special Session 119: PDEs and Variational Problems in Physical and Biological Sciences

Stable spikes for some reaction diffusion systems modeling color pattern formation and consumer chain

Weiwei Ao Wuhan University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we will consider some reaction diffusion system with three components: one is a system for color pattern formation with two activators and one inhibitor, the other is consumer chain model which contains one consumer feeding on the producer and a second consumer feeding on the first consumer. We prove rigorous results on the existence and stability of some spikes solutions for these two models.

Diameter estimate of solutions to the even $L_p$ Aleksandrov problem for $-1

Yibin Feng Lanzhou University Peoples Rep of China Co-Author(s):    Shengnan Hu; Yuanyuan Li; Honglin Lv

Abstract: Our recent work on diameter estimate of solutions to the even $L_p$ Aleksandrov problem for $-1

Critical Elliptic Boundary Value Problems with Singular Trudinger-Moser Nonlinearities

Shiqiu Fu Jiangsu University Peoples Rep of China Co-Author(s):    Kanishka Perera

Abstract: In this talk, we give the existence results of solutions for two classes of eliptic problems that are critical with respect to singular Trudinger-Moser embedding. The proofs are based on compactness and regularity arguments.

Existence of normalized solutions for nonlinear Choquard equations

Qiuping Geng Jiangsu University Peoples Rep of China Co-Author(s):    Jun Wang

Abstract: We study, using variational methods, the existence of normalized solutions to nonlinear Choquard equations under some parameter conditions.

Uniqueness and multiplicity for bistable equations in unbounded domains

Cle Graham University of Wisconsin-Madison USA Co-Author(s):    Henri Berestycki, Juncheng Wei.

Abstract: In this talk, we explore the influence of geometry on bistable elliptic equations with Dirichlet boundary in unbounded domains. We present a surprising dichotomy between epigraphs that are bounded from below and domains containing a cone of aperture greater than $\pi$. The former admit exactly one positive bounded solution, while the latter support infinitely many. In the first case, we strengthen the method of moving planes to treat noncompact sublevel sets. In the second, we exploit a connection with Delaunay surfaces in differential geometry.

On the bifurcation diagram for free boundary problems arising in plasma physics

Aleks Jevnikar University of Udine Italy Co-Author(s):    D. Bartolucci, J. Wei, R. Wu

Abstract: We are concerned with qualitative properties of the bifurcation diagram of a free boundary problem arising in plasma physics, showing in particular uniqueness and monotonicity of its solutions. We then discuss a new approach to study the Rabinowitz continuum of classical Gelfand problems.

Maximizing Reaction via Spatial Localization

Bo Li University of California San Diego USA Co-Author(s):    Yasir Khan

Abstract: We study a PDE-constrained variational problem modeling spatial distributions of enzymes that maximize their reaction with substrates in biological cells. With a given enzyme concentration, the substrate concentration is determined by a reaction-diffusion equation. The reaction functional of enzyme concentrations is defined to be the total amount of reaction flux that is a nonlinear form of product of enzyme and substrate concentrations. We construct reaction-maximizing sequences through localization of enzymes, calculate the first and second variations of the reaction functional, and show the nonexistence of local and global maximizers. We further propose regularized reaction functionals and study their variational properties. Applications to biological cells are discussed. This is joint work with Yasir Khan.

Game-Theoretic Modeling of Adaptive Behavioral Responses in Epidemics

Tangjuan Li Jiangsu University Peoples Rep of China Co-Author(s):    Yanni Xiao, Jane M Heffernan

Abstract: \begin{abstract} During disease outbreaks, individuals adopt preventive measures based on perceived infection risk and the associated costs. These behavioral responses, in turn, influence disease transmission dynamics. In this talk, we propose a co-evolutionary model that integrates adaptive behavioral changes into a classical epidemic framework to investigate their interplay. We show that such adaptive behavior can give rise to complex dynamical phenomena, including Hopf bifurcations, sustained periodic oscillations, and even some counterintuitive outcomes. Furthermore, by incorporating real-world behavioral data, we validate the proposed model and demonstrate its ability to capture key features of observed epidemic patterns. \end{abstract}

Regularity from $p$-harmonic potentials to $\infty$-harmonic potentials in convex rings

Fa Peng School of Mathematical Sciences, Beihang University Peoples Rep of China Co-Author(s):

Abstract: The investigation of problems in shape metamorphism, surface reconstruction, and image interpolation raises fundamental questions concerning the higher-order regularity of $\infty$-potentials(a class of $\infty$-harmonic functions) and their approximation by $p$-harmonic potentials. In this talk, we establish interior $C^1$ and Sobolev regularity for $\infty$-harmonic potentials in arbitrary convex ring domains, contributing to the core theory of $\infty$-Laplace equations and $L^\infty$-variational problems. This is joint work with Prof. Yi Ru-Ya Zhang and Yuan Zhou.

A Priori Estimates for Fully Nonlinear Equations

Guohuan Qiu Academy of Mathematics and Systems Science Peoples Rep of China Co-Author(s):

Abstract: This presentation introduces some progress in the interior regularity theory of fully nonlinear equations. For the equation of positive scalar curvature (also known as the 2-Hessian equation), we completely resolve the interior second-order derivative estimates in the three-dimensional case and for convex solutions. In higher dimensions, for the special Lagrangian curvature equation, we establish a priori interior curvature estimates in both the critical phase case and the convex case.

Analysis of a parabolic type system with singular initial data: existence, multiplicity and stability

Jun Wang Jiangsu University Peoples Rep of China Co-Author(s):    Qiuping Geng and Xuan Wang

Abstract: In this talk, we present results on the existence and multiplicity of positive solutions and examine the stability of a parabolic Lane-Emden type system, with particular emphasis on cases involving singular initial data. On the other hand, we also investigates the asymptotic behavior, blow-up phenomena, and decay rates for the Henon-Hardy Lane-Emden type system.

Sign-changing solutions to discrete nonlinear logarithmic Kirchhoff equations

Lidan Wang Jiangsu University Peoples Rep of China Co-Author(s):

Abstract: We study the discrete logarithmic Kirchhoff equation on$\mathbb{Z}^3$: $$ -\left(a + b \sum_{x \in \mathbb{Z}^3} |\nabla u(x)|^2\right) \Delta u(x) + (\lambda + 1)u(x) = |u|^{p-2}u\log|u|^2, $$ where$a,b>0$,$p>6$, and$\lambda>0$. By using the Nehari manifold method, we prove the existence of least energy sign-changing solutions under suitable$\lambda$ assumptions. We also analyze their asymptotic behavior as$\lambda\to\infty$ by introducing a limiting equation on bounded domains. Our results extend to$n$-dimensional lattice graphs$\mathbb{Z}^n$, with clear physical motivation from wave propagation and elastic vibration problems.

Global convergence of Gursky-Malchiodi flow

Juncheng Wei Chinese University of Hong Kong Hong Kong Co-Author(s):    Liuwei Gong, Sanghoon Lee

Abstract: In their seminal work [JEMS 2015], Gursky and Malchiodi introduced a non-local conformal flow in dimensions $n \geq 5$ to resolve the constant $Q$-curvature problem. They proved {\bf sequential convergence} of the flow for initial metrics with positive scalar curvature and $Q$-curvature, provided the initial energy is sufficiently {\bf small}. The question of global convergence for large initial energy has remained open. In this talk, we resolve this problem by proving {\bf global convergence} of the flow for {\bf arbitrary} initial energy under the same positivity assumptions. Our approach centers on establishing a non-local version of the \L{}ojasiewicz-Simon inequality for the Paneitz-Sobolev quotient along the flow. We construct test bubbles and estimate their Paneitz-Sobolev quotients, a strategy that was carried out in the celebrated work of Brendle (Invent 2006) in the context of the Yamabe flow. We develop a more geometric and systematic proof that addresses the algebraic and computational complexity inherent in the $Q$-curvature and the Paneitz operator. Along the way, we derive a stability inequality for the Paneitz-Sobolev quotient using a higher-order Koiso-Bochner formula established in recent work of Bahuaud, Guenther, Isenberg, and Mazzeo (2025).

Spatial Pattern Formation and Spike Dynamics in a Three-Component Ecosystem Model

Shuangquan Xie Hunan University Peoples Rep of China Co-Author(s):

Abstract: We introduce a three-component extension of the classical Klausmeire-Gray-Scott model, termed the WPAE model, to explore how water, vegetation, and allelopathic inhibition interact to generate complex spatial patterns in ecosystems. Departing from the classical two-component Klausmeier model, where limited resources tend to induce vegetation aggregation, our modified model demonstrates that allelopathic inhibition serves as a critical constraint, localizing vegetation populations specifically in water-abundant regimes. Moreover, we find that elevating the reaction rate of the allelopathic inhibitor leads to the emergence of robust traveling multi-spike solutions. Through systematic analysis, we observe a spectrum of complex spike interactions, including merging, repulsive bouncing, and more sophisticated dynamics involving splitting and competition. This work aims to unravel the underlying mechanisms driving these phenomena. Our approach involves first constructing a single stationary spike solution using matched asymptotic expansions and examining its spectral stability. We further derive explicit traveling spike solutions and obtain governing equations for their propagation speed. The validity of our theoretical results is confirmed via numerical simulations, which show strong consistency with our analytical findings.