Hiroshi Matano(Meiji University)

Title: Front propagation and blocking in metric graphs

Abstract: Reaction-diffusion equations on metric graphs are receiving growing attention these days. Roughly speaking, a metric graph is a directed graph whose edges have a length scale. In this lecture, I will talk about a bistable reaction-diffusion equation on a metric graph of rather an arbitrary configuration, and discuss the propagation and blocking of the solution fronts. I first focus on general principles that apply to a large class of metric graphs, then I discuss some specific examples. This is joint work with Shuichi Jimbo of Hokkaido University.

Jongmin Han(Kyung Hee University) 

Title: On the self-dual Einstein-Maxwell-Higgs equations

Abstract: In this talk, we consider the self-dual equations arising from the Einstein-Maxwell-Higgs model and discuss recent progress on the existence of solutions. Solutions on R2 are classified into three categories: topolgical, type I nontopological, and type II nontopological and some known results will be presented. Solutions on compact surfaces are either topological or nontopolocial according to the asymptotic behavior of solutions as the coupling parameter tends to zero. We present the existence of topological and nontopological solutions when the coupling parameter is small.

Norihisa Ikoma(Keio University) 

Title: Ground state solutions to a Born-Infeld type equation

Abstract: This talk is devoted to the existence of ground state solutionsto a Born-Infeld type equation. Here ground state solutions mean a solution which has the least energy among all nontrivial solutions to the equation. The Born-Infeld operator is a singular elliptic operator and the corresponding energy functional is not smooth. Thus, the arguments used for the scalar field equations (smooth case)are not applicable straightforwardly. In particular, even if there is a minimizer on the Pohozaev set, it is not immediate whether this minimizer is a solution to the equation due to the singularity of the operator (or the nonsmoothness of the functional). Here we give an idea how to prove that the minimizers satisfy the equation. This talk is based on joint work with Bartosz Bieganowski (University of Warsaw) andJarosław Mederski (Institute of Mathematics, Polish Academy of Sciences).

Futoshi Takahashi(Osaka Metropolitan University)

Title: The Hardy inequality with logarithmic weights in a critical case

Abstract: In this talk, we study the Neumann version of the weighted Hardy inequality in the critical case, which involves the logarithmic weights. We show several results concerning the positivity, attainability,and non-attainability of the best constant of the inequality, according to the range of the exponents of logarithmic weightsinvolved in the inequality. This talk is based on the recent joint work with Jaeyoung Byeon (KAIST) and Megumi Sano (Nara Women's University).

Sun-Sig Byun(Seoul National University)

Title: Density and Calderón–Zygmund estimates for matrix-weighted generalized double phase equations

Abstract: We study elliptic equations with generalized double phase growth under matrix-weighted anisotropy. The matrix weight describes the underlying geometry, while the generalized double phase structure extends the classical (p,q)-growth framework. I will present two main results: the density of smooth functions in the associated energy space, excluding the Lavrentiev phenomenon, and local Calderón–Zygmund estimates for weak solutions. The proofs combine logarithmic freezing of the matrix geometry, weighted Sobolev–Poincaré inequalities, and comparison arguments leading to autonomous Lipschitz problems.

Andrea Malchiodi(Scuola Normale Superiore) 

Title: Yamabe metrics on conical manifolds

Abstract: We prove existence of Yamabe metrics on singular manifolds with conical points and conical links of Einstein type that include orbifold structures. We deal with metrics of generic type and derive a counterpart of Aubin’s classical result. The singular nature of the metric determines a different condition on the dimension, compared to the regular case. In dimension four, we can also find solutions of min-max type in presence of Z_2-orbifold points. Some perspectives for other singular manifolds will be discussed. This is joint work with Mattia Freguglia and Francesco Malizia.

Zhi-Qiang Wang(Utah State University)

Title: Radial solutions with phase separations for $N$-systems of repulsive coupling

Abstract: We present recent work on multiple radial solutions for repulsively coupled nonlinear elliptic systems. Solutions exhibit segregation features, which provides further classification of solutions. These solutions are constructed from a variational framework using the associated parabolic flow which works for both positive and nodal solutions.

Kyeongsu Choi(KIAS)

Title: Zoo of ancient curve shortening flows

Abstract: The curve shortening flow is an evolution of curves by geometric heat equation. We call a flow is an ancient solution if it exists from negative infinite time. Thus, by the parabolic Liouville theory, ancient flows can be classified. In this talk, we address several examples of ancient curve shortening flows, and their classification results.

Luciano Mari(University of Milan) 

Title: On hypersurfaces with prescribed Lorentzian meancurvature, and the Born-Infeld model

Abstract: abstract_Luciano Mari.pdf 

Monica Musso(University of Bath) 

Title: Type II non-rotationally symmetric ancient solutions to the Yamabe flow on the sphere.

Abstract: We construct uncountably many families of non-rotationally symmetric ancient solutions to the Yamabe flow on spheres in dimensions three and higher. These are type II solutions whose Ricci curvature changes sign for negative times, and whose limiting space is a wedge of identical round spheres. This shows that positive ancient Yamabe flows are far more diverse than in related settings, such as the two-dimensional Ricci flow, where all compact ancient solutions are rotationally symmetric, and the elliptic Yamabe equation in Euclidean space, which does not admit non-radial multi-bubble solutions. The construction is based on reformulating the problem in Euclidean space and using a gluing method that combines local and global analysis. By working in suitable weighted function spaces, we obtain precise control of error terms and can verify the geometric properties of the resulting flows without relying on symmetry reductions. This is in collaboration with Haixia Chen and Seunghyeok Kim.

Younghun Hong(Chung-Ang University) 

Title: Derivation of ground states for the discrete NLS in the tight-binding limit

Abstract: We present a rigorous derivation of ground states for the discrete nonlinear Schrödinger equation from the continuous nonlinear Schrödinger equation with a deep periodic potential. In the tight-binding regime, low-energy states concentrate near the potential wells, and the leading-order interaction between neighboring wells gives rise to an effective discrete Laplacian. Using a renormalized variational formulation, we show that the continuous ground state energy converges to the discrete ground state energy. The main focus is the critical mass phenomenon for the DNLS in two and three dimensions. We show that the same critical mass determines the existence of ground states for the continuous problem in the tight-binding limit. For masses above the threshold, continuous ground states exist and converge, after a suitable discretization and up to natural symmetries, to ground states of the DNLS.

Maolin Zhou(Nankai University) 

Title: Some Results on Porous Medium Equations with Nonlinear Terms

Abstract: In this talk, we will discuss about the porous medium equation with reaction terms, especially on the convergence of solutions and the propagation of free boundaries.

Ki-ahm Lee(Seoul National University) 

Title: Capacity and Compatibility Conditions for Nonlinear Partial Differential Equations

Abstract: In this talk, we will discuss the concepts of capacity and compatibility conditions for various nonlinear partial differential equations. We will then present several applications to homogenization theory involving hard and soft inclusions, as well as boundary value problems.

Kazunaga Tanaka(Waseda University)

Title: A Lagrangian Approach for Normalized Solutions

Abstract: abstract-kt-jeju.pdf 

Manuel del Pino(University of Bath) 

Title: Concentrated vorticities in the Gross-Pitaevskii equation

Abstract: We consider the Gross-Pitaevskii equation in the entire plane

$$

iu_t = \Delta u + \varepsilon^{-2}(1-|u|^2)u

$$

In collaboration with Rowan Juneman and Monica Musso, we construct solutions with $k$ zeros of local degree $\pm1$ that asymptotically evolve according to the Helmholtz-Kirchhoff law, with highly concentrated vorticity near those points, as formally identified by Neu. In addition, we identify the next term in the vortex dynamics determined as an outer radiation term that solves a linear wave equation, as formally found by Ochnivilkov-Segal

We discuss the analogy to vortex evolution in the incompressible Euler equations and describe recent results on long-term dynamics.

Seungyeal Ha(Seoul National University) 

Title: Weak flocking of the spatially extended kinetic Cucker-Smale model

Abstract: In this talk, we discuss the emergent behaviors of the weak solutions to the kinetic Cucker-Smale (in short KCS) model in a non-compact spatial-velocity support setting. Unlike the compact support situation, non-compact support of a weak solution can cause a communication weight to have zero lower bound. This causes previous approach based on the nonlinear functional approach for spatial and velocity diameters to break down. To overcome this difficulty, we derive refined estimates on the upper bounds for the second-order spatial-velocity moments and show the uniqueness of the weak solution using the estimate on the deviation of particle trajectories. For the estimate of emergent dynamics, we consider two classes of distributions functions with decaying properties (an exponential decay or polynomial decay) in phase space, and then verify that the second moment for the velocity deviation from an average velocity tends to zero asymptotically fast, while the second moment for spatial deviation from the center of mass remains bounded uniformly in time. This illustrates the robustness of the mono-cluster flocking dynamics of the KCS model even for non-compact support settings in phase space and generalizes earlier results on flocking dynamics in a compact support setting. This is a joint work with Dr. Xinyu Wang (Seoul National University).

Angela Pistoia(Sapienza University of Rome) 

Title: Sign-changing Solutions to a Critical Hamiltonian System

Abstract: In this talk, I will discuss sign-changing solutions for critical Hamiltonian elliptic systems in the whole euclidean space, focusing on the construction of infinitely many geometrically distinct non-radial solutions. The results are obtained in collaboration with Yuxia Guo (Tsinghua University) , Seunghyeok Kim (Hanyang University) , and Shusen Yan (Central China Normal University) . The problem lies in the critical regime associated with the hyperbolic condition on the exponents, where compactness is lost and standard variational techniques are not directly applicable.

Hyeonbae Kang(Inha University) 

Title: A decomposition theorem of surface vector fields and applications to spectral theory of the Neumann-Poincare operator in elasticity

Abstract: I will talk about a decomposition theorem of surface vector fields which asserts that vector fields on the boundary of a bounded domain in three dimensions are decomposed into three: the first one extends to the inside the domain as a divergence-free and curl-free vector field, the second to the outside as a divergence-free and curl-free vector field, and the third to both the inside and the outside as a divergence-free harmonic vector field. This decomposition theorem has an excellent application to the spectral theory of the Neumann-Poincar\'e operator in elasticity. I will talk about it. This talk is based on joint papers with K. Ando, S. Fukushima, Y.-G. Ji, and Y. Miyanishi.

Ken-Ichi Nakamura(Meiji University) 

Title: On the sign of the propagation speed of bistable traveling waves in diffusive Lotka–Volterra systems

Abstract: We study the propagation speed of bistable traveling waves in the classical two-component diffusive Lotka–Volterra system under strong competition. The sign of the speed determines the long-term outcome of species competition and biological invasion. Using comparison arguments, we establish refined sufficient conditions determining the sign of the speed. Specifically, in the symmetric case where species differ only in diffusion rates, the faster diffuser prevails over a substantially broader parameter range than previously established.

Elaine Crooks(Swansea University) 

Title: Self-similar fast-reaction limits of reaction-diffusion systems with nonlinear diffusion

Abstract: This talk is concerned with the characterisation of fast-reaction limits of systems with nonlinear diffusion, when there are either two reaction-diffusion equations or one reaction-diffusion equation and one ordinary differential equation, on unbounded domains. The ideas used extend previous results in the linear diffusion case and show that in the fast-reaction limit, spatial segregation leads to the two components of the original systems each converging to the positive and negative parts of a self-similar limit profile that satisfies one of four ordinary-differential systems. The position of the free boundary separating where such self-similar profiles are positive from where they are negative provides information on the rate of penetration of one substance into the other and for specific forms of nonlinear diffusion, some results will be presented on the relationship between the form of the nonlinear diffusion and the position of this free boundary. This is joint work with Yini Du.

Beomjun Choi(KAIST)

Title: Bifurcation, Equilibria and Stability

Abstract: I will discuss two classes of nonlinear diffusion equations arising in interacting particle systems. The first part concerns bifurcation and the emergence of nonhomogeneous steady states in Keller–Segel-type models. The second part concerns the local stability of mixed Nash equilibria for mean-field Langevin descent–ascent. Despite the non-gradient structure and non-self-adjoint linearization of the latter system, exponential convergence near equilibrium can be established through Wasserstein and spectral methods. I will also briefly discuss its finite-particle approximation.

Oshita Yoshihito(Okayama University) 

Title: Segregation pattern in a four-component reaction-diffusion system with mass conservation

Abstract: We deal with a four-component reaction-diffusion system with mass conservation in a bounded domain with the Neumann boundary condition. This system serves as a model describing the segregation pattern which emerges during the maintenance phase of asymmetric cell devision. By utilizing the mass conservation, the stationary problem of the system is reduced to a two-component elliptic system with nonlocal terms,formulated as the Euler-Lagrange equation of an energy functional.

We first establish the spectral comparison theorem, relating the stability/instability of equilibrium solutions to the four-component system to that of the two-component system. This comparison follows from examining the eigenvalue problems of the linearized operators around equilibrium solutions. Subsequently, with an appropriate scaling, we prove a Γ-convergence of the energy functional. Furthermore, in a cylindrical domain, we prove the existence of equilibrium solutions with monotone profile representing a segregation pattern.

This is achieved by applying the gradient flow and the comparison principle to the reduced two-component system. This talk is based on a joint work with Yoshihisa Morita.

Vitaly Moroz(Swansea University) 

Title: Normalized solutions and limit profiles of the Gross-Pitaevskii-Poisson equation

Abstract: Gross-Pitaevskii-Poisson (GPP) equation is a nonlocal modification of the GrossPitaevskii equation with an attractive Coulomb-like term. It appears in the models of self-gravitating Bose-Einstein condensates proposed in cosmology and astrophysics to describe boson and axion stars and cold dark matter halos. We investigate the existence of prescribed mass (normalised) solutions to the GPP equation, paying special attention to the shape and asymptotic behaviour of the associated mass-energy relation curves and to the limit profiles of solutions at the endpoints of these curves. In particular, we show that after appropriate rescalings, the constructed normalized solutions converge either to a ground state of the Choquard equation, or to a compactly supported radial ground state of the integral Thomas-Fermi equation. In different regimes the constructed solutions include global minima, local but not global minima and unstable mountain-pass type solutions. This is a joint work with Riccardo Molle and Giuseppe Riey.

Injee Jeong(KIAS) 

Title: Stability of multiple Lamb dipoles

Abstract: Lamb dipole is an explicit traveling wave solution of the two-dimensional incompressible Euler equations, described by Lamb back in 1895. Due to difficulties arising from the fact that it is the dipole with "maximal mass" under enstrophy and impulse constraints, its nonlinear orbital stability was proved only in 2022 by Abe and Choi. Our main result is the nonlinear orbital stability of linear superpositions of Lamb dipoles, allowing for dipoles with mixed signs. Such a configuration is not a local extremizer of the kinetic energy, which is the main challenge in applying the variational principle to obtain stability. Furthermore, when dipoles have mixed signs, the impulse is not coercive anymore. These issues are handled by Lagrangian bootstrapping schemes, which carefully track the space-time location of various parts of the solution. This is based on joint works with Ken Abe, Kyudong Choi, Guolin Qin, and Yao Yao.

Seick Kim(Yonsei University) 

Title: A sign-changing Poisson kernel for a non-symmetric elliptic operator in a bounded domain

Abstract: We study the Dirichlet problem in the unit disk for a uniformly elliptic divergence-form operator whose skew-symmetric part has a jump discontinuity and is controlled by a real parameter $k$. Using a first-order Dirac systems method, we obtain explicit solution formulas, $L^2$ non-tangential maximal estimates, and almost-everywhere convergence to the prescribed boundary data. We show that the associated $L^2$ boundary equation undergoes a sharp transition at $|k|=1$, reflected in three natural $L^2$ Riemann--Hilbert branches: one for $|k|<1$, one for $k>1$, and one for $k<-1$. The branch for $|k|<1$ is positivity preserving, whereas the branches for $|k|>1$ give sign-changing Poisson kernels, providing a disk analogue of Axelsson's half-space example. Finally, we show that these kernels can be realized beyond the $L^2$ class for suitable data, and that their non-uniqueness is intrinsic to the Riemann--Hilbert branch structure rather than to the $L^2$ threshold $|k|=1$.