Yoshinori Furuto (Tohoku University)   

Abstract: "We consider the parabolic Lam\'{e} system on a bounded domain. We focus on two types of inequalities for higher-order derivatives of solutions. The first concerns local-in-time $L^p$-$L^p$ estimates in the Lebesgue space setting. The second provides space-time $L_t^q L_x^p$ estimates in terms of the Besov norm of the initial data. Both results include the endpoint cases $p=1$ and $p=\infty$."

Fuya Hiroi (University of Tokyo)   

Abstract:We consider the curve diffusion flow for open curves whose endpoints lie on skew lines. In the study of moving boundary problems for geometric gradient flows, it is common to assume that the curves meet the boundary at right angles in order to simplify the analysis. In particular, previous results proving convergence of solutions imposed the right-angle boundary condition. The aim of this poster is to generalize the right-angle condition and establish convergence of the curve diffusion flow under the general contact-angle condition.

Kotaro Horimoto (Tohoku University)   

Abstract: This poster is concerned with the Riemann problem for the two-dimensional barotropic compressible Euler equations. The Riemann data considered here are such that the corresponding one-dimensional self-similar solution consists solely of a contact discontinuity. By means of convex integration, it is shown that the same initial data admit infinitely many admissible weak solutions. This extends the non-uniqueness result of Krupa and Sz\'ekelyhidi (2025), which was proved for a pressure law whose explicit form was not specified, to general strictly increasing pressure laws.

Yoshiki Kaiho (Tohoku University)   

Abstract: We discuss a higher integrability result for very weak solutions, which have lower integrability than standard weak solutions, of elliptic systems with nonstandard growth. This result generalizes the recent work by Baasandorj, Byun and Kim (Trans. Amer. Math. Soc., 2023) with respect to the derivative order and related structural conditions. The key ingredients are a higher-order Lipschitz truncation adapted to the double phase structure and Sobolev--Poincar\'e inequalities associated with the double phase operator.

Rikuya Kakinuma (Tohoku University)   

Abstract: This talk is concerned with nonlinear scalar field equations with power-type absorption terms under $L^2$ constraints. The main result establishes the existence of nontrivial solutions to the $L^2$-constrained problem under certain assumptions, which are closer to the so-called Berestycki--Lions condition than those in previous works, and in some aspects may even go beyond it.

Yuki Kaneko (Kanto Gakuin University)   

Abstract: We discuss a free boundary problem of a multi-stable reaction-diffusion equation. The solution can converge, in a whole space, to a leftward and/or rightward propagating terrace when the free boundaries go to infinity and Stefan coefficients are large enough. We also give sufficient conditions and profiles of the terraces.

Kenta Kumagai (University of Tokyo)   

Abstract: The bifurcation structure of a supercritial elliptic equation is completely clarified by Joseph and Lundgren (1972/73) for power nonlinearities and the exponential nonlinearity in a ball. For theses nonlinearities, the bifurcation curve has infinitely many turning points when the growth rate of nonlinearity is smaller than Joseph-Lundgren critical, while the bifurcation curve has no turning point when the growth rate is greater than or equal to Joseph-Lundgren critical.
In this poster, we give a new type of nonlinearity such that the growth rate is greater than or equal to Joseph-Lundgren critical, while the bifurcation curve has infinitely many turning points. This result shows that the bifurcation structure is not determined solely by the growth rate of the nonlinearity. In fact, we find a general criterion which determines the bifurcation structure.

Ryunosuke Kusaba (Tohoku University)   

Abstract: "In this poster, we investigate the optimal decay rate of global solutions to the convection-diffusion equation with zero-mass initial data. We derive an improved decay estimate of the global solutions for small zero-mass initial data. Moreover, we establish the first-order asymptotic expansion, which provides necessary and sufficient conditions for attaining the improved decay rate. This result shows that the decay rate is optimal. This poster is based on joint work with Professor Yi C. Huang (Nanjing Normal University)."

Kyogo Murai (Tohoku University)   

Abstract: In this poster, we prove the global existence of weak solutions to quasilinear Keller--Segel systems with nonlinear mobility by minimizing movements (JKO scheme) in the product space of the weighted Wasserstein space and L2 space. In particular, we newly show the global existence of weak solutions to the Keller--Segel system with the degenerate diffusion in the critical case. The advantage of our approach is that we can connect the global existence of weak solutions to the Keller--Segel systems with the boundedness from below of a suitable functional. While minimizing movements for Keller--Segel systems with linear mobility are adapted in the product space of the Wasserstein space and L2 space, due to the nonlinearity of mobility, we need to use the weighted Wasserstein space instead of the Wasserstein space. Moreover, since the mobility function is not Lipschitz, we first find solutions to the Keller--Segel systems whose mobility is approximated by a Lipschitz function, and then we establish additional uniform estimates and convergences to derive solutions to the Keller--Segel systems.

Shozo Ogino (Tohoku University)   

Abstract: We study the quasi-neutral limit of the compressible Navier--Stokes--Poisson equations with initial data near a constant equilibrium state in the whole space. We deal with the two physical parameters which come from the effect of pressure and electric potential. We show that the compressible Navier--Stokes--Poisson equations converge to the incompressible flow strongly in the critical Besov spaces. It turned out that whether it oscillates or converges vary depending on the regularity imposed on the functional spaces.

Yusuke Oka (Tohoku University)   

Abstract: We study the Cauchy problem for the fractional semilinear heat equation with distributional inhomogeneous terms. By introducing the Lorentz--Morrey spaces, we overcome limitations of real interpolation in the classical local Morrey spaces and obtain a sharp integral estimate for the nonlinear term. Moreover, in terms of Besov-type spaces, we give necessary conditions and sufficient conditions on inhomogeneous terms for the local-in-time existence of solutions belonging to Lorentz--Morrey spaces.

Taiki Okazaki (Tohoku University)   

Abstract: "We consider the uniqueness of the solution of the dissipative surface quasi-geostrophic equation, without assuming time-continuity and smallness of the solutions. We show that the uniqueness holds in the scale-critical Lebesgue spaces. The proof is based on the energy method, inspired by the approach introduced by Lions and Masmoudi (2001) in the study of uniqueness for the Navier-Stokes equations. A key ingredient of the argument is the justification of the energy inequality via the smoothing effect of the fractional heat semigroup together with an iteration scheme based on the structure of the integral equation. "

Masahiro Sakoda (Tohoku University)   

Abstract: This poster discusses planar closed elastic curves with density-modulated stiffness, inspired by experiments on biological membranes. In 2023, Brazda et al. characterized the heterogeneous elastic curves as the critical points of a geometric functional, defined as the sum of a generalized bending energy with density-modulated stiffness and a Dirichlet energy for the density, under constraints on the total length and the total mass. Since any elastica with constant density is a trivial critical point in this model, the functional can be regarded as a generalization of the classical bending energy. On the other hand, if the coefficient in the generalized elastic energy is non-smooth, the existence of heterogeneous elastic curves with non-classical shapes can be anticipated. The purpose of this study is to prove (i) the existence of infinitely many heterogeneous closed elastic curves with non-classical shapes, and (ii) the existence of weak gradient trajectories connecting the classical elastica to the heterogeneous elastic curves obtained in (i). This work is based on a joint work with Professor Shinya Okabe of Tohoku University.

Yuina Sato (Tohoku University)   

Abstract: We study viscosity solutions to a singular or degenerate parabolic equation involving the normalized p-Laplacian. This equation generalizes both the standard parabolic p-Laplace equation and non-divergence-form equations involving the normalized p-Laplacian. We investigate the asymptotic behavior of viscosity solutions both as t→∞ and as p→∞. For the Cauchy problem, we derive decay estimates using backward self-similar solutions. We also discuss Lyapunov functions for the limiting equation as p→∞.

Kosuke Shibuya (Tohoku University)   

Abstract: "In my poster, the initial value problem of the Burgers type equation with the half Laplacian is concerned. This problem is critical from the viewpoint of well-posedness and has been extensively studied. On the other hand, it is subcritical in terms of the large-time behavior of solutions. Previous studies have shown that the solution behaves like the Poisson kernel time goes to infinity. However, since the derivative order of the equation are balanced, the equation has a first-order hyperbolic structure as well as a parabolic structure and is expected to have a dissipative structure. The main result clarifies that the dissipative wave emerges in the asymptotic expansion and the dissipative wave is explicitly described, and that this nonlinear profile is uniquely determined by the scaling. In addition, the main result also reveal that the logarithmic shift term appears in the expansion, and that the decay rate is optimal. This poster is based on the joint work with Professor Masakazu Yamamoto (Gunma university). "

Yuri Soga (Tohoku University)   

Abstract: We consider a chemotaxis system in the whole space with emphasis on supercritical dimensions. We establish a sharp threshold phenomenon separating global-in- time existence from finite time blow-up in terms of scaling-critical Morrey norms of the initial data. In particular, we prove the existence of singular stationary solutions and show that their Morrey norm values serve as the critical thresholds determining the long-time behavior of solutions.

Ryoko Tomiyasu (Kyushu University)   

Abstract: "Packing problems arise in various contexts ranging from number theory to biological pattern formation. Phyllotactic patterns generated by the golden-angle method exhibit quasi-periodic structures with high packing efficiency despite its simple generative rules. In [1], we extended the classical framework beyond rotationally symmetric surfaces to general n-dimensional manifolds endowed with n orthogonal vector fields, and obtained a new method for generating uniform point configurations on the manifolds together with closed-form lower bounds for locally defined packing density. This framework provides a unified treatment of periodic and non-periodic packings and yields new applications of algebraic lattice theory to uniform point sets and mesh generation algorithms. [1] S. E. Graiff Zurita, R. Oishi-Tomiyasu, Packing theory derived from phyllotaxis and products of linear forms, Constructive Approximation 60 (2024), 515–545. "

Akira Toyoshima (Tohoku University)   

Abstract: In our research, we focus on the separation property of radial solutions to Hénon type equation on the hyperbolic space. In previous research, Hasegawa (2017) showed the existence of a new critical exponent, which does not appear when considering the same equation in the Euclidean space. In our poster, we introduce the results of separation property in the supercritical case, and the character of a key transformation used in the proof. This is a joint work with Professor Norisuke Ioku (Tohoku University).

Hikaru Yamaguchi (Tohoku University)   

Abstract: The ideal functional, which is defined by the Dirichlet energy of curvature, is related to the well-known Euler spiral. Indeed, the Euler spiral can be regarded as a critical point of the ideal functional with length penalisation. Very recently, Okabe and Wheeler proved the following: (i) the length-penalised $L^2$-gradient flow for the ideal functional blows up in finite time, provided the initial curve has a non-zero rotation number and is sufficiently small in energy; and (ii) the length-penalised ideal functional has infinitely many critical points with rotation number of zero. However, the stability of the critical points with zero rotation number and the existence of critical points with a non-zero rotation number remain open problems. As a tool for dynamical approach to these open problems, we consider the length-penalised $H^3$-gradient flow for the ideal functional. In this poster, we will present a partial result towards our aim of investigating the asymptotic behaviour of the length-penalised $H^3$-gradient flow for the ideal functional. More precisely, the main result of this poster is the global-in-time solvability of the $H^3$-gradient flow for any initial closed curves in $H^3$. This presentation is based on joint work with Professor Shinya Okabe of Tohoku University.

Erbol Zhanpeisov (Tohoku University)   

Abstract:   We study the blow-up rate for solutions of the subcritical semilinear heat equation. We prove type I estimates for sign-changing solutions in possibly non-convex domains, extending previous results that required convexity or positivity assumptions. The proof uses the Giga-Kohn energy together with a geometric inequality controlling the effect of non-convexity. This is based on joint work with Hideyuki Miura and Jin Takahashi.

9:30 -- 10:00   Finn Münnich (Heidelberg University)

Abstract: Reaction-diffusion-ODE systems have emerged as powerful models for biological pattern formation, capturing the interplay between diffusive and non-diffusive nonlinear processes. These systems exhibit a rich variety of spatial structures, including classical Turing patterns and far-from-equilibrium patterns exhibiting jumps. Previous studies have primarily focused on diffusion-driven instability (DDI) generated by instability of the purely non-diffusive subsystem, which destabilizes classical Turing patterns. In contrast, we show that DDI can also arise from subsystems that couple non-diffusive and slowly diffusing components, leading to dynamics involving three distinct spatial scales. Moreover, we prove the existence of far-from-equilibrium patterns in a general reaction-diffusion-ODE framework. As illustrative examples, we apply our results to specific models, demonstrating the broad applicability of the approach.

10:10 -- 11:00   Izumi Takagi (Tohoku University)

Looking back on half a century of the research on pattern formation in reaction-diffusion systems.

Abstract: It has been more than fifty years since the activator-inhibitor model was proposed by Gierer and Meinhardt, which stimulated mathematical studies on qualitative properties of solutions to nonlinear partial differential equations. In this talk I would like to sketch the (very personalized) history of studies on reaction-diffusion equations and related fields based on what I witnessed.

11:15 -- 12:30   Group Discussions

12:35   Closing

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