Program

Program book

• TBA

Title & Abstract

• Byun, Sun-Sig (Seoul National University)
  
  Gradient Estimates for Mixed Local–Nonlocal Elliptic Equations
  We present Calderón–Zygmund estimates for nonlinear mixed local–nonlocal elliptic equations with possibly degenerate or unbounded matrix weights. The estimates hold on intrinsic scales with constants independent of the weight’s spectral size. The proof combines weighted fractional estimates with comparison to homogeneous local equations under small oscillation assumptions.
  
  

• Chen, Shibing (University of Science and Technology of China)
  
  TBA
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• Choi, Kyeongsu (Korea Institute for Advanced Study)
  
  TBA
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• Guo, Hongjun (Tongji University)
  
  Reaction-diffusion equations in periodic media: convergence to pulsating fronts
  This talk is concerned with reaction-diffusion-advection equations in spatially periodic media. Under an assumption of weak stability of the constant states 0 and 1, and of existence of pulsating traveling fronts connecting them, we show that fronts’ profiles appear, along sequences of times and points, in the large-time dynamics of the solutions of the Cauchy problem, whether their initial supports are bounded or unbounded. The types of equations that fit into our assumptions are the combustion and the bistable ones. We also show a generalized Freidlin–Gärtner formula and other geometrical properties of the asymptotic invasion shapes, or spreading sets, of invading solutions, and we relate these sets to the upper level sets of the solutions. This talk is based on the joint work with F. Hamel and L. Rossi.
  
  

• Hisa, Kotaro (Fukuoka University)
  
  Infinite-time blow-up and slow decay for the six-dimensional energy-critical heat equation
  In this talk, we consider the six dimensional energy-critical semilinear heat equation with self-similarly decaying initial data. Our main result shows the existence of sign-changing solutions that exhibit infinite-time blow-up and nonnegative solutions that decay strictly more slowly than the self-similar rate. Moreover, the blow-up and decay rates are not uniquely determined by the decay rate of the initial data, but exhibit a certain flexibility depending on the construction. The proof is based on gluing suitably rescaled bubbles to forward self-similar solutions. This talk is based on joint work with J. Takahashi (Science Tokyo) and E. Zhanpeisov (Tohoku Univ.).
  
  

• Ishige, Kazuhiro (The University of Tokyo)
  
  Critical threshold and asymptotic behavior for a one-phase Stefan problem with nonlinear boundary reaction
  We study the large-time behavior of solutions to a one-dimensional one-phase Stefan problem for the heat equation with a nonlinear boundary reaction. It was previously shown that there exist two threshold parameters $\lambda_*$ and $\lambda^*$ with $0<\lambda_*\le\lambda^*<\infty$, which divide the parameter range into global existence with exponential decay, global existence with non-exponential decay, and finite-time blow-up regimes. In this paper, we prove that these two thresholds coincide, and hence that the non-exponential decay regime consists of a single critical parameter. We also identify the rescaled asymptotic profile of the critical solution in the $C^2$ topology and derive the precise growth rate of the moving boundary, together with the sharp $L^\infty$-decay rate of the solution, at the critical parameter. This talk is based on a joint work with Mr. Kensho Araya.
  
  

• Ishiwata, Michinori (The University of Osaka)
  
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• Jeon, Seongmin (Hanyang University)
  
  Existence theory for non-variational systems with free boundaries
  In this talk, we discuss existence results for elliptic and parabolic systems with free boundaries and potentially singular right-hand sides in smooth domains. The approach is based on general “meta-theorems” that yield solutions as limits of smooth approximations under suitable assumptions on the operators and their solutions. We will discuss applications to fully nonlinear and $p$-Laplacian operators, including systems combining these operators, with an emphasis on verifying the properties required by this framework. This is based on joint work with Layan El Hajj and Henrik Shahgholian.
  
  

• Kwon, Dohyun (Yonsei University)
  
  On the Long-Time Behavior of the Volume-Preserving Mean Curvature Flows
  We study the evolution of sets under mean curvature flow subject to a volume constraint. We prove the exponential convergence of the area-preserving anisotropic flat flow to a disjoint union of equal Wulff shapes, which are the critical points of the anisotropic perimeter functional. We also show that the flat flow preserves certain reflection symmetries, which enables us to obtain uniform bounds on the distance between the evolving profile and the initial data. In addition, we present some recent results on the flow in the presence of multiple connected components.
  
  

• Kumagai, Kenta (The University of Tokyo)
  
  Classification of bifurcation diagrams for semilinear elliptic equations in a ball
  The bifurcation structure of semilinear elliptic equations was completely clarified by Joseph–Lundgren (1972/73) for power nonlinearities and the exponential nonlinearity in a ball. In these model cases, the bifurcation curve exhibits three different types, depending on the growth rate of the nonlinearity $f$ and the dimension $N$.
  
  In this talk, we investigate the key factors that determine the bifurcation structure for general supercritical nonlinearities. For $N \ge 3$, we show that unlike in the model cases, the bifurcation structure is determined not by the precise growth rate of $f$ but by a new quantity associated with $f$. This leads to a general classification of the bifurcation structure for $N \ge 3$. We find a unified viewpoint for the bifurcation structures in two and higher dimensions, which is hidden in the model cases, and clarify the bifurcation structure in two dimensions for general supercritical nonlinearities. Part of this work is joint work with Professor Hiroaki Kikuchi (Tsuda University).
  
  

• Lee, Se-Chan (Korea Institute for Advanced Study)
  
  Time derivative estimates for degenerate and normalized parabolic $p$-Laplace equations
  In this talk, I will discuss time regularity for the degenerate parabolic $p$-Laplace equation $u_t=\Delta_p u$ for $p>2$, and its normalized counterpart $u_t=\Delta_p^N u$ for $1<p<\infty$. For both equations, we establish the local boundedness of the time derivative $u_t$. For the degenerate $p$-Laplace equation, the proof is based on the Bernstein technique combined with a suitable regularization. As a consequence, we obtain an optimal regularity result with a connection to the well-known $C^{p’}$-conjecture in the elliptic setting. For the normalized equation, the critical nature requires a logarithmic Bernstein argument together with quantitative stability and an iteration between different regularizations. This talk is based on joint works with Yuanyuan Lian, Hyungsung Yun, and Kai Zhang.
  
  

• Zeng, Chongchun (Georgia Institute of Technology)
  
  Stable and unstable manifolds for capillary gravity water waves and a class of nonlinear PDEs
  Invariant manifold theory is a fundamental tool in the study of local dynamics near invariant structures in smooth evolution systems. It ensures the existence of nonlinearly invariant structures from linear ones. The theory has been well developed for diffeomorphisms, ODEs, semilinear PDEs, and some quasilinear parabolic PDEs. However, it becomes subtle for quasilinear or more nonlinear PDEs due to regularity issues when there is no smoothing effect. In this talk, we consider a class of nonlinear PDEs whose linearizations satisfy certain energy estimates. We prove that the linear exponential dichotomy implies the existence of local stable/unstable manifolds of the equilibria. In particular the result applies to a class of nonlinear Hamiltonian PDEs including the capillary gravity water waves of one or two fluids, quasilinear wave and Schrödinger equations, KdV type equations, etc., for which the linear analysis is also discussed. Basically, for such systems under certain conditions, spectral instability implies the existence of stable and unstable manifolds, which in particular yields the nonlinear instability in rough Sobolev norms and/or the existence of solutions decaying in high Sobolev norms. This is a joint work with Jalal Shatah.
  
  

• Zhanpeisov, Erbol (Tohoku University)
  
  Blow-up rate for the subcritical semilinear heat equation in non-convex domains
  We study the blow-up rate for solutions of the subcritical semilinear heat equation. Type I blow-up means that the rate agrees with that of the associated ODE. 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 (Institute of Science Tokyo).
  
  

• Zhou, Maolin (Nankai University)
  
  Blow-up Profiles and Types of Keller-Segel Model
  In this talk, we will discuss the blow-up phenomena of Keller–Segel model, especially sharp description about the blow-up profiles and classify the blow-up types in higher dimensional cases.