# Registered Data

## [00135] Nonlinear PDEs and related diffusion phenomena

**Session Date & Time**:- 00135 (1/3) : 4E (Aug.24, 17:40-19:20)
- 00135 (2/3) : 5B (Aug.25, 10:40-12:20)
- 00135 (3/3) : 5C (Aug.25, 13:20-15:00)

**Type**: Proposal of Minisymposium**Abstract**: Diffusion equations have a primary role in the description and modeling of several physical phenomena. A classical prototype is the heat equation, deriving from Fourier’s law, which is by now a widely studied topic within the mathematical community, both in Euclidean and non-Euclidean frameworks such as manifolds or metric-measure spaces. In the last decades, many nonlinear and nonlocal versions of this equation and related ones have been proposed and analyzed, which gave rise to challenging mathematical problems. We aim at gathering international experts and talented young researchers that will discuss the most recent advances on the subject.**Organizer(s)**: Kazuhiro Ishige, Tatsuki Kawakami, Matteo Muratori**Classification**:__35Kxx__,__35Rxx__,__39Bxx__,__58Jxx__,__60Hxx__**Speakers Info**:- Kazuhiro Ishige (University of Tokyo)
- Matteo Bonforte (Universidad Autónoma de Madrid)
- Ki-Ahm Lee (Seoul National University)
- Elvise Berchio (Politecnico di Torino)
- Yohei Fujishima (Shizuoka University)
- Masahiko Shimojo (Tokyo Metropolitan University)
- Yannick Sire (Johns Hopkins University)
- Megumi Sano (Hiroshima University)
- Giulia Meglioli (Politecnico di Milano (from December at Bielefeld University, Germany))
- Qing Liu (Okinawa Institute of Science and Technology Graduate University)
- Davide Bianchi (Harbin Institute of Technology)
- Reika Fukuizumi (Tohoku University)

**Talks in Minisymposium**:**[02461] The generalized porous medium equation on graphs: Existence and uniqueness of solutions with l^1 data****Author(s)**:**Davide Bianchi**(Harbin Institute of Technology (Shenzhen))

**Abstract**: We study solutions of the generalized porous medium equation on infinite graphs. For nonnegative or nonpositive integrable data, we prove the existence and uniqueness of mild solutions on any graph. For changing sign integrable data, we show existence and uniqueness under extra assumptions such as local finiteness or a uniform lower bound on the node measure.

**[03203] Radial solutions to a semilinear equation on Riemannian models****Author(s)**:**Elvise Berchio**(Politecnico di Torino)- Alberto Ferrero (Università del Piemonte Orientale)
- Debdip Ganguly (Indian Institute of Technology Delhi)
- Prasun Roychowdhury (National Center for Theoretical Sciences)

**Abstract**: We provide a classification with respect to asymptotic behaviour, stability and intersections properties of radial smooth solutions to the equation $\Delta_g u=e^u$ on Riemannian models. Our assumptions include Riemannian manifolds with sectional curvatures bounded or unbounded from below. As it is well-known in the Euclidean case, intersection and stability properties are influenced by the dimension; here the analysis highlights properties of solutions that cannot be observed in the flat case.

**[03225] Spreading and extinction of solutions to the logarithmic diffusion with a logistic reaction****Author(s)**:**Masahiko Shimojo**(Tokyo Metropolitan University, Department of Mathematical Sciences)

**Abstract**: Logarithmic diffusion is observed in several fields of science, such as the central limit approximation of Carleman’s model based on the Boltzmann equation, a model for long Van-der-Waals interactions in thin fluid films, and the evolution of conformal metric under the Ricci flow on the plane. We focus on the spreading and extinction phenomena of the solution to the logarithmic diffusion equation on a line, in the presence of a logistic reaction term. A Liouville-type theorem will be introduced to understand the extinction and interfacial phenomena from the point of entire solutions.

**[03236] Global regularity estimates for the Poisson equation on complete manifolds****Author(s)**:**Ludovico Marini**(University of Milano-Bicocca (soon at Fukuoka University))- Stefano Meda (University of Milano-Bicocca)
- Stefano Pigola (University of Milano-Bicocca)
- Giona Veronelli (University of Milano-Bicocca)

**Abstract**: In this talk, we investigate the validity of first and second-order, global $L^p$ estimates for the solutions of the Poisson equation. While these estimates always hold on $\mathbb{R}^n$, on complete non-compact manifolds their validity is strongly influenced by the large-scale geometry. I will present some positive results and discuss the sharpness of certain assumptions through counterexamples. This is a joint work with Stefano Meda, Stefano Pigola and Giona Veronelli of the University of Milano-Bicocca.

**[03273] First order fully nonlinear nonlocal evolution equations****Author(s)**:- Takashi Kagaya (Muroran Institute of Technology)
**Qing Liu**(Okinawa Institute of Science and Technology)- Hiroyoshi Mitake (University of Tokyo)

**Abstract**: This talk is concerned with geometric motion of a closed surface whose normal velocity depends on a nonlocal quantity of the enclosed region. Using the level set formulation, we study a class of first order nonlocal evolution equations in the framework of viscosity solution theory. We prove the uniqueness of solutions by establishing a comparison principle. Our existence result is based on careful analysis on parallel surfaces and an optimal control interpretation. We also mention several properties of the solution such as quasiconvexity preserving, fattening phenomenon and large time behavior.

**[03392] Boundary Regularity of Local and Nonlocal Equations****Author(s)**:**Ki-Ahm Lee**(Seoul National University)

**Abstract**: In this talk, we are going to discuss boundary regularities of various degenerate local equations and nonlocal equations. Diffusion rates deform undefined geometry related to diffusion and the corresponding distance function makes an important role in the theory of regularity. And then we will also discuss the possible applications.

**[03474] Characterization of F-concavity preserved by the Dirichlet heat flow****Author(s)**:- Asuka Takatsu (Tokyo Metropolitan University)
- Paolo Salani (University of Florence)
**Kazuhiro Ishige**(The University of Tokyo)

**Abstract**: F-concavity is a generalization of power concavity and, actually, the largest available generalization of the notion of concavity. We characterize the F-concavities preserved by the Dirichlet heat flow in convex domains on Euclidean space, and complete the study of preservation of concavity properties by the Dirichlet heat flow, started by Brascamp and Lieb in 1976 and developed in some recent papers.

**[03548] Quasi self-similarity and its application to the global in time solvability of a superlinear heat equation****Author(s)**:**Yohei Fujishima**(Shizuoka University)

**Abstract**: We discuss the global in time existence of solutions for a superlinear heat equation. In particular, we determine the critical decay rate of initial functions for the global existence of solutions by introducing a quasi self-similar solution for the problem.

**[03585] Weighted Trudinger-Moser inequalities in the subcritical Sobolev spaces****Author(s)**:**Megumi Sano**(Hiroshima University)

**Abstract**: Inspired by Ni's result about the H\'enon equation with nonlinear term which has the strong polynomial growth beyond the Sobolev critical growth, we consider the exponential growth beyond the polynomial growth in a maximization problem. Also we discuss the optimality and the attainability of our maximization problem. Our inequalities are regarded as subcritical versions of the Trudinger-Moser inequalities in the critical Sobolev spaces. This is a joint work with Masahiro Ikeda(RIKEN/Keio Univ.) and Koichi Taniguchi(Tohoku Univ.).

**[03604] Stochastically perturbed log diffusion equations****Author(s)**:**Reika Fukuizumi**(Waseda University)

**Abstract**: We will present a result on the existence and uniqueness of the solution for the stochastic fast logarithmic equation with a Stratonovich multiplicative noise in R^d for d \ge 3. We overcome several technical difficulties due to the degeneracy properties of the logarithm and to the fact that the problem is treated in an unbounded domain. This is a joint work with Ioana Ciotir (INSA Rouen, France) and Dan Goreac (Univ Paris Est, France and Shandong University, China).

**[04047] Results on the Stokes eigenvalue problem under Navier boundary conditions****Author(s)**:**Alessio Falocchi**(Politecnico di Milano)- Filippo Gazzola (Politecnico di Milano)

**Abstract**: We study the Stokes eigenvalue problem under Navier boundary conditions in 2D or 3D bounded domains with connected boundary of class $C^1$. Differently from the Dirichlet boundary conditions, zero may be the least eigenvalue. We fully characterize the domains where this happens. We then consider the general version of the problem in any space dimension with $n\geq2$, characterizing the kernel of the strain tensor for solenoidal vector fields with homogeneous normal trace.

**[04308] Well-posedness with large data for a weighted porous medium equation****Author(s)**:**Troy Petitt**(Politecnico di Milano)- Matteo Muratori (Politecnico di Milano (Italy))

**Abstract**: The large data problem for the porous medium equation is to determine the largest class of initial data for which local well-posedness is guaranteed for the Cauchy problem. We review the classical results by Widder for the heat equation $u_t=\Delta u$. The corresponding problem for the porous medium equation $u_t=\Delta u^m$ for $m>1$ was solved in the 1980s. We extend these results for weighted equations $\rho(x)u_t=\Delta u^m$ for $\rho(x)≅|x|^{-\gamma}$ for $\gamma\in(0,2)$.