Geometric PDE and Applied Analysis Seminar (June 19, 2024)

Talk 1: 15:00-16:00

Speaker: Prof. Tatsuki Kawakami (Ryukoku University)

Title: Asymptotic expansions of solutions to fractional diffusion equations

Abstract: Inhomogeneous fractional diffusion equation appears in the study of various nonlinear problems with anomalous diffusion, the Laplace equation with a dynamical boundary condition, and so on. Under suitable integrability conditions on the inhomogeneous term, the solution behaves like a suitable multiple of the fundamental solution to the linear fractional diffusion equation asymptotically with respect to time. In this talk we give the higher order asymptotic expansions (HOAE) of the large time behavior of the solution. Furthermore, we also give the precise description of the large time behavior of solutions to the Cauchy problem for nonlinear fractional diffusion equations. This talk is based on several joint works with Kazuhiro Ishige (Univ. of Tokyo).

Talk 2: 16:00-17:00

Speaker: Prof. Keisuke Takasao (Kyoto University)

Title: Phase field method for mean curvature flow with obstacles

Abstract: In this talk, we show the global existence of Brakke's mean curvature flow with obstacles and with a right-angle condition, when the obstacles have \(C^{1,1}\)boundaries. We use the Allen-Cahn equation with forcing term representing obstacles and we prove the convergence of the Radon measures given by the energy of the equation to Brakke's mean curvature flow in the sense of varifolds. This talk is based on a joint work with Katerina Nik (TU Delft).