| 概要 |
In this talk we shall study the gradient flow associated with the Sobolev inequality, called the p-Sobolev flow, which is described as a doubly nonlinear degenerate and singular parabolic equation of fast-diffusive type. The equation enjoys the interaction between fast-diffusive and ODE effects. We consider Cauchy-Dirichlet problem for the p-Sobolev flow, and we show a boundedness, a positivity, a regularity of the solution and the asymptotic behavior at infinite-time of the p-Sobolev flow. For the proof we study a local boundedness, the so-called positivity-expansion being valid for doubly nonlinear parabolic equations. Our global existence of the p-Sobolev flow is based on the scaling transformation intrinsic to the doubly nonlinear parabolic equation and this approach also eventually leads to an application to the finite-time extinction-behavior for the so-called fast diffusive doubly nonlinear parabolic equation. This is partially based on a collaborative work with Tuomo Kuusi in University of Helsinki, Finland and Kenta Nakamura in Kumamoto University.
References: T. Kuusi, M. Misawa, K. Nakamura: J. Geom. Anal. 30 no.2 (2020); J. Differ. Eq. 279 (2021); M. Misawa, K. Nakamura: Adv. Calc. Var. (2021); J. Geom. Anal. 33 no.1: 33 (2023); M. Misawa, K. Nakamura, Md Abu Hanif Sarkar: Nonlinear Differ. Eqn. Appl. 30 no.3: 43 (2023); M. Misawa: Calc. Var. 62 (2023), no. 9: 265; M. Misawa, Y. Yamaura: Math. Annalen (2025); T. Kuusi, M. Misawa, K. Nakamura: submitted
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