Functional inequalities via a logarithmic transformation and its application to PDE

2021 Fiscal Year Final Research Report

• Project/Area Number: 18K13441
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Tohoku University (2019-2021); Ehime University (2018)
• Principal Investigator: IOKU NORISUKE 東北大学, 理学研究科, 准教授 (50624607)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Keywords: 非線形スケール不変性 / 擬スケール不変性 / 自己相似性 / 半線形熱方程式 / 関数不等式

Outline of Final Research Achievements

Inspired by a relationship between Tsallis statics and Bolzmann statics, we proposed a scale invariant form of Sobolev inequalities which converges to the limiting case of Sobolev inequalities (Alvino's inequality). This result means that the critical problem can be approximated via a direct limiting procedure of a subcritical problem in the sense of scale invariant structure.
Furthermore, we concerned semilinear heat equations with a general nonlinearity and revealed by focusing a quasi scale invariance that an existence of a singular stationary solution, an optimal singularity of an initial data such that an instant blow-up occurs, the Fujita exponent for a general nonlinearity, a second exponent in the sense of a quasi scaling.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

本研究では一貫して非線形問題のスケール不変構造に着目し，劣臨界・臨界関数不等式の橋渡しを与えるとともに，モデルケースである冪乗型非線形項に対して得られていた半線形熱方程式に対する結果を一般の非線形項に拡張した．スケール不変性の観点から非線形構造の主要部と剰余項を分ける方法が構築された点が意義深い．また，背後にTsallis, Bolzmann統計力学が潜むことから，関連諸分野への波及効果も期待できる．