Optimal well-posedness and forward self-similar solution for the Hardy–Hénon parabolic equation in critical weighted Lebesgue spaces. (September 2022)

• Record Type: Journal Article
• Authors: Chikami, Noboru; Ikeda, Masahiro; Taniguchi, Koichi
• Is Part Of: Nonlinear analysis. Volume 222(2022)
• Publication Date: 2022-09
• Subjects: primary 35K05 -- secondary 35B40; Hardy–Hénon parabolic equation -- Well-posedness -- Global existence -- Nonexistence -- Self-similar solution; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112931
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: The Cauchy problem for the Hardy–Hénon parabolic equation is studied in the critical and subcritical weighted Lebesgue spaces on the Euclidean space R d . In earlier works, the well-posedness of singular initial data and the existence of non-radial forward self-similar solutions to the problem were shown for the Hardy and Fujita cases ( γ ≤ 0 ). The weighted spaces are used to treat the potential | x | γ as an increase or decrease in the weight, which enables us to prove the well-posedness of the problem for all γ, with − min { 2, d } < γ, including the Hénon case ( γ > 0 ). As a by-product of global existence, self-similar solutions to the problem are established for all γ without restrictions. Furthermore, the non-existence of a local solution for supercritical data is also shown. Therefore, our critical exponent, s c is optimal with regard to solvability.