Optimal well-posedness and forward self-similar solution for the Hardy–Hénon parabolic equation in critical weighted Lebesgue spaces. (September 2022) - Chikami, Noboru; Ikeda, Masahiro; Taniguchi, Koichi
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.