Well-posedness and global dynamics for the critical Hardy–Sobolev parabolic equation. (20th October 2021)

• Record Type: Journal Article
• Authors: Chikami, Noboru; Ikeda, Masahiro; Taniguchi, Koichi
• Is Part Of: Nonlinearity. Volume 34:Number 11(2021)
• Page Start: 8094
• Page End: 8142
• Publication Date: 2021-10-20
• Subjects: Hardy–Sobolev parabolic equation -- well-posedness -- global existence -- dissipation -- blow-up; Primary 35K05 -- Secondary 35B40; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ac2c90
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: We study the Cauchy problem for the semilinear heat equation with the singular potential, called the Hardy–Sobolev parabolic equation, in the energy space. The aim of this paper is to determine a necessary and sufficient condition on initial data below or at the ground state, under which the behavior of solutions is completely dichotomized. More precisely, the solution exists globally in time and its energy decays to zero in time, or it blows up in finite or infinite time. The result on the dichotomy for the corresponding Dirichlet problem is also shown as a by-product via comparison principle.