Well-posedness and global dynamics for the critical Hardy-Sobolev parabolic equation

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.

Automated summary

The study focuses on the Cauchy problem for the semilinear heat equation with the singular potential in the energy space, known as the Hardy-Sobolev parabolic equation. The paper aims to establish a necessary and sufficient condition for initial data below or at the ground state to completely dichotomize the behavior of solutions. The results show that the solution will either exist globally in time with energy decaying to zero, or blow up in finite or infinite time, with the dichotomy also demonstrated for the corresponding Dirichlet problem through a comparison principle.