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

TL;DR

This paper analyzes the well-posedness and long-term behavior of solutions to the critical Hardy-Sobolev parabolic equation, establishing conditions for global existence, decay, or blow-up based on initial data in the energy space.

Contribution

It provides a necessary and sufficient criterion for solution behavior below or at the ground state, including a dichotomy of global existence versus blow-up, extending to Dirichlet problems.

Findings

Solutions either exist globally and decay or blow up, depending on initial data.

Established a dichotomy criterion for solution behavior.

Extended results to Dirichlet boundary conditions.

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.