Well-posedness and global existence for Hardy--Hénon parabolic equations in Herz spaces

Abstract

In this paper, we study the Hardy-Hénon parabolic equation \begin{equation*} \partial_t u=Δu+a|x|^{-γ}|u|^βu, \qquad t>0,\; x\in\mathbb{R}^{n}\setminus\{0\}, \quad β>0,\; a\in\mathbb{R}, \end{equation*} under suitable assumptions on the parameter $\gamma$. We establish the local and global well-posedness of mild solutions in homogeneous Herz spaces.