Well-posedness and global existence for Hardy--H\'{e}non parabolic equations in Herz spaces

In this paper, we study the Hardy-H'{e}non parabolic equation \begin{equation*} \partial_t u=\Delta u+a|x|^{-\gamma}|u|^{\beta}u, \qquad t>0,; x\in$\mathbb{R}$^{n}\setminus{0}, \quad \beta>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.