Optimal well-posedness and forward self-similar solution for the Hardy-Hénon parabolic equation in critical weighted Lebesgue spaces

The Cauchy problem for the Hardy-Hénon parabolic equation is studied in the critical and subcritical regime in weighted Lebesgue spaces on the Euclidean space \(\mathbb{R}^d\). Well-posedness for singular initial data and existence of non-radial forward self-similar solution of the problem are previously shown only for the Hardy and Fujita cases (\(\gamma\le 0\)) in earlier works. The weighted spaces enable us to treat the potential \(|x|^{\gamma}\) as an increase or decrease of the weight, thereby we can prove well-posedness to the problem for all \(\gamma\) with \(-\min\{2,d\}0\)). As a byproduct of the well-posedness, the self-similar solutions to the problem are also constructed for all \(\gamma\) without restrictions. A non-existence result of local solution for supercritical data is also shown. Therefore our critical exponent \(s_c\) turns out to be optimal in regards to the solvability.