On the generalized parabolic Hardy-Hénon equation: existence, blow-up, self-similarity and large-time asymptotic behavior. (Q2076584)

In this paper there is investigated the Cauchy problem for the Hardy-Hénon equation \begin{align*}u_t+(-\Delta)^m u&= |x|^{-\alpha}F(u),\quad(x,t)\in\mathbb{R}^n\times(0,\infty),\\ u(0,x)&= u_0(x),\quad x\in\mathbb{R}^n,\end{align*} where \(m\in(0,1)\cup\mathbb{N}\), \(\alpha\in\mathbb{R}\) and \((-\Delta)^m\) is the power \(m>0\) of the minus Laplacian operator. In the paper the considered problem in different functional settings is analysed. The author gives conditions for the parameters so that to ensure local well-posedness, global well-posedness and non-existence of mild solutions. In the paper there are investigated some intrinsic properties of the globally defined solutions such as: self-similarity, radial symmetry, positivity and large-time behavior. For the proof of the main results, the author establishes estimates for the singular polyharmonic heat semigroup in the framework of Lorentz spaces.