Well-posedness of the supercritical Lane-Emden heat flow in Morrey spaces (Q2835363)

The authors consider the Lane-Emden heat flow NEWLINE\[NEWLINEu_t-\Delta u=|u|^{p-2}uNEWLINE\]NEWLINE on \(\Omega \times ]0, T[\), where \(\Omega\) is a smooth bounded domain of \(\mathbb{R}^n\), \(n\geq 3\) and \(p > 2^\ast=\frac{2n}{n-2}\). They establish local and global well-posedness results for the initial value problem with suitably small initial data \(u\) in the Morrey space \(L^{2,\lambda}(\Omega)\), where \(\lambda = \frac{4}{p-2}\). This result is the counterpart of the instantaneous complete blow-up of the flow for certain large data in this space, similar to ill-posedness results proved by Galaktionov-Vazquez for the Lane-Emden flow.