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

Abstract

For any smoothly bounded domain $\Omega\subset\mathbb R^n$, $n\geq 3$, and any exponent $p>2^*=2n/(n-2)$ we study the Lane-Emden heat flow $u_t-\Delta u = |u|^{p-2}u$ on $\Omega\times]0,\infty[$ and establish local and global well-posedness results for the initial value problem with suitably small initial data $u\big|_{t=0}=u_0$ in the Morrey space $L^{2,\lambda}(\Omega)$, where $\lambda=4/(p-2)$. We contrast our results with results on instantaneous complete blow-up of the flow for certain large data in this space, similar to ill-posedness results of Galaktionov-Vazquez for the Lane-Emden flow on $\mathbb R^n$.