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

Abstract

For any smoothly bounded domain $\Omega \subset {\mathbb{R}}^n$, $n\ge 3$, and any exponent $p>2^{\ast }=2n/(n-2)$ 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 $u{|}_{t=0}=u_0$ 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 ${\mathbb{R}}^n$.

11 pages

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