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

Blatt, Simon; Struwe, Michael

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1370-1381.

For any smoothly bounded domain $\Omega \subset {ℝ}^n$, $n\ge 3$, and any exponent $p>2^{*}=2n/(n-2)$ we study the Lane–Emden heat flow $u_t-\Delta u={\left|u\right|}^{p-2}u$ on $\Omega \times ]0,T[$ 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 $L^{2,\lambda }\left(\Omega \right)$ for suitable $T\le \infty$, 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 ${ℝ}^n$.

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