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

Blatt, Simon¹ ; Struwe, Michael² 

¹ Fachbereich Mathematik, Universität Salzburg, Hellbrunner Str. 34, 5020 Salzburg, Austria
² Departement Mathematik, ETH-Zürich, 8092 Zürich, Switzerland

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

Résumé

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 ${ℝ}^n$.

Détail

DOI : 10.1051/cocv/2016041

Classification : 35K55
Keywords: Nonlinear parabolic equations, well-posedness of initial-boundary value problem