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The natural setting for the Lane–Emden equation − Δu =| u |  p −2  u  on a domain   $\Omega \subset \mathbb {R}^n$  ,  n ≥3, for supercritical exponents  p &gt;2*=2 n /( n −2) is identified as the space of functions   $u\in H^1_0\cap L^p(\Omega)$   with finite scale-invariant Morrey norms. We show that this Morrey regularity is propagated by the heat flow associated with this equation, and we study the blow-up profiles.

An Analytic Framework for the Supercritical Lane–Emden Equation and its Gradient Flow