Boundary regularity for the supercritical Lane-Emden heat flow

We establish a Pacard-type monotonicity formula and Morrey bounds up to the boundary for smooth solutions of the Lane-Emden heat flow $$u_t-\Delta u = |u|^{p-2}u$$ u t - Δ u = | u | p - 2 u on a general, smoothly bounded domain $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n , $$n\ge 3$$ n ≥ 3 , for exponents $$p>2^*=2n/(n-2)$$ p > 2 ∗ = 2 n / ( n - 2 ) , extending our previous work on the problem. As a consequence we obtain partially regular, self-similar tangent maps at any first blow-up point of the flow, and partial regularity at the blow-up time if the energy is uniformly bounded from below