Boundary regularity for the supercritical Lane-Emden heat flow (Q745592)

The authors considers the Lane-Emden heat flow \(u_t-\Delta u=|u|^{p-2}u\) on a smooth bounded domain \(\Omega\subset \mathbb{R}^n\) with \(n\geq3\), in the supercritical case \(p>2^{*}=\frac{2n}{n-2}\). Extending their previous results on the problem, they establish a Pacard-type monotonicity formula and Morrey bounds up to the boundary. As an important consequence, they 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.