Michael Struwe (ETH Zürich)
As a model problem for the study of supercritical partial differential equations, on a smoothly bounded domain Ω⊂Rn, ≥3n, for an exponent >=−*222npn, the critical exponent for Sobolev's embedding, we consider the boundary value problem 1 −−Δ=Ω=∂Ω2|| in ,0 on puuuu, and the associated heat flow 2 −−Δ=Ω=∂Ω==20|| in ,0 on , at 0ptuuuuuuut for given smooth initial data 0u. Similar to standard variational approaches to the existence of positive solutions to 1, the flow2 may be regarded as a (negative) gradient flow for the energy functionall associated with equation 1, whose rest points exactly correspond to the solutions of 1. However, solutions to the flow 2 may blow up in finite time. We partially extend the pioneering analysis of the possible blow up profiles of 2 by Giga-Kohn to the supercritical range >*2p and discuss possible consequences for the existence of positive solutions to 1 on topologically non-nontrivial domains.