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

¹ Fachbereich Mathematik, Universität Salzburg, Hellbrunner Str. 34, 5020 Salzburg, Austria
simon.blatt@sbg.ac.at
² Departement Mathematik, ETH-Zürich, 8092 Zürich, Switzerland
michael.struwe@math.ethz.ch

Received: 25 November 2015
Accepted: 7 June 2016

Abstract

For any smoothly bounded domain Ω ⊂ ℝⁿ, n ≥ 3, and any exponent p > 2^∗ = 2n/ (n − 2) we study the Lane–Emden heat flow u_t − Δu = | u | ^{p − 2}u on Ω × ] 0,T [ and establish local and global well-posedness results for the initial value problem with suitably small initial data u|_{t = 0} = u₀ in the Morrey space L^2,λ(Ω) for suitable T ≤ ∞, where λ = 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 ℝⁿ.