Volume 22, Number 4, October-December 2016
Special Issue in honor of Jean-Michel Coron for his 60th birthday
|Page(s)||1370 - 1381|
|Published online||05 August 2016|
Well-posedness of the supercritical Lane–Emden heat flow in Morrey spaces
1 Fachbereich Mathematik, Universität Salzburg, Hellbrunner Str. 34, 5020 Salzburg, Austria
2 Departement Mathematik, ETH-Zürich, 8092 Zürich, Switzerland
Received: 25 November 2015
Accepted: 7 June 2016
For any smoothly bounded domain Ω ⊂ ℝn, n ≥ 3, and any exponent p > 2∗ = 2n/ (n − 2) we study the Lane–Emden heat flow ut − Δu = | u | p − 2u 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 L2,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.
Mathematics Subject Classification: 35K55
Key words: Nonlinear parabolic equations / well-posedness of initial-boundary value problem
© EDP Sciences, SMAI 2016
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