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

Blatt, Simon; Struwe, Michael

doi:10.1051/cocv/2016041

Blatt, Simon ¹ ; Struwe, Michael ²

¹ Fachbereich Mathematik, Universität Salzburg, Hellbrunner Str. 34, 5020 Salzburg, Austria
² Departement Mathematik, ETH-Zürich, 8092 Zürich, Switzerland

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1370-1381.

Résumé

For any smoothly bounded domain $Ω \subset ℝ^{n}$ , $n \geq 3$ , and any exponent $p > 2^{*} = 2 n / (n - 2)$ we study the Lane–Emden heat flow $u_{t} - Δ u = {| u |}^{p - 2} u$ on $Ω \times] 0, T [$ and establish local and global well-posedness results for the initial value problem with suitably small initial data ${u |}_{t = 0} = u_{0}$ in the Morrey space $L^{2, λ} (Ω)$ for suitable $T \leq \infty$ , 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 $ℝ^{n}$ .

MR Zbl

DOI : 10.1051/cocv/2016041

Classification : 35K55
Mots-clés : Nonlinear parabolic equations, well-posedness of initial-boundary value problem

Affiliations des auteurs :

Blatt, Simon ¹ ; Struwe, Michael ²

¹ Fachbereich Mathematik, Universität Salzburg, Hellbrunner Str. 34, 5020 Salzburg, Austria
² Departement Mathematik, ETH-Zürich, 8092 Zürich, Switzerland

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     author = {Blatt, Simon and Struwe, Michael},
     title = {Well-posedness of the supercritical {Lane{\textendash}Emden} heat flow in {Morrey} spaces},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1370--1381},
     publisher = {EDP-Sciences},
     volume = {22},
     number = {4},
     year = {2016},
     doi = {10.1051/cocv/2016041},
     zbl = {1364.35129},
     mrnumber = {3570506},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv/2016041/}
}

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Blatt, Simon; Struwe, Michael. Well-posedness of the supercritical Lane–Emden heat flow in Morrey spaces. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1370-1381. doi : 10.1051/cocv/2016041. https://www.numdam.org/articles/10.1051/cocv/2016041/

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