On self-similar blow up for the energy supercritical semilinear wave equation

Kim, Jihoi

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 1483-1542

We analyse the energy supercritical semilinear wave equation

$${\Phi }_{tt}-\Delta \Phi -{\left|\Phi \right|}^{p-1}\Phi =0$$

in ${ℝ}^d$ space. We first prove in a suitable regime of parameters the existence of a countable family of self-similar profiles which bifurcate from the soliton solution. We then prove the non-radial finite codimensional stability of these profiles by adapting the functional setting of [22].

Nous analysons l’équation d’onde semi-linéaire supercritique en énergie

$${\Phi }_{tt}-\Delta \Phi -{\left|\Phi \right|}^{p-1}\Phi =0$$

dans l’espace ${ℝ}^d$. Nous prouvons d’abord, dans un régime approprié de paramètres, l’existence d’une famille dénombrable de profils auto-similaires qui bifurquent à partir de la solution du soliton. Nous prouvons ensuite la stabilité non radiale en codimension finie de ces profils en adaptant le cadre fonctionnel de [22].

Classification : 35B44, 35C06, 35L05, 35L71
Keywords: Semi-linear wave equation, self-similar solution, blow up, focusing, energy super-critical, finite codimensional stability
Mots-clés : Équation d’onde semi-linéaire, solution auto-similaire, explosion, supercritique en énergie, stabilité en codimension finie

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