Solution to the semilinear wave equation with a pyramid-shaped blow-up surface

Merle, Frank; Zaag, Hatem

doi:10.5802/slsedp.104

Solution to the semilinear wave equation with a pyramid-shaped blow-up surface

Merle, Frank ¹ ; Zaag, Hatem ²

¹ Université de Cergy Pontoise Département de mathématiques 2 avenue Adolphe Chauvin BP 222 95302 Cergy Pontoise cedex France
² Université Paris 13, Institut Galilée Laboratoire Analyse, Géométrie et Applications, CNRS UMR 7539 99 avenue J.B. Clément 93430 Villetaneuse France

Séminaire Laurent Schwartz — EDP et applications (2016-2017), Exposé no. 6, 13 p.

Résumé

We consider the semilinear wave equation with subconformal power nonlinearity in two space dimensions. We construct a finite-time blow-up solution with an isolated characteristic blow-up point at the origin, and a blow-up surface which is centered at the origin and has the shape of a stylized pyramid, whose edges follow the bisectrices of the axes in $ℝ^{2}$ . The blow-up surface is differentiable outside the bisectrices. As for the asymptotic behavior in similarity variables, the solution converges to the classical one-dimensional soliton outside the bisectrices. On the bisectrices outside the origin, it converges (up to a subsequence) to a genuinely two-dimensional stationary solution, whose existence is a by-product of the proof. At the origin, it behaves like the sum of 4 solitons localized on the two axes, with opposite signs for neighbors.

This is the first example of a blow-up solution with a characteristic point in higher dimensions, showing a really two-dimensional behavior. Moreover, the points of the bisectrices outside the origin give us the first example of non-characteristic points where the blow-up surface is non-differentiable.

This note gives only the main ideas. For details, see [52.

Publié le : 2017-11-20

DOI : 10.5802/slsedp.104

Affiliations des auteurs :

Merle, Frank ¹ ; Zaag, Hatem ²

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     title = {Solution to the semilinear wave equation with a~pyramid-shaped blow-up surface},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:6},
     pages = {1--13},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2016-2017},
     doi = {10.5802/slsedp.104},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/slsedp.104/}
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Merle, Frank; Zaag, Hatem. Solution to the semilinear wave equation with a pyramid-shaped blow-up surface. Séminaire Laurent Schwartz — EDP et applications (2016-2017), Exposé no. 6, 13 p. doi : 10.5802/slsedp.104. http://www.numdam.org/articles/10.5802/slsedp.104/

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