De Hatem Zaag
We consider the semilinear wave equation with subconformal power nonlinearity in two space dimensions. We construct a finite-time blow-up solution with a pyramid-shaped blow-up surface and an isolated characteristic blow-up point at the origin. Our solution is symmetric with respect to both axes, and anti-symmetric with respect to both bisectrices. The blow-up surface is differentiable outside the bisecrtices. On the bisectrices, it only has directional derivatives. As for the asymptotic behavior in similariy variables, the solution converges to the classical one-dimensional soliton outside the bisectrices, and to a genuinely two dimensional stationary solution, on the bisectrices, outside the origin. 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.
De Patrick Gerard
De Hatem Zaag
De Thomas Kappeler
De Michal Kowalczyk
De Cécile Huneau
De Mihalis Dafermos
De Andrea Nahmod
De Benjamin Schlein
De Chris Sogge
De Alexandru Ionescu
De Vlad Vicol
De Thomas Alazard
De Robert Jerrard
De Camillo de Lellis
De Chao Li
De Albert Schwarz
De Albert Schwarz
De Stephen Coombes , Peter Liddle
De Victor Kleptsyn
De François-Xavier Vialard
