Blow-up solutions to the semilinear wave equation with a stylized pyramid as a blow-up surface

摘要

We consider the semilinear wave equation with subconformal power nonlinearityin two space dimensions. We construct a finite-time blow-up solution with anisolated characteristic blow-up point at the origin, and a blow-up surfacewhich is centered at the origin and has the shape of a stylized pyramid, whoseedges follow the bisectrices of the axes in $R^2$. The blow-up surface isdifferentiable outside the bisecrtices. As for the asymptotic behavior insimilariy variables, the solution converges to the classical one-dimensionalsoliton outside the bisectrices. On the bisectrices outside the origin, itconverges (up to a subsequence) to a genuinely two-dimensional stationarysolution, whose existence is a by-product of the proof. At the origin, it behaves like the sum of 4 solitons localized on the twoaxes, with opposite signs for neighbors. This is the first example of a blow-upsolution with a characteristic point in higher dimensions, showing a reallytwo-dimensional behavior. Moreover, the points of the bisectrices outside the origin give us the firstexample of non-characteristic points where the blow-up surface isnon-differentiable.