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

TL;DR

This paper constructs a novel finite-time blow-up solution for a two-dimensional semilinear wave equation, featuring a pyramid-shaped blow-up surface and complex asymptotic behaviors, including a characteristic point and non-differentiable surface points.

Contribution

It presents the first example of a higher-dimensional blow-up solution with a characteristic point and a non-differentiable blow-up surface at non-characteristic points.

Findings

Blow-up surface shaped like a stylized pyramid with differentiability outside bisectrices.

Solution converges to a 1D soliton outside bisectrices and to a 2D stationary solution on bisectrices.

At the origin, the solution behaves like four localized solitons with opposite signs.

Abstract

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 bisecrtices. As for the asymptotic behavior in similariy 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.