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arXiv · 2606.23605

Formation of singularities for a family of one-dimensional quasilinear wave equations beyond the variational case

Yuusuke Sugiyama
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Abstract

We consider finite-time singularity formation for classical solutions to the following parameterized nonlinear wave equation: \[ u_{tt}=c(u)^2u_{xx}+λc(u)c'(u)(u_x)^2, \] where \(λ\in[0,2]\) is a parameter. In previous works, it is known that finite-time blow-up solutions exist for \(0<λ\le1\) and for \(λ=2\). The cases \(λ=1\) and \(λ=2\) are known as the variational wave equation and the \(p\)-system, respectively, and they possess symmetric structures or conservation laws. For \(0<λ<1\), the blow-up construction is based on the fact that, after decomposing the wave into two Riemann variables, one component can be kept sufficiently small and hence does not prevent the other component from blowing up. In the present paper, we treat the remaining intermediate case \(1<λ<2\). This case is essentially different from the previous one, since both Riemann variables may grow. The key idea is to introduce suitable supremum functions for the Riemann variables and to derive a comparison principle through right Dini derivatives. This allows us to control the relative size of the two components and to obtain a Riccati-type differential inequality for the dominant supremum function.

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Yuusuke Sugiyama. 2026-06-22. Formation of singularities for a family of one-dimensional quasilinear wave equations beyond the variational case. https://arxiv.org/abs/2606.23605

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