On the Blow-up of the Solutions of a Quasilinear Wave Equation With a Semilinear Source Term

In this paper, following the ideas of Lax, we prove a blow-up result for a class of solutions of the equation ϕ_tt−ϕ_xx−ϕ²_xϕ_xx−ϕ³ = 0, corresponding, in certain cases, to the development of a singularity in the second derivatives of ϕ. These solutions solve locally (in time) the Cauchy problem for smooth initial data belonging to the uniformly local Sobolev spaces considered by Kato and by Majda.