Blowup of classical solutions for a class of 3-D quasilinear wave equations with small initial data

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This paper is concerned with the small smooth data problem for the 3-D nonlinear wave equation ∂ₜ²u- (1+u+ₜ u)Δu=0. This equation is prototypical of the more general equation ∑ᵢ,ⱼ₌₀³gij(u, ∇ u)∂iju=0, where x₀=t and gij(u, ∇ u)=cij+diju+∑ₖ₌₀³eijᵏ∂ₖu+O(|u|²+|∇ u|²) are smooth functions of their arguments, with cij, dij and eijᵏ being constants, and dij≠0 for some (i,j); moreover, ∑ᵢ,ⱼ,ₖ₌₀³eijᵏ(∂ₖu) u does not fulfill the null condition. For the 3-D nonlinear wave equations ∂ₜ²u- (1+u)Δu=0 and ∂ₜ²u- (1+∂ₜ u)Δu=0, H. Lindblad, S. Alinhac, and F. John proved and disproved, respectively, the global existence of small smooth data solutions. For radial initial data, we show that the small smooth data solution of ∂ₜ²u-(1+u+∂ₜ u)Δu=0 blows up in finite time. The explicit expression of the asymptotic lifespan Tε as ε→0+ is also given.