On the critical exponent pc of the 3D quasilinear wave equation [formula omitted] with short pulse initial data. I, global existence

Abstract

For the 3D quasilinear wave equation −(1+(∂tϕ)p)∂t2ϕ+Δϕ=0 with the short pulse initial data (ϕ,∂tϕ)(1,x)=(δ2−ε0ϕ0(r−1δ,ω),δ1−ε0ϕ1(r−1δ,ω)), where p∈N, p≥2, 0<ε0<1, r=|x|, ω=xr∈S2, and δ>0 is sufficiently small, under the outgoing constraint condition (∂t+∂r)kϕ(1,x)=O(δ2−ε0−kmax⁡{0,1−(1−ε0)p}) for k=1,2, we will establish the global existence of smooth large data solution ϕ when p>pc with pc=11−ε0 being the critical exponent. In the forthcoming paper, when 1≤p≤pc, we show the formation of the outgoing shock before the time t=2 under the same outgoing constraint condition and the other suitable assumption of (ϕ0,ϕ1).